Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching

An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School

Wang Wei Soumlnnerhed CUL-doktorand Enheten foumlr Laumlrande och Undervisning IPD Goumlteborgs universitet Planeringsseminarium 2009-04-29

1

Planeringsseminariumsunderlag 2009 Wang Wei Soumlnnerhed April 6 2009

1 AIM AND BACKGROUND

11 Introduction On a personal level my research interest started with visits to some Swedish mathematics classrooms a few years ago when I was a teacher student studying to become a mathematics teacher at upper-secondary school During these visits the students were studying the use of the quadratic formula for solving quadratic equations usually called the PQ formula in school Teaching focus seemed to be put much on how to use PQ formula to solve quadratic equations which is quite different from my own mathematics learning experience in China When I studied the same content in middle school in China I was taught to try the factorization method first and then use PQ formula later if it did not work with factorization for solving a quadratic equation The different teaching focus on using different methods to solve quadratic equations lays the ground of my interest for this research In mathematics solving quadratic equations belongs to the field of algebra There are different methods to solve quadratic equations such as the geometrical method through completing squares using quadratic formula PQ formula factorization using graphs of quadratic functions to solve quadratic equations This is a rich but complicated area in algebra How these specific mathematical contents are presented in secondary-school teaching and what sources the teachers have based their lessons on are questions relating to mathematical content and pedagogical-content issues With the starting point of teaching this research needs to consider both the field of algebra as a subject and the subject related resources available for teaching in this area These two research focuses are components of Pedagogical Content Knowledge (Shulman 1986 p 9) One category of PCK according to Shulman is ldquothe ways of representing and formulating the subject that make it comprehensible to othersrdquo (p 9) Hence an important question arises What forms of representation of a specific lesson topic are available for teachers This is an essential point for what this research is about In order to find different forms of representation and different methods for solving quadratic equations I have chosen to investigate and analyze mathematics textbooks in the Mathematics B course and the related teacherrsquos guides for Swedish upper-secondary school Analyzing mathematics textbooks and teaching material is not just to investigate the facts or concepts in the field of algebra concerning different solving methods but also to find the pedagogical structure of this subject The research therefore has its purpose to explore pedagogical content knowledge in the textbooks and teaching material which is related to a teacherrsquos knowledge

12 Why is it about algebra School algebra is a difficult area in the Swedish studentsrsquo studies of mathematics (Haumlggstroumlm 2006) Why is algebra difficult According to Haumlggstroumlm the algebraic circle (Bergsten et al 2002) shows that students first ought to translate problems expressed in daily words into algebraic structures by using algebraic symbols Thereafter the students should formulate the algebraic structures with certain given rules Then comes solving the problems All of these

2

three steps require that the students are able to handle symbols and concepts and that they have the necessary skills needed for operations as long as they understand the contents of the problems Operations are based on the early knowledge of arithmetic (Haumlggstroumlm 2006) Swedish students learn elementary algebra at the beginning of upper-secondary school The subject mathematics is divided into five levels from A to E At A level called Mathematics A students are required to be able to translate simplify and reform expressions of quadratic equations according to the Swedish syllabus for Mathematics A (Skolverket 2000) At the B level students should go more deeply into the learning of algebra and functions During this course the students should be able to tolka foumlrenkla och omforma uttryck av andra graden samt loumlsa andragradsekvationer och tillaumlmpa kunskaperna vid problemloumlsning (Skolverket 2000 p 83) The aim of Mathematics B emphasizes the importance of solving quadratic equations It is during this study period that students build up most of their knowledge of algebra At upper-secondary schools in Sweden students are taught to use the approaches of completing the square and the quadratic formula called PQ formula to informally solve quadratic equations as well as factorization The application of the graphical method is introduced in the chapter on functions at the mathematics B course Applying PQ formula is an efficient way when solving all kinds of different quadratic equations On the one hand quadratic formula has become a powerful mathematics instrument for students who can use it freely without paying much attention to equationsrsquo structures and operational procedures but on the other hand as a result knowing why and how to use other methods to solve quadratic equations may become less important in practice something which might reduce studentsrsquo opportunities to think mathematically The problem with solving a quadratic equation by either quadratic formula or calculators without knowing why and how causes a dilemma in mathematics learning and teaching From teachersrsquo point of views it is important to make students understand how to solve quadratic equations in order to develop their mathematics thinking But for students quickly and easily finding solutions for quadratic equations might be their goals The need for finding correct and efficient methods may change the character of school mathematics and therefore lead mathematics teaching to a more practical character than pure mathematical science Compared to quadratic formula factorization in algebra content may not serve as an efficient tool for solving quadratic equations but can probably increase studentsrsquo algebra structure sense Factorization and quadratic formula belong to this dilemma area

13 The choice of which mathematics textbooks to investigate Some international studies have compared the effectiveness of using different methods to solve quadratic equations from a mathematics-didactics point of view How many methods are usually taught for solving quadratic equations in Swedish upper-secondary mathematics classrooms How do these different methods relate to each other and thus influence studentsrsquo understanding of quadratic equations and algebra structure Why are some methods emphasized in teaching at the same time as some others are not These questions concern two fields one is mathematics as a scientific discipline and the other is mathematics didacticsndashmainly normative didactics which ldquoinvolves discussions about the educational goals choice of content and methods but should also include justifications and recommendationsrdquo (Johansson 2006 p 12) To be able to answer these questions I need to find resources covering these two fields That means mathematics textbooks The previous Swedish study

3

(Johansson 2006) has found that the mathematics textbook is the most influential factor in classroom teaching and learning According to Johansson (2006) textbooks seem to dictate the teaching of mathematics in many aspects The use of textbooks is a very important framework in mathematics teaching Textbooks influence not only what kind of tasks students are working with and the examples presented by the teachers but also how mathematics is portrayed in terms of the concepts and the features that are related to the subject (p 26) From previously done research Johansson (2006) has found the importance of analyzing the content of textbooks because of the following reasons ldquomathematical topics in textbooks are most likely presented by the teachersrdquo ldquoTeachersrsquo pedagogical strategies are often influenced by the instructional approach of the materialrdquo ldquoTeachers sequence of instruction are often parallels to that of the textbookrdquo and ldquoTeachers report that textbooks are a primary information source in deciding how to present contentrdquo (p 48) The same mathematical topics can be emphasized in one book but might be overlooked in another An analysis of mathematics textbooks reveals the implied beliefs of what mathematics is and how it can be taught and learned It also reflects on the influence of the educational culture from a specific country It is obvious that mathematics textbooks play important roles in Swedish classrooms though not all teachers are the slaves of the textbooks Since mathematics textbooks are important sources for teaching and learning mathematics in Swedish classrooms I have decided to start my study by investigating eight mathematics textbooks used for the course Mathematics B in Swedish upper-secondary schools and by analyzing three of them in detail The related teaching guide material will be investigated in next step

14 Research Aim

The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3

2 Mathematics background in the field of algebra

21 Algebra history and its development Quadratic equations and their unknowns as well as polynomials belong to the field of algebra in mathematics In Swedish mathematics textbooks for mathematics course B there is one chapter on algebra Many critical elements of algebra have been presented in every book but there are no descriptions or definitions of the word ldquoalgebrardquo What is algebra Colin Maclaurin in his 1748 algebra text defined it as ldquoAlgebra is a general Method of Computation by certain Signs and Symbols which have been contrived for this Purpose and found convenient It is called an Universal Arithmetic and proceeds by Operations and Rules similar to those in Common Arithmetic founded upon the same Principlesrdquo (Katz 2006 p 185) Leonhard Euler in his own algebra text in 1770 defined algebra as ldquoThe science

4

which teaches how to determine unknowns quantities by means of those that are knownrdquo (Katz 2006 p 185) In this part I will mainly give a survey of the historical development of algebra including solving quadratic equations according to four stages the rhetorical stage the syncopated stage the symbolic stage and the purely abstract stage The purpose is to try to find out why algebra in school is like a threshold which hinders the students in their understanding Can a historical perspective on algebra explain this obstacle Algebra is an old science and its historical background is complicated Algebra procedures have developed slowly There are different opinions about where the evolution of the term ldquoalgebrardquo started It is commonly believed that algebra first appeared among the Egyptians the Babylonians the Greeks or the Arabs The geometrical influence on algebraic reasoning was strong in ancient Greece However the word algebra originated in Baghdad where the Arabic scientist al-Khwarizmi (AD 780-850) published a short book about calculating with the help from al-jabr and al-muqabala a book on the solution of an equation as a rule (Kvasz 2006) Todayrsquos algebra has its root in Arabic algebra Western mathematics tended to turn algebraic operations into symbols and later developed abstract algebra The process of algebra development was slow and its whole history lasted 4000 years

The rhetorical stage originated from geometry ideas

Historically algebra developed in three stages the rhetorical stage the syncopated stage and the symbolic stage According to Katz (2006) algebra development can also be categorized into four conceptual stages ldquothe geometric stage where most of the concepts of algebra are geometric the static equation-solving stage where the goal is to find numbers satisfying certain relationship the dynamic function stage where motion seems to be an underlying idea and finally the abstract stage where structure is the goalrdquo (p 186) Algebra began very early in recorded history Algebra texts date from the first half of the second millennium BCE 37 or 38 centuries ago and were written by people living in Mesopotamia and Egypt (Derbyshire 2006) During the Hammurabi period from about 1790 to 1600 BCE the Babylonians started their civilization by pressing written words in patterns called cuneiform or wedge-shaped stylus into wet clay Many tablets in cuneiform had a mathematical-algebraic content Their mathematical texts were of two kinds table texts and problem texts The table texts were lists of multiplication tables tables of squares and cubes as well as advanced lists like the famous Plimpton 322 tablet which is about Pythagorean triples The number system of the Babyloniansrsquo time was based on 60 digits rather than todayrsquos 10 digits for example our number of 37 denotes three sixties and seven ones which is our number 187 The problem was that the Babylonians had neither defined zero nor negative numbers The Babylonians of Hammurabirsquos era had no proper algebraic symbolism All mathematical problems were expressed in words for example unknown quantity in Sumerianrsquos Akkadian text was expressed as igum (length) and igibum (width) as reciprocal The application of algebra might have its origin in the need for measuring land areas At the rhetorical stage all mathematical statements and arguments were expressed in words and sentences (Derbyshire 2006) Babylonian mathematics had two roots one is accountancy problems and the other one is a ldquocut-and-pasterdquo geometry probably developed for understanding the division of land Many old-Babylonian clay tablets contain quadratic problems of which the goal was to find such geometric quantities as the length and width of a rectangle As an example we are given that the sum of the length and width of a rectangle is

5

216 and the area of the rectangle is

217 (Kvatz 2006 p 187 and Derbyshire 2006

pp 25-27) What is the length and the width of this rectangle The tablet described in detail

the steps the writer went through First the writer halves 216 to get

413 Next he squares

413

to get 16910 From this area he subtracts the given area

217 giving

1613 The square root of

this number is extracted431 Finally the length is 5

431

413 =+ while the width is

211

431

413 =minus (Kvatz 2006) The whole process can be translated into parts of a quadratic

formula that is 2172

216

431

2

minus⎟⎠⎞

⎜⎝⎛ divide= The Babylonians did not know anything about

negative numbers the only solution for them was in positive numbers and hence their algorithm did not deliver the two solutions to the quadratic equation so their formula is slightly different from quadratic formula (Derbyshire 2006) Their formula is

⎟⎠⎞

⎜⎝⎛ divideplusmnminus⎟

⎠⎞

⎜⎝⎛ divide= 2

216

2172

216

2

x If we denote the sum of the length and the width of the

rectangle as b and the given area as c this formula will be ⎟⎠⎞

⎜⎝⎛plusmnminus⎟

⎠⎞

⎜⎝⎛=

22

2 bcbx although it is

not exactly like the modern quadratic formula There are different interpretations of Neugebauer and Sachesrsquo translation of the Babylonianrsquos tablets for this text on finding the length and width of a rectangle However it is very clear that the text from the tablets is dealing with a geometric procedure The problem was solved in words but with geometric ideas This was the beginning of algebra The Greek mathematician Euclid (300 BC) in his Book II of Elements solved some algebraic problems by manipulating geometric figures but based them on clearly stated axioms The geometrical method is in Kvatzrsquos opinion more explicit in another work of his Data The following example illustrates how Euclid solved a quadratic equation by a geometrical method Euclid defined ldquoproposition 1rdquo which is like axiom 1 If two straight lines contain a given area in a given angle and if the sum of them be given then shall each of them be given (Kvatz 2006 p 189) Euclid set up a rectangle with one side x = AS and y = AC Then a line was drawn so that BS = AC and the completed rectangle was ACDB Suppose that

was given and the area of rectangle ACFS (Figure 1) was given denoted as c What were the two sides AS (x) and AC (y) of the rectangle

byxAB =+=

A x S y B

y y

C F D

Figure 1

6

In order to find the length and the width of the rectangle Euclid bisected AB at E constructed the square on BE and then claimed that this square was equal to the sum of the rectangle ACFS and the small shaded square at the bottom (Figure 2)

A E S B

x

y y F

C G D Figure 2 According to Euclid the area of the rectangle ACFS was given which was c and the area of

the new square EGDB was also given which was ( )22b because 222byxSBASEB =

+=

+=

The equivalent relationship between the areas can be formulated as an quadratic equation 22

22⎟⎠⎞

⎜⎝⎛ minus+=⎟

⎠⎞

⎜⎝⎛ bxcb or

22

22⎟⎠⎞

⎜⎝⎛ minus+=⎟

⎠⎞

⎜⎝⎛ ybcb Euclid found out this equivalent relationship

geometrically and took use of this relationship in finding the solutions of the problem so the

length and width of the rectangle ACFS are cbbx minus⎟⎠⎞

⎜⎝⎛+=

2

22 and cbby minus⎟

⎠⎞

⎜⎝⎛minus=

2

22

These two formulas are almost identical with the Babyloniansrsquo solutions in rhetoric expressions The difference is that Greek algebra was based on geometric manipulation while Babylonian algebra was based on rhetoric manipulation with geometrical ideas In general the early stage of algebra from ancient Babylon and Egypt to Greek was mainly geometrical The syncopated stage ndash the beginning of the static equation-solving stage by using In Roman Egypt in probably the second or third century CE the algebra stage was at the syncopated stage which means written algebraic texts were expressed in words but involved special symbols-abbreviations According to recorded mathematics history one the of pioneers who used these special symbols to solve equations with only numbers but no connection with geometry was Diophantus who lived in Alexandria in Egypt around the third century Diophantus used the Greek alphabetic system for writing numbers He wrote a treatise titled Arithmetica of which less than half has been maintained today The surviving part of his work consists of 189 problems in which the object is to find numbers or families of numbers satisfying certain conditions In mathematics today Diophantusrsquo mathematical analysis is known as number theoryndashnot algebra However he used only number symbols to solve equations without the help of geometry Diophantus wrote the coefficient after the variable instead of before it as we do He used Greek ς for unknown quantity our modern x Most of his book deals with indeterminate equations which contain more than one unknown and a potentially infinite number of solutions His problem is that he could not represent more than one unknown instead he solved quadratic equations with two unknowns through substituting one by another and then the quadratic equations became the ones with one unknown At that time negative numbers were regarded to be absurd but he knew how to

7

bring a term from one side of an equation to the other gather up like terms for simplification and some elementary principles of expansion and factorization (Derbyshire 2006) Diophantus made his own literal symbolism with the use of special letter symbols for the unknown and its powers for subtraction and equality From Diophantus algebra history moved into another conceptual stage the equation-solving stage according to Katz (2006) In India quadratic formula appeared without any geometric support Brahmaggupta (598-665) was one of the first mathematicians who could systematically handle negative numbers and zero He gave a general solution to quadratic equations and realized that there were two roots for a quadratic equation It was possible that one of the roots was a negative number Baskharacharya (1114-1185) solved mathematics problems with the use of quadratic equations in his book Siddhanta Siromani (ldquoMathematical Pearlsrdquo) He presented an algorithm to reduce a quadratic equation to a first-degree equation (Olteanu 2007) It is commonly believed that the first true algebra text is the work on al-jabr and al-muqabala by Mohanmmad ibn Musa al-Khwarizmi (780-850) written in Baghdad around 825 (Katz 2006) The word algebra came from the title of this work The word al-jabr means restoration or reestablishment that is to eliminate negative terms through adding the same terms to both sides of equations The word of al-muqabalas means balance meaning to divide every term in a quadratic equation by the coefficient of the second degreersquos term (Olteanu 2007) The first part of his book is a manual for solving linear and quadratic equations Al-Khwarizimi classified equations into six types three of which were mixed quadratic equations For each type he presented an algorithm for its solution Five of the six types of equations were quadratic equations which can be expressed in modern form

cbxaxbxcaxcbxaxcaxbxax +==+=+== 22222 Here is an example of solving the equation ldquoTake the half of the number of the things that is five and multiply it by itself you obtain twenty-five Add this to thirty-nine you get sixty-four Take the square root or eight and subtract from it one half of the number of things which is five The result three is the thingrdquo (Kvasz 2006 p 292) Like Babylonian mathematicians al-Khwarizimirsquos algorithm is entirely verbal On the other hand al-Khwarizimi sometimes completed his algorithm by using geometrical explanations which can be translated as todayrsquos square-completing method Using the example of solving the quadratic equation the completed geometrical procedures are illustrated on next page in figures 3 4 and 5

39102 =+ xx

39102 =+ xx

8

x

xsup2

Figure 3 (Olteanu 2007 p 30) 5x2

25x

xsup2

25x

5x2

Figure 4 (Olteanu 2007 p 30) 254 254

39

254 254

Figure 5 (Olteanu 2007 p 30) According to Olteanursquos (2007) translation al-Khwarizimi started with a square whose sides are x and area is xsup2 (see Figure 3) Then he added four equal rectangles whose areas in total

was 10x along each side of the square that is xx sdotsdot=25410 Each rectanglersquos area is thus x

25

with its length x and its width 52 (see Figure 4) The sum of the big square and four rectangles was given which was 39 The equivalence relationship was Finally figure 4 was completed by adding four small equal squares which had an area the size of

39102 =+ xx

425

25

25

=sdot for each small square and the sum of those was 25 Through adding this sum to

both sides of the equation the area of the biggest square obtained in figure 5 was 64 The

equation translation is 425439

4254102 sdot+=sdot++ xx The side of the biggest square was 8 and

had its relation with other sides of different squares expressed in the first degree equation

9

25

258 ++= x Then x was 3 By ldquocut-and-pasterdquo geometry (Katz 2006 p 191)

al-Khwarizimi reduced the second degree of an equation to the first degree and thereafter solved it Unlike his Babylonian predecessors al-Khwarizimi always presented his problem abstractly rather than geometrically relating to lengths and widths The symbolic stage At this stage of algebra ldquoall numbers operations relationships are expressed through a set of easily recognized symbols and manipulations on the symbols take place according to well-understood rulesrdquo (Katz 2006 p 186) The ancient algebra and geometry had developed sophisticatedly in Egypt Persia Greece India and China After Medieval Islamic scholars gave us the word ldquoalgebrardquo Western Europe began the struggle for the development of algebra starting from some algebraists from Italy Italian mathematician Leonardo Pisano later known as Fibonacci traveled in the 12th and 13th centuries to Persia India and China When he returned to Italy he had wider knowledge of arithmetic and algebra His book Liber abbaci was the best math textbook since the end of Ancient world His book is credited with having introduced Indian numerals including zero to the West But his algebraic skills had been shown in two other works after this one With the introduction of printed books during the second half of the 15th century the development of algebra was sped up Several Italian mathematicians including Girolamo Cardano had figured out how to solve cubic and quadratic equations Algebra became purely abstract with the exception of an English mathematician named Robert Recorde who lived in the 16th century and created quadratic problems from real world experience (Derbyshire 2006) It was in France that algebra had developed into a well organized literal symbolism In his work In artem analyticem isagoge French mathematician Franςois Viegravete (1540-1603) in the late 16th century became the first mathematician to use letters representing numbers systematically and effectively (Derbyshire 2006) He made a range of letters available for many different quantities This was the beginning of modern literal symbolism Viegravetersquos unknown quantity was divided into two classes unknown quantities (meaning ldquothings soughtrdquo) denoted by A E I O U and Y while ldquothings givenrdquo was denoted by constants like B C Dhellip For example his A is our unknown x Viegravete was a pioneer in the study of equations His two papers on the theory of equations were published twelve years after his death In the second paper titled ldquoDe equationem emendationerdquo (ldquoOn the perfecting of equationsrdquo) Viegravete opened up the line of inquiry that led to the study of the symmetries of an equationrsquos solutions to Galois theory the theory of groups and all of modern algebra He found the relationship between the solutions of the equation and the coefficients for the first five degrees of equations in a single unknown To explain this in our modern symbols we suppose that the two solutions of the quadratic equation are 02 =++ qpxx α and β which means βα == xx Based on this logic the following thing must be true 0))(( =minusminus βα xx since only α andβ and no other values of x make this equation true This form of equation is just a rewritten form of the same equation If we multiply out those parentheses this rewritten equation turns to be Compared to the original equation the relationships between the solutions and the coefficients we obtain

0)(2 =++minus αββα xxqp =minus=+ αββα

(Derbyshire 2006) It is said that Viegravete discovered the solution formula called quadratic

10

formula today for general quadratic equations which is a

acbbx2

42

21minusplusmnminus

= for a general

quadratic equation (Olteanu 2007) 002 ne=++ acbxax Another French mathematician and philosopher who had strong influence in the history of algebra was Reneacute Descartes (1596-1650) His idea to use the Cartesian system of coordinates which was derived from his Latin name developed both algebra and geometry In his work La geacuteomeacutetrie written in 1637 Descartesndashthrough using numbers to indentify points in a Cartesian coordinatesndashconnected geometrical objects to algebraic numbers and made the classical geometry become analytical geometry He took up the plus and minus sign from the German Cossists of the previous century and also the square-root sign From Descartes the symbol of an unknown was represented by x which led to the modern system of literal symbolism (Derbyshire 2006) Descartes developed the idea of functions in Cartesian coordinates although the idea of function can be traced back to the Islamic mathematician Sharaf al-Din al-Tusi from Persia (Katz 2006) and Klaudius Ptolemaios about 2000 years ago (Olteanu 2007) Descartes declared that every curve in a Cartesian coordinate system has an equivalent equation which can represent the points on the curve or vice versa every equation containing x and y can be represented by a curve through its coordinate points However the word function was introduced by Gottfried Wilhelm Leibniz in 1693 and the definition of function was defined by Leonhard Euler in his work Introductio analysin infinitorium in 1748 Euler had also given the symbol of a denotation as a relying variable (Olteanu 2007) In the 18)(xf th century after the discovery of calculus by Newton and Leibniz algebra went into an area of analysis ndashthe study of limits infinite sequences and series functions derivatives and integrals (Derbyshire 2006) The purely abstract stage-algebra structure Since 17th century algebra has not anymore solely been about finding solutions for different degrees of equations mathematicians have started to integrate algebra with astronomy and physics Johann Kepler and Galileo Galilei were interested in curves and finding a mechanism for representing motion instead of quantity-like numbers But their arguments were not algebraic-symbolic but geometrical It was Fermat and Descartes who were regarded as the fathers of analytic geometry and who showed how to represent a curve described verbally through algebra analytic geometry and gave a mechanism for representing motion Newton in his Principia picked up on Fermat and Descartesrsquo representation and developed the calculus With the invention of the calculus mathematical problems were solved by curves not just points Algebra grew more and more to present paths of motion In order to be able to judge if the algebraic manipulations were correct axioms were formulated for arithmetic applied for algebraic manipulations In the late 18th century Lagrange introduced the idea of permutations into the search for solutions Therafter Galois developed methods for determining what conditions polynomial equations need to meet to be solvable New abstract algebraic concepts were found like ldquofieldrdquo and ldquogrouprdquo during the early 19th century In 1854 Ceyley gave an axiomatic definition of a group During the 1890s this definition entered textbooks along with the axiomatic definition of a field an idea which had roots in the work by Galois By the beginning of the 20th century algebra became less about finding solutions to equations and more about looking for common structures in many mathematical objects defined by sets of axioms (Katz 2006)

11

In short algebra development took 4000 years It took about 3500 years for mathematicians to start to use algebra symbols in systems Just almost 100 years ago algebra developed into a purely abstract science The long history may give both mathematics educators and students some explanations to why it is so difficult to teach and learn algebra

22 Three solving methods for quadratic equations Already in the previous part I have presented geometrically and rhetorically solving quadratic equations from a historical perspective In order to get more explicit understanding of quadratic equations quadratic formula and factorization from mathematics didactic point of view this part presents three common methods for solving quadratic equations 1) Completing the square 2) Factorization 3) Quadratic formula The reason to present these three methods is that they are the common topics for discussion in the articles searched in this area concerning mathematics education at upper-secondary school 1) Completing the square In two articles Solving quadratic equations by completing squares (Vinogradova 2007) and Geometric approaches to quadratic equations from other times and places (Allaire amp Bradley 2001) the authors advise mathematics teachers to use the ideas of ancient mathematician al-Khwarizmi and his method of completing quadratic equations by geometrical algebra and simplifying it in order to represent it to the students By doing so the teachers relate mathematics quantity to physical objects in a visual way Didactically teachers may start with a concrete example in which students can understand the content visually Later from the concrete example the teachers lead the students to another example expressed in algebraic symbols for example to study a rectangle whose area is )10( +xx and supposing this area is 39 this becomes 39)10( =+xx Four steps are followed to build a new square which is actually called completing the square x 10

Figure 6 x 5

Figure 7

x

x

5 5

12

A Begin with a rectangle of the area )10( +xx that is with the short side as x and the long side )10( +x The shaded area is 10bullx (Figure 6)

B Divide the shaded rectangle into two small shaded rectangles with the size 5x each and move them to each adjacent side of the xsup2 square (Figure 7) The total area is still

x10

)10(39 += xxC In Figure 7 a new square is shaped in the lower-right hand corner with the side 5 or

area 25 By adding this small square to the diagram the large square with an area of is ldquocompletedrdquo Therefore we can write 25)5(22 ++ xx 25)10(2539 ++=+ xx or

The large squarersquos area is now because the side has a length of Therefore

6425)5(22 =++ xx 2)5( +x5+x 38564)5( 2 =rArr=+rArr=+ xxx

After these three steps a solution to this quadratic equation is derived by completing the square (Allaire amp Bradley 2001) 2) Factorization Hoffman (1976) presents three approaches (A B and C) for factorizing quadratic expressions through observing and grouping He takes a second-degree polynomial as an example

273 2 ++ xx

a The inspection method 1) suggests factors of and 23x x3 x 2) 2 suggests factors of 1 and 2 or ndash1 and ndash2 3) Checking the linear term shows that the correct factorization is x7 )2)(13(273 2 ++=++ xxxx

This method according to Hoffman (1976) is based on the studentrsquos ability to simplify by inspecting the indicated product of a pair of linear expressions

b The method of decomposition of the linear term 1) Multiply and 2 to get 23x 26x2) Decompose into the sum of two terms whose product is x7 26x

This gives xxx += 67 3) Factorize by grouping 2)6(3 2 +++ xxx

2)6(3 2 +++ xxx = )

)2()63( 2 +++ xxx

= 2(1)2(3 +++ xxx= )13)(2( ++ xx

c Third method 1) Multiply 3 and 2 to get 6 2) Find two numbers whose product is 6 and whose sum is 7 The numbers are 6 and 1 3) 273 2 ++ xx

=3

)13)(63( ++ xx

= )13)(2( ++ xx

13

These three approaches can not actually be divided as three unrelated approaches especially the third method is seemingly unclear What Hoffman (1976) wants to demonstrate is that all the three approaches are based on observing the relationship among the coefficients and the constant of the polynomial Through finding the product and the sum of two factors the polynomial can be factorized in the product of two binomials In the authorrsquos opinion factorization is heavily dependent on skill in multiplication and using distributive law reversely He means that ldquostudents who do not have reasonably strong skills in multiplication should not be expected to develop strong skills in factorizationrdquo (p 55) Factorizing polynomials is not only for simplifying high-degree polynomials (for example the third or fourth-degree polynomials) but can also be applied for solving quadratic equations Supposing this polynomial forming a quadratic equation Solving this equation is actually to decompose the polynomial into a factoringrsquos form

0273 2 =++ xx0)13)(2( =++ xx

which is equivalent to the polynomial form The essential step to solve this equation is to follow the zero-factor propertyndashalso called null-factor law (nollprodukt in Swedish)ndashthat is for real numbers p and q p bull q = 0 if (and only if) p = 0 or q = 0 It means that either of the binomials on the left side of the equation has to be equal to zero in order to satisfy the

equivalent relation of this quadratic equation The solutions of two roots or 2minus=x31

minus=x

are obtained through making either 02 =+x or 013 =+x 3) Quadratic formula Solving quadratics equations by factorization is constrained within the simple quadratic equations over the whole integers and rational numbers as coefficients What happens when quadratic equations have coefficients that belong to an irrational domain or big quantity The quadratic formula is a solution to such a situation As mentioned previously historically the French mathematician Franccedilois Vieacutete discovered the quadratic formula

aacbbx

242 minusplusmnminus

= ) which is applicable for all kinds of quadratic equations In

Swedish textbooks this formula is written in PQ form by making for the equation

that is

0( nea

1=a

02 =++ qpxx qppx minus⎟⎠⎞

⎜⎝⎛plusmnminus=

2

22 with

abp = and

acq = In Oleatursquos study (2007)

studentsrsquo difficulties in using algebraic symbols no longer is the problem But on the other hand handling the parameters or coefficients in quadratic equations like and rewrite them in the equivalent form with p and q being real numbers becomes obstacles for the students when they study the algebra course at upper-secondary school

02 =++ cbxax02 =++ qpxx

The quadratic formula a

acbbx2

42 minusplusmnminus= has been regarded as the standard method

according to US mathematics teachers Obermeyer (1982) presents a way of how to derive this formula by completing a square through 11 steps (I will not present them here) At the same time an alternate method is also introduced according to the same principle All the methods mentioned so far are based on the square rules The standard method or PQ form might be regarded as an efficient and direct method but it may

222 2)( bababa +plusmn=plusmn

14

lead students to solve quadratic equations in a mechanical way besides having problems with the troublesome plusmn symbolism (Stover 1978) Both Stover (1978) and Olteanu (2007) have suggested that using a graph of quadratic functions to solve quadratic equations is another alternative method

23 Mathematical background of factorization Factorization in algebra structure Among the three solving methods presented above both the geometrical method and quadratic formula can be traced back to algebra history in section 21 What is the trace of factorization in algebra history Factorization was at the last stage in algebra history and belonged to algebra structure (Derbyshire 2006) In Swedish mathematics textbooks the chapter on algebra often includes polynomial factorization and quadratic equations But the factorization in the textbooks is not at the same abstract level as the factorization of algebra structure although they share the same axioms and there are certain connections between these two different levels of factorization How are polynomial and quadratic equations as well as factorization related to each other In order to find the theoretical background of factorizing polynomials this part constitutes a mathematical theoretical review by referring to two books concerning algebra structure (Vretblad 2000 Durbin 1992) and a research study (Bosseacute and Nandakumar 2005) A quadratic equation can be regarded as polynomial equation Solving quadratic equations has much to do with ldquounique factorization theoremrdquo The polynomial

is a second-degree polynomial A polynomial consists of coefficients in whole numbers rational numbers real numbers or complex numbers and variables (sometimes called unknowns) in different degrees as well as constants A polynomial

has symbols as coefficients which are real numbers If all are zero and so

)0(02 ne=++ acbxax

cbxax ++2

nnxaxaxaaxf ++++= )( 2

210 naaa 10

ka 1gek 0)( axf = is constant If all are zero then we say the is a zero polynomial If meaning that is not a zero polynomial and n is the biggest number and then we say that the polynomial is at degrees A zero polynomial has no degree at all According to the definition (Vredblad 2000) if is a polynomial the equation is called an algebraic equation or polynomial equation A solution or a root of this equation is a number

kaf 0nef f

0nena f nf

0)( =xfα so that 0)( =αf Such a number is even

called a zero point (nollstaumllle in Swedish) of the polynomial which means when f α=x the value of the polynomial is zero (p 161) Every polynomial whose degree is equal or greater than 1 can be written as a product of irreducible polynomial In this way through polynomial division a polynomial with n degrees can be expressed from a sum of terms into a product of its denominator and its quotient The process of changing the expressions is factorization for

instance 12

22

+=minusminusminus x

xxx The factoring form is )1)(2(22 minusminus=minusminus xxxx

Factorization of polynomials is analogue to factorization of whole numbers which is very much related to the concept of ring A ring consists of a set with two operations which are a sum and a product of two elements over a field Any polynomial from zero polynomial to n degree-polynomials can be written or operated through addition and multiplication of the

15

elements over a field F including whole numbers or integers rational numbers real numbers and complex numbers In a ring you can multiply any two elements but you canrsquot always divide Factorization is a way to deal with this fact Here are some examples In the ring of integers Z is an example of factorization in some polynomial rings for instance Q[x] the ring of polynomial in one variable with rational numbers as coefficients such as

7214 sdot=

)451)(2

51(8

52

251 2 +minus=minus+ xxxx is factorization Z[x] the ring of polynomial in one

variable with whole numbers as integers such as is factorization )1)(2(22 minusminus=minusminus xxxx The polynomial ring R[x] means the ring of polynomial in one variable with real numbers as coefficients Factorizing of such polynomials implies that the polynomial factors are the elements in the ring of R[x] for instance )2)(2(22 minus+=minus xxx The same thing is true for C[x] the ring of polynomial in one variable with complex numbers as coefficients Thus it is coefficients of polynomials that determine which ring a polynomial belongs to and if it is possible to be factorized in this ring For example number 6 can be factorized in two ways

in the ring of Z and 32times )51)(51( ii +minus in the ring of C (complex numbers) However there are cases in which divisions can not be done within the ring for example 52 or

)3()52( 2 ++minus xxx which means that no element in the ring can multiply with a denominator obtaining the numerator Therefore neither 5 nor can be factorized 522 +minus xx If a polynomial of degree at least one has no other divisors then it is considered to be irreducible or prime (Durbin 1992) Of every polynomial of degree at least one can be written as a product of irreducible polynomials according to unique factorization theorem each polynomial of degree at least one over a field F can be written as an element of F times a product of monic irreducible polynomials over F and except for the order in which these irreducible polynomials are written this can be done in only one way (p 221) Thus a factorable polynomial can be written into a product with another polynomial such as In other words the condition that )()()( xqaxxf minus= )( ax minus is a factor of a polynomial

in a variable x is only true if )(xf 0)( =af For example if and

6)( 2 minus+= xxxf0624)2( =minus+=f 0639)3( =minusminus=minusf the factors of are and )(xf )2( minusx )3( +x

The value of the polynomial is zero which is also expressed as the polynomial having a zero point (nollstaumllle in Swedish) when

)(xfα=x

Based on unique factorization theorem finding solutions of an algebraic equation is equivalent to finding the first-degree factors for a polynomial (Vredblad 2000) In this way solving a quadratic equation can be handled through finding the first-degree factors for the second-degree polynomial and there after finding the roots of the quadratic equation This is often called using factorization to solve quadratic equations Besides unique factorization theorem the connection of using factorization to solve a quadratic equation is that this method relates to prime numbers using distributive law backwards commutative law multiplication division and operating fractions As matter of fact it is about application of factorizing integers in polynomials at upper-secondary school level Since quadratic equations in the mathematics B course at Swedish upper-secondary school are limited to real numbers and often within the whole numbers (integers) many quadratic equations should be able to be solved by factorization according to my own

16

judgment Therefore using factorization for solving quadratic equations is very much related to arithmetic operations number concept and algebra structure It may offer students both mathematical conceptual understanding and operational skill For algebra beginners factorization can be traced back to studentsrsquo early years of mathematics education during which students do not only work with multiplication but also on factoring integers for example 56 = 8 middot 7 = 2 middot 2 middot 2 middot 7 = 4 middot 14 = 2 middot 28 In this example two operations are included multiplication and division Multiplication operation may help students to understand the multiplicative structure of the integers while factoring integers may become meaningful when it is related to divisibility divisors and prime numbers These two operational skills or competence are major techniques for simplifying fractions and finding common denominators Being able to understand and operate factoring integers has laid the essential grounds for algebra beginners to study factorization of polynomials Thus factorization reunites studentsrsquo early mathematics knowledge of factoring integers with later knowledge of algebra structure Factorability In a study on factorability by Bosseacute and Nandakumar (2005) the probability of factorability is reasoned through investigating the range of integers as coefficients of quadratic equations In the quadratic equation a b and c are randomly generated integers within a determined range r such as

002 ne=++ acbxaxyrx lele where x and y are integers When is a

perfect square or where

acb 42 minus

)4( 2 acb minus is an integer quadratics are factorable According to the studyrsquos data it shows that as the range for a b and c increases the probability of factorability of a quadratic with randomly selected integer values for a b and c decreases Bosseacute and Nandakumar confirm that as the range for coefficients expands to infinltltinfinminus r the probablility of factorability of a quadratic with randomly selected integer values for coefficients approaches zero The studyrsquos data collected from college algebra courses and textbooks demonstrates that about 15 of quadratics with integer coefficients [ ]1010minusisinr are factorable Thus about 85 of quadratics can not be factorized In spite of the limitation of factorability within the exercises concerning factoring practice chosen from 27 surveyed college algebra textbooks about 94 of the problems were factorable Within the textbooks 55 of the quadratics to be factored were within the range [-10 10] Even if many quadratic equations are factorable choosing the right factor pairs is time consuming and unnecessary for example

has to be factorized through choosing the right pairs of factors among nine pairs for 36 and 8 pairs of factors for 24 Comparing the three different methods for solving quadratic equations the study suggests that completing the square and quadratic formula are more effective informative and useful than factorization

0245936 2 =++ xx

231 Different kinds of factorizations for solving different quadratic equations How can we find the binomial factors of a quadratic equation or a second-degree polynomial without knowing one of the divisors as a binomial factor The answer is that we have to observe the relations among all coefficients and constants as well as the positive and negative signs before them under the condition of being factorable Most of the coefficients in quadratic equations or polynomials among exercises provided for practicing factoring are integers (here I borrow the same notation r as the one used by Bosseacute and Nandakumar) with

17

their range between ndash10 and 10 that is 10ler In this case it is possible for a student to make a judgment of whether or not a quadratic equation is factorable Five different kinds of factorization depending on five kinds of quadratic equations Factoring quadratic expressions can be divided into five different kinds depending on the type of quadratic equations The coefficients a b p q and constant c illustrated in the following quadratic equations are defined not to be equal to zero thus 000 nenene cba The first kind of quadratic equations is type 1 the obviously factorable type (Bosseacute and Nandakumar 2004) then the equation is factorized as

where the roots are

0002 nene=+ babxax

0)( =+ baxxabxx minus== 21 0

The second kind of quadratic equations is type 2 The factors of this second degree polynomial are so the roots to this type of quadratic

equations are

0)( 22 =minusbax))(()( 22 baxbaxbax minus+=minus

bax plusmn=21 This kind of factorization is based on the difference-of-squares

formula for example can be solved by factorizing into 049 2 =minusx 0)23)(23( =minus+ xx Thus

the roots are 32

32

21 =minus= xx This kind of factorization is also obviously factorable

The third kind of quadratic equations is type 3 This type of equations can be solved directly by factoring into two identical binomials

The double roots obtained from this

factorization are

02)( 22 =+plusmn babxax

0))(()(2)( 222 =plusmnplusmn=plusmn=+plusmn baxbaxbaxbabxax

abx plusmn=21 This method has utilized square rules which are quite easily

observed The forth kind of quadratic equations is type 4 In this equation the coefficient of the -term is 1 with p and q as positive or negative integers Type 4 equations can always be solved by the approaches of completing the square and quadratic formula However they can often be solved by factorization too since the quadratic expressions are factorable by using distributive laws reversely The roots are obtained through finding two factors for q At the same time these two factors become satisfying with the relation so that the sum of factors is equal to That way the two factors become two roots of a quadratic equation Denoting the two factors or two roots as the relationship between coefficient and constant is

02 =++ qpxx2x

qminus

21rr)( 2121 rrprrq +minus=bull= Quadratic equations can be written as the product of

two first-degree binomials or factoring form 0))(()( 212121

22 =minusminus=++minus=++ rxrxrrxrrxqpxx Through the factoring form the roots of this equation are obtained being The core of factorizing type 4 equations is seeking the connections between coefficients and roots which reveal the studentsrsquo basic ability of arithmetic multiplication and addition It is often more effective in this case to use factorization than to use quadratic formula or completing the square Let us have a look at the example In this equation we get

2211 rxrx ==

035122 =+minus xx

18

35)7)(5(12)7()5( =minusminus=minus=minus+minus= qp or 75)75( bull=bullminus= qp (here we can directly get roots) Factorizing the equation we obtain 0)7)(5( =minusminus xx and roots are The procedures of finding factors of the equation often occur in some studentsrsquo minds quickly without writing down the whole computing process but those students often have good skills in arithmetic operations (reference) Compared to using factorization for solving this equation application of quadratic formula seems ineffective A possible problem in this method is how to find the signs before the factors or how to judge if the factors in the parentheses are positive or negative integers In order to check if the solution is correct we can always multiply these two factored binomials by distributive law and see if the expanded polynomial is the same as the original quadratic equation

75 21 == xx

The fifth kind of quadratic equations is type 5 Still the methods of completing squares and quadratic formula can be applied for all type 5 equations but there are cases in which factorization can be used for solving this type of equations within a certain range of integers as mentioned before Sometimes procedures of factorizing such equations can be carried out in our minds quickly like type 4 equations but in many cases they can not As a matter of fact finding two binomial factors for a factorable quadratic equation or a factorable second-degree polynomial is a complicated process and requires systematic work In his article published in Mathematics in School Jackman (2005) demonstrates this systematic procedure via factorizing second-degree polynomials A second-degree polynomial can be written as where are all integers By multiplying out the polynomial becomes thus

0002 nene=++ bacbxax

))((2 srxqpxcbxax ++=++ srqpcbaqsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== Writing the factors of a and c on two lines p q times r s Finding the right pairs of factors for a and c has to be done by listing out the groups of products of different factors for example The factor pairs for 6 are (6 1) (1 6) (3 2) (2 3) and the factor pairs for (-4) are (4 -1) (-4 1) (2 -2) (-2 2) (1 -4) (-1 4) Writing them on two lines p 6 1 3 2 q 4 -4 2 -2 1 -1

456 2 minus+ xx

times r 1 6 2 3 s -1 1 -2 2 -4 4 Then comes calculating the values of ps and qr until the value of b = 5 is obtained The calculating procedures are divided into 4 groups and 6 steps in each group which means 4 times 6 steps in total Here I only give one grouprsquos procedure as an example 6 middot (-1) + 4 middot 1 = -2 6 middot 1 + 1middot (-4) = 2 6 middot (-2) + 1 middot 2 = -10 6 middot 2 + 1middot (-2) = 10 6 middot (-4) + 1 middot 1 = -23 6 middot 4 + 1 middot (-1) = 23 helliphellip After the 24ndashstep calculating procedures two computing procedures obtain the same correct result 3 (ndash1) + 4 2 = 5 (yes) 2 4 + (-1) 3 = 5 (yes) Are they the same From 3 (ndash1) + 4 2 = 5 the polynomial can be factorized into )12)(43( minus+ xx while from 2 4 + (ndash1) 3 = 5 the polynomial can be factorized into )43)(12( +minus xx According to communicative law these two binomial products are the same The shortcoming of this

19

method is that it is too ineffective and complicated The addition of matrixes is almost impossible and unbearable Nevertheless based on the same idea I suggest finding factors by guessing analyzing and checking Using the same example here we can find that one of the binomials must have a negative integer because of ndash4 which implies that 5 is the sum of a positive integer and a negative integer Then looking at a and c the factors of 6 can be (1 6) and (2 3) while the factors of 4 can be (1 4) and (2 2) when we regard them as absolute values We know that the factoring form of the polynomial will be like

and the connections of all coefficients are

456 2 minus+ xx

qsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== The coefficient 5=b is the difference of two products of factors for 6 and 4 First we try the biggest number of product such as 2446 =bull then 524 =minus the unknown is which is a prime number the only factors for 19 are (19 1) There is not any combination of products of factors for 6 and 4 satisfying 19 Therefore the multiplications of factors (1 6) and (1 4) are not included in this case So we move to factors (2 2) (2 3) and (1 4)

19 =

In the second step we try The result is negative it is obviously

4264466)23)(22( 22 minusminus=minus+minus=+minus xxxxxxx2 lt 5 which implies that the combination of (2 3) and (2 2) is not

considerable One pair of factors has to be removed and that is (2 2) which does not influence the coefficient of the -term Then we try The result this time is negative too but we can see that

2x 456)43)(12( 2 minusminus=minus+ xxxxb is 5 which guides us to interchanging the positions of

negative and positive signs in the parentheses The next attempt is and it is correct Seeking the correct factors for the polynomial

has taken four major operating steps by trying the combinations of factorsrsquo products and relating the sum of the products with the value of the x-term coefficient The strategy is to make use of distributive law reversely first and then multiply the product of every pair of parentheses in order to check if they are the right pair of polynomial factors Factorizing second-degree polynomials is the major step for solving quadratic equations After factoring polynomials into the product of two binomials as in the example given above the roots can be obtained by making each binomial equal to zero according to the theory of zero point (ldquonollstaumlllerdquo in Swedish) of a polynomial equation originating from unique factorization theorem

456)43)(12( 2 minus+=+minus xxxx456 2 minus+ xx

Factorization by using the technique of vertically and crosswise sutra Binomial expansion or multiplication of binomials by distributive law and factorization of the second-degree polynomials by using distributive law backwards have an extremely symmetrical effect and the results of two kinds of operations are equivalent It is crucial for the students to be able to perceive mathematics structure and be acquainted with arithmetic operations as well as to understand the relationship between quantities when they carry out factoring procedures The expansion of binomials by distributive law has been named as the method FOIL which stands for first outside inside and last according to multiplication orders of two binomials for example (Nataraj and Thomas 2006) Many students are used to this procedure Another technique to multiply two binomials is called vertically and crosswise sutra origin from the ancient Indian Vedic mathematics Vertically and crosswise sutra and FOIL are much the same in principle but it makes use of

824)4)(2( 2 +++=++ xxxxx

20

the power of visualization and improves facility with the expansion of binomials and the factorization of quadratic expressions In this method two binomials are written into two lines Using the same example the two lines are written as 824)4)(2( 2 +++=++ xxxxx

x + 2 x + 4 ______________ xsup2 4x + 2x 8

The order of multiplication is as shown below x + 2 x + 4

4 3 2 1 (the numbers 1 2 3 and 4 refer to steps) xsup2 + 2x + 4x + 8 Figure 3 The vertically and crosswise method of multiplication of binomials Step 2 and 3 can be interchangeable If we put this form into a line expanded according to the order above would be like

)4)(2( ++ xxxbullrarrbull 442 clockwise and xxx bullrarrbull2 in

counterclockwise direction According to sutra we can imagine drawing a thread starting from 2 then going through 4 and x clockwise and then draw the second thread starting from the same place 2 later going through x in the second bracket and finally x in the first bracket counterclockwise Learning expansion of binomials by the methods FOIL and Vertical and clockwise sutra is the basic process before students study factorization In this study Nataraj and Thomas (2006) have found that the students appear to do factorization using guess-and-check as well as decomposition or the vertical and clockwise approach Their finding is that afterwards the students performed better on algebra questions especially on the factorizations They conclude ldquohellipthe values of the method may lie in what it adds to the studentsrsquo overall algebraic conceptions and knowledge of mathematical structurerdquo (p 16) In general solving these five kinds of quadratic equations by factorization can be traced back to unique factorization theorem The important character in this method is that you can always check if factorization is correct or not through multiplying out the two pairs of a parenthesis and see if the expanded polynomial is the given one or not Using factorization is not only about method but also about fostering the studentsrsquo mathematical thinking The solving processes are more important than solutions

21

3 Review of related literature and research Besides the focus on mathematics content another focus in the background work for this research is about previous studies and research in the field of mathematics didactic concerning different methods for solving quadratic equations and the use of factorization In order to study the previous research and studies in the related field three scientific databases and a search engine have been used and 113 articles have been investigated Topics such as solving quadratic expressions and equations are often discussed as teaching ideas and strategies shared by mathematics educators in different countries from lower and upper-secondary schools to colleges according to the findings in the article search These didactic topics from mathematics teachersrsquo perspectives include for example using geometric approaches to solve quadratic equations (Allaire amp Bradley 2001) utilizing completing squares (Vinogradova 2007) factoring quadratics (Hoffman 1976 Cheng 1980) and using factorization completing the square graphical methods to solve quadratic equations (Macdonald 1986) Kostsopoulos (2007) links cognitive reasoning to pedagogical problems appearing in the mathematics classrooms in her article published in Australian Mathematics Teacher Based on her teaching experiences Kostsopoulos is aware of her pedagogical strategies lacking insight on dealing with studentsrsquo difficulties of multiplication and factorization of quadratics She assumes that ldquostudentsrsquo problems with factorization and with identifying varied representations of the same quadratic relationshiprdquo may be linked to the ways in which the brain constructs cognitive representations (p 19) Factoring of quadratics is the rewriting of polynomials as a product of polynomials It requires students to have both a conceptual understanding of multiplication of polynomials and the procedural knowledge to retrieve basic multiplication facts effectively Based on the theory about the order matters in the brainrsquos ability to retrieve number facts (Phenix amp Campbell 2001) Kostsopoulos conjectures that studentsrsquo ability to access the appropriate long-term semantic memory is limited when they are confused by mixing the order and varied forms of quadratics Although this article is not about a research study it seeks the reason why factorizing quadratics is difficult and therefore might provide math teachers with alternative explanations for similar pedagogical frustrations With Von Glasersfeldrsquos radical constructivism and Peter Gaumlrdenforsrsquo epistemic semantics theory as his theoretical frameworks Rauff (1994) suggests that belief-based teaching can be successful in teaching factoring According to Rauff constructivist learning theory takes the view that students construct their own beliefs and knowledge of mathematics Learning occurs when the students change their belief set Referring to belief set theory from Gaumlrdenfors (1988) Rauff (1994) claims that ldquoa belief set is expanded when a new belief is added to it and contracted when a proposition is no longer believed and is removed from the setrdquo (p 421) Expansion contraction and revision are three components to comprise basic mechanisms of changes of belief according to Rauff When examining and comparing studentsrsquo (ages 18-20) own definitions of factoring Rauff analyzes the errors of their definitions with constructivism and epistemic semantics as analyzing tools The result of this study shows that the approach to teaching factoring rely heavily upon what a student believes about factoring is quite productive in two dimensions First it reveals the source of nonstandard factoring Second it provides a starting point for modification of the studentrsquos underlying conceptions of factoring Rauff proposes that belief-based teaching can also be

22

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

Planeringsseminariumsunderlag 2009 Wang Wei Soumlnnerhed April 6 2009

1 AIM AND BACKGROUND

11 Introduction On a personal level my research interest started with visits to some Swedish mathematics classrooms a few years ago when I was a teacher student studying to become a mathematics teacher at upper-secondary school During these visits the students were studying the use of the quadratic formula for solving quadratic equations usually called the PQ formula in school Teaching focus seemed to be put much on how to use PQ formula to solve quadratic equations which is quite different from my own mathematics learning experience in China When I studied the same content in middle school in China I was taught to try the factorization method first and then use PQ formula later if it did not work with factorization for solving a quadratic equation The different teaching focus on using different methods to solve quadratic equations lays the ground of my interest for this research In mathematics solving quadratic equations belongs to the field of algebra There are different methods to solve quadratic equations such as the geometrical method through completing squares using quadratic formula PQ formula factorization using graphs of quadratic functions to solve quadratic equations This is a rich but complicated area in algebra How these specific mathematical contents are presented in secondary-school teaching and what sources the teachers have based their lessons on are questions relating to mathematical content and pedagogical-content issues With the starting point of teaching this research needs to consider both the field of algebra as a subject and the subject related resources available for teaching in this area These two research focuses are components of Pedagogical Content Knowledge (Shulman 1986 p 9) One category of PCK according to Shulman is ldquothe ways of representing and formulating the subject that make it comprehensible to othersrdquo (p 9) Hence an important question arises What forms of representation of a specific lesson topic are available for teachers This is an essential point for what this research is about In order to find different forms of representation and different methods for solving quadratic equations I have chosen to investigate and analyze mathematics textbooks in the Mathematics B course and the related teacherrsquos guides for Swedish upper-secondary school Analyzing mathematics textbooks and teaching material is not just to investigate the facts or concepts in the field of algebra concerning different solving methods but also to find the pedagogical structure of this subject The research therefore has its purpose to explore pedagogical content knowledge in the textbooks and teaching material which is related to a teacherrsquos knowledge

12 Why is it about algebra School algebra is a difficult area in the Swedish studentsrsquo studies of mathematics (Haumlggstroumlm 2006) Why is algebra difficult According to Haumlggstroumlm the algebraic circle (Bergsten et al 2002) shows that students first ought to translate problems expressed in daily words into algebraic structures by using algebraic symbols Thereafter the students should formulate the algebraic structures with certain given rules Then comes solving the problems All of these

2

three steps require that the students are able to handle symbols and concepts and that they have the necessary skills needed for operations as long as they understand the contents of the problems Operations are based on the early knowledge of arithmetic (Haumlggstroumlm 2006) Swedish students learn elementary algebra at the beginning of upper-secondary school The subject mathematics is divided into five levels from A to E At A level called Mathematics A students are required to be able to translate simplify and reform expressions of quadratic equations according to the Swedish syllabus for Mathematics A (Skolverket 2000) At the B level students should go more deeply into the learning of algebra and functions During this course the students should be able to tolka foumlrenkla och omforma uttryck av andra graden samt loumlsa andragradsekvationer och tillaumlmpa kunskaperna vid problemloumlsning (Skolverket 2000 p 83) The aim of Mathematics B emphasizes the importance of solving quadratic equations It is during this study period that students build up most of their knowledge of algebra At upper-secondary schools in Sweden students are taught to use the approaches of completing the square and the quadratic formula called PQ formula to informally solve quadratic equations as well as factorization The application of the graphical method is introduced in the chapter on functions at the mathematics B course Applying PQ formula is an efficient way when solving all kinds of different quadratic equations On the one hand quadratic formula has become a powerful mathematics instrument for students who can use it freely without paying much attention to equationsrsquo structures and operational procedures but on the other hand as a result knowing why and how to use other methods to solve quadratic equations may become less important in practice something which might reduce studentsrsquo opportunities to think mathematically The problem with solving a quadratic equation by either quadratic formula or calculators without knowing why and how causes a dilemma in mathematics learning and teaching From teachersrsquo point of views it is important to make students understand how to solve quadratic equations in order to develop their mathematics thinking But for students quickly and easily finding solutions for quadratic equations might be their goals The need for finding correct and efficient methods may change the character of school mathematics and therefore lead mathematics teaching to a more practical character than pure mathematical science Compared to quadratic formula factorization in algebra content may not serve as an efficient tool for solving quadratic equations but can probably increase studentsrsquo algebra structure sense Factorization and quadratic formula belong to this dilemma area

13 The choice of which mathematics textbooks to investigate Some international studies have compared the effectiveness of using different methods to solve quadratic equations from a mathematics-didactics point of view How many methods are usually taught for solving quadratic equations in Swedish upper-secondary mathematics classrooms How do these different methods relate to each other and thus influence studentsrsquo understanding of quadratic equations and algebra structure Why are some methods emphasized in teaching at the same time as some others are not These questions concern two fields one is mathematics as a scientific discipline and the other is mathematics didacticsndashmainly normative didactics which ldquoinvolves discussions about the educational goals choice of content and methods but should also include justifications and recommendationsrdquo (Johansson 2006 p 12) To be able to answer these questions I need to find resources covering these two fields That means mathematics textbooks The previous Swedish study

3

(Johansson 2006) has found that the mathematics textbook is the most influential factor in classroom teaching and learning According to Johansson (2006) textbooks seem to dictate the teaching of mathematics in many aspects The use of textbooks is a very important framework in mathematics teaching Textbooks influence not only what kind of tasks students are working with and the examples presented by the teachers but also how mathematics is portrayed in terms of the concepts and the features that are related to the subject (p 26) From previously done research Johansson (2006) has found the importance of analyzing the content of textbooks because of the following reasons ldquomathematical topics in textbooks are most likely presented by the teachersrdquo ldquoTeachersrsquo pedagogical strategies are often influenced by the instructional approach of the materialrdquo ldquoTeachers sequence of instruction are often parallels to that of the textbookrdquo and ldquoTeachers report that textbooks are a primary information source in deciding how to present contentrdquo (p 48) The same mathematical topics can be emphasized in one book but might be overlooked in another An analysis of mathematics textbooks reveals the implied beliefs of what mathematics is and how it can be taught and learned It also reflects on the influence of the educational culture from a specific country It is obvious that mathematics textbooks play important roles in Swedish classrooms though not all teachers are the slaves of the textbooks Since mathematics textbooks are important sources for teaching and learning mathematics in Swedish classrooms I have decided to start my study by investigating eight mathematics textbooks used for the course Mathematics B in Swedish upper-secondary schools and by analyzing three of them in detail The related teaching guide material will be investigated in next step

14 Research Aim

The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3

2 Mathematics background in the field of algebra

21 Algebra history and its development Quadratic equations and their unknowns as well as polynomials belong to the field of algebra in mathematics In Swedish mathematics textbooks for mathematics course B there is one chapter on algebra Many critical elements of algebra have been presented in every book but there are no descriptions or definitions of the word ldquoalgebrardquo What is algebra Colin Maclaurin in his 1748 algebra text defined it as ldquoAlgebra is a general Method of Computation by certain Signs and Symbols which have been contrived for this Purpose and found convenient It is called an Universal Arithmetic and proceeds by Operations and Rules similar to those in Common Arithmetic founded upon the same Principlesrdquo (Katz 2006 p 185) Leonhard Euler in his own algebra text in 1770 defined algebra as ldquoThe science

4

which teaches how to determine unknowns quantities by means of those that are knownrdquo (Katz 2006 p 185) In this part I will mainly give a survey of the historical development of algebra including solving quadratic equations according to four stages the rhetorical stage the syncopated stage the symbolic stage and the purely abstract stage The purpose is to try to find out why algebra in school is like a threshold which hinders the students in their understanding Can a historical perspective on algebra explain this obstacle Algebra is an old science and its historical background is complicated Algebra procedures have developed slowly There are different opinions about where the evolution of the term ldquoalgebrardquo started It is commonly believed that algebra first appeared among the Egyptians the Babylonians the Greeks or the Arabs The geometrical influence on algebraic reasoning was strong in ancient Greece However the word algebra originated in Baghdad where the Arabic scientist al-Khwarizmi (AD 780-850) published a short book about calculating with the help from al-jabr and al-muqabala a book on the solution of an equation as a rule (Kvasz 2006) Todayrsquos algebra has its root in Arabic algebra Western mathematics tended to turn algebraic operations into symbols and later developed abstract algebra The process of algebra development was slow and its whole history lasted 4000 years

The rhetorical stage originated from geometry ideas

Historically algebra developed in three stages the rhetorical stage the syncopated stage and the symbolic stage According to Katz (2006) algebra development can also be categorized into four conceptual stages ldquothe geometric stage where most of the concepts of algebra are geometric the static equation-solving stage where the goal is to find numbers satisfying certain relationship the dynamic function stage where motion seems to be an underlying idea and finally the abstract stage where structure is the goalrdquo (p 186) Algebra began very early in recorded history Algebra texts date from the first half of the second millennium BCE 37 or 38 centuries ago and were written by people living in Mesopotamia and Egypt (Derbyshire 2006) During the Hammurabi period from about 1790 to 1600 BCE the Babylonians started their civilization by pressing written words in patterns called cuneiform or wedge-shaped stylus into wet clay Many tablets in cuneiform had a mathematical-algebraic content Their mathematical texts were of two kinds table texts and problem texts The table texts were lists of multiplication tables tables of squares and cubes as well as advanced lists like the famous Plimpton 322 tablet which is about Pythagorean triples The number system of the Babyloniansrsquo time was based on 60 digits rather than todayrsquos 10 digits for example our number of 37 denotes three sixties and seven ones which is our number 187 The problem was that the Babylonians had neither defined zero nor negative numbers The Babylonians of Hammurabirsquos era had no proper algebraic symbolism All mathematical problems were expressed in words for example unknown quantity in Sumerianrsquos Akkadian text was expressed as igum (length) and igibum (width) as reciprocal The application of algebra might have its origin in the need for measuring land areas At the rhetorical stage all mathematical statements and arguments were expressed in words and sentences (Derbyshire 2006) Babylonian mathematics had two roots one is accountancy problems and the other one is a ldquocut-and-pasterdquo geometry probably developed for understanding the division of land Many old-Babylonian clay tablets contain quadratic problems of which the goal was to find such geometric quantities as the length and width of a rectangle As an example we are given that the sum of the length and width of a rectangle is

5

216 and the area of the rectangle is

217 (Kvatz 2006 p 187 and Derbyshire 2006

pp 25-27) What is the length and the width of this rectangle The tablet described in detail

the steps the writer went through First the writer halves 216 to get

413 Next he squares

413

to get 16910 From this area he subtracts the given area

217 giving

1613 The square root of

this number is extracted431 Finally the length is 5

431

413 =+ while the width is

211

431

413 =minus (Kvatz 2006) The whole process can be translated into parts of a quadratic

formula that is 2172

216

431

2

minus⎟⎠⎞

⎜⎝⎛ divide= The Babylonians did not know anything about

negative numbers the only solution for them was in positive numbers and hence their algorithm did not deliver the two solutions to the quadratic equation so their formula is slightly different from quadratic formula (Derbyshire 2006) Their formula is

⎟⎠⎞

⎜⎝⎛ divideplusmnminus⎟

⎠⎞

⎜⎝⎛ divide= 2

216

2172

216

2

x If we denote the sum of the length and the width of the

rectangle as b and the given area as c this formula will be ⎟⎠⎞

⎜⎝⎛plusmnminus⎟

⎠⎞

⎜⎝⎛=

22

2 bcbx although it is

not exactly like the modern quadratic formula There are different interpretations of Neugebauer and Sachesrsquo translation of the Babylonianrsquos tablets for this text on finding the length and width of a rectangle However it is very clear that the text from the tablets is dealing with a geometric procedure The problem was solved in words but with geometric ideas This was the beginning of algebra The Greek mathematician Euclid (300 BC) in his Book II of Elements solved some algebraic problems by manipulating geometric figures but based them on clearly stated axioms The geometrical method is in Kvatzrsquos opinion more explicit in another work of his Data The following example illustrates how Euclid solved a quadratic equation by a geometrical method Euclid defined ldquoproposition 1rdquo which is like axiom 1 If two straight lines contain a given area in a given angle and if the sum of them be given then shall each of them be given (Kvatz 2006 p 189) Euclid set up a rectangle with one side x = AS and y = AC Then a line was drawn so that BS = AC and the completed rectangle was ACDB Suppose that

was given and the area of rectangle ACFS (Figure 1) was given denoted as c What were the two sides AS (x) and AC (y) of the rectangle

byxAB =+=

A x S y B

y y

C F D

Figure 1

6

In order to find the length and the width of the rectangle Euclid bisected AB at E constructed the square on BE and then claimed that this square was equal to the sum of the rectangle ACFS and the small shaded square at the bottom (Figure 2)

A E S B

x

y y F

C G D Figure 2 According to Euclid the area of the rectangle ACFS was given which was c and the area of

the new square EGDB was also given which was ( )22b because 222byxSBASEB =

+=

+=

The equivalent relationship between the areas can be formulated as an quadratic equation 22

22⎟⎠⎞

⎜⎝⎛ minus+=⎟

⎠⎞

⎜⎝⎛ bxcb or

22

22⎟⎠⎞

⎜⎝⎛ minus+=⎟

⎠⎞

⎜⎝⎛ ybcb Euclid found out this equivalent relationship

geometrically and took use of this relationship in finding the solutions of the problem so the

length and width of the rectangle ACFS are cbbx minus⎟⎠⎞

⎜⎝⎛+=

2

22 and cbby minus⎟

⎠⎞

⎜⎝⎛minus=

2

22

These two formulas are almost identical with the Babyloniansrsquo solutions in rhetoric expressions The difference is that Greek algebra was based on geometric manipulation while Babylonian algebra was based on rhetoric manipulation with geometrical ideas In general the early stage of algebra from ancient Babylon and Egypt to Greek was mainly geometrical The syncopated stage ndash the beginning of the static equation-solving stage by using In Roman Egypt in probably the second or third century CE the algebra stage was at the syncopated stage which means written algebraic texts were expressed in words but involved special symbols-abbreviations According to recorded mathematics history one the of pioneers who used these special symbols to solve equations with only numbers but no connection with geometry was Diophantus who lived in Alexandria in Egypt around the third century Diophantus used the Greek alphabetic system for writing numbers He wrote a treatise titled Arithmetica of which less than half has been maintained today The surviving part of his work consists of 189 problems in which the object is to find numbers or families of numbers satisfying certain conditions In mathematics today Diophantusrsquo mathematical analysis is known as number theoryndashnot algebra However he used only number symbols to solve equations without the help of geometry Diophantus wrote the coefficient after the variable instead of before it as we do He used Greek ς for unknown quantity our modern x Most of his book deals with indeterminate equations which contain more than one unknown and a potentially infinite number of solutions His problem is that he could not represent more than one unknown instead he solved quadratic equations with two unknowns through substituting one by another and then the quadratic equations became the ones with one unknown At that time negative numbers were regarded to be absurd but he knew how to

7

bring a term from one side of an equation to the other gather up like terms for simplification and some elementary principles of expansion and factorization (Derbyshire 2006) Diophantus made his own literal symbolism with the use of special letter symbols for the unknown and its powers for subtraction and equality From Diophantus algebra history moved into another conceptual stage the equation-solving stage according to Katz (2006) In India quadratic formula appeared without any geometric support Brahmaggupta (598-665) was one of the first mathematicians who could systematically handle negative numbers and zero He gave a general solution to quadratic equations and realized that there were two roots for a quadratic equation It was possible that one of the roots was a negative number Baskharacharya (1114-1185) solved mathematics problems with the use of quadratic equations in his book Siddhanta Siromani (ldquoMathematical Pearlsrdquo) He presented an algorithm to reduce a quadratic equation to a first-degree equation (Olteanu 2007) It is commonly believed that the first true algebra text is the work on al-jabr and al-muqabala by Mohanmmad ibn Musa al-Khwarizmi (780-850) written in Baghdad around 825 (Katz 2006) The word algebra came from the title of this work The word al-jabr means restoration or reestablishment that is to eliminate negative terms through adding the same terms to both sides of equations The word of al-muqabalas means balance meaning to divide every term in a quadratic equation by the coefficient of the second degreersquos term (Olteanu 2007) The first part of his book is a manual for solving linear and quadratic equations Al-Khwarizimi classified equations into six types three of which were mixed quadratic equations For each type he presented an algorithm for its solution Five of the six types of equations were quadratic equations which can be expressed in modern form

cbxaxbxcaxcbxaxcaxbxax +==+=+== 22222 Here is an example of solving the equation ldquoTake the half of the number of the things that is five and multiply it by itself you obtain twenty-five Add this to thirty-nine you get sixty-four Take the square root or eight and subtract from it one half of the number of things which is five The result three is the thingrdquo (Kvasz 2006 p 292) Like Babylonian mathematicians al-Khwarizimirsquos algorithm is entirely verbal On the other hand al-Khwarizimi sometimes completed his algorithm by using geometrical explanations which can be translated as todayrsquos square-completing method Using the example of solving the quadratic equation the completed geometrical procedures are illustrated on next page in figures 3 4 and 5

39102 =+ xx

39102 =+ xx

8

x

xsup2

Figure 3 (Olteanu 2007 p 30) 5x2

25x

xsup2

25x

5x2

Figure 4 (Olteanu 2007 p 30) 254 254

39

254 254

Figure 5 (Olteanu 2007 p 30) According to Olteanursquos (2007) translation al-Khwarizimi started with a square whose sides are x and area is xsup2 (see Figure 3) Then he added four equal rectangles whose areas in total

was 10x along each side of the square that is xx sdotsdot=25410 Each rectanglersquos area is thus x

25

with its length x and its width 52 (see Figure 4) The sum of the big square and four rectangles was given which was 39 The equivalence relationship was Finally figure 4 was completed by adding four small equal squares which had an area the size of

39102 =+ xx

425

25

25

=sdot for each small square and the sum of those was 25 Through adding this sum to

both sides of the equation the area of the biggest square obtained in figure 5 was 64 The

equation translation is 425439

4254102 sdot+=sdot++ xx The side of the biggest square was 8 and

had its relation with other sides of different squares expressed in the first degree equation

9

25

258 ++= x Then x was 3 By ldquocut-and-pasterdquo geometry (Katz 2006 p 191)

al-Khwarizimi reduced the second degree of an equation to the first degree and thereafter solved it Unlike his Babylonian predecessors al-Khwarizimi always presented his problem abstractly rather than geometrically relating to lengths and widths The symbolic stage At this stage of algebra ldquoall numbers operations relationships are expressed through a set of easily recognized symbols and manipulations on the symbols take place according to well-understood rulesrdquo (Katz 2006 p 186) The ancient algebra and geometry had developed sophisticatedly in Egypt Persia Greece India and China After Medieval Islamic scholars gave us the word ldquoalgebrardquo Western Europe began the struggle for the development of algebra starting from some algebraists from Italy Italian mathematician Leonardo Pisano later known as Fibonacci traveled in the 12th and 13th centuries to Persia India and China When he returned to Italy he had wider knowledge of arithmetic and algebra His book Liber abbaci was the best math textbook since the end of Ancient world His book is credited with having introduced Indian numerals including zero to the West But his algebraic skills had been shown in two other works after this one With the introduction of printed books during the second half of the 15th century the development of algebra was sped up Several Italian mathematicians including Girolamo Cardano had figured out how to solve cubic and quadratic equations Algebra became purely abstract with the exception of an English mathematician named Robert Recorde who lived in the 16th century and created quadratic problems from real world experience (Derbyshire 2006) It was in France that algebra had developed into a well organized literal symbolism In his work In artem analyticem isagoge French mathematician Franςois Viegravete (1540-1603) in the late 16th century became the first mathematician to use letters representing numbers systematically and effectively (Derbyshire 2006) He made a range of letters available for many different quantities This was the beginning of modern literal symbolism Viegravetersquos unknown quantity was divided into two classes unknown quantities (meaning ldquothings soughtrdquo) denoted by A E I O U and Y while ldquothings givenrdquo was denoted by constants like B C Dhellip For example his A is our unknown x Viegravete was a pioneer in the study of equations His two papers on the theory of equations were published twelve years after his death In the second paper titled ldquoDe equationem emendationerdquo (ldquoOn the perfecting of equationsrdquo) Viegravete opened up the line of inquiry that led to the study of the symmetries of an equationrsquos solutions to Galois theory the theory of groups and all of modern algebra He found the relationship between the solutions of the equation and the coefficients for the first five degrees of equations in a single unknown To explain this in our modern symbols we suppose that the two solutions of the quadratic equation are 02 =++ qpxx α and β which means βα == xx Based on this logic the following thing must be true 0))(( =minusminus βα xx since only α andβ and no other values of x make this equation true This form of equation is just a rewritten form of the same equation If we multiply out those parentheses this rewritten equation turns to be Compared to the original equation the relationships between the solutions and the coefficients we obtain

0)(2 =++minus αββα xxqp =minus=+ αββα

(Derbyshire 2006) It is said that Viegravete discovered the solution formula called quadratic

10

formula today for general quadratic equations which is a

acbbx2

42

21minusplusmnminus

= for a general

quadratic equation (Olteanu 2007) 002 ne=++ acbxax Another French mathematician and philosopher who had strong influence in the history of algebra was Reneacute Descartes (1596-1650) His idea to use the Cartesian system of coordinates which was derived from his Latin name developed both algebra and geometry In his work La geacuteomeacutetrie written in 1637 Descartesndashthrough using numbers to indentify points in a Cartesian coordinatesndashconnected geometrical objects to algebraic numbers and made the classical geometry become analytical geometry He took up the plus and minus sign from the German Cossists of the previous century and also the square-root sign From Descartes the symbol of an unknown was represented by x which led to the modern system of literal symbolism (Derbyshire 2006) Descartes developed the idea of functions in Cartesian coordinates although the idea of function can be traced back to the Islamic mathematician Sharaf al-Din al-Tusi from Persia (Katz 2006) and Klaudius Ptolemaios about 2000 years ago (Olteanu 2007) Descartes declared that every curve in a Cartesian coordinate system has an equivalent equation which can represent the points on the curve or vice versa every equation containing x and y can be represented by a curve through its coordinate points However the word function was introduced by Gottfried Wilhelm Leibniz in 1693 and the definition of function was defined by Leonhard Euler in his work Introductio analysin infinitorium in 1748 Euler had also given the symbol of a denotation as a relying variable (Olteanu 2007) In the 18)(xf th century after the discovery of calculus by Newton and Leibniz algebra went into an area of analysis ndashthe study of limits infinite sequences and series functions derivatives and integrals (Derbyshire 2006) The purely abstract stage-algebra structure Since 17th century algebra has not anymore solely been about finding solutions for different degrees of equations mathematicians have started to integrate algebra with astronomy and physics Johann Kepler and Galileo Galilei were interested in curves and finding a mechanism for representing motion instead of quantity-like numbers But their arguments were not algebraic-symbolic but geometrical It was Fermat and Descartes who were regarded as the fathers of analytic geometry and who showed how to represent a curve described verbally through algebra analytic geometry and gave a mechanism for representing motion Newton in his Principia picked up on Fermat and Descartesrsquo representation and developed the calculus With the invention of the calculus mathematical problems were solved by curves not just points Algebra grew more and more to present paths of motion In order to be able to judge if the algebraic manipulations were correct axioms were formulated for arithmetic applied for algebraic manipulations In the late 18th century Lagrange introduced the idea of permutations into the search for solutions Therafter Galois developed methods for determining what conditions polynomial equations need to meet to be solvable New abstract algebraic concepts were found like ldquofieldrdquo and ldquogrouprdquo during the early 19th century In 1854 Ceyley gave an axiomatic definition of a group During the 1890s this definition entered textbooks along with the axiomatic definition of a field an idea which had roots in the work by Galois By the beginning of the 20th century algebra became less about finding solutions to equations and more about looking for common structures in many mathematical objects defined by sets of axioms (Katz 2006)

11

In short algebra development took 4000 years It took about 3500 years for mathematicians to start to use algebra symbols in systems Just almost 100 years ago algebra developed into a purely abstract science The long history may give both mathematics educators and students some explanations to why it is so difficult to teach and learn algebra

22 Three solving methods for quadratic equations Already in the previous part I have presented geometrically and rhetorically solving quadratic equations from a historical perspective In order to get more explicit understanding of quadratic equations quadratic formula and factorization from mathematics didactic point of view this part presents three common methods for solving quadratic equations 1) Completing the square 2) Factorization 3) Quadratic formula The reason to present these three methods is that they are the common topics for discussion in the articles searched in this area concerning mathematics education at upper-secondary school 1) Completing the square In two articles Solving quadratic equations by completing squares (Vinogradova 2007) and Geometric approaches to quadratic equations from other times and places (Allaire amp Bradley 2001) the authors advise mathematics teachers to use the ideas of ancient mathematician al-Khwarizmi and his method of completing quadratic equations by geometrical algebra and simplifying it in order to represent it to the students By doing so the teachers relate mathematics quantity to physical objects in a visual way Didactically teachers may start with a concrete example in which students can understand the content visually Later from the concrete example the teachers lead the students to another example expressed in algebraic symbols for example to study a rectangle whose area is )10( +xx and supposing this area is 39 this becomes 39)10( =+xx Four steps are followed to build a new square which is actually called completing the square x 10

Figure 6 x 5

Figure 7

x

x

5 5

12

A Begin with a rectangle of the area )10( +xx that is with the short side as x and the long side )10( +x The shaded area is 10bullx (Figure 6)

B Divide the shaded rectangle into two small shaded rectangles with the size 5x each and move them to each adjacent side of the xsup2 square (Figure 7) The total area is still

x10

)10(39 += xxC In Figure 7 a new square is shaped in the lower-right hand corner with the side 5 or

area 25 By adding this small square to the diagram the large square with an area of is ldquocompletedrdquo Therefore we can write 25)5(22 ++ xx 25)10(2539 ++=+ xx or

The large squarersquos area is now because the side has a length of Therefore

6425)5(22 =++ xx 2)5( +x5+x 38564)5( 2 =rArr=+rArr=+ xxx

After these three steps a solution to this quadratic equation is derived by completing the square (Allaire amp Bradley 2001) 2) Factorization Hoffman (1976) presents three approaches (A B and C) for factorizing quadratic expressions through observing and grouping He takes a second-degree polynomial as an example

273 2 ++ xx

a The inspection method 1) suggests factors of and 23x x3 x 2) 2 suggests factors of 1 and 2 or ndash1 and ndash2 3) Checking the linear term shows that the correct factorization is x7 )2)(13(273 2 ++=++ xxxx

This method according to Hoffman (1976) is based on the studentrsquos ability to simplify by inspecting the indicated product of a pair of linear expressions

b The method of decomposition of the linear term 1) Multiply and 2 to get 23x 26x2) Decompose into the sum of two terms whose product is x7 26x

This gives xxx += 67 3) Factorize by grouping 2)6(3 2 +++ xxx

2)6(3 2 +++ xxx = )

)2()63( 2 +++ xxx

= 2(1)2(3 +++ xxx= )13)(2( ++ xx

c Third method 1) Multiply 3 and 2 to get 6 2) Find two numbers whose product is 6 and whose sum is 7 The numbers are 6 and 1 3) 273 2 ++ xx

=3

)13)(63( ++ xx

= )13)(2( ++ xx

13

These three approaches can not actually be divided as three unrelated approaches especially the third method is seemingly unclear What Hoffman (1976) wants to demonstrate is that all the three approaches are based on observing the relationship among the coefficients and the constant of the polynomial Through finding the product and the sum of two factors the polynomial can be factorized in the product of two binomials In the authorrsquos opinion factorization is heavily dependent on skill in multiplication and using distributive law reversely He means that ldquostudents who do not have reasonably strong skills in multiplication should not be expected to develop strong skills in factorizationrdquo (p 55) Factorizing polynomials is not only for simplifying high-degree polynomials (for example the third or fourth-degree polynomials) but can also be applied for solving quadratic equations Supposing this polynomial forming a quadratic equation Solving this equation is actually to decompose the polynomial into a factoringrsquos form

0273 2 =++ xx0)13)(2( =++ xx

which is equivalent to the polynomial form The essential step to solve this equation is to follow the zero-factor propertyndashalso called null-factor law (nollprodukt in Swedish)ndashthat is for real numbers p and q p bull q = 0 if (and only if) p = 0 or q = 0 It means that either of the binomials on the left side of the equation has to be equal to zero in order to satisfy the

equivalent relation of this quadratic equation The solutions of two roots or 2minus=x31

minus=x

are obtained through making either 02 =+x or 013 =+x 3) Quadratic formula Solving quadratics equations by factorization is constrained within the simple quadratic equations over the whole integers and rational numbers as coefficients What happens when quadratic equations have coefficients that belong to an irrational domain or big quantity The quadratic formula is a solution to such a situation As mentioned previously historically the French mathematician Franccedilois Vieacutete discovered the quadratic formula

aacbbx

242 minusplusmnminus

= ) which is applicable for all kinds of quadratic equations In

Swedish textbooks this formula is written in PQ form by making for the equation

that is

0( nea

1=a

02 =++ qpxx qppx minus⎟⎠⎞

⎜⎝⎛plusmnminus=

2

22 with

abp = and

acq = In Oleatursquos study (2007)

studentsrsquo difficulties in using algebraic symbols no longer is the problem But on the other hand handling the parameters or coefficients in quadratic equations like and rewrite them in the equivalent form with p and q being real numbers becomes obstacles for the students when they study the algebra course at upper-secondary school

02 =++ cbxax02 =++ qpxx

The quadratic formula a

acbbx2

42 minusplusmnminus= has been regarded as the standard method

according to US mathematics teachers Obermeyer (1982) presents a way of how to derive this formula by completing a square through 11 steps (I will not present them here) At the same time an alternate method is also introduced according to the same principle All the methods mentioned so far are based on the square rules The standard method or PQ form might be regarded as an efficient and direct method but it may

222 2)( bababa +plusmn=plusmn

14

lead students to solve quadratic equations in a mechanical way besides having problems with the troublesome plusmn symbolism (Stover 1978) Both Stover (1978) and Olteanu (2007) have suggested that using a graph of quadratic functions to solve quadratic equations is another alternative method

23 Mathematical background of factorization Factorization in algebra structure Among the three solving methods presented above both the geometrical method and quadratic formula can be traced back to algebra history in section 21 What is the trace of factorization in algebra history Factorization was at the last stage in algebra history and belonged to algebra structure (Derbyshire 2006) In Swedish mathematics textbooks the chapter on algebra often includes polynomial factorization and quadratic equations But the factorization in the textbooks is not at the same abstract level as the factorization of algebra structure although they share the same axioms and there are certain connections between these two different levels of factorization How are polynomial and quadratic equations as well as factorization related to each other In order to find the theoretical background of factorizing polynomials this part constitutes a mathematical theoretical review by referring to two books concerning algebra structure (Vretblad 2000 Durbin 1992) and a research study (Bosseacute and Nandakumar 2005) A quadratic equation can be regarded as polynomial equation Solving quadratic equations has much to do with ldquounique factorization theoremrdquo The polynomial

is a second-degree polynomial A polynomial consists of coefficients in whole numbers rational numbers real numbers or complex numbers and variables (sometimes called unknowns) in different degrees as well as constants A polynomial

has symbols as coefficients which are real numbers If all are zero and so

)0(02 ne=++ acbxax

cbxax ++2

nnxaxaxaaxf ++++= )( 2

210 naaa 10

ka 1gek 0)( axf = is constant If all are zero then we say the is a zero polynomial If meaning that is not a zero polynomial and n is the biggest number and then we say that the polynomial is at degrees A zero polynomial has no degree at all According to the definition (Vredblad 2000) if is a polynomial the equation is called an algebraic equation or polynomial equation A solution or a root of this equation is a number

kaf 0nef f

0nena f nf

0)( =xfα so that 0)( =αf Such a number is even

called a zero point (nollstaumllle in Swedish) of the polynomial which means when f α=x the value of the polynomial is zero (p 161) Every polynomial whose degree is equal or greater than 1 can be written as a product of irreducible polynomial In this way through polynomial division a polynomial with n degrees can be expressed from a sum of terms into a product of its denominator and its quotient The process of changing the expressions is factorization for

instance 12

22

+=minusminusminus x

xxx The factoring form is )1)(2(22 minusminus=minusminus xxxx

Factorization of polynomials is analogue to factorization of whole numbers which is very much related to the concept of ring A ring consists of a set with two operations which are a sum and a product of two elements over a field Any polynomial from zero polynomial to n degree-polynomials can be written or operated through addition and multiplication of the

15

elements over a field F including whole numbers or integers rational numbers real numbers and complex numbers In a ring you can multiply any two elements but you canrsquot always divide Factorization is a way to deal with this fact Here are some examples In the ring of integers Z is an example of factorization in some polynomial rings for instance Q[x] the ring of polynomial in one variable with rational numbers as coefficients such as

7214 sdot=

)451)(2

51(8

52

251 2 +minus=minus+ xxxx is factorization Z[x] the ring of polynomial in one

variable with whole numbers as integers such as is factorization )1)(2(22 minusminus=minusminus xxxx The polynomial ring R[x] means the ring of polynomial in one variable with real numbers as coefficients Factorizing of such polynomials implies that the polynomial factors are the elements in the ring of R[x] for instance )2)(2(22 minus+=minus xxx The same thing is true for C[x] the ring of polynomial in one variable with complex numbers as coefficients Thus it is coefficients of polynomials that determine which ring a polynomial belongs to and if it is possible to be factorized in this ring For example number 6 can be factorized in two ways

in the ring of Z and 32times )51)(51( ii +minus in the ring of C (complex numbers) However there are cases in which divisions can not be done within the ring for example 52 or

)3()52( 2 ++minus xxx which means that no element in the ring can multiply with a denominator obtaining the numerator Therefore neither 5 nor can be factorized 522 +minus xx If a polynomial of degree at least one has no other divisors then it is considered to be irreducible or prime (Durbin 1992) Of every polynomial of degree at least one can be written as a product of irreducible polynomials according to unique factorization theorem each polynomial of degree at least one over a field F can be written as an element of F times a product of monic irreducible polynomials over F and except for the order in which these irreducible polynomials are written this can be done in only one way (p 221) Thus a factorable polynomial can be written into a product with another polynomial such as In other words the condition that )()()( xqaxxf minus= )( ax minus is a factor of a polynomial

in a variable x is only true if )(xf 0)( =af For example if and

6)( 2 minus+= xxxf0624)2( =minus+=f 0639)3( =minusminus=minusf the factors of are and )(xf )2( minusx )3( +x

The value of the polynomial is zero which is also expressed as the polynomial having a zero point (nollstaumllle in Swedish) when

)(xfα=x

Based on unique factorization theorem finding solutions of an algebraic equation is equivalent to finding the first-degree factors for a polynomial (Vredblad 2000) In this way solving a quadratic equation can be handled through finding the first-degree factors for the second-degree polynomial and there after finding the roots of the quadratic equation This is often called using factorization to solve quadratic equations Besides unique factorization theorem the connection of using factorization to solve a quadratic equation is that this method relates to prime numbers using distributive law backwards commutative law multiplication division and operating fractions As matter of fact it is about application of factorizing integers in polynomials at upper-secondary school level Since quadratic equations in the mathematics B course at Swedish upper-secondary school are limited to real numbers and often within the whole numbers (integers) many quadratic equations should be able to be solved by factorization according to my own

16

judgment Therefore using factorization for solving quadratic equations is very much related to arithmetic operations number concept and algebra structure It may offer students both mathematical conceptual understanding and operational skill For algebra beginners factorization can be traced back to studentsrsquo early years of mathematics education during which students do not only work with multiplication but also on factoring integers for example 56 = 8 middot 7 = 2 middot 2 middot 2 middot 7 = 4 middot 14 = 2 middot 28 In this example two operations are included multiplication and division Multiplication operation may help students to understand the multiplicative structure of the integers while factoring integers may become meaningful when it is related to divisibility divisors and prime numbers These two operational skills or competence are major techniques for simplifying fractions and finding common denominators Being able to understand and operate factoring integers has laid the essential grounds for algebra beginners to study factorization of polynomials Thus factorization reunites studentsrsquo early mathematics knowledge of factoring integers with later knowledge of algebra structure Factorability In a study on factorability by Bosseacute and Nandakumar (2005) the probability of factorability is reasoned through investigating the range of integers as coefficients of quadratic equations In the quadratic equation a b and c are randomly generated integers within a determined range r such as

002 ne=++ acbxaxyrx lele where x and y are integers When is a

perfect square or where

acb 42 minus

)4( 2 acb minus is an integer quadratics are factorable According to the studyrsquos data it shows that as the range for a b and c increases the probability of factorability of a quadratic with randomly selected integer values for a b and c decreases Bosseacute and Nandakumar confirm that as the range for coefficients expands to infinltltinfinminus r the probablility of factorability of a quadratic with randomly selected integer values for coefficients approaches zero The studyrsquos data collected from college algebra courses and textbooks demonstrates that about 15 of quadratics with integer coefficients [ ]1010minusisinr are factorable Thus about 85 of quadratics can not be factorized In spite of the limitation of factorability within the exercises concerning factoring practice chosen from 27 surveyed college algebra textbooks about 94 of the problems were factorable Within the textbooks 55 of the quadratics to be factored were within the range [-10 10] Even if many quadratic equations are factorable choosing the right factor pairs is time consuming and unnecessary for example

has to be factorized through choosing the right pairs of factors among nine pairs for 36 and 8 pairs of factors for 24 Comparing the three different methods for solving quadratic equations the study suggests that completing the square and quadratic formula are more effective informative and useful than factorization

0245936 2 =++ xx

231 Different kinds of factorizations for solving different quadratic equations How can we find the binomial factors of a quadratic equation or a second-degree polynomial without knowing one of the divisors as a binomial factor The answer is that we have to observe the relations among all coefficients and constants as well as the positive and negative signs before them under the condition of being factorable Most of the coefficients in quadratic equations or polynomials among exercises provided for practicing factoring are integers (here I borrow the same notation r as the one used by Bosseacute and Nandakumar) with

17

their range between ndash10 and 10 that is 10ler In this case it is possible for a student to make a judgment of whether or not a quadratic equation is factorable Five different kinds of factorization depending on five kinds of quadratic equations Factoring quadratic expressions can be divided into five different kinds depending on the type of quadratic equations The coefficients a b p q and constant c illustrated in the following quadratic equations are defined not to be equal to zero thus 000 nenene cba The first kind of quadratic equations is type 1 the obviously factorable type (Bosseacute and Nandakumar 2004) then the equation is factorized as

where the roots are

0002 nene=+ babxax

0)( =+ baxxabxx minus== 21 0

The second kind of quadratic equations is type 2 The factors of this second degree polynomial are so the roots to this type of quadratic

equations are

0)( 22 =minusbax))(()( 22 baxbaxbax minus+=minus

bax plusmn=21 This kind of factorization is based on the difference-of-squares

formula for example can be solved by factorizing into 049 2 =minusx 0)23)(23( =minus+ xx Thus

the roots are 32

32

21 =minus= xx This kind of factorization is also obviously factorable

The third kind of quadratic equations is type 3 This type of equations can be solved directly by factoring into two identical binomials

The double roots obtained from this

factorization are

02)( 22 =+plusmn babxax

0))(()(2)( 222 =plusmnplusmn=plusmn=+plusmn baxbaxbaxbabxax

abx plusmn=21 This method has utilized square rules which are quite easily

observed The forth kind of quadratic equations is type 4 In this equation the coefficient of the -term is 1 with p and q as positive or negative integers Type 4 equations can always be solved by the approaches of completing the square and quadratic formula However they can often be solved by factorization too since the quadratic expressions are factorable by using distributive laws reversely The roots are obtained through finding two factors for q At the same time these two factors become satisfying with the relation so that the sum of factors is equal to That way the two factors become two roots of a quadratic equation Denoting the two factors or two roots as the relationship between coefficient and constant is

02 =++ qpxx2x

qminus

21rr)( 2121 rrprrq +minus=bull= Quadratic equations can be written as the product of

two first-degree binomials or factoring form 0))(()( 212121

22 =minusminus=++minus=++ rxrxrrxrrxqpxx Through the factoring form the roots of this equation are obtained being The core of factorizing type 4 equations is seeking the connections between coefficients and roots which reveal the studentsrsquo basic ability of arithmetic multiplication and addition It is often more effective in this case to use factorization than to use quadratic formula or completing the square Let us have a look at the example In this equation we get

2211 rxrx ==

035122 =+minus xx

18

35)7)(5(12)7()5( =minusminus=minus=minus+minus= qp or 75)75( bull=bullminus= qp (here we can directly get roots) Factorizing the equation we obtain 0)7)(5( =minusminus xx and roots are The procedures of finding factors of the equation often occur in some studentsrsquo minds quickly without writing down the whole computing process but those students often have good skills in arithmetic operations (reference) Compared to using factorization for solving this equation application of quadratic formula seems ineffective A possible problem in this method is how to find the signs before the factors or how to judge if the factors in the parentheses are positive or negative integers In order to check if the solution is correct we can always multiply these two factored binomials by distributive law and see if the expanded polynomial is the same as the original quadratic equation

75 21 == xx

The fifth kind of quadratic equations is type 5 Still the methods of completing squares and quadratic formula can be applied for all type 5 equations but there are cases in which factorization can be used for solving this type of equations within a certain range of integers as mentioned before Sometimes procedures of factorizing such equations can be carried out in our minds quickly like type 4 equations but in many cases they can not As a matter of fact finding two binomial factors for a factorable quadratic equation or a factorable second-degree polynomial is a complicated process and requires systematic work In his article published in Mathematics in School Jackman (2005) demonstrates this systematic procedure via factorizing second-degree polynomials A second-degree polynomial can be written as where are all integers By multiplying out the polynomial becomes thus

0002 nene=++ bacbxax

))((2 srxqpxcbxax ++=++ srqpcbaqsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== Writing the factors of a and c on two lines p q times r s Finding the right pairs of factors for a and c has to be done by listing out the groups of products of different factors for example The factor pairs for 6 are (6 1) (1 6) (3 2) (2 3) and the factor pairs for (-4) are (4 -1) (-4 1) (2 -2) (-2 2) (1 -4) (-1 4) Writing them on two lines p 6 1 3 2 q 4 -4 2 -2 1 -1

456 2 minus+ xx

times r 1 6 2 3 s -1 1 -2 2 -4 4 Then comes calculating the values of ps and qr until the value of b = 5 is obtained The calculating procedures are divided into 4 groups and 6 steps in each group which means 4 times 6 steps in total Here I only give one grouprsquos procedure as an example 6 middot (-1) + 4 middot 1 = -2 6 middot 1 + 1middot (-4) = 2 6 middot (-2) + 1 middot 2 = -10 6 middot 2 + 1middot (-2) = 10 6 middot (-4) + 1 middot 1 = -23 6 middot 4 + 1 middot (-1) = 23 helliphellip After the 24ndashstep calculating procedures two computing procedures obtain the same correct result 3 (ndash1) + 4 2 = 5 (yes) 2 4 + (-1) 3 = 5 (yes) Are they the same From 3 (ndash1) + 4 2 = 5 the polynomial can be factorized into )12)(43( minus+ xx while from 2 4 + (ndash1) 3 = 5 the polynomial can be factorized into )43)(12( +minus xx According to communicative law these two binomial products are the same The shortcoming of this

19

method is that it is too ineffective and complicated The addition of matrixes is almost impossible and unbearable Nevertheless based on the same idea I suggest finding factors by guessing analyzing and checking Using the same example here we can find that one of the binomials must have a negative integer because of ndash4 which implies that 5 is the sum of a positive integer and a negative integer Then looking at a and c the factors of 6 can be (1 6) and (2 3) while the factors of 4 can be (1 4) and (2 2) when we regard them as absolute values We know that the factoring form of the polynomial will be like

and the connections of all coefficients are

456 2 minus+ xx

qsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== The coefficient 5=b is the difference of two products of factors for 6 and 4 First we try the biggest number of product such as 2446 =bull then 524 =minus the unknown is which is a prime number the only factors for 19 are (19 1) There is not any combination of products of factors for 6 and 4 satisfying 19 Therefore the multiplications of factors (1 6) and (1 4) are not included in this case So we move to factors (2 2) (2 3) and (1 4)

19 =

In the second step we try The result is negative it is obviously

4264466)23)(22( 22 minusminus=minus+minus=+minus xxxxxxx2 lt 5 which implies that the combination of (2 3) and (2 2) is not

considerable One pair of factors has to be removed and that is (2 2) which does not influence the coefficient of the -term Then we try The result this time is negative too but we can see that

2x 456)43)(12( 2 minusminus=minus+ xxxxb is 5 which guides us to interchanging the positions of

negative and positive signs in the parentheses The next attempt is and it is correct Seeking the correct factors for the polynomial

has taken four major operating steps by trying the combinations of factorsrsquo products and relating the sum of the products with the value of the x-term coefficient The strategy is to make use of distributive law reversely first and then multiply the product of every pair of parentheses in order to check if they are the right pair of polynomial factors Factorizing second-degree polynomials is the major step for solving quadratic equations After factoring polynomials into the product of two binomials as in the example given above the roots can be obtained by making each binomial equal to zero according to the theory of zero point (ldquonollstaumlllerdquo in Swedish) of a polynomial equation originating from unique factorization theorem

456)43)(12( 2 minus+=+minus xxxx456 2 minus+ xx

Factorization by using the technique of vertically and crosswise sutra Binomial expansion or multiplication of binomials by distributive law and factorization of the second-degree polynomials by using distributive law backwards have an extremely symmetrical effect and the results of two kinds of operations are equivalent It is crucial for the students to be able to perceive mathematics structure and be acquainted with arithmetic operations as well as to understand the relationship between quantities when they carry out factoring procedures The expansion of binomials by distributive law has been named as the method FOIL which stands for first outside inside and last according to multiplication orders of two binomials for example (Nataraj and Thomas 2006) Many students are used to this procedure Another technique to multiply two binomials is called vertically and crosswise sutra origin from the ancient Indian Vedic mathematics Vertically and crosswise sutra and FOIL are much the same in principle but it makes use of

824)4)(2( 2 +++=++ xxxxx

20

the power of visualization and improves facility with the expansion of binomials and the factorization of quadratic expressions In this method two binomials are written into two lines Using the same example the two lines are written as 824)4)(2( 2 +++=++ xxxxx

x + 2 x + 4 ______________ xsup2 4x + 2x 8

The order of multiplication is as shown below x + 2 x + 4

4 3 2 1 (the numbers 1 2 3 and 4 refer to steps) xsup2 + 2x + 4x + 8 Figure 3 The vertically and crosswise method of multiplication of binomials Step 2 and 3 can be interchangeable If we put this form into a line expanded according to the order above would be like

)4)(2( ++ xxxbullrarrbull 442 clockwise and xxx bullrarrbull2 in

counterclockwise direction According to sutra we can imagine drawing a thread starting from 2 then going through 4 and x clockwise and then draw the second thread starting from the same place 2 later going through x in the second bracket and finally x in the first bracket counterclockwise Learning expansion of binomials by the methods FOIL and Vertical and clockwise sutra is the basic process before students study factorization In this study Nataraj and Thomas (2006) have found that the students appear to do factorization using guess-and-check as well as decomposition or the vertical and clockwise approach Their finding is that afterwards the students performed better on algebra questions especially on the factorizations They conclude ldquohellipthe values of the method may lie in what it adds to the studentsrsquo overall algebraic conceptions and knowledge of mathematical structurerdquo (p 16) In general solving these five kinds of quadratic equations by factorization can be traced back to unique factorization theorem The important character in this method is that you can always check if factorization is correct or not through multiplying out the two pairs of a parenthesis and see if the expanded polynomial is the given one or not Using factorization is not only about method but also about fostering the studentsrsquo mathematical thinking The solving processes are more important than solutions

21

3 Review of related literature and research Besides the focus on mathematics content another focus in the background work for this research is about previous studies and research in the field of mathematics didactic concerning different methods for solving quadratic equations and the use of factorization In order to study the previous research and studies in the related field three scientific databases and a search engine have been used and 113 articles have been investigated Topics such as solving quadratic expressions and equations are often discussed as teaching ideas and strategies shared by mathematics educators in different countries from lower and upper-secondary schools to colleges according to the findings in the article search These didactic topics from mathematics teachersrsquo perspectives include for example using geometric approaches to solve quadratic equations (Allaire amp Bradley 2001) utilizing completing squares (Vinogradova 2007) factoring quadratics (Hoffman 1976 Cheng 1980) and using factorization completing the square graphical methods to solve quadratic equations (Macdonald 1986) Kostsopoulos (2007) links cognitive reasoning to pedagogical problems appearing in the mathematics classrooms in her article published in Australian Mathematics Teacher Based on her teaching experiences Kostsopoulos is aware of her pedagogical strategies lacking insight on dealing with studentsrsquo difficulties of multiplication and factorization of quadratics She assumes that ldquostudentsrsquo problems with factorization and with identifying varied representations of the same quadratic relationshiprdquo may be linked to the ways in which the brain constructs cognitive representations (p 19) Factoring of quadratics is the rewriting of polynomials as a product of polynomials It requires students to have both a conceptual understanding of multiplication of polynomials and the procedural knowledge to retrieve basic multiplication facts effectively Based on the theory about the order matters in the brainrsquos ability to retrieve number facts (Phenix amp Campbell 2001) Kostsopoulos conjectures that studentsrsquo ability to access the appropriate long-term semantic memory is limited when they are confused by mixing the order and varied forms of quadratics Although this article is not about a research study it seeks the reason why factorizing quadratics is difficult and therefore might provide math teachers with alternative explanations for similar pedagogical frustrations With Von Glasersfeldrsquos radical constructivism and Peter Gaumlrdenforsrsquo epistemic semantics theory as his theoretical frameworks Rauff (1994) suggests that belief-based teaching can be successful in teaching factoring According to Rauff constructivist learning theory takes the view that students construct their own beliefs and knowledge of mathematics Learning occurs when the students change their belief set Referring to belief set theory from Gaumlrdenfors (1988) Rauff (1994) claims that ldquoa belief set is expanded when a new belief is added to it and contracted when a proposition is no longer believed and is removed from the setrdquo (p 421) Expansion contraction and revision are three components to comprise basic mechanisms of changes of belief according to Rauff When examining and comparing studentsrsquo (ages 18-20) own definitions of factoring Rauff analyzes the errors of their definitions with constructivism and epistemic semantics as analyzing tools The result of this study shows that the approach to teaching factoring rely heavily upon what a student believes about factoring is quite productive in two dimensions First it reveals the source of nonstandard factoring Second it provides a starting point for modification of the studentrsquos underlying conceptions of factoring Rauff proposes that belief-based teaching can also be

22

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

three steps require that the students are able to handle symbols and concepts and that they have the necessary skills needed for operations as long as they understand the contents of the problems Operations are based on the early knowledge of arithmetic (Haumlggstroumlm 2006) Swedish students learn elementary algebra at the beginning of upper-secondary school The subject mathematics is divided into five levels from A to E At A level called Mathematics A students are required to be able to translate simplify and reform expressions of quadratic equations according to the Swedish syllabus for Mathematics A (Skolverket 2000) At the B level students should go more deeply into the learning of algebra and functions During this course the students should be able to tolka foumlrenkla och omforma uttryck av andra graden samt loumlsa andragradsekvationer och tillaumlmpa kunskaperna vid problemloumlsning (Skolverket 2000 p 83) The aim of Mathematics B emphasizes the importance of solving quadratic equations It is during this study period that students build up most of their knowledge of algebra At upper-secondary schools in Sweden students are taught to use the approaches of completing the square and the quadratic formula called PQ formula to informally solve quadratic equations as well as factorization The application of the graphical method is introduced in the chapter on functions at the mathematics B course Applying PQ formula is an efficient way when solving all kinds of different quadratic equations On the one hand quadratic formula has become a powerful mathematics instrument for students who can use it freely without paying much attention to equationsrsquo structures and operational procedures but on the other hand as a result knowing why and how to use other methods to solve quadratic equations may become less important in practice something which might reduce studentsrsquo opportunities to think mathematically The problem with solving a quadratic equation by either quadratic formula or calculators without knowing why and how causes a dilemma in mathematics learning and teaching From teachersrsquo point of views it is important to make students understand how to solve quadratic equations in order to develop their mathematics thinking But for students quickly and easily finding solutions for quadratic equations might be their goals The need for finding correct and efficient methods may change the character of school mathematics and therefore lead mathematics teaching to a more practical character than pure mathematical science Compared to quadratic formula factorization in algebra content may not serve as an efficient tool for solving quadratic equations but can probably increase studentsrsquo algebra structure sense Factorization and quadratic formula belong to this dilemma area

13 The choice of which mathematics textbooks to investigate Some international studies have compared the effectiveness of using different methods to solve quadratic equations from a mathematics-didactics point of view How many methods are usually taught for solving quadratic equations in Swedish upper-secondary mathematics classrooms How do these different methods relate to each other and thus influence studentsrsquo understanding of quadratic equations and algebra structure Why are some methods emphasized in teaching at the same time as some others are not These questions concern two fields one is mathematics as a scientific discipline and the other is mathematics didacticsndashmainly normative didactics which ldquoinvolves discussions about the educational goals choice of content and methods but should also include justifications and recommendationsrdquo (Johansson 2006 p 12) To be able to answer these questions I need to find resources covering these two fields That means mathematics textbooks The previous Swedish study

3

(Johansson 2006) has found that the mathematics textbook is the most influential factor in classroom teaching and learning According to Johansson (2006) textbooks seem to dictate the teaching of mathematics in many aspects The use of textbooks is a very important framework in mathematics teaching Textbooks influence not only what kind of tasks students are working with and the examples presented by the teachers but also how mathematics is portrayed in terms of the concepts and the features that are related to the subject (p 26) From previously done research Johansson (2006) has found the importance of analyzing the content of textbooks because of the following reasons ldquomathematical topics in textbooks are most likely presented by the teachersrdquo ldquoTeachersrsquo pedagogical strategies are often influenced by the instructional approach of the materialrdquo ldquoTeachers sequence of instruction are often parallels to that of the textbookrdquo and ldquoTeachers report that textbooks are a primary information source in deciding how to present contentrdquo (p 48) The same mathematical topics can be emphasized in one book but might be overlooked in another An analysis of mathematics textbooks reveals the implied beliefs of what mathematics is and how it can be taught and learned It also reflects on the influence of the educational culture from a specific country It is obvious that mathematics textbooks play important roles in Swedish classrooms though not all teachers are the slaves of the textbooks Since mathematics textbooks are important sources for teaching and learning mathematics in Swedish classrooms I have decided to start my study by investigating eight mathematics textbooks used for the course Mathematics B in Swedish upper-secondary schools and by analyzing three of them in detail The related teaching guide material will be investigated in next step

14 Research Aim

The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3

2 Mathematics background in the field of algebra

21 Algebra history and its development Quadratic equations and their unknowns as well as polynomials belong to the field of algebra in mathematics In Swedish mathematics textbooks for mathematics course B there is one chapter on algebra Many critical elements of algebra have been presented in every book but there are no descriptions or definitions of the word ldquoalgebrardquo What is algebra Colin Maclaurin in his 1748 algebra text defined it as ldquoAlgebra is a general Method of Computation by certain Signs and Symbols which have been contrived for this Purpose and found convenient It is called an Universal Arithmetic and proceeds by Operations and Rules similar to those in Common Arithmetic founded upon the same Principlesrdquo (Katz 2006 p 185) Leonhard Euler in his own algebra text in 1770 defined algebra as ldquoThe science

4

which teaches how to determine unknowns quantities by means of those that are knownrdquo (Katz 2006 p 185) In this part I will mainly give a survey of the historical development of algebra including solving quadratic equations according to four stages the rhetorical stage the syncopated stage the symbolic stage and the purely abstract stage The purpose is to try to find out why algebra in school is like a threshold which hinders the students in their understanding Can a historical perspective on algebra explain this obstacle Algebra is an old science and its historical background is complicated Algebra procedures have developed slowly There are different opinions about where the evolution of the term ldquoalgebrardquo started It is commonly believed that algebra first appeared among the Egyptians the Babylonians the Greeks or the Arabs The geometrical influence on algebraic reasoning was strong in ancient Greece However the word algebra originated in Baghdad where the Arabic scientist al-Khwarizmi (AD 780-850) published a short book about calculating with the help from al-jabr and al-muqabala a book on the solution of an equation as a rule (Kvasz 2006) Todayrsquos algebra has its root in Arabic algebra Western mathematics tended to turn algebraic operations into symbols and later developed abstract algebra The process of algebra development was slow and its whole history lasted 4000 years

The rhetorical stage originated from geometry ideas

Historically algebra developed in three stages the rhetorical stage the syncopated stage and the symbolic stage According to Katz (2006) algebra development can also be categorized into four conceptual stages ldquothe geometric stage where most of the concepts of algebra are geometric the static equation-solving stage where the goal is to find numbers satisfying certain relationship the dynamic function stage where motion seems to be an underlying idea and finally the abstract stage where structure is the goalrdquo (p 186) Algebra began very early in recorded history Algebra texts date from the first half of the second millennium BCE 37 or 38 centuries ago and were written by people living in Mesopotamia and Egypt (Derbyshire 2006) During the Hammurabi period from about 1790 to 1600 BCE the Babylonians started their civilization by pressing written words in patterns called cuneiform or wedge-shaped stylus into wet clay Many tablets in cuneiform had a mathematical-algebraic content Their mathematical texts were of two kinds table texts and problem texts The table texts were lists of multiplication tables tables of squares and cubes as well as advanced lists like the famous Plimpton 322 tablet which is about Pythagorean triples The number system of the Babyloniansrsquo time was based on 60 digits rather than todayrsquos 10 digits for example our number of 37 denotes three sixties and seven ones which is our number 187 The problem was that the Babylonians had neither defined zero nor negative numbers The Babylonians of Hammurabirsquos era had no proper algebraic symbolism All mathematical problems were expressed in words for example unknown quantity in Sumerianrsquos Akkadian text was expressed as igum (length) and igibum (width) as reciprocal The application of algebra might have its origin in the need for measuring land areas At the rhetorical stage all mathematical statements and arguments were expressed in words and sentences (Derbyshire 2006) Babylonian mathematics had two roots one is accountancy problems and the other one is a ldquocut-and-pasterdquo geometry probably developed for understanding the division of land Many old-Babylonian clay tablets contain quadratic problems of which the goal was to find such geometric quantities as the length and width of a rectangle As an example we are given that the sum of the length and width of a rectangle is

5

216 and the area of the rectangle is

217 (Kvatz 2006 p 187 and Derbyshire 2006

pp 25-27) What is the length and the width of this rectangle The tablet described in detail

the steps the writer went through First the writer halves 216 to get

413 Next he squares

413

to get 16910 From this area he subtracts the given area

217 giving

1613 The square root of

this number is extracted431 Finally the length is 5

431

413 =+ while the width is

211

431

413 =minus (Kvatz 2006) The whole process can be translated into parts of a quadratic

formula that is 2172

216

431

2

minus⎟⎠⎞

⎜⎝⎛ divide= The Babylonians did not know anything about

negative numbers the only solution for them was in positive numbers and hence their algorithm did not deliver the two solutions to the quadratic equation so their formula is slightly different from quadratic formula (Derbyshire 2006) Their formula is

⎟⎠⎞

⎜⎝⎛ divideplusmnminus⎟

⎠⎞

⎜⎝⎛ divide= 2

216

2172

216

2

x If we denote the sum of the length and the width of the

rectangle as b and the given area as c this formula will be ⎟⎠⎞

⎜⎝⎛plusmnminus⎟

⎠⎞

⎜⎝⎛=

22

2 bcbx although it is

not exactly like the modern quadratic formula There are different interpretations of Neugebauer and Sachesrsquo translation of the Babylonianrsquos tablets for this text on finding the length and width of a rectangle However it is very clear that the text from the tablets is dealing with a geometric procedure The problem was solved in words but with geometric ideas This was the beginning of algebra The Greek mathematician Euclid (300 BC) in his Book II of Elements solved some algebraic problems by manipulating geometric figures but based them on clearly stated axioms The geometrical method is in Kvatzrsquos opinion more explicit in another work of his Data The following example illustrates how Euclid solved a quadratic equation by a geometrical method Euclid defined ldquoproposition 1rdquo which is like axiom 1 If two straight lines contain a given area in a given angle and if the sum of them be given then shall each of them be given (Kvatz 2006 p 189) Euclid set up a rectangle with one side x = AS and y = AC Then a line was drawn so that BS = AC and the completed rectangle was ACDB Suppose that

was given and the area of rectangle ACFS (Figure 1) was given denoted as c What were the two sides AS (x) and AC (y) of the rectangle

byxAB =+=

A x S y B

y y

C F D

Figure 1

6

In order to find the length and the width of the rectangle Euclid bisected AB at E constructed the square on BE and then claimed that this square was equal to the sum of the rectangle ACFS and the small shaded square at the bottom (Figure 2)

A E S B

x

y y F

C G D Figure 2 According to Euclid the area of the rectangle ACFS was given which was c and the area of

the new square EGDB was also given which was ( )22b because 222byxSBASEB =

+=

+=

The equivalent relationship between the areas can be formulated as an quadratic equation 22

22⎟⎠⎞

⎜⎝⎛ minus+=⎟

⎠⎞

⎜⎝⎛ bxcb or

22

22⎟⎠⎞

⎜⎝⎛ minus+=⎟

⎠⎞

⎜⎝⎛ ybcb Euclid found out this equivalent relationship

geometrically and took use of this relationship in finding the solutions of the problem so the

length and width of the rectangle ACFS are cbbx minus⎟⎠⎞

⎜⎝⎛+=

2

22 and cbby minus⎟

⎠⎞

⎜⎝⎛minus=

2

22

These two formulas are almost identical with the Babyloniansrsquo solutions in rhetoric expressions The difference is that Greek algebra was based on geometric manipulation while Babylonian algebra was based on rhetoric manipulation with geometrical ideas In general the early stage of algebra from ancient Babylon and Egypt to Greek was mainly geometrical The syncopated stage ndash the beginning of the static equation-solving stage by using In Roman Egypt in probably the second or third century CE the algebra stage was at the syncopated stage which means written algebraic texts were expressed in words but involved special symbols-abbreviations According to recorded mathematics history one the of pioneers who used these special symbols to solve equations with only numbers but no connection with geometry was Diophantus who lived in Alexandria in Egypt around the third century Diophantus used the Greek alphabetic system for writing numbers He wrote a treatise titled Arithmetica of which less than half has been maintained today The surviving part of his work consists of 189 problems in which the object is to find numbers or families of numbers satisfying certain conditions In mathematics today Diophantusrsquo mathematical analysis is known as number theoryndashnot algebra However he used only number symbols to solve equations without the help of geometry Diophantus wrote the coefficient after the variable instead of before it as we do He used Greek ς for unknown quantity our modern x Most of his book deals with indeterminate equations which contain more than one unknown and a potentially infinite number of solutions His problem is that he could not represent more than one unknown instead he solved quadratic equations with two unknowns through substituting one by another and then the quadratic equations became the ones with one unknown At that time negative numbers were regarded to be absurd but he knew how to

7

bring a term from one side of an equation to the other gather up like terms for simplification and some elementary principles of expansion and factorization (Derbyshire 2006) Diophantus made his own literal symbolism with the use of special letter symbols for the unknown and its powers for subtraction and equality From Diophantus algebra history moved into another conceptual stage the equation-solving stage according to Katz (2006) In India quadratic formula appeared without any geometric support Brahmaggupta (598-665) was one of the first mathematicians who could systematically handle negative numbers and zero He gave a general solution to quadratic equations and realized that there were two roots for a quadratic equation It was possible that one of the roots was a negative number Baskharacharya (1114-1185) solved mathematics problems with the use of quadratic equations in his book Siddhanta Siromani (ldquoMathematical Pearlsrdquo) He presented an algorithm to reduce a quadratic equation to a first-degree equation (Olteanu 2007) It is commonly believed that the first true algebra text is the work on al-jabr and al-muqabala by Mohanmmad ibn Musa al-Khwarizmi (780-850) written in Baghdad around 825 (Katz 2006) The word algebra came from the title of this work The word al-jabr means restoration or reestablishment that is to eliminate negative terms through adding the same terms to both sides of equations The word of al-muqabalas means balance meaning to divide every term in a quadratic equation by the coefficient of the second degreersquos term (Olteanu 2007) The first part of his book is a manual for solving linear and quadratic equations Al-Khwarizimi classified equations into six types three of which were mixed quadratic equations For each type he presented an algorithm for its solution Five of the six types of equations were quadratic equations which can be expressed in modern form

cbxaxbxcaxcbxaxcaxbxax +==+=+== 22222 Here is an example of solving the equation ldquoTake the half of the number of the things that is five and multiply it by itself you obtain twenty-five Add this to thirty-nine you get sixty-four Take the square root or eight and subtract from it one half of the number of things which is five The result three is the thingrdquo (Kvasz 2006 p 292) Like Babylonian mathematicians al-Khwarizimirsquos algorithm is entirely verbal On the other hand al-Khwarizimi sometimes completed his algorithm by using geometrical explanations which can be translated as todayrsquos square-completing method Using the example of solving the quadratic equation the completed geometrical procedures are illustrated on next page in figures 3 4 and 5

39102 =+ xx

39102 =+ xx

8

x

xsup2

Figure 3 (Olteanu 2007 p 30) 5x2

25x

xsup2

25x

5x2

Figure 4 (Olteanu 2007 p 30) 254 254

39

254 254

Figure 5 (Olteanu 2007 p 30) According to Olteanursquos (2007) translation al-Khwarizimi started with a square whose sides are x and area is xsup2 (see Figure 3) Then he added four equal rectangles whose areas in total

was 10x along each side of the square that is xx sdotsdot=25410 Each rectanglersquos area is thus x

25

with its length x and its width 52 (see Figure 4) The sum of the big square and four rectangles was given which was 39 The equivalence relationship was Finally figure 4 was completed by adding four small equal squares which had an area the size of

39102 =+ xx

425

25

25

=sdot for each small square and the sum of those was 25 Through adding this sum to

both sides of the equation the area of the biggest square obtained in figure 5 was 64 The

equation translation is 425439

4254102 sdot+=sdot++ xx The side of the biggest square was 8 and

had its relation with other sides of different squares expressed in the first degree equation

9

25

258 ++= x Then x was 3 By ldquocut-and-pasterdquo geometry (Katz 2006 p 191)

al-Khwarizimi reduced the second degree of an equation to the first degree and thereafter solved it Unlike his Babylonian predecessors al-Khwarizimi always presented his problem abstractly rather than geometrically relating to lengths and widths The symbolic stage At this stage of algebra ldquoall numbers operations relationships are expressed through a set of easily recognized symbols and manipulations on the symbols take place according to well-understood rulesrdquo (Katz 2006 p 186) The ancient algebra and geometry had developed sophisticatedly in Egypt Persia Greece India and China After Medieval Islamic scholars gave us the word ldquoalgebrardquo Western Europe began the struggle for the development of algebra starting from some algebraists from Italy Italian mathematician Leonardo Pisano later known as Fibonacci traveled in the 12th and 13th centuries to Persia India and China When he returned to Italy he had wider knowledge of arithmetic and algebra His book Liber abbaci was the best math textbook since the end of Ancient world His book is credited with having introduced Indian numerals including zero to the West But his algebraic skills had been shown in two other works after this one With the introduction of printed books during the second half of the 15th century the development of algebra was sped up Several Italian mathematicians including Girolamo Cardano had figured out how to solve cubic and quadratic equations Algebra became purely abstract with the exception of an English mathematician named Robert Recorde who lived in the 16th century and created quadratic problems from real world experience (Derbyshire 2006) It was in France that algebra had developed into a well organized literal symbolism In his work In artem analyticem isagoge French mathematician Franςois Viegravete (1540-1603) in the late 16th century became the first mathematician to use letters representing numbers systematically and effectively (Derbyshire 2006) He made a range of letters available for many different quantities This was the beginning of modern literal symbolism Viegravetersquos unknown quantity was divided into two classes unknown quantities (meaning ldquothings soughtrdquo) denoted by A E I O U and Y while ldquothings givenrdquo was denoted by constants like B C Dhellip For example his A is our unknown x Viegravete was a pioneer in the study of equations His two papers on the theory of equations were published twelve years after his death In the second paper titled ldquoDe equationem emendationerdquo (ldquoOn the perfecting of equationsrdquo) Viegravete opened up the line of inquiry that led to the study of the symmetries of an equationrsquos solutions to Galois theory the theory of groups and all of modern algebra He found the relationship between the solutions of the equation and the coefficients for the first five degrees of equations in a single unknown To explain this in our modern symbols we suppose that the two solutions of the quadratic equation are 02 =++ qpxx α and β which means βα == xx Based on this logic the following thing must be true 0))(( =minusminus βα xx since only α andβ and no other values of x make this equation true This form of equation is just a rewritten form of the same equation If we multiply out those parentheses this rewritten equation turns to be Compared to the original equation the relationships between the solutions and the coefficients we obtain

0)(2 =++minus αββα xxqp =minus=+ αββα

(Derbyshire 2006) It is said that Viegravete discovered the solution formula called quadratic

10

formula today for general quadratic equations which is a

acbbx2

42

21minusplusmnminus

= for a general

quadratic equation (Olteanu 2007) 002 ne=++ acbxax Another French mathematician and philosopher who had strong influence in the history of algebra was Reneacute Descartes (1596-1650) His idea to use the Cartesian system of coordinates which was derived from his Latin name developed both algebra and geometry In his work La geacuteomeacutetrie written in 1637 Descartesndashthrough using numbers to indentify points in a Cartesian coordinatesndashconnected geometrical objects to algebraic numbers and made the classical geometry become analytical geometry He took up the plus and minus sign from the German Cossists of the previous century and also the square-root sign From Descartes the symbol of an unknown was represented by x which led to the modern system of literal symbolism (Derbyshire 2006) Descartes developed the idea of functions in Cartesian coordinates although the idea of function can be traced back to the Islamic mathematician Sharaf al-Din al-Tusi from Persia (Katz 2006) and Klaudius Ptolemaios about 2000 years ago (Olteanu 2007) Descartes declared that every curve in a Cartesian coordinate system has an equivalent equation which can represent the points on the curve or vice versa every equation containing x and y can be represented by a curve through its coordinate points However the word function was introduced by Gottfried Wilhelm Leibniz in 1693 and the definition of function was defined by Leonhard Euler in his work Introductio analysin infinitorium in 1748 Euler had also given the symbol of a denotation as a relying variable (Olteanu 2007) In the 18)(xf th century after the discovery of calculus by Newton and Leibniz algebra went into an area of analysis ndashthe study of limits infinite sequences and series functions derivatives and integrals (Derbyshire 2006) The purely abstract stage-algebra structure Since 17th century algebra has not anymore solely been about finding solutions for different degrees of equations mathematicians have started to integrate algebra with astronomy and physics Johann Kepler and Galileo Galilei were interested in curves and finding a mechanism for representing motion instead of quantity-like numbers But their arguments were not algebraic-symbolic but geometrical It was Fermat and Descartes who were regarded as the fathers of analytic geometry and who showed how to represent a curve described verbally through algebra analytic geometry and gave a mechanism for representing motion Newton in his Principia picked up on Fermat and Descartesrsquo representation and developed the calculus With the invention of the calculus mathematical problems were solved by curves not just points Algebra grew more and more to present paths of motion In order to be able to judge if the algebraic manipulations were correct axioms were formulated for arithmetic applied for algebraic manipulations In the late 18th century Lagrange introduced the idea of permutations into the search for solutions Therafter Galois developed methods for determining what conditions polynomial equations need to meet to be solvable New abstract algebraic concepts were found like ldquofieldrdquo and ldquogrouprdquo during the early 19th century In 1854 Ceyley gave an axiomatic definition of a group During the 1890s this definition entered textbooks along with the axiomatic definition of a field an idea which had roots in the work by Galois By the beginning of the 20th century algebra became less about finding solutions to equations and more about looking for common structures in many mathematical objects defined by sets of axioms (Katz 2006)

11

In short algebra development took 4000 years It took about 3500 years for mathematicians to start to use algebra symbols in systems Just almost 100 years ago algebra developed into a purely abstract science The long history may give both mathematics educators and students some explanations to why it is so difficult to teach and learn algebra

22 Three solving methods for quadratic equations Already in the previous part I have presented geometrically and rhetorically solving quadratic equations from a historical perspective In order to get more explicit understanding of quadratic equations quadratic formula and factorization from mathematics didactic point of view this part presents three common methods for solving quadratic equations 1) Completing the square 2) Factorization 3) Quadratic formula The reason to present these three methods is that they are the common topics for discussion in the articles searched in this area concerning mathematics education at upper-secondary school 1) Completing the square In two articles Solving quadratic equations by completing squares (Vinogradova 2007) and Geometric approaches to quadratic equations from other times and places (Allaire amp Bradley 2001) the authors advise mathematics teachers to use the ideas of ancient mathematician al-Khwarizmi and his method of completing quadratic equations by geometrical algebra and simplifying it in order to represent it to the students By doing so the teachers relate mathematics quantity to physical objects in a visual way Didactically teachers may start with a concrete example in which students can understand the content visually Later from the concrete example the teachers lead the students to another example expressed in algebraic symbols for example to study a rectangle whose area is )10( +xx and supposing this area is 39 this becomes 39)10( =+xx Four steps are followed to build a new square which is actually called completing the square x 10

Figure 6 x 5

Figure 7

x

x

5 5

12

A Begin with a rectangle of the area )10( +xx that is with the short side as x and the long side )10( +x The shaded area is 10bullx (Figure 6)

B Divide the shaded rectangle into two small shaded rectangles with the size 5x each and move them to each adjacent side of the xsup2 square (Figure 7) The total area is still

x10

)10(39 += xxC In Figure 7 a new square is shaped in the lower-right hand corner with the side 5 or

area 25 By adding this small square to the diagram the large square with an area of is ldquocompletedrdquo Therefore we can write 25)5(22 ++ xx 25)10(2539 ++=+ xx or

The large squarersquos area is now because the side has a length of Therefore

6425)5(22 =++ xx 2)5( +x5+x 38564)5( 2 =rArr=+rArr=+ xxx

After these three steps a solution to this quadratic equation is derived by completing the square (Allaire amp Bradley 2001) 2) Factorization Hoffman (1976) presents three approaches (A B and C) for factorizing quadratic expressions through observing and grouping He takes a second-degree polynomial as an example

273 2 ++ xx

a The inspection method 1) suggests factors of and 23x x3 x 2) 2 suggests factors of 1 and 2 or ndash1 and ndash2 3) Checking the linear term shows that the correct factorization is x7 )2)(13(273 2 ++=++ xxxx

This method according to Hoffman (1976) is based on the studentrsquos ability to simplify by inspecting the indicated product of a pair of linear expressions

b The method of decomposition of the linear term 1) Multiply and 2 to get 23x 26x2) Decompose into the sum of two terms whose product is x7 26x

This gives xxx += 67 3) Factorize by grouping 2)6(3 2 +++ xxx

2)6(3 2 +++ xxx = )

)2()63( 2 +++ xxx

= 2(1)2(3 +++ xxx= )13)(2( ++ xx

c Third method 1) Multiply 3 and 2 to get 6 2) Find two numbers whose product is 6 and whose sum is 7 The numbers are 6 and 1 3) 273 2 ++ xx

=3

)13)(63( ++ xx

= )13)(2( ++ xx

13

These three approaches can not actually be divided as three unrelated approaches especially the third method is seemingly unclear What Hoffman (1976) wants to demonstrate is that all the three approaches are based on observing the relationship among the coefficients and the constant of the polynomial Through finding the product and the sum of two factors the polynomial can be factorized in the product of two binomials In the authorrsquos opinion factorization is heavily dependent on skill in multiplication and using distributive law reversely He means that ldquostudents who do not have reasonably strong skills in multiplication should not be expected to develop strong skills in factorizationrdquo (p 55) Factorizing polynomials is not only for simplifying high-degree polynomials (for example the third or fourth-degree polynomials) but can also be applied for solving quadratic equations Supposing this polynomial forming a quadratic equation Solving this equation is actually to decompose the polynomial into a factoringrsquos form

0273 2 =++ xx0)13)(2( =++ xx

which is equivalent to the polynomial form The essential step to solve this equation is to follow the zero-factor propertyndashalso called null-factor law (nollprodukt in Swedish)ndashthat is for real numbers p and q p bull q = 0 if (and only if) p = 0 or q = 0 It means that either of the binomials on the left side of the equation has to be equal to zero in order to satisfy the

equivalent relation of this quadratic equation The solutions of two roots or 2minus=x31

minus=x

are obtained through making either 02 =+x or 013 =+x 3) Quadratic formula Solving quadratics equations by factorization is constrained within the simple quadratic equations over the whole integers and rational numbers as coefficients What happens when quadratic equations have coefficients that belong to an irrational domain or big quantity The quadratic formula is a solution to such a situation As mentioned previously historically the French mathematician Franccedilois Vieacutete discovered the quadratic formula

aacbbx

242 minusplusmnminus

= ) which is applicable for all kinds of quadratic equations In

Swedish textbooks this formula is written in PQ form by making for the equation

that is

0( nea

1=a

02 =++ qpxx qppx minus⎟⎠⎞

⎜⎝⎛plusmnminus=

2

22 with

abp = and

acq = In Oleatursquos study (2007)

studentsrsquo difficulties in using algebraic symbols no longer is the problem But on the other hand handling the parameters or coefficients in quadratic equations like and rewrite them in the equivalent form with p and q being real numbers becomes obstacles for the students when they study the algebra course at upper-secondary school

02 =++ cbxax02 =++ qpxx

The quadratic formula a

acbbx2

42 minusplusmnminus= has been regarded as the standard method

according to US mathematics teachers Obermeyer (1982) presents a way of how to derive this formula by completing a square through 11 steps (I will not present them here) At the same time an alternate method is also introduced according to the same principle All the methods mentioned so far are based on the square rules The standard method or PQ form might be regarded as an efficient and direct method but it may

222 2)( bababa +plusmn=plusmn

14

lead students to solve quadratic equations in a mechanical way besides having problems with the troublesome plusmn symbolism (Stover 1978) Both Stover (1978) and Olteanu (2007) have suggested that using a graph of quadratic functions to solve quadratic equations is another alternative method

23 Mathematical background of factorization Factorization in algebra structure Among the three solving methods presented above both the geometrical method and quadratic formula can be traced back to algebra history in section 21 What is the trace of factorization in algebra history Factorization was at the last stage in algebra history and belonged to algebra structure (Derbyshire 2006) In Swedish mathematics textbooks the chapter on algebra often includes polynomial factorization and quadratic equations But the factorization in the textbooks is not at the same abstract level as the factorization of algebra structure although they share the same axioms and there are certain connections between these two different levels of factorization How are polynomial and quadratic equations as well as factorization related to each other In order to find the theoretical background of factorizing polynomials this part constitutes a mathematical theoretical review by referring to two books concerning algebra structure (Vretblad 2000 Durbin 1992) and a research study (Bosseacute and Nandakumar 2005) A quadratic equation can be regarded as polynomial equation Solving quadratic equations has much to do with ldquounique factorization theoremrdquo The polynomial

is a second-degree polynomial A polynomial consists of coefficients in whole numbers rational numbers real numbers or complex numbers and variables (sometimes called unknowns) in different degrees as well as constants A polynomial

has symbols as coefficients which are real numbers If all are zero and so

)0(02 ne=++ acbxax

cbxax ++2

nnxaxaxaaxf ++++= )( 2

210 naaa 10

ka 1gek 0)( axf = is constant If all are zero then we say the is a zero polynomial If meaning that is not a zero polynomial and n is the biggest number and then we say that the polynomial is at degrees A zero polynomial has no degree at all According to the definition (Vredblad 2000) if is a polynomial the equation is called an algebraic equation or polynomial equation A solution or a root of this equation is a number

kaf 0nef f

0nena f nf

0)( =xfα so that 0)( =αf Such a number is even

called a zero point (nollstaumllle in Swedish) of the polynomial which means when f α=x the value of the polynomial is zero (p 161) Every polynomial whose degree is equal or greater than 1 can be written as a product of irreducible polynomial In this way through polynomial division a polynomial with n degrees can be expressed from a sum of terms into a product of its denominator and its quotient The process of changing the expressions is factorization for

instance 12

22

+=minusminusminus x

xxx The factoring form is )1)(2(22 minusminus=minusminus xxxx

Factorization of polynomials is analogue to factorization of whole numbers which is very much related to the concept of ring A ring consists of a set with two operations which are a sum and a product of two elements over a field Any polynomial from zero polynomial to n degree-polynomials can be written or operated through addition and multiplication of the

15

elements over a field F including whole numbers or integers rational numbers real numbers and complex numbers In a ring you can multiply any two elements but you canrsquot always divide Factorization is a way to deal with this fact Here are some examples In the ring of integers Z is an example of factorization in some polynomial rings for instance Q[x] the ring of polynomial in one variable with rational numbers as coefficients such as

7214 sdot=

)451)(2

51(8

52

251 2 +minus=minus+ xxxx is factorization Z[x] the ring of polynomial in one

variable with whole numbers as integers such as is factorization )1)(2(22 minusminus=minusminus xxxx The polynomial ring R[x] means the ring of polynomial in one variable with real numbers as coefficients Factorizing of such polynomials implies that the polynomial factors are the elements in the ring of R[x] for instance )2)(2(22 minus+=minus xxx The same thing is true for C[x] the ring of polynomial in one variable with complex numbers as coefficients Thus it is coefficients of polynomials that determine which ring a polynomial belongs to and if it is possible to be factorized in this ring For example number 6 can be factorized in two ways

in the ring of Z and 32times )51)(51( ii +minus in the ring of C (complex numbers) However there are cases in which divisions can not be done within the ring for example 52 or

)3()52( 2 ++minus xxx which means that no element in the ring can multiply with a denominator obtaining the numerator Therefore neither 5 nor can be factorized 522 +minus xx If a polynomial of degree at least one has no other divisors then it is considered to be irreducible or prime (Durbin 1992) Of every polynomial of degree at least one can be written as a product of irreducible polynomials according to unique factorization theorem each polynomial of degree at least one over a field F can be written as an element of F times a product of monic irreducible polynomials over F and except for the order in which these irreducible polynomials are written this can be done in only one way (p 221) Thus a factorable polynomial can be written into a product with another polynomial such as In other words the condition that )()()( xqaxxf minus= )( ax minus is a factor of a polynomial

in a variable x is only true if )(xf 0)( =af For example if and

6)( 2 minus+= xxxf0624)2( =minus+=f 0639)3( =minusminus=minusf the factors of are and )(xf )2( minusx )3( +x

The value of the polynomial is zero which is also expressed as the polynomial having a zero point (nollstaumllle in Swedish) when

)(xfα=x

Based on unique factorization theorem finding solutions of an algebraic equation is equivalent to finding the first-degree factors for a polynomial (Vredblad 2000) In this way solving a quadratic equation can be handled through finding the first-degree factors for the second-degree polynomial and there after finding the roots of the quadratic equation This is often called using factorization to solve quadratic equations Besides unique factorization theorem the connection of using factorization to solve a quadratic equation is that this method relates to prime numbers using distributive law backwards commutative law multiplication division and operating fractions As matter of fact it is about application of factorizing integers in polynomials at upper-secondary school level Since quadratic equations in the mathematics B course at Swedish upper-secondary school are limited to real numbers and often within the whole numbers (integers) many quadratic equations should be able to be solved by factorization according to my own

16

judgment Therefore using factorization for solving quadratic equations is very much related to arithmetic operations number concept and algebra structure It may offer students both mathematical conceptual understanding and operational skill For algebra beginners factorization can be traced back to studentsrsquo early years of mathematics education during which students do not only work with multiplication but also on factoring integers for example 56 = 8 middot 7 = 2 middot 2 middot 2 middot 7 = 4 middot 14 = 2 middot 28 In this example two operations are included multiplication and division Multiplication operation may help students to understand the multiplicative structure of the integers while factoring integers may become meaningful when it is related to divisibility divisors and prime numbers These two operational skills or competence are major techniques for simplifying fractions and finding common denominators Being able to understand and operate factoring integers has laid the essential grounds for algebra beginners to study factorization of polynomials Thus factorization reunites studentsrsquo early mathematics knowledge of factoring integers with later knowledge of algebra structure Factorability In a study on factorability by Bosseacute and Nandakumar (2005) the probability of factorability is reasoned through investigating the range of integers as coefficients of quadratic equations In the quadratic equation a b and c are randomly generated integers within a determined range r such as

002 ne=++ acbxaxyrx lele where x and y are integers When is a

perfect square or where

acb 42 minus

)4( 2 acb minus is an integer quadratics are factorable According to the studyrsquos data it shows that as the range for a b and c increases the probability of factorability of a quadratic with randomly selected integer values for a b and c decreases Bosseacute and Nandakumar confirm that as the range for coefficients expands to infinltltinfinminus r the probablility of factorability of a quadratic with randomly selected integer values for coefficients approaches zero The studyrsquos data collected from college algebra courses and textbooks demonstrates that about 15 of quadratics with integer coefficients [ ]1010minusisinr are factorable Thus about 85 of quadratics can not be factorized In spite of the limitation of factorability within the exercises concerning factoring practice chosen from 27 surveyed college algebra textbooks about 94 of the problems were factorable Within the textbooks 55 of the quadratics to be factored were within the range [-10 10] Even if many quadratic equations are factorable choosing the right factor pairs is time consuming and unnecessary for example

has to be factorized through choosing the right pairs of factors among nine pairs for 36 and 8 pairs of factors for 24 Comparing the three different methods for solving quadratic equations the study suggests that completing the square and quadratic formula are more effective informative and useful than factorization

0245936 2 =++ xx

231 Different kinds of factorizations for solving different quadratic equations How can we find the binomial factors of a quadratic equation or a second-degree polynomial without knowing one of the divisors as a binomial factor The answer is that we have to observe the relations among all coefficients and constants as well as the positive and negative signs before them under the condition of being factorable Most of the coefficients in quadratic equations or polynomials among exercises provided for practicing factoring are integers (here I borrow the same notation r as the one used by Bosseacute and Nandakumar) with

17

their range between ndash10 and 10 that is 10ler In this case it is possible for a student to make a judgment of whether or not a quadratic equation is factorable Five different kinds of factorization depending on five kinds of quadratic equations Factoring quadratic expressions can be divided into five different kinds depending on the type of quadratic equations The coefficients a b p q and constant c illustrated in the following quadratic equations are defined not to be equal to zero thus 000 nenene cba The first kind of quadratic equations is type 1 the obviously factorable type (Bosseacute and Nandakumar 2004) then the equation is factorized as

where the roots are

0002 nene=+ babxax

0)( =+ baxxabxx minus== 21 0

The second kind of quadratic equations is type 2 The factors of this second degree polynomial are so the roots to this type of quadratic

equations are

0)( 22 =minusbax))(()( 22 baxbaxbax minus+=minus

bax plusmn=21 This kind of factorization is based on the difference-of-squares

formula for example can be solved by factorizing into 049 2 =minusx 0)23)(23( =minus+ xx Thus

the roots are 32

32

21 =minus= xx This kind of factorization is also obviously factorable

The third kind of quadratic equations is type 3 This type of equations can be solved directly by factoring into two identical binomials

The double roots obtained from this

factorization are

02)( 22 =+plusmn babxax

0))(()(2)( 222 =plusmnplusmn=plusmn=+plusmn baxbaxbaxbabxax

abx plusmn=21 This method has utilized square rules which are quite easily

observed The forth kind of quadratic equations is type 4 In this equation the coefficient of the -term is 1 with p and q as positive or negative integers Type 4 equations can always be solved by the approaches of completing the square and quadratic formula However they can often be solved by factorization too since the quadratic expressions are factorable by using distributive laws reversely The roots are obtained through finding two factors for q At the same time these two factors become satisfying with the relation so that the sum of factors is equal to That way the two factors become two roots of a quadratic equation Denoting the two factors or two roots as the relationship between coefficient and constant is

02 =++ qpxx2x

qminus

21rr)( 2121 rrprrq +minus=bull= Quadratic equations can be written as the product of

two first-degree binomials or factoring form 0))(()( 212121

22 =minusminus=++minus=++ rxrxrrxrrxqpxx Through the factoring form the roots of this equation are obtained being The core of factorizing type 4 equations is seeking the connections between coefficients and roots which reveal the studentsrsquo basic ability of arithmetic multiplication and addition It is often more effective in this case to use factorization than to use quadratic formula or completing the square Let us have a look at the example In this equation we get

2211 rxrx ==

035122 =+minus xx

18

35)7)(5(12)7()5( =minusminus=minus=minus+minus= qp or 75)75( bull=bullminus= qp (here we can directly get roots) Factorizing the equation we obtain 0)7)(5( =minusminus xx and roots are The procedures of finding factors of the equation often occur in some studentsrsquo minds quickly without writing down the whole computing process but those students often have good skills in arithmetic operations (reference) Compared to using factorization for solving this equation application of quadratic formula seems ineffective A possible problem in this method is how to find the signs before the factors or how to judge if the factors in the parentheses are positive or negative integers In order to check if the solution is correct we can always multiply these two factored binomials by distributive law and see if the expanded polynomial is the same as the original quadratic equation

75 21 == xx

The fifth kind of quadratic equations is type 5 Still the methods of completing squares and quadratic formula can be applied for all type 5 equations but there are cases in which factorization can be used for solving this type of equations within a certain range of integers as mentioned before Sometimes procedures of factorizing such equations can be carried out in our minds quickly like type 4 equations but in many cases they can not As a matter of fact finding two binomial factors for a factorable quadratic equation or a factorable second-degree polynomial is a complicated process and requires systematic work In his article published in Mathematics in School Jackman (2005) demonstrates this systematic procedure via factorizing second-degree polynomials A second-degree polynomial can be written as where are all integers By multiplying out the polynomial becomes thus

0002 nene=++ bacbxax

))((2 srxqpxcbxax ++=++ srqpcbaqsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== Writing the factors of a and c on two lines p q times r s Finding the right pairs of factors for a and c has to be done by listing out the groups of products of different factors for example The factor pairs for 6 are (6 1) (1 6) (3 2) (2 3) and the factor pairs for (-4) are (4 -1) (-4 1) (2 -2) (-2 2) (1 -4) (-1 4) Writing them on two lines p 6 1 3 2 q 4 -4 2 -2 1 -1

456 2 minus+ xx

times r 1 6 2 3 s -1 1 -2 2 -4 4 Then comes calculating the values of ps and qr until the value of b = 5 is obtained The calculating procedures are divided into 4 groups and 6 steps in each group which means 4 times 6 steps in total Here I only give one grouprsquos procedure as an example 6 middot (-1) + 4 middot 1 = -2 6 middot 1 + 1middot (-4) = 2 6 middot (-2) + 1 middot 2 = -10 6 middot 2 + 1middot (-2) = 10 6 middot (-4) + 1 middot 1 = -23 6 middot 4 + 1 middot (-1) = 23 helliphellip After the 24ndashstep calculating procedures two computing procedures obtain the same correct result 3 (ndash1) + 4 2 = 5 (yes) 2 4 + (-1) 3 = 5 (yes) Are they the same From 3 (ndash1) + 4 2 = 5 the polynomial can be factorized into )12)(43( minus+ xx while from 2 4 + (ndash1) 3 = 5 the polynomial can be factorized into )43)(12( +minus xx According to communicative law these two binomial products are the same The shortcoming of this

19

method is that it is too ineffective and complicated The addition of matrixes is almost impossible and unbearable Nevertheless based on the same idea I suggest finding factors by guessing analyzing and checking Using the same example here we can find that one of the binomials must have a negative integer because of ndash4 which implies that 5 is the sum of a positive integer and a negative integer Then looking at a and c the factors of 6 can be (1 6) and (2 3) while the factors of 4 can be (1 4) and (2 2) when we regard them as absolute values We know that the factoring form of the polynomial will be like

and the connections of all coefficients are

456 2 minus+ xx

qsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== The coefficient 5=b is the difference of two products of factors for 6 and 4 First we try the biggest number of product such as 2446 =bull then 524 =minus the unknown is which is a prime number the only factors for 19 are (19 1) There is not any combination of products of factors for 6 and 4 satisfying 19 Therefore the multiplications of factors (1 6) and (1 4) are not included in this case So we move to factors (2 2) (2 3) and (1 4)

19 =

In the second step we try The result is negative it is obviously

4264466)23)(22( 22 minusminus=minus+minus=+minus xxxxxxx2 lt 5 which implies that the combination of (2 3) and (2 2) is not

considerable One pair of factors has to be removed and that is (2 2) which does not influence the coefficient of the -term Then we try The result this time is negative too but we can see that

2x 456)43)(12( 2 minusminus=minus+ xxxxb is 5 which guides us to interchanging the positions of

negative and positive signs in the parentheses The next attempt is and it is correct Seeking the correct factors for the polynomial

has taken four major operating steps by trying the combinations of factorsrsquo products and relating the sum of the products with the value of the x-term coefficient The strategy is to make use of distributive law reversely first and then multiply the product of every pair of parentheses in order to check if they are the right pair of polynomial factors Factorizing second-degree polynomials is the major step for solving quadratic equations After factoring polynomials into the product of two binomials as in the example given above the roots can be obtained by making each binomial equal to zero according to the theory of zero point (ldquonollstaumlllerdquo in Swedish) of a polynomial equation originating from unique factorization theorem

456)43)(12( 2 minus+=+minus xxxx456 2 minus+ xx

Factorization by using the technique of vertically and crosswise sutra Binomial expansion or multiplication of binomials by distributive law and factorization of the second-degree polynomials by using distributive law backwards have an extremely symmetrical effect and the results of two kinds of operations are equivalent It is crucial for the students to be able to perceive mathematics structure and be acquainted with arithmetic operations as well as to understand the relationship between quantities when they carry out factoring procedures The expansion of binomials by distributive law has been named as the method FOIL which stands for first outside inside and last according to multiplication orders of two binomials for example (Nataraj and Thomas 2006) Many students are used to this procedure Another technique to multiply two binomials is called vertically and crosswise sutra origin from the ancient Indian Vedic mathematics Vertically and crosswise sutra and FOIL are much the same in principle but it makes use of

824)4)(2( 2 +++=++ xxxxx

20

the power of visualization and improves facility with the expansion of binomials and the factorization of quadratic expressions In this method two binomials are written into two lines Using the same example the two lines are written as 824)4)(2( 2 +++=++ xxxxx

x + 2 x + 4 ______________ xsup2 4x + 2x 8

The order of multiplication is as shown below x + 2 x + 4

4 3 2 1 (the numbers 1 2 3 and 4 refer to steps) xsup2 + 2x + 4x + 8 Figure 3 The vertically and crosswise method of multiplication of binomials Step 2 and 3 can be interchangeable If we put this form into a line expanded according to the order above would be like

)4)(2( ++ xxxbullrarrbull 442 clockwise and xxx bullrarrbull2 in

counterclockwise direction According to sutra we can imagine drawing a thread starting from 2 then going through 4 and x clockwise and then draw the second thread starting from the same place 2 later going through x in the second bracket and finally x in the first bracket counterclockwise Learning expansion of binomials by the methods FOIL and Vertical and clockwise sutra is the basic process before students study factorization In this study Nataraj and Thomas (2006) have found that the students appear to do factorization using guess-and-check as well as decomposition or the vertical and clockwise approach Their finding is that afterwards the students performed better on algebra questions especially on the factorizations They conclude ldquohellipthe values of the method may lie in what it adds to the studentsrsquo overall algebraic conceptions and knowledge of mathematical structurerdquo (p 16) In general solving these five kinds of quadratic equations by factorization can be traced back to unique factorization theorem The important character in this method is that you can always check if factorization is correct or not through multiplying out the two pairs of a parenthesis and see if the expanded polynomial is the given one or not Using factorization is not only about method but also about fostering the studentsrsquo mathematical thinking The solving processes are more important than solutions

21

3 Review of related literature and research Besides the focus on mathematics content another focus in the background work for this research is about previous studies and research in the field of mathematics didactic concerning different methods for solving quadratic equations and the use of factorization In order to study the previous research and studies in the related field three scientific databases and a search engine have been used and 113 articles have been investigated Topics such as solving quadratic expressions and equations are often discussed as teaching ideas and strategies shared by mathematics educators in different countries from lower and upper-secondary schools to colleges according to the findings in the article search These didactic topics from mathematics teachersrsquo perspectives include for example using geometric approaches to solve quadratic equations (Allaire amp Bradley 2001) utilizing completing squares (Vinogradova 2007) factoring quadratics (Hoffman 1976 Cheng 1980) and using factorization completing the square graphical methods to solve quadratic equations (Macdonald 1986) Kostsopoulos (2007) links cognitive reasoning to pedagogical problems appearing in the mathematics classrooms in her article published in Australian Mathematics Teacher Based on her teaching experiences Kostsopoulos is aware of her pedagogical strategies lacking insight on dealing with studentsrsquo difficulties of multiplication and factorization of quadratics She assumes that ldquostudentsrsquo problems with factorization and with identifying varied representations of the same quadratic relationshiprdquo may be linked to the ways in which the brain constructs cognitive representations (p 19) Factoring of quadratics is the rewriting of polynomials as a product of polynomials It requires students to have both a conceptual understanding of multiplication of polynomials and the procedural knowledge to retrieve basic multiplication facts effectively Based on the theory about the order matters in the brainrsquos ability to retrieve number facts (Phenix amp Campbell 2001) Kostsopoulos conjectures that studentsrsquo ability to access the appropriate long-term semantic memory is limited when they are confused by mixing the order and varied forms of quadratics Although this article is not about a research study it seeks the reason why factorizing quadratics is difficult and therefore might provide math teachers with alternative explanations for similar pedagogical frustrations With Von Glasersfeldrsquos radical constructivism and Peter Gaumlrdenforsrsquo epistemic semantics theory as his theoretical frameworks Rauff (1994) suggests that belief-based teaching can be successful in teaching factoring According to Rauff constructivist learning theory takes the view that students construct their own beliefs and knowledge of mathematics Learning occurs when the students change their belief set Referring to belief set theory from Gaumlrdenfors (1988) Rauff (1994) claims that ldquoa belief set is expanded when a new belief is added to it and contracted when a proposition is no longer believed and is removed from the setrdquo (p 421) Expansion contraction and revision are three components to comprise basic mechanisms of changes of belief according to Rauff When examining and comparing studentsrsquo (ages 18-20) own definitions of factoring Rauff analyzes the errors of their definitions with constructivism and epistemic semantics as analyzing tools The result of this study shows that the approach to teaching factoring rely heavily upon what a student believes about factoring is quite productive in two dimensions First it reveals the source of nonstandard factoring Second it provides a starting point for modification of the studentrsquos underlying conceptions of factoring Rauff proposes that belief-based teaching can also be

22

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

(Johansson 2006) has found that the mathematics textbook is the most influential factor in classroom teaching and learning According to Johansson (2006) textbooks seem to dictate the teaching of mathematics in many aspects The use of textbooks is a very important framework in mathematics teaching Textbooks influence not only what kind of tasks students are working with and the examples presented by the teachers but also how mathematics is portrayed in terms of the concepts and the features that are related to the subject (p 26) From previously done research Johansson (2006) has found the importance of analyzing the content of textbooks because of the following reasons ldquomathematical topics in textbooks are most likely presented by the teachersrdquo ldquoTeachersrsquo pedagogical strategies are often influenced by the instructional approach of the materialrdquo ldquoTeachers sequence of instruction are often parallels to that of the textbookrdquo and ldquoTeachers report that textbooks are a primary information source in deciding how to present contentrdquo (p 48) The same mathematical topics can be emphasized in one book but might be overlooked in another An analysis of mathematics textbooks reveals the implied beliefs of what mathematics is and how it can be taught and learned It also reflects on the influence of the educational culture from a specific country It is obvious that mathematics textbooks play important roles in Swedish classrooms though not all teachers are the slaves of the textbooks Since mathematics textbooks are important sources for teaching and learning mathematics in Swedish classrooms I have decided to start my study by investigating eight mathematics textbooks used for the course Mathematics B in Swedish upper-secondary schools and by analyzing three of them in detail The related teaching guide material will be investigated in next step

14 Research Aim

The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3

2 Mathematics background in the field of algebra

21 Algebra history and its development Quadratic equations and their unknowns as well as polynomials belong to the field of algebra in mathematics In Swedish mathematics textbooks for mathematics course B there is one chapter on algebra Many critical elements of algebra have been presented in every book but there are no descriptions or definitions of the word ldquoalgebrardquo What is algebra Colin Maclaurin in his 1748 algebra text defined it as ldquoAlgebra is a general Method of Computation by certain Signs and Symbols which have been contrived for this Purpose and found convenient It is called an Universal Arithmetic and proceeds by Operations and Rules similar to those in Common Arithmetic founded upon the same Principlesrdquo (Katz 2006 p 185) Leonhard Euler in his own algebra text in 1770 defined algebra as ldquoThe science

4

which teaches how to determine unknowns quantities by means of those that are knownrdquo (Katz 2006 p 185) In this part I will mainly give a survey of the historical development of algebra including solving quadratic equations according to four stages the rhetorical stage the syncopated stage the symbolic stage and the purely abstract stage The purpose is to try to find out why algebra in school is like a threshold which hinders the students in their understanding Can a historical perspective on algebra explain this obstacle Algebra is an old science and its historical background is complicated Algebra procedures have developed slowly There are different opinions about where the evolution of the term ldquoalgebrardquo started It is commonly believed that algebra first appeared among the Egyptians the Babylonians the Greeks or the Arabs The geometrical influence on algebraic reasoning was strong in ancient Greece However the word algebra originated in Baghdad where the Arabic scientist al-Khwarizmi (AD 780-850) published a short book about calculating with the help from al-jabr and al-muqabala a book on the solution of an equation as a rule (Kvasz 2006) Todayrsquos algebra has its root in Arabic algebra Western mathematics tended to turn algebraic operations into symbols and later developed abstract algebra The process of algebra development was slow and its whole history lasted 4000 years

The rhetorical stage originated from geometry ideas

Historically algebra developed in three stages the rhetorical stage the syncopated stage and the symbolic stage According to Katz (2006) algebra development can also be categorized into four conceptual stages ldquothe geometric stage where most of the concepts of algebra are geometric the static equation-solving stage where the goal is to find numbers satisfying certain relationship the dynamic function stage where motion seems to be an underlying idea and finally the abstract stage where structure is the goalrdquo (p 186) Algebra began very early in recorded history Algebra texts date from the first half of the second millennium BCE 37 or 38 centuries ago and were written by people living in Mesopotamia and Egypt (Derbyshire 2006) During the Hammurabi period from about 1790 to 1600 BCE the Babylonians started their civilization by pressing written words in patterns called cuneiform or wedge-shaped stylus into wet clay Many tablets in cuneiform had a mathematical-algebraic content Their mathematical texts were of two kinds table texts and problem texts The table texts were lists of multiplication tables tables of squares and cubes as well as advanced lists like the famous Plimpton 322 tablet which is about Pythagorean triples The number system of the Babyloniansrsquo time was based on 60 digits rather than todayrsquos 10 digits for example our number of 37 denotes three sixties and seven ones which is our number 187 The problem was that the Babylonians had neither defined zero nor negative numbers The Babylonians of Hammurabirsquos era had no proper algebraic symbolism All mathematical problems were expressed in words for example unknown quantity in Sumerianrsquos Akkadian text was expressed as igum (length) and igibum (width) as reciprocal The application of algebra might have its origin in the need for measuring land areas At the rhetorical stage all mathematical statements and arguments were expressed in words and sentences (Derbyshire 2006) Babylonian mathematics had two roots one is accountancy problems and the other one is a ldquocut-and-pasterdquo geometry probably developed for understanding the division of land Many old-Babylonian clay tablets contain quadratic problems of which the goal was to find such geometric quantities as the length and width of a rectangle As an example we are given that the sum of the length and width of a rectangle is

5

216 and the area of the rectangle is

217 (Kvatz 2006 p 187 and Derbyshire 2006

pp 25-27) What is the length and the width of this rectangle The tablet described in detail

the steps the writer went through First the writer halves 216 to get

413 Next he squares

413

to get 16910 From this area he subtracts the given area

217 giving

1613 The square root of

this number is extracted431 Finally the length is 5

431

413 =+ while the width is

211

431

413 =minus (Kvatz 2006) The whole process can be translated into parts of a quadratic

formula that is 2172

216

431

2

minus⎟⎠⎞

⎜⎝⎛ divide= The Babylonians did not know anything about

negative numbers the only solution for them was in positive numbers and hence their algorithm did not deliver the two solutions to the quadratic equation so their formula is slightly different from quadratic formula (Derbyshire 2006) Their formula is

⎟⎠⎞

⎜⎝⎛ divideplusmnminus⎟

⎠⎞

⎜⎝⎛ divide= 2

216

2172

216

2

x If we denote the sum of the length and the width of the

rectangle as b and the given area as c this formula will be ⎟⎠⎞

⎜⎝⎛plusmnminus⎟

⎠⎞

⎜⎝⎛=

22

2 bcbx although it is

not exactly like the modern quadratic formula There are different interpretations of Neugebauer and Sachesrsquo translation of the Babylonianrsquos tablets for this text on finding the length and width of a rectangle However it is very clear that the text from the tablets is dealing with a geometric procedure The problem was solved in words but with geometric ideas This was the beginning of algebra The Greek mathematician Euclid (300 BC) in his Book II of Elements solved some algebraic problems by manipulating geometric figures but based them on clearly stated axioms The geometrical method is in Kvatzrsquos opinion more explicit in another work of his Data The following example illustrates how Euclid solved a quadratic equation by a geometrical method Euclid defined ldquoproposition 1rdquo which is like axiom 1 If two straight lines contain a given area in a given angle and if the sum of them be given then shall each of them be given (Kvatz 2006 p 189) Euclid set up a rectangle with one side x = AS and y = AC Then a line was drawn so that BS = AC and the completed rectangle was ACDB Suppose that

was given and the area of rectangle ACFS (Figure 1) was given denoted as c What were the two sides AS (x) and AC (y) of the rectangle

byxAB =+=

A x S y B

y y

C F D

Figure 1

6

In order to find the length and the width of the rectangle Euclid bisected AB at E constructed the square on BE and then claimed that this square was equal to the sum of the rectangle ACFS and the small shaded square at the bottom (Figure 2)

A E S B

x

y y F

C G D Figure 2 According to Euclid the area of the rectangle ACFS was given which was c and the area of

the new square EGDB was also given which was ( )22b because 222byxSBASEB =

+=

+=

The equivalent relationship between the areas can be formulated as an quadratic equation 22

22⎟⎠⎞

⎜⎝⎛ minus+=⎟

⎠⎞

⎜⎝⎛ bxcb or

22

22⎟⎠⎞

⎜⎝⎛ minus+=⎟

⎠⎞

⎜⎝⎛ ybcb Euclid found out this equivalent relationship

geometrically and took use of this relationship in finding the solutions of the problem so the

length and width of the rectangle ACFS are cbbx minus⎟⎠⎞

⎜⎝⎛+=

2

22 and cbby minus⎟

⎠⎞

⎜⎝⎛minus=

2

22

These two formulas are almost identical with the Babyloniansrsquo solutions in rhetoric expressions The difference is that Greek algebra was based on geometric manipulation while Babylonian algebra was based on rhetoric manipulation with geometrical ideas In general the early stage of algebra from ancient Babylon and Egypt to Greek was mainly geometrical The syncopated stage ndash the beginning of the static equation-solving stage by using In Roman Egypt in probably the second or third century CE the algebra stage was at the syncopated stage which means written algebraic texts were expressed in words but involved special symbols-abbreviations According to recorded mathematics history one the of pioneers who used these special symbols to solve equations with only numbers but no connection with geometry was Diophantus who lived in Alexandria in Egypt around the third century Diophantus used the Greek alphabetic system for writing numbers He wrote a treatise titled Arithmetica of which less than half has been maintained today The surviving part of his work consists of 189 problems in which the object is to find numbers or families of numbers satisfying certain conditions In mathematics today Diophantusrsquo mathematical analysis is known as number theoryndashnot algebra However he used only number symbols to solve equations without the help of geometry Diophantus wrote the coefficient after the variable instead of before it as we do He used Greek ς for unknown quantity our modern x Most of his book deals with indeterminate equations which contain more than one unknown and a potentially infinite number of solutions His problem is that he could not represent more than one unknown instead he solved quadratic equations with two unknowns through substituting one by another and then the quadratic equations became the ones with one unknown At that time negative numbers were regarded to be absurd but he knew how to

7

bring a term from one side of an equation to the other gather up like terms for simplification and some elementary principles of expansion and factorization (Derbyshire 2006) Diophantus made his own literal symbolism with the use of special letter symbols for the unknown and its powers for subtraction and equality From Diophantus algebra history moved into another conceptual stage the equation-solving stage according to Katz (2006) In India quadratic formula appeared without any geometric support Brahmaggupta (598-665) was one of the first mathematicians who could systematically handle negative numbers and zero He gave a general solution to quadratic equations and realized that there were two roots for a quadratic equation It was possible that one of the roots was a negative number Baskharacharya (1114-1185) solved mathematics problems with the use of quadratic equations in his book Siddhanta Siromani (ldquoMathematical Pearlsrdquo) He presented an algorithm to reduce a quadratic equation to a first-degree equation (Olteanu 2007) It is commonly believed that the first true algebra text is the work on al-jabr and al-muqabala by Mohanmmad ibn Musa al-Khwarizmi (780-850) written in Baghdad around 825 (Katz 2006) The word algebra came from the title of this work The word al-jabr means restoration or reestablishment that is to eliminate negative terms through adding the same terms to both sides of equations The word of al-muqabalas means balance meaning to divide every term in a quadratic equation by the coefficient of the second degreersquos term (Olteanu 2007) The first part of his book is a manual for solving linear and quadratic equations Al-Khwarizimi classified equations into six types three of which were mixed quadratic equations For each type he presented an algorithm for its solution Five of the six types of equations were quadratic equations which can be expressed in modern form

cbxaxbxcaxcbxaxcaxbxax +==+=+== 22222 Here is an example of solving the equation ldquoTake the half of the number of the things that is five and multiply it by itself you obtain twenty-five Add this to thirty-nine you get sixty-four Take the square root or eight and subtract from it one half of the number of things which is five The result three is the thingrdquo (Kvasz 2006 p 292) Like Babylonian mathematicians al-Khwarizimirsquos algorithm is entirely verbal On the other hand al-Khwarizimi sometimes completed his algorithm by using geometrical explanations which can be translated as todayrsquos square-completing method Using the example of solving the quadratic equation the completed geometrical procedures are illustrated on next page in figures 3 4 and 5

39102 =+ xx

39102 =+ xx

8

x

xsup2

Figure 3 (Olteanu 2007 p 30) 5x2

25x

xsup2

25x

5x2

Figure 4 (Olteanu 2007 p 30) 254 254

39

254 254

Figure 5 (Olteanu 2007 p 30) According to Olteanursquos (2007) translation al-Khwarizimi started with a square whose sides are x and area is xsup2 (see Figure 3) Then he added four equal rectangles whose areas in total

was 10x along each side of the square that is xx sdotsdot=25410 Each rectanglersquos area is thus x

25

with its length x and its width 52 (see Figure 4) The sum of the big square and four rectangles was given which was 39 The equivalence relationship was Finally figure 4 was completed by adding four small equal squares which had an area the size of

39102 =+ xx

425

25

25

=sdot for each small square and the sum of those was 25 Through adding this sum to

both sides of the equation the area of the biggest square obtained in figure 5 was 64 The

equation translation is 425439

4254102 sdot+=sdot++ xx The side of the biggest square was 8 and

had its relation with other sides of different squares expressed in the first degree equation

9

25

258 ++= x Then x was 3 By ldquocut-and-pasterdquo geometry (Katz 2006 p 191)

al-Khwarizimi reduced the second degree of an equation to the first degree and thereafter solved it Unlike his Babylonian predecessors al-Khwarizimi always presented his problem abstractly rather than geometrically relating to lengths and widths The symbolic stage At this stage of algebra ldquoall numbers operations relationships are expressed through a set of easily recognized symbols and manipulations on the symbols take place according to well-understood rulesrdquo (Katz 2006 p 186) The ancient algebra and geometry had developed sophisticatedly in Egypt Persia Greece India and China After Medieval Islamic scholars gave us the word ldquoalgebrardquo Western Europe began the struggle for the development of algebra starting from some algebraists from Italy Italian mathematician Leonardo Pisano later known as Fibonacci traveled in the 12th and 13th centuries to Persia India and China When he returned to Italy he had wider knowledge of arithmetic and algebra His book Liber abbaci was the best math textbook since the end of Ancient world His book is credited with having introduced Indian numerals including zero to the West But his algebraic skills had been shown in two other works after this one With the introduction of printed books during the second half of the 15th century the development of algebra was sped up Several Italian mathematicians including Girolamo Cardano had figured out how to solve cubic and quadratic equations Algebra became purely abstract with the exception of an English mathematician named Robert Recorde who lived in the 16th century and created quadratic problems from real world experience (Derbyshire 2006) It was in France that algebra had developed into a well organized literal symbolism In his work In artem analyticem isagoge French mathematician Franςois Viegravete (1540-1603) in the late 16th century became the first mathematician to use letters representing numbers systematically and effectively (Derbyshire 2006) He made a range of letters available for many different quantities This was the beginning of modern literal symbolism Viegravetersquos unknown quantity was divided into two classes unknown quantities (meaning ldquothings soughtrdquo) denoted by A E I O U and Y while ldquothings givenrdquo was denoted by constants like B C Dhellip For example his A is our unknown x Viegravete was a pioneer in the study of equations His two papers on the theory of equations were published twelve years after his death In the second paper titled ldquoDe equationem emendationerdquo (ldquoOn the perfecting of equationsrdquo) Viegravete opened up the line of inquiry that led to the study of the symmetries of an equationrsquos solutions to Galois theory the theory of groups and all of modern algebra He found the relationship between the solutions of the equation and the coefficients for the first five degrees of equations in a single unknown To explain this in our modern symbols we suppose that the two solutions of the quadratic equation are 02 =++ qpxx α and β which means βα == xx Based on this logic the following thing must be true 0))(( =minusminus βα xx since only α andβ and no other values of x make this equation true This form of equation is just a rewritten form of the same equation If we multiply out those parentheses this rewritten equation turns to be Compared to the original equation the relationships between the solutions and the coefficients we obtain

0)(2 =++minus αββα xxqp =minus=+ αββα

(Derbyshire 2006) It is said that Viegravete discovered the solution formula called quadratic

10

formula today for general quadratic equations which is a

acbbx2

42

21minusplusmnminus

= for a general

quadratic equation (Olteanu 2007) 002 ne=++ acbxax Another French mathematician and philosopher who had strong influence in the history of algebra was Reneacute Descartes (1596-1650) His idea to use the Cartesian system of coordinates which was derived from his Latin name developed both algebra and geometry In his work La geacuteomeacutetrie written in 1637 Descartesndashthrough using numbers to indentify points in a Cartesian coordinatesndashconnected geometrical objects to algebraic numbers and made the classical geometry become analytical geometry He took up the plus and minus sign from the German Cossists of the previous century and also the square-root sign From Descartes the symbol of an unknown was represented by x which led to the modern system of literal symbolism (Derbyshire 2006) Descartes developed the idea of functions in Cartesian coordinates although the idea of function can be traced back to the Islamic mathematician Sharaf al-Din al-Tusi from Persia (Katz 2006) and Klaudius Ptolemaios about 2000 years ago (Olteanu 2007) Descartes declared that every curve in a Cartesian coordinate system has an equivalent equation which can represent the points on the curve or vice versa every equation containing x and y can be represented by a curve through its coordinate points However the word function was introduced by Gottfried Wilhelm Leibniz in 1693 and the definition of function was defined by Leonhard Euler in his work Introductio analysin infinitorium in 1748 Euler had also given the symbol of a denotation as a relying variable (Olteanu 2007) In the 18)(xf th century after the discovery of calculus by Newton and Leibniz algebra went into an area of analysis ndashthe study of limits infinite sequences and series functions derivatives and integrals (Derbyshire 2006) The purely abstract stage-algebra structure Since 17th century algebra has not anymore solely been about finding solutions for different degrees of equations mathematicians have started to integrate algebra with astronomy and physics Johann Kepler and Galileo Galilei were interested in curves and finding a mechanism for representing motion instead of quantity-like numbers But their arguments were not algebraic-symbolic but geometrical It was Fermat and Descartes who were regarded as the fathers of analytic geometry and who showed how to represent a curve described verbally through algebra analytic geometry and gave a mechanism for representing motion Newton in his Principia picked up on Fermat and Descartesrsquo representation and developed the calculus With the invention of the calculus mathematical problems were solved by curves not just points Algebra grew more and more to present paths of motion In order to be able to judge if the algebraic manipulations were correct axioms were formulated for arithmetic applied for algebraic manipulations In the late 18th century Lagrange introduced the idea of permutations into the search for solutions Therafter Galois developed methods for determining what conditions polynomial equations need to meet to be solvable New abstract algebraic concepts were found like ldquofieldrdquo and ldquogrouprdquo during the early 19th century In 1854 Ceyley gave an axiomatic definition of a group During the 1890s this definition entered textbooks along with the axiomatic definition of a field an idea which had roots in the work by Galois By the beginning of the 20th century algebra became less about finding solutions to equations and more about looking for common structures in many mathematical objects defined by sets of axioms (Katz 2006)

11

In short algebra development took 4000 years It took about 3500 years for mathematicians to start to use algebra symbols in systems Just almost 100 years ago algebra developed into a purely abstract science The long history may give both mathematics educators and students some explanations to why it is so difficult to teach and learn algebra

22 Three solving methods for quadratic equations Already in the previous part I have presented geometrically and rhetorically solving quadratic equations from a historical perspective In order to get more explicit understanding of quadratic equations quadratic formula and factorization from mathematics didactic point of view this part presents three common methods for solving quadratic equations 1) Completing the square 2) Factorization 3) Quadratic formula The reason to present these three methods is that they are the common topics for discussion in the articles searched in this area concerning mathematics education at upper-secondary school 1) Completing the square In two articles Solving quadratic equations by completing squares (Vinogradova 2007) and Geometric approaches to quadratic equations from other times and places (Allaire amp Bradley 2001) the authors advise mathematics teachers to use the ideas of ancient mathematician al-Khwarizmi and his method of completing quadratic equations by geometrical algebra and simplifying it in order to represent it to the students By doing so the teachers relate mathematics quantity to physical objects in a visual way Didactically teachers may start with a concrete example in which students can understand the content visually Later from the concrete example the teachers lead the students to another example expressed in algebraic symbols for example to study a rectangle whose area is )10( +xx and supposing this area is 39 this becomes 39)10( =+xx Four steps are followed to build a new square which is actually called completing the square x 10

Figure 6 x 5

Figure 7

x

x

5 5

12

A Begin with a rectangle of the area )10( +xx that is with the short side as x and the long side )10( +x The shaded area is 10bullx (Figure 6)

B Divide the shaded rectangle into two small shaded rectangles with the size 5x each and move them to each adjacent side of the xsup2 square (Figure 7) The total area is still

x10

)10(39 += xxC In Figure 7 a new square is shaped in the lower-right hand corner with the side 5 or

area 25 By adding this small square to the diagram the large square with an area of is ldquocompletedrdquo Therefore we can write 25)5(22 ++ xx 25)10(2539 ++=+ xx or

The large squarersquos area is now because the side has a length of Therefore

6425)5(22 =++ xx 2)5( +x5+x 38564)5( 2 =rArr=+rArr=+ xxx

After these three steps a solution to this quadratic equation is derived by completing the square (Allaire amp Bradley 2001) 2) Factorization Hoffman (1976) presents three approaches (A B and C) for factorizing quadratic expressions through observing and grouping He takes a second-degree polynomial as an example

273 2 ++ xx

a The inspection method 1) suggests factors of and 23x x3 x 2) 2 suggests factors of 1 and 2 or ndash1 and ndash2 3) Checking the linear term shows that the correct factorization is x7 )2)(13(273 2 ++=++ xxxx

This method according to Hoffman (1976) is based on the studentrsquos ability to simplify by inspecting the indicated product of a pair of linear expressions

b The method of decomposition of the linear term 1) Multiply and 2 to get 23x 26x2) Decompose into the sum of two terms whose product is x7 26x

This gives xxx += 67 3) Factorize by grouping 2)6(3 2 +++ xxx

2)6(3 2 +++ xxx = )

)2()63( 2 +++ xxx

= 2(1)2(3 +++ xxx= )13)(2( ++ xx

c Third method 1) Multiply 3 and 2 to get 6 2) Find two numbers whose product is 6 and whose sum is 7 The numbers are 6 and 1 3) 273 2 ++ xx

=3

)13)(63( ++ xx

= )13)(2( ++ xx

13

These three approaches can not actually be divided as three unrelated approaches especially the third method is seemingly unclear What Hoffman (1976) wants to demonstrate is that all the three approaches are based on observing the relationship among the coefficients and the constant of the polynomial Through finding the product and the sum of two factors the polynomial can be factorized in the product of two binomials In the authorrsquos opinion factorization is heavily dependent on skill in multiplication and using distributive law reversely He means that ldquostudents who do not have reasonably strong skills in multiplication should not be expected to develop strong skills in factorizationrdquo (p 55) Factorizing polynomials is not only for simplifying high-degree polynomials (for example the third or fourth-degree polynomials) but can also be applied for solving quadratic equations Supposing this polynomial forming a quadratic equation Solving this equation is actually to decompose the polynomial into a factoringrsquos form

0273 2 =++ xx0)13)(2( =++ xx

which is equivalent to the polynomial form The essential step to solve this equation is to follow the zero-factor propertyndashalso called null-factor law (nollprodukt in Swedish)ndashthat is for real numbers p and q p bull q = 0 if (and only if) p = 0 or q = 0 It means that either of the binomials on the left side of the equation has to be equal to zero in order to satisfy the

equivalent relation of this quadratic equation The solutions of two roots or 2minus=x31

minus=x

are obtained through making either 02 =+x or 013 =+x 3) Quadratic formula Solving quadratics equations by factorization is constrained within the simple quadratic equations over the whole integers and rational numbers as coefficients What happens when quadratic equations have coefficients that belong to an irrational domain or big quantity The quadratic formula is a solution to such a situation As mentioned previously historically the French mathematician Franccedilois Vieacutete discovered the quadratic formula

aacbbx

242 minusplusmnminus

= ) which is applicable for all kinds of quadratic equations In

Swedish textbooks this formula is written in PQ form by making for the equation

that is

0( nea

1=a

02 =++ qpxx qppx minus⎟⎠⎞

⎜⎝⎛plusmnminus=

2

22 with

abp = and

acq = In Oleatursquos study (2007)

studentsrsquo difficulties in using algebraic symbols no longer is the problem But on the other hand handling the parameters or coefficients in quadratic equations like and rewrite them in the equivalent form with p and q being real numbers becomes obstacles for the students when they study the algebra course at upper-secondary school

02 =++ cbxax02 =++ qpxx

The quadratic formula a

acbbx2

42 minusplusmnminus= has been regarded as the standard method

according to US mathematics teachers Obermeyer (1982) presents a way of how to derive this formula by completing a square through 11 steps (I will not present them here) At the same time an alternate method is also introduced according to the same principle All the methods mentioned so far are based on the square rules The standard method or PQ form might be regarded as an efficient and direct method but it may

222 2)( bababa +plusmn=plusmn

14

lead students to solve quadratic equations in a mechanical way besides having problems with the troublesome plusmn symbolism (Stover 1978) Both Stover (1978) and Olteanu (2007) have suggested that using a graph of quadratic functions to solve quadratic equations is another alternative method

23 Mathematical background of factorization Factorization in algebra structure Among the three solving methods presented above both the geometrical method and quadratic formula can be traced back to algebra history in section 21 What is the trace of factorization in algebra history Factorization was at the last stage in algebra history and belonged to algebra structure (Derbyshire 2006) In Swedish mathematics textbooks the chapter on algebra often includes polynomial factorization and quadratic equations But the factorization in the textbooks is not at the same abstract level as the factorization of algebra structure although they share the same axioms and there are certain connections between these two different levels of factorization How are polynomial and quadratic equations as well as factorization related to each other In order to find the theoretical background of factorizing polynomials this part constitutes a mathematical theoretical review by referring to two books concerning algebra structure (Vretblad 2000 Durbin 1992) and a research study (Bosseacute and Nandakumar 2005) A quadratic equation can be regarded as polynomial equation Solving quadratic equations has much to do with ldquounique factorization theoremrdquo The polynomial

is a second-degree polynomial A polynomial consists of coefficients in whole numbers rational numbers real numbers or complex numbers and variables (sometimes called unknowns) in different degrees as well as constants A polynomial

has symbols as coefficients which are real numbers If all are zero and so

)0(02 ne=++ acbxax

cbxax ++2

nnxaxaxaaxf ++++= )( 2

210 naaa 10

ka 1gek 0)( axf = is constant If all are zero then we say the is a zero polynomial If meaning that is not a zero polynomial and n is the biggest number and then we say that the polynomial is at degrees A zero polynomial has no degree at all According to the definition (Vredblad 2000) if is a polynomial the equation is called an algebraic equation or polynomial equation A solution or a root of this equation is a number

kaf 0nef f

0nena f nf

0)( =xfα so that 0)( =αf Such a number is even

called a zero point (nollstaumllle in Swedish) of the polynomial which means when f α=x the value of the polynomial is zero (p 161) Every polynomial whose degree is equal or greater than 1 can be written as a product of irreducible polynomial In this way through polynomial division a polynomial with n degrees can be expressed from a sum of terms into a product of its denominator and its quotient The process of changing the expressions is factorization for

instance 12

22

+=minusminusminus x

xxx The factoring form is )1)(2(22 minusminus=minusminus xxxx

Factorization of polynomials is analogue to factorization of whole numbers which is very much related to the concept of ring A ring consists of a set with two operations which are a sum and a product of two elements over a field Any polynomial from zero polynomial to n degree-polynomials can be written or operated through addition and multiplication of the

15

elements over a field F including whole numbers or integers rational numbers real numbers and complex numbers In a ring you can multiply any two elements but you canrsquot always divide Factorization is a way to deal with this fact Here are some examples In the ring of integers Z is an example of factorization in some polynomial rings for instance Q[x] the ring of polynomial in one variable with rational numbers as coefficients such as

7214 sdot=

)451)(2

51(8

52

251 2 +minus=minus+ xxxx is factorization Z[x] the ring of polynomial in one

variable with whole numbers as integers such as is factorization )1)(2(22 minusminus=minusminus xxxx The polynomial ring R[x] means the ring of polynomial in one variable with real numbers as coefficients Factorizing of such polynomials implies that the polynomial factors are the elements in the ring of R[x] for instance )2)(2(22 minus+=minus xxx The same thing is true for C[x] the ring of polynomial in one variable with complex numbers as coefficients Thus it is coefficients of polynomials that determine which ring a polynomial belongs to and if it is possible to be factorized in this ring For example number 6 can be factorized in two ways

in the ring of Z and 32times )51)(51( ii +minus in the ring of C (complex numbers) However there are cases in which divisions can not be done within the ring for example 52 or

)3()52( 2 ++minus xxx which means that no element in the ring can multiply with a denominator obtaining the numerator Therefore neither 5 nor can be factorized 522 +minus xx If a polynomial of degree at least one has no other divisors then it is considered to be irreducible or prime (Durbin 1992) Of every polynomial of degree at least one can be written as a product of irreducible polynomials according to unique factorization theorem each polynomial of degree at least one over a field F can be written as an element of F times a product of monic irreducible polynomials over F and except for the order in which these irreducible polynomials are written this can be done in only one way (p 221) Thus a factorable polynomial can be written into a product with another polynomial such as In other words the condition that )()()( xqaxxf minus= )( ax minus is a factor of a polynomial

in a variable x is only true if )(xf 0)( =af For example if and

6)( 2 minus+= xxxf0624)2( =minus+=f 0639)3( =minusminus=minusf the factors of are and )(xf )2( minusx )3( +x

The value of the polynomial is zero which is also expressed as the polynomial having a zero point (nollstaumllle in Swedish) when

)(xfα=x

Based on unique factorization theorem finding solutions of an algebraic equation is equivalent to finding the first-degree factors for a polynomial (Vredblad 2000) In this way solving a quadratic equation can be handled through finding the first-degree factors for the second-degree polynomial and there after finding the roots of the quadratic equation This is often called using factorization to solve quadratic equations Besides unique factorization theorem the connection of using factorization to solve a quadratic equation is that this method relates to prime numbers using distributive law backwards commutative law multiplication division and operating fractions As matter of fact it is about application of factorizing integers in polynomials at upper-secondary school level Since quadratic equations in the mathematics B course at Swedish upper-secondary school are limited to real numbers and often within the whole numbers (integers) many quadratic equations should be able to be solved by factorization according to my own

16

judgment Therefore using factorization for solving quadratic equations is very much related to arithmetic operations number concept and algebra structure It may offer students both mathematical conceptual understanding and operational skill For algebra beginners factorization can be traced back to studentsrsquo early years of mathematics education during which students do not only work with multiplication but also on factoring integers for example 56 = 8 middot 7 = 2 middot 2 middot 2 middot 7 = 4 middot 14 = 2 middot 28 In this example two operations are included multiplication and division Multiplication operation may help students to understand the multiplicative structure of the integers while factoring integers may become meaningful when it is related to divisibility divisors and prime numbers These two operational skills or competence are major techniques for simplifying fractions and finding common denominators Being able to understand and operate factoring integers has laid the essential grounds for algebra beginners to study factorization of polynomials Thus factorization reunites studentsrsquo early mathematics knowledge of factoring integers with later knowledge of algebra structure Factorability In a study on factorability by Bosseacute and Nandakumar (2005) the probability of factorability is reasoned through investigating the range of integers as coefficients of quadratic equations In the quadratic equation a b and c are randomly generated integers within a determined range r such as

002 ne=++ acbxaxyrx lele where x and y are integers When is a

perfect square or where

acb 42 minus

)4( 2 acb minus is an integer quadratics are factorable According to the studyrsquos data it shows that as the range for a b and c increases the probability of factorability of a quadratic with randomly selected integer values for a b and c decreases Bosseacute and Nandakumar confirm that as the range for coefficients expands to infinltltinfinminus r the probablility of factorability of a quadratic with randomly selected integer values for coefficients approaches zero The studyrsquos data collected from college algebra courses and textbooks demonstrates that about 15 of quadratics with integer coefficients [ ]1010minusisinr are factorable Thus about 85 of quadratics can not be factorized In spite of the limitation of factorability within the exercises concerning factoring practice chosen from 27 surveyed college algebra textbooks about 94 of the problems were factorable Within the textbooks 55 of the quadratics to be factored were within the range [-10 10] Even if many quadratic equations are factorable choosing the right factor pairs is time consuming and unnecessary for example

has to be factorized through choosing the right pairs of factors among nine pairs for 36 and 8 pairs of factors for 24 Comparing the three different methods for solving quadratic equations the study suggests that completing the square and quadratic formula are more effective informative and useful than factorization

0245936 2 =++ xx

231 Different kinds of factorizations for solving different quadratic equations How can we find the binomial factors of a quadratic equation or a second-degree polynomial without knowing one of the divisors as a binomial factor The answer is that we have to observe the relations among all coefficients and constants as well as the positive and negative signs before them under the condition of being factorable Most of the coefficients in quadratic equations or polynomials among exercises provided for practicing factoring are integers (here I borrow the same notation r as the one used by Bosseacute and Nandakumar) with

17

their range between ndash10 and 10 that is 10ler In this case it is possible for a student to make a judgment of whether or not a quadratic equation is factorable Five different kinds of factorization depending on five kinds of quadratic equations Factoring quadratic expressions can be divided into five different kinds depending on the type of quadratic equations The coefficients a b p q and constant c illustrated in the following quadratic equations are defined not to be equal to zero thus 000 nenene cba The first kind of quadratic equations is type 1 the obviously factorable type (Bosseacute and Nandakumar 2004) then the equation is factorized as

where the roots are

0002 nene=+ babxax

0)( =+ baxxabxx minus== 21 0

The second kind of quadratic equations is type 2 The factors of this second degree polynomial are so the roots to this type of quadratic

equations are

0)( 22 =minusbax))(()( 22 baxbaxbax minus+=minus

bax plusmn=21 This kind of factorization is based on the difference-of-squares

formula for example can be solved by factorizing into 049 2 =minusx 0)23)(23( =minus+ xx Thus

the roots are 32

32

21 =minus= xx This kind of factorization is also obviously factorable

The third kind of quadratic equations is type 3 This type of equations can be solved directly by factoring into two identical binomials

The double roots obtained from this

factorization are

02)( 22 =+plusmn babxax

0))(()(2)( 222 =plusmnplusmn=plusmn=+plusmn baxbaxbaxbabxax

abx plusmn=21 This method has utilized square rules which are quite easily

observed The forth kind of quadratic equations is type 4 In this equation the coefficient of the -term is 1 with p and q as positive or negative integers Type 4 equations can always be solved by the approaches of completing the square and quadratic formula However they can often be solved by factorization too since the quadratic expressions are factorable by using distributive laws reversely The roots are obtained through finding two factors for q At the same time these two factors become satisfying with the relation so that the sum of factors is equal to That way the two factors become two roots of a quadratic equation Denoting the two factors or two roots as the relationship between coefficient and constant is

02 =++ qpxx2x

qminus

21rr)( 2121 rrprrq +minus=bull= Quadratic equations can be written as the product of

two first-degree binomials or factoring form 0))(()( 212121

22 =minusminus=++minus=++ rxrxrrxrrxqpxx Through the factoring form the roots of this equation are obtained being The core of factorizing type 4 equations is seeking the connections between coefficients and roots which reveal the studentsrsquo basic ability of arithmetic multiplication and addition It is often more effective in this case to use factorization than to use quadratic formula or completing the square Let us have a look at the example In this equation we get

2211 rxrx ==

035122 =+minus xx

18

35)7)(5(12)7()5( =minusminus=minus=minus+minus= qp or 75)75( bull=bullminus= qp (here we can directly get roots) Factorizing the equation we obtain 0)7)(5( =minusminus xx and roots are The procedures of finding factors of the equation often occur in some studentsrsquo minds quickly without writing down the whole computing process but those students often have good skills in arithmetic operations (reference) Compared to using factorization for solving this equation application of quadratic formula seems ineffective A possible problem in this method is how to find the signs before the factors or how to judge if the factors in the parentheses are positive or negative integers In order to check if the solution is correct we can always multiply these two factored binomials by distributive law and see if the expanded polynomial is the same as the original quadratic equation

75 21 == xx

The fifth kind of quadratic equations is type 5 Still the methods of completing squares and quadratic formula can be applied for all type 5 equations but there are cases in which factorization can be used for solving this type of equations within a certain range of integers as mentioned before Sometimes procedures of factorizing such equations can be carried out in our minds quickly like type 4 equations but in many cases they can not As a matter of fact finding two binomial factors for a factorable quadratic equation or a factorable second-degree polynomial is a complicated process and requires systematic work In his article published in Mathematics in School Jackman (2005) demonstrates this systematic procedure via factorizing second-degree polynomials A second-degree polynomial can be written as where are all integers By multiplying out the polynomial becomes thus

0002 nene=++ bacbxax

))((2 srxqpxcbxax ++=++ srqpcbaqsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== Writing the factors of a and c on two lines p q times r s Finding the right pairs of factors for a and c has to be done by listing out the groups of products of different factors for example The factor pairs for 6 are (6 1) (1 6) (3 2) (2 3) and the factor pairs for (-4) are (4 -1) (-4 1) (2 -2) (-2 2) (1 -4) (-1 4) Writing them on two lines p 6 1 3 2 q 4 -4 2 -2 1 -1

456 2 minus+ xx

times r 1 6 2 3 s -1 1 -2 2 -4 4 Then comes calculating the values of ps and qr until the value of b = 5 is obtained The calculating procedures are divided into 4 groups and 6 steps in each group which means 4 times 6 steps in total Here I only give one grouprsquos procedure as an example 6 middot (-1) + 4 middot 1 = -2 6 middot 1 + 1middot (-4) = 2 6 middot (-2) + 1 middot 2 = -10 6 middot 2 + 1middot (-2) = 10 6 middot (-4) + 1 middot 1 = -23 6 middot 4 + 1 middot (-1) = 23 helliphellip After the 24ndashstep calculating procedures two computing procedures obtain the same correct result 3 (ndash1) + 4 2 = 5 (yes) 2 4 + (-1) 3 = 5 (yes) Are they the same From 3 (ndash1) + 4 2 = 5 the polynomial can be factorized into )12)(43( minus+ xx while from 2 4 + (ndash1) 3 = 5 the polynomial can be factorized into )43)(12( +minus xx According to communicative law these two binomial products are the same The shortcoming of this

19

method is that it is too ineffective and complicated The addition of matrixes is almost impossible and unbearable Nevertheless based on the same idea I suggest finding factors by guessing analyzing and checking Using the same example here we can find that one of the binomials must have a negative integer because of ndash4 which implies that 5 is the sum of a positive integer and a negative integer Then looking at a and c the factors of 6 can be (1 6) and (2 3) while the factors of 4 can be (1 4) and (2 2) when we regard them as absolute values We know that the factoring form of the polynomial will be like

and the connections of all coefficients are

456 2 minus+ xx

qsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== The coefficient 5=b is the difference of two products of factors for 6 and 4 First we try the biggest number of product such as 2446 =bull then 524 =minus the unknown is which is a prime number the only factors for 19 are (19 1) There is not any combination of products of factors for 6 and 4 satisfying 19 Therefore the multiplications of factors (1 6) and (1 4) are not included in this case So we move to factors (2 2) (2 3) and (1 4)

19 =

In the second step we try The result is negative it is obviously

4264466)23)(22( 22 minusminus=minus+minus=+minus xxxxxxx2 lt 5 which implies that the combination of (2 3) and (2 2) is not

considerable One pair of factors has to be removed and that is (2 2) which does not influence the coefficient of the -term Then we try The result this time is negative too but we can see that

2x 456)43)(12( 2 minusminus=minus+ xxxxb is 5 which guides us to interchanging the positions of

negative and positive signs in the parentheses The next attempt is and it is correct Seeking the correct factors for the polynomial

has taken four major operating steps by trying the combinations of factorsrsquo products and relating the sum of the products with the value of the x-term coefficient The strategy is to make use of distributive law reversely first and then multiply the product of every pair of parentheses in order to check if they are the right pair of polynomial factors Factorizing second-degree polynomials is the major step for solving quadratic equations After factoring polynomials into the product of two binomials as in the example given above the roots can be obtained by making each binomial equal to zero according to the theory of zero point (ldquonollstaumlllerdquo in Swedish) of a polynomial equation originating from unique factorization theorem

456)43)(12( 2 minus+=+minus xxxx456 2 minus+ xx

Factorization by using the technique of vertically and crosswise sutra Binomial expansion or multiplication of binomials by distributive law and factorization of the second-degree polynomials by using distributive law backwards have an extremely symmetrical effect and the results of two kinds of operations are equivalent It is crucial for the students to be able to perceive mathematics structure and be acquainted with arithmetic operations as well as to understand the relationship between quantities when they carry out factoring procedures The expansion of binomials by distributive law has been named as the method FOIL which stands for first outside inside and last according to multiplication orders of two binomials for example (Nataraj and Thomas 2006) Many students are used to this procedure Another technique to multiply two binomials is called vertically and crosswise sutra origin from the ancient Indian Vedic mathematics Vertically and crosswise sutra and FOIL are much the same in principle but it makes use of

824)4)(2( 2 +++=++ xxxxx

20

the power of visualization and improves facility with the expansion of binomials and the factorization of quadratic expressions In this method two binomials are written into two lines Using the same example the two lines are written as 824)4)(2( 2 +++=++ xxxxx

x + 2 x + 4 ______________ xsup2 4x + 2x 8

The order of multiplication is as shown below x + 2 x + 4

4 3 2 1 (the numbers 1 2 3 and 4 refer to steps) xsup2 + 2x + 4x + 8 Figure 3 The vertically and crosswise method of multiplication of binomials Step 2 and 3 can be interchangeable If we put this form into a line expanded according to the order above would be like

)4)(2( ++ xxxbullrarrbull 442 clockwise and xxx bullrarrbull2 in

counterclockwise direction According to sutra we can imagine drawing a thread starting from 2 then going through 4 and x clockwise and then draw the second thread starting from the same place 2 later going through x in the second bracket and finally x in the first bracket counterclockwise Learning expansion of binomials by the methods FOIL and Vertical and clockwise sutra is the basic process before students study factorization In this study Nataraj and Thomas (2006) have found that the students appear to do factorization using guess-and-check as well as decomposition or the vertical and clockwise approach Their finding is that afterwards the students performed better on algebra questions especially on the factorizations They conclude ldquohellipthe values of the method may lie in what it adds to the studentsrsquo overall algebraic conceptions and knowledge of mathematical structurerdquo (p 16) In general solving these five kinds of quadratic equations by factorization can be traced back to unique factorization theorem The important character in this method is that you can always check if factorization is correct or not through multiplying out the two pairs of a parenthesis and see if the expanded polynomial is the given one or not Using factorization is not only about method but also about fostering the studentsrsquo mathematical thinking The solving processes are more important than solutions

21

3 Review of related literature and research Besides the focus on mathematics content another focus in the background work for this research is about previous studies and research in the field of mathematics didactic concerning different methods for solving quadratic equations and the use of factorization In order to study the previous research and studies in the related field three scientific databases and a search engine have been used and 113 articles have been investigated Topics such as solving quadratic expressions and equations are often discussed as teaching ideas and strategies shared by mathematics educators in different countries from lower and upper-secondary schools to colleges according to the findings in the article search These didactic topics from mathematics teachersrsquo perspectives include for example using geometric approaches to solve quadratic equations (Allaire amp Bradley 2001) utilizing completing squares (Vinogradova 2007) factoring quadratics (Hoffman 1976 Cheng 1980) and using factorization completing the square graphical methods to solve quadratic equations (Macdonald 1986) Kostsopoulos (2007) links cognitive reasoning to pedagogical problems appearing in the mathematics classrooms in her article published in Australian Mathematics Teacher Based on her teaching experiences Kostsopoulos is aware of her pedagogical strategies lacking insight on dealing with studentsrsquo difficulties of multiplication and factorization of quadratics She assumes that ldquostudentsrsquo problems with factorization and with identifying varied representations of the same quadratic relationshiprdquo may be linked to the ways in which the brain constructs cognitive representations (p 19) Factoring of quadratics is the rewriting of polynomials as a product of polynomials It requires students to have both a conceptual understanding of multiplication of polynomials and the procedural knowledge to retrieve basic multiplication facts effectively Based on the theory about the order matters in the brainrsquos ability to retrieve number facts (Phenix amp Campbell 2001) Kostsopoulos conjectures that studentsrsquo ability to access the appropriate long-term semantic memory is limited when they are confused by mixing the order and varied forms of quadratics Although this article is not about a research study it seeks the reason why factorizing quadratics is difficult and therefore might provide math teachers with alternative explanations for similar pedagogical frustrations With Von Glasersfeldrsquos radical constructivism and Peter Gaumlrdenforsrsquo epistemic semantics theory as his theoretical frameworks Rauff (1994) suggests that belief-based teaching can be successful in teaching factoring According to Rauff constructivist learning theory takes the view that students construct their own beliefs and knowledge of mathematics Learning occurs when the students change their belief set Referring to belief set theory from Gaumlrdenfors (1988) Rauff (1994) claims that ldquoa belief set is expanded when a new belief is added to it and contracted when a proposition is no longer believed and is removed from the setrdquo (p 421) Expansion contraction and revision are three components to comprise basic mechanisms of changes of belief according to Rauff When examining and comparing studentsrsquo (ages 18-20) own definitions of factoring Rauff analyzes the errors of their definitions with constructivism and epistemic semantics as analyzing tools The result of this study shows that the approach to teaching factoring rely heavily upon what a student believes about factoring is quite productive in two dimensions First it reveals the source of nonstandard factoring Second it provides a starting point for modification of the studentrsquos underlying conceptions of factoring Rauff proposes that belief-based teaching can also be

22

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

which teaches how to determine unknowns quantities by means of those that are knownrdquo (Katz 2006 p 185) In this part I will mainly give a survey of the historical development of algebra including solving quadratic equations according to four stages the rhetorical stage the syncopated stage the symbolic stage and the purely abstract stage The purpose is to try to find out why algebra in school is like a threshold which hinders the students in their understanding Can a historical perspective on algebra explain this obstacle Algebra is an old science and its historical background is complicated Algebra procedures have developed slowly There are different opinions about where the evolution of the term ldquoalgebrardquo started It is commonly believed that algebra first appeared among the Egyptians the Babylonians the Greeks or the Arabs The geometrical influence on algebraic reasoning was strong in ancient Greece However the word algebra originated in Baghdad where the Arabic scientist al-Khwarizmi (AD 780-850) published a short book about calculating with the help from al-jabr and al-muqabala a book on the solution of an equation as a rule (Kvasz 2006) Todayrsquos algebra has its root in Arabic algebra Western mathematics tended to turn algebraic operations into symbols and later developed abstract algebra The process of algebra development was slow and its whole history lasted 4000 years

The rhetorical stage originated from geometry ideas

Historically algebra developed in three stages the rhetorical stage the syncopated stage and the symbolic stage According to Katz (2006) algebra development can also be categorized into four conceptual stages ldquothe geometric stage where most of the concepts of algebra are geometric the static equation-solving stage where the goal is to find numbers satisfying certain relationship the dynamic function stage where motion seems to be an underlying idea and finally the abstract stage where structure is the goalrdquo (p 186) Algebra began very early in recorded history Algebra texts date from the first half of the second millennium BCE 37 or 38 centuries ago and were written by people living in Mesopotamia and Egypt (Derbyshire 2006) During the Hammurabi period from about 1790 to 1600 BCE the Babylonians started their civilization by pressing written words in patterns called cuneiform or wedge-shaped stylus into wet clay Many tablets in cuneiform had a mathematical-algebraic content Their mathematical texts were of two kinds table texts and problem texts The table texts were lists of multiplication tables tables of squares and cubes as well as advanced lists like the famous Plimpton 322 tablet which is about Pythagorean triples The number system of the Babyloniansrsquo time was based on 60 digits rather than todayrsquos 10 digits for example our number of 37 denotes three sixties and seven ones which is our number 187 The problem was that the Babylonians had neither defined zero nor negative numbers The Babylonians of Hammurabirsquos era had no proper algebraic symbolism All mathematical problems were expressed in words for example unknown quantity in Sumerianrsquos Akkadian text was expressed as igum (length) and igibum (width) as reciprocal The application of algebra might have its origin in the need for measuring land areas At the rhetorical stage all mathematical statements and arguments were expressed in words and sentences (Derbyshire 2006) Babylonian mathematics had two roots one is accountancy problems and the other one is a ldquocut-and-pasterdquo geometry probably developed for understanding the division of land Many old-Babylonian clay tablets contain quadratic problems of which the goal was to find such geometric quantities as the length and width of a rectangle As an example we are given that the sum of the length and width of a rectangle is

5

216 and the area of the rectangle is

217 (Kvatz 2006 p 187 and Derbyshire 2006

pp 25-27) What is the length and the width of this rectangle The tablet described in detail

the steps the writer went through First the writer halves 216 to get

413 Next he squares

413

to get 16910 From this area he subtracts the given area

217 giving

1613 The square root of

this number is extracted431 Finally the length is 5

431

413 =+ while the width is

211

431

413 =minus (Kvatz 2006) The whole process can be translated into parts of a quadratic

formula that is 2172

216

431

2

minus⎟⎠⎞

⎜⎝⎛ divide= The Babylonians did not know anything about

negative numbers the only solution for them was in positive numbers and hence their algorithm did not deliver the two solutions to the quadratic equation so their formula is slightly different from quadratic formula (Derbyshire 2006) Their formula is

⎟⎠⎞

⎜⎝⎛ divideplusmnminus⎟

⎠⎞

⎜⎝⎛ divide= 2

216

2172

216

2

x If we denote the sum of the length and the width of the

rectangle as b and the given area as c this formula will be ⎟⎠⎞

⎜⎝⎛plusmnminus⎟

⎠⎞

⎜⎝⎛=

22

2 bcbx although it is

not exactly like the modern quadratic formula There are different interpretations of Neugebauer and Sachesrsquo translation of the Babylonianrsquos tablets for this text on finding the length and width of a rectangle However it is very clear that the text from the tablets is dealing with a geometric procedure The problem was solved in words but with geometric ideas This was the beginning of algebra The Greek mathematician Euclid (300 BC) in his Book II of Elements solved some algebraic problems by manipulating geometric figures but based them on clearly stated axioms The geometrical method is in Kvatzrsquos opinion more explicit in another work of his Data The following example illustrates how Euclid solved a quadratic equation by a geometrical method Euclid defined ldquoproposition 1rdquo which is like axiom 1 If two straight lines contain a given area in a given angle and if the sum of them be given then shall each of them be given (Kvatz 2006 p 189) Euclid set up a rectangle with one side x = AS and y = AC Then a line was drawn so that BS = AC and the completed rectangle was ACDB Suppose that

was given and the area of rectangle ACFS (Figure 1) was given denoted as c What were the two sides AS (x) and AC (y) of the rectangle

byxAB =+=

A x S y B

y y

C F D

Figure 1

6

In order to find the length and the width of the rectangle Euclid bisected AB at E constructed the square on BE and then claimed that this square was equal to the sum of the rectangle ACFS and the small shaded square at the bottom (Figure 2)

A E S B

x

y y F

C G D Figure 2 According to Euclid the area of the rectangle ACFS was given which was c and the area of

the new square EGDB was also given which was ( )22b because 222byxSBASEB =

+=

+=

The equivalent relationship between the areas can be formulated as an quadratic equation 22

22⎟⎠⎞

⎜⎝⎛ minus+=⎟

⎠⎞

⎜⎝⎛ bxcb or

22

22⎟⎠⎞

⎜⎝⎛ minus+=⎟

⎠⎞

⎜⎝⎛ ybcb Euclid found out this equivalent relationship

geometrically and took use of this relationship in finding the solutions of the problem so the

length and width of the rectangle ACFS are cbbx minus⎟⎠⎞

⎜⎝⎛+=

2

22 and cbby minus⎟

⎠⎞

⎜⎝⎛minus=

2

22

These two formulas are almost identical with the Babyloniansrsquo solutions in rhetoric expressions The difference is that Greek algebra was based on geometric manipulation while Babylonian algebra was based on rhetoric manipulation with geometrical ideas In general the early stage of algebra from ancient Babylon and Egypt to Greek was mainly geometrical The syncopated stage ndash the beginning of the static equation-solving stage by using In Roman Egypt in probably the second or third century CE the algebra stage was at the syncopated stage which means written algebraic texts were expressed in words but involved special symbols-abbreviations According to recorded mathematics history one the of pioneers who used these special symbols to solve equations with only numbers but no connection with geometry was Diophantus who lived in Alexandria in Egypt around the third century Diophantus used the Greek alphabetic system for writing numbers He wrote a treatise titled Arithmetica of which less than half has been maintained today The surviving part of his work consists of 189 problems in which the object is to find numbers or families of numbers satisfying certain conditions In mathematics today Diophantusrsquo mathematical analysis is known as number theoryndashnot algebra However he used only number symbols to solve equations without the help of geometry Diophantus wrote the coefficient after the variable instead of before it as we do He used Greek ς for unknown quantity our modern x Most of his book deals with indeterminate equations which contain more than one unknown and a potentially infinite number of solutions His problem is that he could not represent more than one unknown instead he solved quadratic equations with two unknowns through substituting one by another and then the quadratic equations became the ones with one unknown At that time negative numbers were regarded to be absurd but he knew how to

7

bring a term from one side of an equation to the other gather up like terms for simplification and some elementary principles of expansion and factorization (Derbyshire 2006) Diophantus made his own literal symbolism with the use of special letter symbols for the unknown and its powers for subtraction and equality From Diophantus algebra history moved into another conceptual stage the equation-solving stage according to Katz (2006) In India quadratic formula appeared without any geometric support Brahmaggupta (598-665) was one of the first mathematicians who could systematically handle negative numbers and zero He gave a general solution to quadratic equations and realized that there were two roots for a quadratic equation It was possible that one of the roots was a negative number Baskharacharya (1114-1185) solved mathematics problems with the use of quadratic equations in his book Siddhanta Siromani (ldquoMathematical Pearlsrdquo) He presented an algorithm to reduce a quadratic equation to a first-degree equation (Olteanu 2007) It is commonly believed that the first true algebra text is the work on al-jabr and al-muqabala by Mohanmmad ibn Musa al-Khwarizmi (780-850) written in Baghdad around 825 (Katz 2006) The word algebra came from the title of this work The word al-jabr means restoration or reestablishment that is to eliminate negative terms through adding the same terms to both sides of equations The word of al-muqabalas means balance meaning to divide every term in a quadratic equation by the coefficient of the second degreersquos term (Olteanu 2007) The first part of his book is a manual for solving linear and quadratic equations Al-Khwarizimi classified equations into six types three of which were mixed quadratic equations For each type he presented an algorithm for its solution Five of the six types of equations were quadratic equations which can be expressed in modern form

cbxaxbxcaxcbxaxcaxbxax +==+=+== 22222 Here is an example of solving the equation ldquoTake the half of the number of the things that is five and multiply it by itself you obtain twenty-five Add this to thirty-nine you get sixty-four Take the square root or eight and subtract from it one half of the number of things which is five The result three is the thingrdquo (Kvasz 2006 p 292) Like Babylonian mathematicians al-Khwarizimirsquos algorithm is entirely verbal On the other hand al-Khwarizimi sometimes completed his algorithm by using geometrical explanations which can be translated as todayrsquos square-completing method Using the example of solving the quadratic equation the completed geometrical procedures are illustrated on next page in figures 3 4 and 5

39102 =+ xx

39102 =+ xx

8

x

xsup2

Figure 3 (Olteanu 2007 p 30) 5x2

25x

xsup2

25x

5x2

Figure 4 (Olteanu 2007 p 30) 254 254

39

254 254

Figure 5 (Olteanu 2007 p 30) According to Olteanursquos (2007) translation al-Khwarizimi started with a square whose sides are x and area is xsup2 (see Figure 3) Then he added four equal rectangles whose areas in total

was 10x along each side of the square that is xx sdotsdot=25410 Each rectanglersquos area is thus x

25

with its length x and its width 52 (see Figure 4) The sum of the big square and four rectangles was given which was 39 The equivalence relationship was Finally figure 4 was completed by adding four small equal squares which had an area the size of

39102 =+ xx

425

25

25

=sdot for each small square and the sum of those was 25 Through adding this sum to

both sides of the equation the area of the biggest square obtained in figure 5 was 64 The

equation translation is 425439

4254102 sdot+=sdot++ xx The side of the biggest square was 8 and

had its relation with other sides of different squares expressed in the first degree equation

9

25

258 ++= x Then x was 3 By ldquocut-and-pasterdquo geometry (Katz 2006 p 191)

al-Khwarizimi reduced the second degree of an equation to the first degree and thereafter solved it Unlike his Babylonian predecessors al-Khwarizimi always presented his problem abstractly rather than geometrically relating to lengths and widths The symbolic stage At this stage of algebra ldquoall numbers operations relationships are expressed through a set of easily recognized symbols and manipulations on the symbols take place according to well-understood rulesrdquo (Katz 2006 p 186) The ancient algebra and geometry had developed sophisticatedly in Egypt Persia Greece India and China After Medieval Islamic scholars gave us the word ldquoalgebrardquo Western Europe began the struggle for the development of algebra starting from some algebraists from Italy Italian mathematician Leonardo Pisano later known as Fibonacci traveled in the 12th and 13th centuries to Persia India and China When he returned to Italy he had wider knowledge of arithmetic and algebra His book Liber abbaci was the best math textbook since the end of Ancient world His book is credited with having introduced Indian numerals including zero to the West But his algebraic skills had been shown in two other works after this one With the introduction of printed books during the second half of the 15th century the development of algebra was sped up Several Italian mathematicians including Girolamo Cardano had figured out how to solve cubic and quadratic equations Algebra became purely abstract with the exception of an English mathematician named Robert Recorde who lived in the 16th century and created quadratic problems from real world experience (Derbyshire 2006) It was in France that algebra had developed into a well organized literal symbolism In his work In artem analyticem isagoge French mathematician Franςois Viegravete (1540-1603) in the late 16th century became the first mathematician to use letters representing numbers systematically and effectively (Derbyshire 2006) He made a range of letters available for many different quantities This was the beginning of modern literal symbolism Viegravetersquos unknown quantity was divided into two classes unknown quantities (meaning ldquothings soughtrdquo) denoted by A E I O U and Y while ldquothings givenrdquo was denoted by constants like B C Dhellip For example his A is our unknown x Viegravete was a pioneer in the study of equations His two papers on the theory of equations were published twelve years after his death In the second paper titled ldquoDe equationem emendationerdquo (ldquoOn the perfecting of equationsrdquo) Viegravete opened up the line of inquiry that led to the study of the symmetries of an equationrsquos solutions to Galois theory the theory of groups and all of modern algebra He found the relationship between the solutions of the equation and the coefficients for the first five degrees of equations in a single unknown To explain this in our modern symbols we suppose that the two solutions of the quadratic equation are 02 =++ qpxx α and β which means βα == xx Based on this logic the following thing must be true 0))(( =minusminus βα xx since only α andβ and no other values of x make this equation true This form of equation is just a rewritten form of the same equation If we multiply out those parentheses this rewritten equation turns to be Compared to the original equation the relationships between the solutions and the coefficients we obtain

0)(2 =++minus αββα xxqp =minus=+ αββα

(Derbyshire 2006) It is said that Viegravete discovered the solution formula called quadratic

10

formula today for general quadratic equations which is a

acbbx2

42

21minusplusmnminus

= for a general

quadratic equation (Olteanu 2007) 002 ne=++ acbxax Another French mathematician and philosopher who had strong influence in the history of algebra was Reneacute Descartes (1596-1650) His idea to use the Cartesian system of coordinates which was derived from his Latin name developed both algebra and geometry In his work La geacuteomeacutetrie written in 1637 Descartesndashthrough using numbers to indentify points in a Cartesian coordinatesndashconnected geometrical objects to algebraic numbers and made the classical geometry become analytical geometry He took up the plus and minus sign from the German Cossists of the previous century and also the square-root sign From Descartes the symbol of an unknown was represented by x which led to the modern system of literal symbolism (Derbyshire 2006) Descartes developed the idea of functions in Cartesian coordinates although the idea of function can be traced back to the Islamic mathematician Sharaf al-Din al-Tusi from Persia (Katz 2006) and Klaudius Ptolemaios about 2000 years ago (Olteanu 2007) Descartes declared that every curve in a Cartesian coordinate system has an equivalent equation which can represent the points on the curve or vice versa every equation containing x and y can be represented by a curve through its coordinate points However the word function was introduced by Gottfried Wilhelm Leibniz in 1693 and the definition of function was defined by Leonhard Euler in his work Introductio analysin infinitorium in 1748 Euler had also given the symbol of a denotation as a relying variable (Olteanu 2007) In the 18)(xf th century after the discovery of calculus by Newton and Leibniz algebra went into an area of analysis ndashthe study of limits infinite sequences and series functions derivatives and integrals (Derbyshire 2006) The purely abstract stage-algebra structure Since 17th century algebra has not anymore solely been about finding solutions for different degrees of equations mathematicians have started to integrate algebra with astronomy and physics Johann Kepler and Galileo Galilei were interested in curves and finding a mechanism for representing motion instead of quantity-like numbers But their arguments were not algebraic-symbolic but geometrical It was Fermat and Descartes who were regarded as the fathers of analytic geometry and who showed how to represent a curve described verbally through algebra analytic geometry and gave a mechanism for representing motion Newton in his Principia picked up on Fermat and Descartesrsquo representation and developed the calculus With the invention of the calculus mathematical problems were solved by curves not just points Algebra grew more and more to present paths of motion In order to be able to judge if the algebraic manipulations were correct axioms were formulated for arithmetic applied for algebraic manipulations In the late 18th century Lagrange introduced the idea of permutations into the search for solutions Therafter Galois developed methods for determining what conditions polynomial equations need to meet to be solvable New abstract algebraic concepts were found like ldquofieldrdquo and ldquogrouprdquo during the early 19th century In 1854 Ceyley gave an axiomatic definition of a group During the 1890s this definition entered textbooks along with the axiomatic definition of a field an idea which had roots in the work by Galois By the beginning of the 20th century algebra became less about finding solutions to equations and more about looking for common structures in many mathematical objects defined by sets of axioms (Katz 2006)

11

In short algebra development took 4000 years It took about 3500 years for mathematicians to start to use algebra symbols in systems Just almost 100 years ago algebra developed into a purely abstract science The long history may give both mathematics educators and students some explanations to why it is so difficult to teach and learn algebra

22 Three solving methods for quadratic equations Already in the previous part I have presented geometrically and rhetorically solving quadratic equations from a historical perspective In order to get more explicit understanding of quadratic equations quadratic formula and factorization from mathematics didactic point of view this part presents three common methods for solving quadratic equations 1) Completing the square 2) Factorization 3) Quadratic formula The reason to present these three methods is that they are the common topics for discussion in the articles searched in this area concerning mathematics education at upper-secondary school 1) Completing the square In two articles Solving quadratic equations by completing squares (Vinogradova 2007) and Geometric approaches to quadratic equations from other times and places (Allaire amp Bradley 2001) the authors advise mathematics teachers to use the ideas of ancient mathematician al-Khwarizmi and his method of completing quadratic equations by geometrical algebra and simplifying it in order to represent it to the students By doing so the teachers relate mathematics quantity to physical objects in a visual way Didactically teachers may start with a concrete example in which students can understand the content visually Later from the concrete example the teachers lead the students to another example expressed in algebraic symbols for example to study a rectangle whose area is )10( +xx and supposing this area is 39 this becomes 39)10( =+xx Four steps are followed to build a new square which is actually called completing the square x 10

Figure 6 x 5

Figure 7

x

x

5 5

12

A Begin with a rectangle of the area )10( +xx that is with the short side as x and the long side )10( +x The shaded area is 10bullx (Figure 6)

B Divide the shaded rectangle into two small shaded rectangles with the size 5x each and move them to each adjacent side of the xsup2 square (Figure 7) The total area is still

x10

)10(39 += xxC In Figure 7 a new square is shaped in the lower-right hand corner with the side 5 or

area 25 By adding this small square to the diagram the large square with an area of is ldquocompletedrdquo Therefore we can write 25)5(22 ++ xx 25)10(2539 ++=+ xx or

The large squarersquos area is now because the side has a length of Therefore

6425)5(22 =++ xx 2)5( +x5+x 38564)5( 2 =rArr=+rArr=+ xxx

After these three steps a solution to this quadratic equation is derived by completing the square (Allaire amp Bradley 2001) 2) Factorization Hoffman (1976) presents three approaches (A B and C) for factorizing quadratic expressions through observing and grouping He takes a second-degree polynomial as an example

273 2 ++ xx

a The inspection method 1) suggests factors of and 23x x3 x 2) 2 suggests factors of 1 and 2 or ndash1 and ndash2 3) Checking the linear term shows that the correct factorization is x7 )2)(13(273 2 ++=++ xxxx

This method according to Hoffman (1976) is based on the studentrsquos ability to simplify by inspecting the indicated product of a pair of linear expressions

b The method of decomposition of the linear term 1) Multiply and 2 to get 23x 26x2) Decompose into the sum of two terms whose product is x7 26x

This gives xxx += 67 3) Factorize by grouping 2)6(3 2 +++ xxx

2)6(3 2 +++ xxx = )

)2()63( 2 +++ xxx

= 2(1)2(3 +++ xxx= )13)(2( ++ xx

c Third method 1) Multiply 3 and 2 to get 6 2) Find two numbers whose product is 6 and whose sum is 7 The numbers are 6 and 1 3) 273 2 ++ xx

=3

)13)(63( ++ xx

= )13)(2( ++ xx

13

These three approaches can not actually be divided as three unrelated approaches especially the third method is seemingly unclear What Hoffman (1976) wants to demonstrate is that all the three approaches are based on observing the relationship among the coefficients and the constant of the polynomial Through finding the product and the sum of two factors the polynomial can be factorized in the product of two binomials In the authorrsquos opinion factorization is heavily dependent on skill in multiplication and using distributive law reversely He means that ldquostudents who do not have reasonably strong skills in multiplication should not be expected to develop strong skills in factorizationrdquo (p 55) Factorizing polynomials is not only for simplifying high-degree polynomials (for example the third or fourth-degree polynomials) but can also be applied for solving quadratic equations Supposing this polynomial forming a quadratic equation Solving this equation is actually to decompose the polynomial into a factoringrsquos form

0273 2 =++ xx0)13)(2( =++ xx

which is equivalent to the polynomial form The essential step to solve this equation is to follow the zero-factor propertyndashalso called null-factor law (nollprodukt in Swedish)ndashthat is for real numbers p and q p bull q = 0 if (and only if) p = 0 or q = 0 It means that either of the binomials on the left side of the equation has to be equal to zero in order to satisfy the

equivalent relation of this quadratic equation The solutions of two roots or 2minus=x31

minus=x

are obtained through making either 02 =+x or 013 =+x 3) Quadratic formula Solving quadratics equations by factorization is constrained within the simple quadratic equations over the whole integers and rational numbers as coefficients What happens when quadratic equations have coefficients that belong to an irrational domain or big quantity The quadratic formula is a solution to such a situation As mentioned previously historically the French mathematician Franccedilois Vieacutete discovered the quadratic formula

aacbbx

242 minusplusmnminus

= ) which is applicable for all kinds of quadratic equations In

Swedish textbooks this formula is written in PQ form by making for the equation

that is

0( nea

1=a

02 =++ qpxx qppx minus⎟⎠⎞

⎜⎝⎛plusmnminus=

2

22 with

abp = and

acq = In Oleatursquos study (2007)

studentsrsquo difficulties in using algebraic symbols no longer is the problem But on the other hand handling the parameters or coefficients in quadratic equations like and rewrite them in the equivalent form with p and q being real numbers becomes obstacles for the students when they study the algebra course at upper-secondary school

02 =++ cbxax02 =++ qpxx

The quadratic formula a

acbbx2

42 minusplusmnminus= has been regarded as the standard method

according to US mathematics teachers Obermeyer (1982) presents a way of how to derive this formula by completing a square through 11 steps (I will not present them here) At the same time an alternate method is also introduced according to the same principle All the methods mentioned so far are based on the square rules The standard method or PQ form might be regarded as an efficient and direct method but it may

222 2)( bababa +plusmn=plusmn

14

lead students to solve quadratic equations in a mechanical way besides having problems with the troublesome plusmn symbolism (Stover 1978) Both Stover (1978) and Olteanu (2007) have suggested that using a graph of quadratic functions to solve quadratic equations is another alternative method

23 Mathematical background of factorization Factorization in algebra structure Among the three solving methods presented above both the geometrical method and quadratic formula can be traced back to algebra history in section 21 What is the trace of factorization in algebra history Factorization was at the last stage in algebra history and belonged to algebra structure (Derbyshire 2006) In Swedish mathematics textbooks the chapter on algebra often includes polynomial factorization and quadratic equations But the factorization in the textbooks is not at the same abstract level as the factorization of algebra structure although they share the same axioms and there are certain connections between these two different levels of factorization How are polynomial and quadratic equations as well as factorization related to each other In order to find the theoretical background of factorizing polynomials this part constitutes a mathematical theoretical review by referring to two books concerning algebra structure (Vretblad 2000 Durbin 1992) and a research study (Bosseacute and Nandakumar 2005) A quadratic equation can be regarded as polynomial equation Solving quadratic equations has much to do with ldquounique factorization theoremrdquo The polynomial

is a second-degree polynomial A polynomial consists of coefficients in whole numbers rational numbers real numbers or complex numbers and variables (sometimes called unknowns) in different degrees as well as constants A polynomial

has symbols as coefficients which are real numbers If all are zero and so

)0(02 ne=++ acbxax

cbxax ++2

nnxaxaxaaxf ++++= )( 2

210 naaa 10

ka 1gek 0)( axf = is constant If all are zero then we say the is a zero polynomial If meaning that is not a zero polynomial and n is the biggest number and then we say that the polynomial is at degrees A zero polynomial has no degree at all According to the definition (Vredblad 2000) if is a polynomial the equation is called an algebraic equation or polynomial equation A solution or a root of this equation is a number

kaf 0nef f

0nena f nf

0)( =xfα so that 0)( =αf Such a number is even

called a zero point (nollstaumllle in Swedish) of the polynomial which means when f α=x the value of the polynomial is zero (p 161) Every polynomial whose degree is equal or greater than 1 can be written as a product of irreducible polynomial In this way through polynomial division a polynomial with n degrees can be expressed from a sum of terms into a product of its denominator and its quotient The process of changing the expressions is factorization for

instance 12

22

+=minusminusminus x

xxx The factoring form is )1)(2(22 minusminus=minusminus xxxx

Factorization of polynomials is analogue to factorization of whole numbers which is very much related to the concept of ring A ring consists of a set with two operations which are a sum and a product of two elements over a field Any polynomial from zero polynomial to n degree-polynomials can be written or operated through addition and multiplication of the

15

elements over a field F including whole numbers or integers rational numbers real numbers and complex numbers In a ring you can multiply any two elements but you canrsquot always divide Factorization is a way to deal with this fact Here are some examples In the ring of integers Z is an example of factorization in some polynomial rings for instance Q[x] the ring of polynomial in one variable with rational numbers as coefficients such as

7214 sdot=

)451)(2

51(8

52

251 2 +minus=minus+ xxxx is factorization Z[x] the ring of polynomial in one

variable with whole numbers as integers such as is factorization )1)(2(22 minusminus=minusminus xxxx The polynomial ring R[x] means the ring of polynomial in one variable with real numbers as coefficients Factorizing of such polynomials implies that the polynomial factors are the elements in the ring of R[x] for instance )2)(2(22 minus+=minus xxx The same thing is true for C[x] the ring of polynomial in one variable with complex numbers as coefficients Thus it is coefficients of polynomials that determine which ring a polynomial belongs to and if it is possible to be factorized in this ring For example number 6 can be factorized in two ways

in the ring of Z and 32times )51)(51( ii +minus in the ring of C (complex numbers) However there are cases in which divisions can not be done within the ring for example 52 or

)3()52( 2 ++minus xxx which means that no element in the ring can multiply with a denominator obtaining the numerator Therefore neither 5 nor can be factorized 522 +minus xx If a polynomial of degree at least one has no other divisors then it is considered to be irreducible or prime (Durbin 1992) Of every polynomial of degree at least one can be written as a product of irreducible polynomials according to unique factorization theorem each polynomial of degree at least one over a field F can be written as an element of F times a product of monic irreducible polynomials over F and except for the order in which these irreducible polynomials are written this can be done in only one way (p 221) Thus a factorable polynomial can be written into a product with another polynomial such as In other words the condition that )()()( xqaxxf minus= )( ax minus is a factor of a polynomial

in a variable x is only true if )(xf 0)( =af For example if and

6)( 2 minus+= xxxf0624)2( =minus+=f 0639)3( =minusminus=minusf the factors of are and )(xf )2( minusx )3( +x

The value of the polynomial is zero which is also expressed as the polynomial having a zero point (nollstaumllle in Swedish) when

)(xfα=x

Based on unique factorization theorem finding solutions of an algebraic equation is equivalent to finding the first-degree factors for a polynomial (Vredblad 2000) In this way solving a quadratic equation can be handled through finding the first-degree factors for the second-degree polynomial and there after finding the roots of the quadratic equation This is often called using factorization to solve quadratic equations Besides unique factorization theorem the connection of using factorization to solve a quadratic equation is that this method relates to prime numbers using distributive law backwards commutative law multiplication division and operating fractions As matter of fact it is about application of factorizing integers in polynomials at upper-secondary school level Since quadratic equations in the mathematics B course at Swedish upper-secondary school are limited to real numbers and often within the whole numbers (integers) many quadratic equations should be able to be solved by factorization according to my own

16

judgment Therefore using factorization for solving quadratic equations is very much related to arithmetic operations number concept and algebra structure It may offer students both mathematical conceptual understanding and operational skill For algebra beginners factorization can be traced back to studentsrsquo early years of mathematics education during which students do not only work with multiplication but also on factoring integers for example 56 = 8 middot 7 = 2 middot 2 middot 2 middot 7 = 4 middot 14 = 2 middot 28 In this example two operations are included multiplication and division Multiplication operation may help students to understand the multiplicative structure of the integers while factoring integers may become meaningful when it is related to divisibility divisors and prime numbers These two operational skills or competence are major techniques for simplifying fractions and finding common denominators Being able to understand and operate factoring integers has laid the essential grounds for algebra beginners to study factorization of polynomials Thus factorization reunites studentsrsquo early mathematics knowledge of factoring integers with later knowledge of algebra structure Factorability In a study on factorability by Bosseacute and Nandakumar (2005) the probability of factorability is reasoned through investigating the range of integers as coefficients of quadratic equations In the quadratic equation a b and c are randomly generated integers within a determined range r such as

002 ne=++ acbxaxyrx lele where x and y are integers When is a

perfect square or where

acb 42 minus

)4( 2 acb minus is an integer quadratics are factorable According to the studyrsquos data it shows that as the range for a b and c increases the probability of factorability of a quadratic with randomly selected integer values for a b and c decreases Bosseacute and Nandakumar confirm that as the range for coefficients expands to infinltltinfinminus r the probablility of factorability of a quadratic with randomly selected integer values for coefficients approaches zero The studyrsquos data collected from college algebra courses and textbooks demonstrates that about 15 of quadratics with integer coefficients [ ]1010minusisinr are factorable Thus about 85 of quadratics can not be factorized In spite of the limitation of factorability within the exercises concerning factoring practice chosen from 27 surveyed college algebra textbooks about 94 of the problems were factorable Within the textbooks 55 of the quadratics to be factored were within the range [-10 10] Even if many quadratic equations are factorable choosing the right factor pairs is time consuming and unnecessary for example

has to be factorized through choosing the right pairs of factors among nine pairs for 36 and 8 pairs of factors for 24 Comparing the three different methods for solving quadratic equations the study suggests that completing the square and quadratic formula are more effective informative and useful than factorization

0245936 2 =++ xx

231 Different kinds of factorizations for solving different quadratic equations How can we find the binomial factors of a quadratic equation or a second-degree polynomial without knowing one of the divisors as a binomial factor The answer is that we have to observe the relations among all coefficients and constants as well as the positive and negative signs before them under the condition of being factorable Most of the coefficients in quadratic equations or polynomials among exercises provided for practicing factoring are integers (here I borrow the same notation r as the one used by Bosseacute and Nandakumar) with

17

their range between ndash10 and 10 that is 10ler In this case it is possible for a student to make a judgment of whether or not a quadratic equation is factorable Five different kinds of factorization depending on five kinds of quadratic equations Factoring quadratic expressions can be divided into five different kinds depending on the type of quadratic equations The coefficients a b p q and constant c illustrated in the following quadratic equations are defined not to be equal to zero thus 000 nenene cba The first kind of quadratic equations is type 1 the obviously factorable type (Bosseacute and Nandakumar 2004) then the equation is factorized as

where the roots are

0002 nene=+ babxax

0)( =+ baxxabxx minus== 21 0

The second kind of quadratic equations is type 2 The factors of this second degree polynomial are so the roots to this type of quadratic

equations are

0)( 22 =minusbax))(()( 22 baxbaxbax minus+=minus

bax plusmn=21 This kind of factorization is based on the difference-of-squares

formula for example can be solved by factorizing into 049 2 =minusx 0)23)(23( =minus+ xx Thus

the roots are 32

32

21 =minus= xx This kind of factorization is also obviously factorable

The third kind of quadratic equations is type 3 This type of equations can be solved directly by factoring into two identical binomials

The double roots obtained from this

factorization are

02)( 22 =+plusmn babxax

0))(()(2)( 222 =plusmnplusmn=plusmn=+plusmn baxbaxbaxbabxax

abx plusmn=21 This method has utilized square rules which are quite easily

observed The forth kind of quadratic equations is type 4 In this equation the coefficient of the -term is 1 with p and q as positive or negative integers Type 4 equations can always be solved by the approaches of completing the square and quadratic formula However they can often be solved by factorization too since the quadratic expressions are factorable by using distributive laws reversely The roots are obtained through finding two factors for q At the same time these two factors become satisfying with the relation so that the sum of factors is equal to That way the two factors become two roots of a quadratic equation Denoting the two factors or two roots as the relationship between coefficient and constant is

02 =++ qpxx2x

qminus

21rr)( 2121 rrprrq +minus=bull= Quadratic equations can be written as the product of

two first-degree binomials or factoring form 0))(()( 212121

22 =minusminus=++minus=++ rxrxrrxrrxqpxx Through the factoring form the roots of this equation are obtained being The core of factorizing type 4 equations is seeking the connections between coefficients and roots which reveal the studentsrsquo basic ability of arithmetic multiplication and addition It is often more effective in this case to use factorization than to use quadratic formula or completing the square Let us have a look at the example In this equation we get

2211 rxrx ==

035122 =+minus xx

18

35)7)(5(12)7()5( =minusminus=minus=minus+minus= qp or 75)75( bull=bullminus= qp (here we can directly get roots) Factorizing the equation we obtain 0)7)(5( =minusminus xx and roots are The procedures of finding factors of the equation often occur in some studentsrsquo minds quickly without writing down the whole computing process but those students often have good skills in arithmetic operations (reference) Compared to using factorization for solving this equation application of quadratic formula seems ineffective A possible problem in this method is how to find the signs before the factors or how to judge if the factors in the parentheses are positive or negative integers In order to check if the solution is correct we can always multiply these two factored binomials by distributive law and see if the expanded polynomial is the same as the original quadratic equation

75 21 == xx

The fifth kind of quadratic equations is type 5 Still the methods of completing squares and quadratic formula can be applied for all type 5 equations but there are cases in which factorization can be used for solving this type of equations within a certain range of integers as mentioned before Sometimes procedures of factorizing such equations can be carried out in our minds quickly like type 4 equations but in many cases they can not As a matter of fact finding two binomial factors for a factorable quadratic equation or a factorable second-degree polynomial is a complicated process and requires systematic work In his article published in Mathematics in School Jackman (2005) demonstrates this systematic procedure via factorizing second-degree polynomials A second-degree polynomial can be written as where are all integers By multiplying out the polynomial becomes thus

0002 nene=++ bacbxax

))((2 srxqpxcbxax ++=++ srqpcbaqsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== Writing the factors of a and c on two lines p q times r s Finding the right pairs of factors for a and c has to be done by listing out the groups of products of different factors for example The factor pairs for 6 are (6 1) (1 6) (3 2) (2 3) and the factor pairs for (-4) are (4 -1) (-4 1) (2 -2) (-2 2) (1 -4) (-1 4) Writing them on two lines p 6 1 3 2 q 4 -4 2 -2 1 -1

456 2 minus+ xx

times r 1 6 2 3 s -1 1 -2 2 -4 4 Then comes calculating the values of ps and qr until the value of b = 5 is obtained The calculating procedures are divided into 4 groups and 6 steps in each group which means 4 times 6 steps in total Here I only give one grouprsquos procedure as an example 6 middot (-1) + 4 middot 1 = -2 6 middot 1 + 1middot (-4) = 2 6 middot (-2) + 1 middot 2 = -10 6 middot 2 + 1middot (-2) = 10 6 middot (-4) + 1 middot 1 = -23 6 middot 4 + 1 middot (-1) = 23 helliphellip After the 24ndashstep calculating procedures two computing procedures obtain the same correct result 3 (ndash1) + 4 2 = 5 (yes) 2 4 + (-1) 3 = 5 (yes) Are they the same From 3 (ndash1) + 4 2 = 5 the polynomial can be factorized into )12)(43( minus+ xx while from 2 4 + (ndash1) 3 = 5 the polynomial can be factorized into )43)(12( +minus xx According to communicative law these two binomial products are the same The shortcoming of this

19

method is that it is too ineffective and complicated The addition of matrixes is almost impossible and unbearable Nevertheless based on the same idea I suggest finding factors by guessing analyzing and checking Using the same example here we can find that one of the binomials must have a negative integer because of ndash4 which implies that 5 is the sum of a positive integer and a negative integer Then looking at a and c the factors of 6 can be (1 6) and (2 3) while the factors of 4 can be (1 4) and (2 2) when we regard them as absolute values We know that the factoring form of the polynomial will be like

and the connections of all coefficients are

456 2 minus+ xx

qsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== The coefficient 5=b is the difference of two products of factors for 6 and 4 First we try the biggest number of product such as 2446 =bull then 524 =minus the unknown is which is a prime number the only factors for 19 are (19 1) There is not any combination of products of factors for 6 and 4 satisfying 19 Therefore the multiplications of factors (1 6) and (1 4) are not included in this case So we move to factors (2 2) (2 3) and (1 4)

19 =

In the second step we try The result is negative it is obviously

4264466)23)(22( 22 minusminus=minus+minus=+minus xxxxxxx2 lt 5 which implies that the combination of (2 3) and (2 2) is not

considerable One pair of factors has to be removed and that is (2 2) which does not influence the coefficient of the -term Then we try The result this time is negative too but we can see that

2x 456)43)(12( 2 minusminus=minus+ xxxxb is 5 which guides us to interchanging the positions of

negative and positive signs in the parentheses The next attempt is and it is correct Seeking the correct factors for the polynomial

has taken four major operating steps by trying the combinations of factorsrsquo products and relating the sum of the products with the value of the x-term coefficient The strategy is to make use of distributive law reversely first and then multiply the product of every pair of parentheses in order to check if they are the right pair of polynomial factors Factorizing second-degree polynomials is the major step for solving quadratic equations After factoring polynomials into the product of two binomials as in the example given above the roots can be obtained by making each binomial equal to zero according to the theory of zero point (ldquonollstaumlllerdquo in Swedish) of a polynomial equation originating from unique factorization theorem

456)43)(12( 2 minus+=+minus xxxx456 2 minus+ xx

Factorization by using the technique of vertically and crosswise sutra Binomial expansion or multiplication of binomials by distributive law and factorization of the second-degree polynomials by using distributive law backwards have an extremely symmetrical effect and the results of two kinds of operations are equivalent It is crucial for the students to be able to perceive mathematics structure and be acquainted with arithmetic operations as well as to understand the relationship between quantities when they carry out factoring procedures The expansion of binomials by distributive law has been named as the method FOIL which stands for first outside inside and last according to multiplication orders of two binomials for example (Nataraj and Thomas 2006) Many students are used to this procedure Another technique to multiply two binomials is called vertically and crosswise sutra origin from the ancient Indian Vedic mathematics Vertically and crosswise sutra and FOIL are much the same in principle but it makes use of

824)4)(2( 2 +++=++ xxxxx

20

the power of visualization and improves facility with the expansion of binomials and the factorization of quadratic expressions In this method two binomials are written into two lines Using the same example the two lines are written as 824)4)(2( 2 +++=++ xxxxx

x + 2 x + 4 ______________ xsup2 4x + 2x 8

The order of multiplication is as shown below x + 2 x + 4

4 3 2 1 (the numbers 1 2 3 and 4 refer to steps) xsup2 + 2x + 4x + 8 Figure 3 The vertically and crosswise method of multiplication of binomials Step 2 and 3 can be interchangeable If we put this form into a line expanded according to the order above would be like

)4)(2( ++ xxxbullrarrbull 442 clockwise and xxx bullrarrbull2 in

counterclockwise direction According to sutra we can imagine drawing a thread starting from 2 then going through 4 and x clockwise and then draw the second thread starting from the same place 2 later going through x in the second bracket and finally x in the first bracket counterclockwise Learning expansion of binomials by the methods FOIL and Vertical and clockwise sutra is the basic process before students study factorization In this study Nataraj and Thomas (2006) have found that the students appear to do factorization using guess-and-check as well as decomposition or the vertical and clockwise approach Their finding is that afterwards the students performed better on algebra questions especially on the factorizations They conclude ldquohellipthe values of the method may lie in what it adds to the studentsrsquo overall algebraic conceptions and knowledge of mathematical structurerdquo (p 16) In general solving these five kinds of quadratic equations by factorization can be traced back to unique factorization theorem The important character in this method is that you can always check if factorization is correct or not through multiplying out the two pairs of a parenthesis and see if the expanded polynomial is the given one or not Using factorization is not only about method but also about fostering the studentsrsquo mathematical thinking The solving processes are more important than solutions

21

3 Review of related literature and research Besides the focus on mathematics content another focus in the background work for this research is about previous studies and research in the field of mathematics didactic concerning different methods for solving quadratic equations and the use of factorization In order to study the previous research and studies in the related field three scientific databases and a search engine have been used and 113 articles have been investigated Topics such as solving quadratic expressions and equations are often discussed as teaching ideas and strategies shared by mathematics educators in different countries from lower and upper-secondary schools to colleges according to the findings in the article search These didactic topics from mathematics teachersrsquo perspectives include for example using geometric approaches to solve quadratic equations (Allaire amp Bradley 2001) utilizing completing squares (Vinogradova 2007) factoring quadratics (Hoffman 1976 Cheng 1980) and using factorization completing the square graphical methods to solve quadratic equations (Macdonald 1986) Kostsopoulos (2007) links cognitive reasoning to pedagogical problems appearing in the mathematics classrooms in her article published in Australian Mathematics Teacher Based on her teaching experiences Kostsopoulos is aware of her pedagogical strategies lacking insight on dealing with studentsrsquo difficulties of multiplication and factorization of quadratics She assumes that ldquostudentsrsquo problems with factorization and with identifying varied representations of the same quadratic relationshiprdquo may be linked to the ways in which the brain constructs cognitive representations (p 19) Factoring of quadratics is the rewriting of polynomials as a product of polynomials It requires students to have both a conceptual understanding of multiplication of polynomials and the procedural knowledge to retrieve basic multiplication facts effectively Based on the theory about the order matters in the brainrsquos ability to retrieve number facts (Phenix amp Campbell 2001) Kostsopoulos conjectures that studentsrsquo ability to access the appropriate long-term semantic memory is limited when they are confused by mixing the order and varied forms of quadratics Although this article is not about a research study it seeks the reason why factorizing quadratics is difficult and therefore might provide math teachers with alternative explanations for similar pedagogical frustrations With Von Glasersfeldrsquos radical constructivism and Peter Gaumlrdenforsrsquo epistemic semantics theory as his theoretical frameworks Rauff (1994) suggests that belief-based teaching can be successful in teaching factoring According to Rauff constructivist learning theory takes the view that students construct their own beliefs and knowledge of mathematics Learning occurs when the students change their belief set Referring to belief set theory from Gaumlrdenfors (1988) Rauff (1994) claims that ldquoa belief set is expanded when a new belief is added to it and contracted when a proposition is no longer believed and is removed from the setrdquo (p 421) Expansion contraction and revision are three components to comprise basic mechanisms of changes of belief according to Rauff When examining and comparing studentsrsquo (ages 18-20) own definitions of factoring Rauff analyzes the errors of their definitions with constructivism and epistemic semantics as analyzing tools The result of this study shows that the approach to teaching factoring rely heavily upon what a student believes about factoring is quite productive in two dimensions First it reveals the source of nonstandard factoring Second it provides a starting point for modification of the studentrsquos underlying conceptions of factoring Rauff proposes that belief-based teaching can also be

22

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

216 and the area of the rectangle is

217 (Kvatz 2006 p 187 and Derbyshire 2006

pp 25-27) What is the length and the width of this rectangle The tablet described in detail

the steps the writer went through First the writer halves 216 to get

413 Next he squares

413

to get 16910 From this area he subtracts the given area

217 giving

1613 The square root of

this number is extracted431 Finally the length is 5

431

413 =+ while the width is

211

431

413 =minus (Kvatz 2006) The whole process can be translated into parts of a quadratic

formula that is 2172

216

431

2

minus⎟⎠⎞

⎜⎝⎛ divide= The Babylonians did not know anything about

negative numbers the only solution for them was in positive numbers and hence their algorithm did not deliver the two solutions to the quadratic equation so their formula is slightly different from quadratic formula (Derbyshire 2006) Their formula is

⎟⎠⎞

⎜⎝⎛ divideplusmnminus⎟

⎠⎞

⎜⎝⎛ divide= 2

216

2172

216

2

x If we denote the sum of the length and the width of the

rectangle as b and the given area as c this formula will be ⎟⎠⎞

⎜⎝⎛plusmnminus⎟

⎠⎞

⎜⎝⎛=

22

2 bcbx although it is

not exactly like the modern quadratic formula There are different interpretations of Neugebauer and Sachesrsquo translation of the Babylonianrsquos tablets for this text on finding the length and width of a rectangle However it is very clear that the text from the tablets is dealing with a geometric procedure The problem was solved in words but with geometric ideas This was the beginning of algebra The Greek mathematician Euclid (300 BC) in his Book II of Elements solved some algebraic problems by manipulating geometric figures but based them on clearly stated axioms The geometrical method is in Kvatzrsquos opinion more explicit in another work of his Data The following example illustrates how Euclid solved a quadratic equation by a geometrical method Euclid defined ldquoproposition 1rdquo which is like axiom 1 If two straight lines contain a given area in a given angle and if the sum of them be given then shall each of them be given (Kvatz 2006 p 189) Euclid set up a rectangle with one side x = AS and y = AC Then a line was drawn so that BS = AC and the completed rectangle was ACDB Suppose that

was given and the area of rectangle ACFS (Figure 1) was given denoted as c What were the two sides AS (x) and AC (y) of the rectangle

byxAB =+=

A x S y B

y y

C F D

Figure 1

6

In order to find the length and the width of the rectangle Euclid bisected AB at E constructed the square on BE and then claimed that this square was equal to the sum of the rectangle ACFS and the small shaded square at the bottom (Figure 2)

A E S B

x

y y F

C G D Figure 2 According to Euclid the area of the rectangle ACFS was given which was c and the area of

the new square EGDB was also given which was ( )22b because 222byxSBASEB =

+=

+=

The equivalent relationship between the areas can be formulated as an quadratic equation 22

22⎟⎠⎞

⎜⎝⎛ minus+=⎟

⎠⎞

⎜⎝⎛ bxcb or

22

22⎟⎠⎞

⎜⎝⎛ minus+=⎟

⎠⎞

⎜⎝⎛ ybcb Euclid found out this equivalent relationship

geometrically and took use of this relationship in finding the solutions of the problem so the

length and width of the rectangle ACFS are cbbx minus⎟⎠⎞

⎜⎝⎛+=

2

22 and cbby minus⎟

⎠⎞

⎜⎝⎛minus=

2

22

These two formulas are almost identical with the Babyloniansrsquo solutions in rhetoric expressions The difference is that Greek algebra was based on geometric manipulation while Babylonian algebra was based on rhetoric manipulation with geometrical ideas In general the early stage of algebra from ancient Babylon and Egypt to Greek was mainly geometrical The syncopated stage ndash the beginning of the static equation-solving stage by using In Roman Egypt in probably the second or third century CE the algebra stage was at the syncopated stage which means written algebraic texts were expressed in words but involved special symbols-abbreviations According to recorded mathematics history one the of pioneers who used these special symbols to solve equations with only numbers but no connection with geometry was Diophantus who lived in Alexandria in Egypt around the third century Diophantus used the Greek alphabetic system for writing numbers He wrote a treatise titled Arithmetica of which less than half has been maintained today The surviving part of his work consists of 189 problems in which the object is to find numbers or families of numbers satisfying certain conditions In mathematics today Diophantusrsquo mathematical analysis is known as number theoryndashnot algebra However he used only number symbols to solve equations without the help of geometry Diophantus wrote the coefficient after the variable instead of before it as we do He used Greek ς for unknown quantity our modern x Most of his book deals with indeterminate equations which contain more than one unknown and a potentially infinite number of solutions His problem is that he could not represent more than one unknown instead he solved quadratic equations with two unknowns through substituting one by another and then the quadratic equations became the ones with one unknown At that time negative numbers were regarded to be absurd but he knew how to

7

bring a term from one side of an equation to the other gather up like terms for simplification and some elementary principles of expansion and factorization (Derbyshire 2006) Diophantus made his own literal symbolism with the use of special letter symbols for the unknown and its powers for subtraction and equality From Diophantus algebra history moved into another conceptual stage the equation-solving stage according to Katz (2006) In India quadratic formula appeared without any geometric support Brahmaggupta (598-665) was one of the first mathematicians who could systematically handle negative numbers and zero He gave a general solution to quadratic equations and realized that there were two roots for a quadratic equation It was possible that one of the roots was a negative number Baskharacharya (1114-1185) solved mathematics problems with the use of quadratic equations in his book Siddhanta Siromani (ldquoMathematical Pearlsrdquo) He presented an algorithm to reduce a quadratic equation to a first-degree equation (Olteanu 2007) It is commonly believed that the first true algebra text is the work on al-jabr and al-muqabala by Mohanmmad ibn Musa al-Khwarizmi (780-850) written in Baghdad around 825 (Katz 2006) The word algebra came from the title of this work The word al-jabr means restoration or reestablishment that is to eliminate negative terms through adding the same terms to both sides of equations The word of al-muqabalas means balance meaning to divide every term in a quadratic equation by the coefficient of the second degreersquos term (Olteanu 2007) The first part of his book is a manual for solving linear and quadratic equations Al-Khwarizimi classified equations into six types three of which were mixed quadratic equations For each type he presented an algorithm for its solution Five of the six types of equations were quadratic equations which can be expressed in modern form

cbxaxbxcaxcbxaxcaxbxax +==+=+== 22222 Here is an example of solving the equation ldquoTake the half of the number of the things that is five and multiply it by itself you obtain twenty-five Add this to thirty-nine you get sixty-four Take the square root or eight and subtract from it one half of the number of things which is five The result three is the thingrdquo (Kvasz 2006 p 292) Like Babylonian mathematicians al-Khwarizimirsquos algorithm is entirely verbal On the other hand al-Khwarizimi sometimes completed his algorithm by using geometrical explanations which can be translated as todayrsquos square-completing method Using the example of solving the quadratic equation the completed geometrical procedures are illustrated on next page in figures 3 4 and 5

39102 =+ xx

39102 =+ xx

8

x

xsup2

Figure 3 (Olteanu 2007 p 30) 5x2

25x

xsup2

25x

5x2

Figure 4 (Olteanu 2007 p 30) 254 254

39

254 254

Figure 5 (Olteanu 2007 p 30) According to Olteanursquos (2007) translation al-Khwarizimi started with a square whose sides are x and area is xsup2 (see Figure 3) Then he added four equal rectangles whose areas in total

was 10x along each side of the square that is xx sdotsdot=25410 Each rectanglersquos area is thus x

25

with its length x and its width 52 (see Figure 4) The sum of the big square and four rectangles was given which was 39 The equivalence relationship was Finally figure 4 was completed by adding four small equal squares which had an area the size of

39102 =+ xx

425

25

25

=sdot for each small square and the sum of those was 25 Through adding this sum to

both sides of the equation the area of the biggest square obtained in figure 5 was 64 The

equation translation is 425439

4254102 sdot+=sdot++ xx The side of the biggest square was 8 and

had its relation with other sides of different squares expressed in the first degree equation

9

25

258 ++= x Then x was 3 By ldquocut-and-pasterdquo geometry (Katz 2006 p 191)

al-Khwarizimi reduced the second degree of an equation to the first degree and thereafter solved it Unlike his Babylonian predecessors al-Khwarizimi always presented his problem abstractly rather than geometrically relating to lengths and widths The symbolic stage At this stage of algebra ldquoall numbers operations relationships are expressed through a set of easily recognized symbols and manipulations on the symbols take place according to well-understood rulesrdquo (Katz 2006 p 186) The ancient algebra and geometry had developed sophisticatedly in Egypt Persia Greece India and China After Medieval Islamic scholars gave us the word ldquoalgebrardquo Western Europe began the struggle for the development of algebra starting from some algebraists from Italy Italian mathematician Leonardo Pisano later known as Fibonacci traveled in the 12th and 13th centuries to Persia India and China When he returned to Italy he had wider knowledge of arithmetic and algebra His book Liber abbaci was the best math textbook since the end of Ancient world His book is credited with having introduced Indian numerals including zero to the West But his algebraic skills had been shown in two other works after this one With the introduction of printed books during the second half of the 15th century the development of algebra was sped up Several Italian mathematicians including Girolamo Cardano had figured out how to solve cubic and quadratic equations Algebra became purely abstract with the exception of an English mathematician named Robert Recorde who lived in the 16th century and created quadratic problems from real world experience (Derbyshire 2006) It was in France that algebra had developed into a well organized literal symbolism In his work In artem analyticem isagoge French mathematician Franςois Viegravete (1540-1603) in the late 16th century became the first mathematician to use letters representing numbers systematically and effectively (Derbyshire 2006) He made a range of letters available for many different quantities This was the beginning of modern literal symbolism Viegravetersquos unknown quantity was divided into two classes unknown quantities (meaning ldquothings soughtrdquo) denoted by A E I O U and Y while ldquothings givenrdquo was denoted by constants like B C Dhellip For example his A is our unknown x Viegravete was a pioneer in the study of equations His two papers on the theory of equations were published twelve years after his death In the second paper titled ldquoDe equationem emendationerdquo (ldquoOn the perfecting of equationsrdquo) Viegravete opened up the line of inquiry that led to the study of the symmetries of an equationrsquos solutions to Galois theory the theory of groups and all of modern algebra He found the relationship between the solutions of the equation and the coefficients for the first five degrees of equations in a single unknown To explain this in our modern symbols we suppose that the two solutions of the quadratic equation are 02 =++ qpxx α and β which means βα == xx Based on this logic the following thing must be true 0))(( =minusminus βα xx since only α andβ and no other values of x make this equation true This form of equation is just a rewritten form of the same equation If we multiply out those parentheses this rewritten equation turns to be Compared to the original equation the relationships between the solutions and the coefficients we obtain

0)(2 =++minus αββα xxqp =minus=+ αββα

(Derbyshire 2006) It is said that Viegravete discovered the solution formula called quadratic

10

formula today for general quadratic equations which is a

acbbx2

42

21minusplusmnminus

= for a general

quadratic equation (Olteanu 2007) 002 ne=++ acbxax Another French mathematician and philosopher who had strong influence in the history of algebra was Reneacute Descartes (1596-1650) His idea to use the Cartesian system of coordinates which was derived from his Latin name developed both algebra and geometry In his work La geacuteomeacutetrie written in 1637 Descartesndashthrough using numbers to indentify points in a Cartesian coordinatesndashconnected geometrical objects to algebraic numbers and made the classical geometry become analytical geometry He took up the plus and minus sign from the German Cossists of the previous century and also the square-root sign From Descartes the symbol of an unknown was represented by x which led to the modern system of literal symbolism (Derbyshire 2006) Descartes developed the idea of functions in Cartesian coordinates although the idea of function can be traced back to the Islamic mathematician Sharaf al-Din al-Tusi from Persia (Katz 2006) and Klaudius Ptolemaios about 2000 years ago (Olteanu 2007) Descartes declared that every curve in a Cartesian coordinate system has an equivalent equation which can represent the points on the curve or vice versa every equation containing x and y can be represented by a curve through its coordinate points However the word function was introduced by Gottfried Wilhelm Leibniz in 1693 and the definition of function was defined by Leonhard Euler in his work Introductio analysin infinitorium in 1748 Euler had also given the symbol of a denotation as a relying variable (Olteanu 2007) In the 18)(xf th century after the discovery of calculus by Newton and Leibniz algebra went into an area of analysis ndashthe study of limits infinite sequences and series functions derivatives and integrals (Derbyshire 2006) The purely abstract stage-algebra structure Since 17th century algebra has not anymore solely been about finding solutions for different degrees of equations mathematicians have started to integrate algebra with astronomy and physics Johann Kepler and Galileo Galilei were interested in curves and finding a mechanism for representing motion instead of quantity-like numbers But their arguments were not algebraic-symbolic but geometrical It was Fermat and Descartes who were regarded as the fathers of analytic geometry and who showed how to represent a curve described verbally through algebra analytic geometry and gave a mechanism for representing motion Newton in his Principia picked up on Fermat and Descartesrsquo representation and developed the calculus With the invention of the calculus mathematical problems were solved by curves not just points Algebra grew more and more to present paths of motion In order to be able to judge if the algebraic manipulations were correct axioms were formulated for arithmetic applied for algebraic manipulations In the late 18th century Lagrange introduced the idea of permutations into the search for solutions Therafter Galois developed methods for determining what conditions polynomial equations need to meet to be solvable New abstract algebraic concepts were found like ldquofieldrdquo and ldquogrouprdquo during the early 19th century In 1854 Ceyley gave an axiomatic definition of a group During the 1890s this definition entered textbooks along with the axiomatic definition of a field an idea which had roots in the work by Galois By the beginning of the 20th century algebra became less about finding solutions to equations and more about looking for common structures in many mathematical objects defined by sets of axioms (Katz 2006)

11

In short algebra development took 4000 years It took about 3500 years for mathematicians to start to use algebra symbols in systems Just almost 100 years ago algebra developed into a purely abstract science The long history may give both mathematics educators and students some explanations to why it is so difficult to teach and learn algebra

22 Three solving methods for quadratic equations Already in the previous part I have presented geometrically and rhetorically solving quadratic equations from a historical perspective In order to get more explicit understanding of quadratic equations quadratic formula and factorization from mathematics didactic point of view this part presents three common methods for solving quadratic equations 1) Completing the square 2) Factorization 3) Quadratic formula The reason to present these three methods is that they are the common topics for discussion in the articles searched in this area concerning mathematics education at upper-secondary school 1) Completing the square In two articles Solving quadratic equations by completing squares (Vinogradova 2007) and Geometric approaches to quadratic equations from other times and places (Allaire amp Bradley 2001) the authors advise mathematics teachers to use the ideas of ancient mathematician al-Khwarizmi and his method of completing quadratic equations by geometrical algebra and simplifying it in order to represent it to the students By doing so the teachers relate mathematics quantity to physical objects in a visual way Didactically teachers may start with a concrete example in which students can understand the content visually Later from the concrete example the teachers lead the students to another example expressed in algebraic symbols for example to study a rectangle whose area is )10( +xx and supposing this area is 39 this becomes 39)10( =+xx Four steps are followed to build a new square which is actually called completing the square x 10

Figure 6 x 5

Figure 7

x

x

5 5

12

A Begin with a rectangle of the area )10( +xx that is with the short side as x and the long side )10( +x The shaded area is 10bullx (Figure 6)

B Divide the shaded rectangle into two small shaded rectangles with the size 5x each and move them to each adjacent side of the xsup2 square (Figure 7) The total area is still

x10

)10(39 += xxC In Figure 7 a new square is shaped in the lower-right hand corner with the side 5 or

area 25 By adding this small square to the diagram the large square with an area of is ldquocompletedrdquo Therefore we can write 25)5(22 ++ xx 25)10(2539 ++=+ xx or

The large squarersquos area is now because the side has a length of Therefore

6425)5(22 =++ xx 2)5( +x5+x 38564)5( 2 =rArr=+rArr=+ xxx

After these three steps a solution to this quadratic equation is derived by completing the square (Allaire amp Bradley 2001) 2) Factorization Hoffman (1976) presents three approaches (A B and C) for factorizing quadratic expressions through observing and grouping He takes a second-degree polynomial as an example

273 2 ++ xx

a The inspection method 1) suggests factors of and 23x x3 x 2) 2 suggests factors of 1 and 2 or ndash1 and ndash2 3) Checking the linear term shows that the correct factorization is x7 )2)(13(273 2 ++=++ xxxx

This method according to Hoffman (1976) is based on the studentrsquos ability to simplify by inspecting the indicated product of a pair of linear expressions

b The method of decomposition of the linear term 1) Multiply and 2 to get 23x 26x2) Decompose into the sum of two terms whose product is x7 26x

This gives xxx += 67 3) Factorize by grouping 2)6(3 2 +++ xxx

2)6(3 2 +++ xxx = )

)2()63( 2 +++ xxx

= 2(1)2(3 +++ xxx= )13)(2( ++ xx

c Third method 1) Multiply 3 and 2 to get 6 2) Find two numbers whose product is 6 and whose sum is 7 The numbers are 6 and 1 3) 273 2 ++ xx

=3

)13)(63( ++ xx

= )13)(2( ++ xx

13

These three approaches can not actually be divided as three unrelated approaches especially the third method is seemingly unclear What Hoffman (1976) wants to demonstrate is that all the three approaches are based on observing the relationship among the coefficients and the constant of the polynomial Through finding the product and the sum of two factors the polynomial can be factorized in the product of two binomials In the authorrsquos opinion factorization is heavily dependent on skill in multiplication and using distributive law reversely He means that ldquostudents who do not have reasonably strong skills in multiplication should not be expected to develop strong skills in factorizationrdquo (p 55) Factorizing polynomials is not only for simplifying high-degree polynomials (for example the third or fourth-degree polynomials) but can also be applied for solving quadratic equations Supposing this polynomial forming a quadratic equation Solving this equation is actually to decompose the polynomial into a factoringrsquos form

0273 2 =++ xx0)13)(2( =++ xx

which is equivalent to the polynomial form The essential step to solve this equation is to follow the zero-factor propertyndashalso called null-factor law (nollprodukt in Swedish)ndashthat is for real numbers p and q p bull q = 0 if (and only if) p = 0 or q = 0 It means that either of the binomials on the left side of the equation has to be equal to zero in order to satisfy the

equivalent relation of this quadratic equation The solutions of two roots or 2minus=x31

minus=x

are obtained through making either 02 =+x or 013 =+x 3) Quadratic formula Solving quadratics equations by factorization is constrained within the simple quadratic equations over the whole integers and rational numbers as coefficients What happens when quadratic equations have coefficients that belong to an irrational domain or big quantity The quadratic formula is a solution to such a situation As mentioned previously historically the French mathematician Franccedilois Vieacutete discovered the quadratic formula

aacbbx

242 minusplusmnminus

= ) which is applicable for all kinds of quadratic equations In

Swedish textbooks this formula is written in PQ form by making for the equation

that is

0( nea

1=a

02 =++ qpxx qppx minus⎟⎠⎞

⎜⎝⎛plusmnminus=

2

22 with

abp = and

acq = In Oleatursquos study (2007)

studentsrsquo difficulties in using algebraic symbols no longer is the problem But on the other hand handling the parameters or coefficients in quadratic equations like and rewrite them in the equivalent form with p and q being real numbers becomes obstacles for the students when they study the algebra course at upper-secondary school

02 =++ cbxax02 =++ qpxx

The quadratic formula a

acbbx2

42 minusplusmnminus= has been regarded as the standard method

according to US mathematics teachers Obermeyer (1982) presents a way of how to derive this formula by completing a square through 11 steps (I will not present them here) At the same time an alternate method is also introduced according to the same principle All the methods mentioned so far are based on the square rules The standard method or PQ form might be regarded as an efficient and direct method but it may

222 2)( bababa +plusmn=plusmn

14

lead students to solve quadratic equations in a mechanical way besides having problems with the troublesome plusmn symbolism (Stover 1978) Both Stover (1978) and Olteanu (2007) have suggested that using a graph of quadratic functions to solve quadratic equations is another alternative method

23 Mathematical background of factorization Factorization in algebra structure Among the three solving methods presented above both the geometrical method and quadratic formula can be traced back to algebra history in section 21 What is the trace of factorization in algebra history Factorization was at the last stage in algebra history and belonged to algebra structure (Derbyshire 2006) In Swedish mathematics textbooks the chapter on algebra often includes polynomial factorization and quadratic equations But the factorization in the textbooks is not at the same abstract level as the factorization of algebra structure although they share the same axioms and there are certain connections between these two different levels of factorization How are polynomial and quadratic equations as well as factorization related to each other In order to find the theoretical background of factorizing polynomials this part constitutes a mathematical theoretical review by referring to two books concerning algebra structure (Vretblad 2000 Durbin 1992) and a research study (Bosseacute and Nandakumar 2005) A quadratic equation can be regarded as polynomial equation Solving quadratic equations has much to do with ldquounique factorization theoremrdquo The polynomial

is a second-degree polynomial A polynomial consists of coefficients in whole numbers rational numbers real numbers or complex numbers and variables (sometimes called unknowns) in different degrees as well as constants A polynomial

has symbols as coefficients which are real numbers If all are zero and so

)0(02 ne=++ acbxax

cbxax ++2

nnxaxaxaaxf ++++= )( 2

210 naaa 10

ka 1gek 0)( axf = is constant If all are zero then we say the is a zero polynomial If meaning that is not a zero polynomial and n is the biggest number and then we say that the polynomial is at degrees A zero polynomial has no degree at all According to the definition (Vredblad 2000) if is a polynomial the equation is called an algebraic equation or polynomial equation A solution or a root of this equation is a number

kaf 0nef f

0nena f nf

0)( =xfα so that 0)( =αf Such a number is even

called a zero point (nollstaumllle in Swedish) of the polynomial which means when f α=x the value of the polynomial is zero (p 161) Every polynomial whose degree is equal or greater than 1 can be written as a product of irreducible polynomial In this way through polynomial division a polynomial with n degrees can be expressed from a sum of terms into a product of its denominator and its quotient The process of changing the expressions is factorization for

instance 12

22

+=minusminusminus x

xxx The factoring form is )1)(2(22 minusminus=minusminus xxxx

Factorization of polynomials is analogue to factorization of whole numbers which is very much related to the concept of ring A ring consists of a set with two operations which are a sum and a product of two elements over a field Any polynomial from zero polynomial to n degree-polynomials can be written or operated through addition and multiplication of the

15

elements over a field F including whole numbers or integers rational numbers real numbers and complex numbers In a ring you can multiply any two elements but you canrsquot always divide Factorization is a way to deal with this fact Here are some examples In the ring of integers Z is an example of factorization in some polynomial rings for instance Q[x] the ring of polynomial in one variable with rational numbers as coefficients such as

7214 sdot=

)451)(2

51(8

52

251 2 +minus=minus+ xxxx is factorization Z[x] the ring of polynomial in one

variable with whole numbers as integers such as is factorization )1)(2(22 minusminus=minusminus xxxx The polynomial ring R[x] means the ring of polynomial in one variable with real numbers as coefficients Factorizing of such polynomials implies that the polynomial factors are the elements in the ring of R[x] for instance )2)(2(22 minus+=minus xxx The same thing is true for C[x] the ring of polynomial in one variable with complex numbers as coefficients Thus it is coefficients of polynomials that determine which ring a polynomial belongs to and if it is possible to be factorized in this ring For example number 6 can be factorized in two ways

in the ring of Z and 32times )51)(51( ii +minus in the ring of C (complex numbers) However there are cases in which divisions can not be done within the ring for example 52 or

)3()52( 2 ++minus xxx which means that no element in the ring can multiply with a denominator obtaining the numerator Therefore neither 5 nor can be factorized 522 +minus xx If a polynomial of degree at least one has no other divisors then it is considered to be irreducible or prime (Durbin 1992) Of every polynomial of degree at least one can be written as a product of irreducible polynomials according to unique factorization theorem each polynomial of degree at least one over a field F can be written as an element of F times a product of monic irreducible polynomials over F and except for the order in which these irreducible polynomials are written this can be done in only one way (p 221) Thus a factorable polynomial can be written into a product with another polynomial such as In other words the condition that )()()( xqaxxf minus= )( ax minus is a factor of a polynomial

in a variable x is only true if )(xf 0)( =af For example if and

6)( 2 minus+= xxxf0624)2( =minus+=f 0639)3( =minusminus=minusf the factors of are and )(xf )2( minusx )3( +x

The value of the polynomial is zero which is also expressed as the polynomial having a zero point (nollstaumllle in Swedish) when

)(xfα=x

Based on unique factorization theorem finding solutions of an algebraic equation is equivalent to finding the first-degree factors for a polynomial (Vredblad 2000) In this way solving a quadratic equation can be handled through finding the first-degree factors for the second-degree polynomial and there after finding the roots of the quadratic equation This is often called using factorization to solve quadratic equations Besides unique factorization theorem the connection of using factorization to solve a quadratic equation is that this method relates to prime numbers using distributive law backwards commutative law multiplication division and operating fractions As matter of fact it is about application of factorizing integers in polynomials at upper-secondary school level Since quadratic equations in the mathematics B course at Swedish upper-secondary school are limited to real numbers and often within the whole numbers (integers) many quadratic equations should be able to be solved by factorization according to my own

16

judgment Therefore using factorization for solving quadratic equations is very much related to arithmetic operations number concept and algebra structure It may offer students both mathematical conceptual understanding and operational skill For algebra beginners factorization can be traced back to studentsrsquo early years of mathematics education during which students do not only work with multiplication but also on factoring integers for example 56 = 8 middot 7 = 2 middot 2 middot 2 middot 7 = 4 middot 14 = 2 middot 28 In this example two operations are included multiplication and division Multiplication operation may help students to understand the multiplicative structure of the integers while factoring integers may become meaningful when it is related to divisibility divisors and prime numbers These two operational skills or competence are major techniques for simplifying fractions and finding common denominators Being able to understand and operate factoring integers has laid the essential grounds for algebra beginners to study factorization of polynomials Thus factorization reunites studentsrsquo early mathematics knowledge of factoring integers with later knowledge of algebra structure Factorability In a study on factorability by Bosseacute and Nandakumar (2005) the probability of factorability is reasoned through investigating the range of integers as coefficients of quadratic equations In the quadratic equation a b and c are randomly generated integers within a determined range r such as

002 ne=++ acbxaxyrx lele where x and y are integers When is a

perfect square or where

acb 42 minus

)4( 2 acb minus is an integer quadratics are factorable According to the studyrsquos data it shows that as the range for a b and c increases the probability of factorability of a quadratic with randomly selected integer values for a b and c decreases Bosseacute and Nandakumar confirm that as the range for coefficients expands to infinltltinfinminus r the probablility of factorability of a quadratic with randomly selected integer values for coefficients approaches zero The studyrsquos data collected from college algebra courses and textbooks demonstrates that about 15 of quadratics with integer coefficients [ ]1010minusisinr are factorable Thus about 85 of quadratics can not be factorized In spite of the limitation of factorability within the exercises concerning factoring practice chosen from 27 surveyed college algebra textbooks about 94 of the problems were factorable Within the textbooks 55 of the quadratics to be factored were within the range [-10 10] Even if many quadratic equations are factorable choosing the right factor pairs is time consuming and unnecessary for example

has to be factorized through choosing the right pairs of factors among nine pairs for 36 and 8 pairs of factors for 24 Comparing the three different methods for solving quadratic equations the study suggests that completing the square and quadratic formula are more effective informative and useful than factorization

0245936 2 =++ xx

231 Different kinds of factorizations for solving different quadratic equations How can we find the binomial factors of a quadratic equation or a second-degree polynomial without knowing one of the divisors as a binomial factor The answer is that we have to observe the relations among all coefficients and constants as well as the positive and negative signs before them under the condition of being factorable Most of the coefficients in quadratic equations or polynomials among exercises provided for practicing factoring are integers (here I borrow the same notation r as the one used by Bosseacute and Nandakumar) with

17

their range between ndash10 and 10 that is 10ler In this case it is possible for a student to make a judgment of whether or not a quadratic equation is factorable Five different kinds of factorization depending on five kinds of quadratic equations Factoring quadratic expressions can be divided into five different kinds depending on the type of quadratic equations The coefficients a b p q and constant c illustrated in the following quadratic equations are defined not to be equal to zero thus 000 nenene cba The first kind of quadratic equations is type 1 the obviously factorable type (Bosseacute and Nandakumar 2004) then the equation is factorized as

where the roots are

0002 nene=+ babxax

0)( =+ baxxabxx minus== 21 0

The second kind of quadratic equations is type 2 The factors of this second degree polynomial are so the roots to this type of quadratic

equations are

0)( 22 =minusbax))(()( 22 baxbaxbax minus+=minus

bax plusmn=21 This kind of factorization is based on the difference-of-squares

formula for example can be solved by factorizing into 049 2 =minusx 0)23)(23( =minus+ xx Thus

the roots are 32

32

21 =minus= xx This kind of factorization is also obviously factorable

The third kind of quadratic equations is type 3 This type of equations can be solved directly by factoring into two identical binomials

The double roots obtained from this

factorization are

02)( 22 =+plusmn babxax

0))(()(2)( 222 =plusmnplusmn=plusmn=+plusmn baxbaxbaxbabxax

abx plusmn=21 This method has utilized square rules which are quite easily

observed The forth kind of quadratic equations is type 4 In this equation the coefficient of the -term is 1 with p and q as positive or negative integers Type 4 equations can always be solved by the approaches of completing the square and quadratic formula However they can often be solved by factorization too since the quadratic expressions are factorable by using distributive laws reversely The roots are obtained through finding two factors for q At the same time these two factors become satisfying with the relation so that the sum of factors is equal to That way the two factors become two roots of a quadratic equation Denoting the two factors or two roots as the relationship between coefficient and constant is

02 =++ qpxx2x

qminus

21rr)( 2121 rrprrq +minus=bull= Quadratic equations can be written as the product of

two first-degree binomials or factoring form 0))(()( 212121

22 =minusminus=++minus=++ rxrxrrxrrxqpxx Through the factoring form the roots of this equation are obtained being The core of factorizing type 4 equations is seeking the connections between coefficients and roots which reveal the studentsrsquo basic ability of arithmetic multiplication and addition It is often more effective in this case to use factorization than to use quadratic formula or completing the square Let us have a look at the example In this equation we get

2211 rxrx ==

035122 =+minus xx

18

35)7)(5(12)7()5( =minusminus=minus=minus+minus= qp or 75)75( bull=bullminus= qp (here we can directly get roots) Factorizing the equation we obtain 0)7)(5( =minusminus xx and roots are The procedures of finding factors of the equation often occur in some studentsrsquo minds quickly without writing down the whole computing process but those students often have good skills in arithmetic operations (reference) Compared to using factorization for solving this equation application of quadratic formula seems ineffective A possible problem in this method is how to find the signs before the factors or how to judge if the factors in the parentheses are positive or negative integers In order to check if the solution is correct we can always multiply these two factored binomials by distributive law and see if the expanded polynomial is the same as the original quadratic equation

75 21 == xx

The fifth kind of quadratic equations is type 5 Still the methods of completing squares and quadratic formula can be applied for all type 5 equations but there are cases in which factorization can be used for solving this type of equations within a certain range of integers as mentioned before Sometimes procedures of factorizing such equations can be carried out in our minds quickly like type 4 equations but in many cases they can not As a matter of fact finding two binomial factors for a factorable quadratic equation or a factorable second-degree polynomial is a complicated process and requires systematic work In his article published in Mathematics in School Jackman (2005) demonstrates this systematic procedure via factorizing second-degree polynomials A second-degree polynomial can be written as where are all integers By multiplying out the polynomial becomes thus

0002 nene=++ bacbxax

))((2 srxqpxcbxax ++=++ srqpcbaqsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== Writing the factors of a and c on two lines p q times r s Finding the right pairs of factors for a and c has to be done by listing out the groups of products of different factors for example The factor pairs for 6 are (6 1) (1 6) (3 2) (2 3) and the factor pairs for (-4) are (4 -1) (-4 1) (2 -2) (-2 2) (1 -4) (-1 4) Writing them on two lines p 6 1 3 2 q 4 -4 2 -2 1 -1

456 2 minus+ xx

times r 1 6 2 3 s -1 1 -2 2 -4 4 Then comes calculating the values of ps and qr until the value of b = 5 is obtained The calculating procedures are divided into 4 groups and 6 steps in each group which means 4 times 6 steps in total Here I only give one grouprsquos procedure as an example 6 middot (-1) + 4 middot 1 = -2 6 middot 1 + 1middot (-4) = 2 6 middot (-2) + 1 middot 2 = -10 6 middot 2 + 1middot (-2) = 10 6 middot (-4) + 1 middot 1 = -23 6 middot 4 + 1 middot (-1) = 23 helliphellip After the 24ndashstep calculating procedures two computing procedures obtain the same correct result 3 (ndash1) + 4 2 = 5 (yes) 2 4 + (-1) 3 = 5 (yes) Are they the same From 3 (ndash1) + 4 2 = 5 the polynomial can be factorized into )12)(43( minus+ xx while from 2 4 + (ndash1) 3 = 5 the polynomial can be factorized into )43)(12( +minus xx According to communicative law these two binomial products are the same The shortcoming of this

19

method is that it is too ineffective and complicated The addition of matrixes is almost impossible and unbearable Nevertheless based on the same idea I suggest finding factors by guessing analyzing and checking Using the same example here we can find that one of the binomials must have a negative integer because of ndash4 which implies that 5 is the sum of a positive integer and a negative integer Then looking at a and c the factors of 6 can be (1 6) and (2 3) while the factors of 4 can be (1 4) and (2 2) when we regard them as absolute values We know that the factoring form of the polynomial will be like

and the connections of all coefficients are

456 2 minus+ xx

qsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== The coefficient 5=b is the difference of two products of factors for 6 and 4 First we try the biggest number of product such as 2446 =bull then 524 =minus the unknown is which is a prime number the only factors for 19 are (19 1) There is not any combination of products of factors for 6 and 4 satisfying 19 Therefore the multiplications of factors (1 6) and (1 4) are not included in this case So we move to factors (2 2) (2 3) and (1 4)

19 =

In the second step we try The result is negative it is obviously

4264466)23)(22( 22 minusminus=minus+minus=+minus xxxxxxx2 lt 5 which implies that the combination of (2 3) and (2 2) is not

considerable One pair of factors has to be removed and that is (2 2) which does not influence the coefficient of the -term Then we try The result this time is negative too but we can see that

2x 456)43)(12( 2 minusminus=minus+ xxxxb is 5 which guides us to interchanging the positions of

negative and positive signs in the parentheses The next attempt is and it is correct Seeking the correct factors for the polynomial

has taken four major operating steps by trying the combinations of factorsrsquo products and relating the sum of the products with the value of the x-term coefficient The strategy is to make use of distributive law reversely first and then multiply the product of every pair of parentheses in order to check if they are the right pair of polynomial factors Factorizing second-degree polynomials is the major step for solving quadratic equations After factoring polynomials into the product of two binomials as in the example given above the roots can be obtained by making each binomial equal to zero according to the theory of zero point (ldquonollstaumlllerdquo in Swedish) of a polynomial equation originating from unique factorization theorem

456)43)(12( 2 minus+=+minus xxxx456 2 minus+ xx

Factorization by using the technique of vertically and crosswise sutra Binomial expansion or multiplication of binomials by distributive law and factorization of the second-degree polynomials by using distributive law backwards have an extremely symmetrical effect and the results of two kinds of operations are equivalent It is crucial for the students to be able to perceive mathematics structure and be acquainted with arithmetic operations as well as to understand the relationship between quantities when they carry out factoring procedures The expansion of binomials by distributive law has been named as the method FOIL which stands for first outside inside and last according to multiplication orders of two binomials for example (Nataraj and Thomas 2006) Many students are used to this procedure Another technique to multiply two binomials is called vertically and crosswise sutra origin from the ancient Indian Vedic mathematics Vertically and crosswise sutra and FOIL are much the same in principle but it makes use of

824)4)(2( 2 +++=++ xxxxx

20

the power of visualization and improves facility with the expansion of binomials and the factorization of quadratic expressions In this method two binomials are written into two lines Using the same example the two lines are written as 824)4)(2( 2 +++=++ xxxxx

x + 2 x + 4 ______________ xsup2 4x + 2x 8

The order of multiplication is as shown below x + 2 x + 4

4 3 2 1 (the numbers 1 2 3 and 4 refer to steps) xsup2 + 2x + 4x + 8 Figure 3 The vertically and crosswise method of multiplication of binomials Step 2 and 3 can be interchangeable If we put this form into a line expanded according to the order above would be like

)4)(2( ++ xxxbullrarrbull 442 clockwise and xxx bullrarrbull2 in

counterclockwise direction According to sutra we can imagine drawing a thread starting from 2 then going through 4 and x clockwise and then draw the second thread starting from the same place 2 later going through x in the second bracket and finally x in the first bracket counterclockwise Learning expansion of binomials by the methods FOIL and Vertical and clockwise sutra is the basic process before students study factorization In this study Nataraj and Thomas (2006) have found that the students appear to do factorization using guess-and-check as well as decomposition or the vertical and clockwise approach Their finding is that afterwards the students performed better on algebra questions especially on the factorizations They conclude ldquohellipthe values of the method may lie in what it adds to the studentsrsquo overall algebraic conceptions and knowledge of mathematical structurerdquo (p 16) In general solving these five kinds of quadratic equations by factorization can be traced back to unique factorization theorem The important character in this method is that you can always check if factorization is correct or not through multiplying out the two pairs of a parenthesis and see if the expanded polynomial is the given one or not Using factorization is not only about method but also about fostering the studentsrsquo mathematical thinking The solving processes are more important than solutions

21

3 Review of related literature and research Besides the focus on mathematics content another focus in the background work for this research is about previous studies and research in the field of mathematics didactic concerning different methods for solving quadratic equations and the use of factorization In order to study the previous research and studies in the related field three scientific databases and a search engine have been used and 113 articles have been investigated Topics such as solving quadratic expressions and equations are often discussed as teaching ideas and strategies shared by mathematics educators in different countries from lower and upper-secondary schools to colleges according to the findings in the article search These didactic topics from mathematics teachersrsquo perspectives include for example using geometric approaches to solve quadratic equations (Allaire amp Bradley 2001) utilizing completing squares (Vinogradova 2007) factoring quadratics (Hoffman 1976 Cheng 1980) and using factorization completing the square graphical methods to solve quadratic equations (Macdonald 1986) Kostsopoulos (2007) links cognitive reasoning to pedagogical problems appearing in the mathematics classrooms in her article published in Australian Mathematics Teacher Based on her teaching experiences Kostsopoulos is aware of her pedagogical strategies lacking insight on dealing with studentsrsquo difficulties of multiplication and factorization of quadratics She assumes that ldquostudentsrsquo problems with factorization and with identifying varied representations of the same quadratic relationshiprdquo may be linked to the ways in which the brain constructs cognitive representations (p 19) Factoring of quadratics is the rewriting of polynomials as a product of polynomials It requires students to have both a conceptual understanding of multiplication of polynomials and the procedural knowledge to retrieve basic multiplication facts effectively Based on the theory about the order matters in the brainrsquos ability to retrieve number facts (Phenix amp Campbell 2001) Kostsopoulos conjectures that studentsrsquo ability to access the appropriate long-term semantic memory is limited when they are confused by mixing the order and varied forms of quadratics Although this article is not about a research study it seeks the reason why factorizing quadratics is difficult and therefore might provide math teachers with alternative explanations for similar pedagogical frustrations With Von Glasersfeldrsquos radical constructivism and Peter Gaumlrdenforsrsquo epistemic semantics theory as his theoretical frameworks Rauff (1994) suggests that belief-based teaching can be successful in teaching factoring According to Rauff constructivist learning theory takes the view that students construct their own beliefs and knowledge of mathematics Learning occurs when the students change their belief set Referring to belief set theory from Gaumlrdenfors (1988) Rauff (1994) claims that ldquoa belief set is expanded when a new belief is added to it and contracted when a proposition is no longer believed and is removed from the setrdquo (p 421) Expansion contraction and revision are three components to comprise basic mechanisms of changes of belief according to Rauff When examining and comparing studentsrsquo (ages 18-20) own definitions of factoring Rauff analyzes the errors of their definitions with constructivism and epistemic semantics as analyzing tools The result of this study shows that the approach to teaching factoring rely heavily upon what a student believes about factoring is quite productive in two dimensions First it reveals the source of nonstandard factoring Second it provides a starting point for modification of the studentrsquos underlying conceptions of factoring Rauff proposes that belief-based teaching can also be

22

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

In order to find the length and the width of the rectangle Euclid bisected AB at E constructed the square on BE and then claimed that this square was equal to the sum of the rectangle ACFS and the small shaded square at the bottom (Figure 2)

A E S B

x

y y F

C G D Figure 2 According to Euclid the area of the rectangle ACFS was given which was c and the area of

the new square EGDB was also given which was ( )22b because 222byxSBASEB =

+=

+=

The equivalent relationship between the areas can be formulated as an quadratic equation 22

22⎟⎠⎞

⎜⎝⎛ minus+=⎟

⎠⎞

⎜⎝⎛ bxcb or

22

22⎟⎠⎞

⎜⎝⎛ minus+=⎟

⎠⎞

⎜⎝⎛ ybcb Euclid found out this equivalent relationship

geometrically and took use of this relationship in finding the solutions of the problem so the

length and width of the rectangle ACFS are cbbx minus⎟⎠⎞

⎜⎝⎛+=

2

22 and cbby minus⎟

⎠⎞

⎜⎝⎛minus=

2

22

These two formulas are almost identical with the Babyloniansrsquo solutions in rhetoric expressions The difference is that Greek algebra was based on geometric manipulation while Babylonian algebra was based on rhetoric manipulation with geometrical ideas In general the early stage of algebra from ancient Babylon and Egypt to Greek was mainly geometrical The syncopated stage ndash the beginning of the static equation-solving stage by using In Roman Egypt in probably the second or third century CE the algebra stage was at the syncopated stage which means written algebraic texts were expressed in words but involved special symbols-abbreviations According to recorded mathematics history one the of pioneers who used these special symbols to solve equations with only numbers but no connection with geometry was Diophantus who lived in Alexandria in Egypt around the third century Diophantus used the Greek alphabetic system for writing numbers He wrote a treatise titled Arithmetica of which less than half has been maintained today The surviving part of his work consists of 189 problems in which the object is to find numbers or families of numbers satisfying certain conditions In mathematics today Diophantusrsquo mathematical analysis is known as number theoryndashnot algebra However he used only number symbols to solve equations without the help of geometry Diophantus wrote the coefficient after the variable instead of before it as we do He used Greek ς for unknown quantity our modern x Most of his book deals with indeterminate equations which contain more than one unknown and a potentially infinite number of solutions His problem is that he could not represent more than one unknown instead he solved quadratic equations with two unknowns through substituting one by another and then the quadratic equations became the ones with one unknown At that time negative numbers were regarded to be absurd but he knew how to

7

bring a term from one side of an equation to the other gather up like terms for simplification and some elementary principles of expansion and factorization (Derbyshire 2006) Diophantus made his own literal symbolism with the use of special letter symbols for the unknown and its powers for subtraction and equality From Diophantus algebra history moved into another conceptual stage the equation-solving stage according to Katz (2006) In India quadratic formula appeared without any geometric support Brahmaggupta (598-665) was one of the first mathematicians who could systematically handle negative numbers and zero He gave a general solution to quadratic equations and realized that there were two roots for a quadratic equation It was possible that one of the roots was a negative number Baskharacharya (1114-1185) solved mathematics problems with the use of quadratic equations in his book Siddhanta Siromani (ldquoMathematical Pearlsrdquo) He presented an algorithm to reduce a quadratic equation to a first-degree equation (Olteanu 2007) It is commonly believed that the first true algebra text is the work on al-jabr and al-muqabala by Mohanmmad ibn Musa al-Khwarizmi (780-850) written in Baghdad around 825 (Katz 2006) The word algebra came from the title of this work The word al-jabr means restoration or reestablishment that is to eliminate negative terms through adding the same terms to both sides of equations The word of al-muqabalas means balance meaning to divide every term in a quadratic equation by the coefficient of the second degreersquos term (Olteanu 2007) The first part of his book is a manual for solving linear and quadratic equations Al-Khwarizimi classified equations into six types three of which were mixed quadratic equations For each type he presented an algorithm for its solution Five of the six types of equations were quadratic equations which can be expressed in modern form

cbxaxbxcaxcbxaxcaxbxax +==+=+== 22222 Here is an example of solving the equation ldquoTake the half of the number of the things that is five and multiply it by itself you obtain twenty-five Add this to thirty-nine you get sixty-four Take the square root or eight and subtract from it one half of the number of things which is five The result three is the thingrdquo (Kvasz 2006 p 292) Like Babylonian mathematicians al-Khwarizimirsquos algorithm is entirely verbal On the other hand al-Khwarizimi sometimes completed his algorithm by using geometrical explanations which can be translated as todayrsquos square-completing method Using the example of solving the quadratic equation the completed geometrical procedures are illustrated on next page in figures 3 4 and 5

39102 =+ xx

39102 =+ xx

8

x

xsup2

Figure 3 (Olteanu 2007 p 30) 5x2

25x

xsup2

25x

5x2

Figure 4 (Olteanu 2007 p 30) 254 254

39

254 254

Figure 5 (Olteanu 2007 p 30) According to Olteanursquos (2007) translation al-Khwarizimi started with a square whose sides are x and area is xsup2 (see Figure 3) Then he added four equal rectangles whose areas in total

was 10x along each side of the square that is xx sdotsdot=25410 Each rectanglersquos area is thus x

25

with its length x and its width 52 (see Figure 4) The sum of the big square and four rectangles was given which was 39 The equivalence relationship was Finally figure 4 was completed by adding four small equal squares which had an area the size of

39102 =+ xx

425

25

25

=sdot for each small square and the sum of those was 25 Through adding this sum to

both sides of the equation the area of the biggest square obtained in figure 5 was 64 The

equation translation is 425439

4254102 sdot+=sdot++ xx The side of the biggest square was 8 and

had its relation with other sides of different squares expressed in the first degree equation

9

25

258 ++= x Then x was 3 By ldquocut-and-pasterdquo geometry (Katz 2006 p 191)

al-Khwarizimi reduced the second degree of an equation to the first degree and thereafter solved it Unlike his Babylonian predecessors al-Khwarizimi always presented his problem abstractly rather than geometrically relating to lengths and widths The symbolic stage At this stage of algebra ldquoall numbers operations relationships are expressed through a set of easily recognized symbols and manipulations on the symbols take place according to well-understood rulesrdquo (Katz 2006 p 186) The ancient algebra and geometry had developed sophisticatedly in Egypt Persia Greece India and China After Medieval Islamic scholars gave us the word ldquoalgebrardquo Western Europe began the struggle for the development of algebra starting from some algebraists from Italy Italian mathematician Leonardo Pisano later known as Fibonacci traveled in the 12th and 13th centuries to Persia India and China When he returned to Italy he had wider knowledge of arithmetic and algebra His book Liber abbaci was the best math textbook since the end of Ancient world His book is credited with having introduced Indian numerals including zero to the West But his algebraic skills had been shown in two other works after this one With the introduction of printed books during the second half of the 15th century the development of algebra was sped up Several Italian mathematicians including Girolamo Cardano had figured out how to solve cubic and quadratic equations Algebra became purely abstract with the exception of an English mathematician named Robert Recorde who lived in the 16th century and created quadratic problems from real world experience (Derbyshire 2006) It was in France that algebra had developed into a well organized literal symbolism In his work In artem analyticem isagoge French mathematician Franςois Viegravete (1540-1603) in the late 16th century became the first mathematician to use letters representing numbers systematically and effectively (Derbyshire 2006) He made a range of letters available for many different quantities This was the beginning of modern literal symbolism Viegravetersquos unknown quantity was divided into two classes unknown quantities (meaning ldquothings soughtrdquo) denoted by A E I O U and Y while ldquothings givenrdquo was denoted by constants like B C Dhellip For example his A is our unknown x Viegravete was a pioneer in the study of equations His two papers on the theory of equations were published twelve years after his death In the second paper titled ldquoDe equationem emendationerdquo (ldquoOn the perfecting of equationsrdquo) Viegravete opened up the line of inquiry that led to the study of the symmetries of an equationrsquos solutions to Galois theory the theory of groups and all of modern algebra He found the relationship between the solutions of the equation and the coefficients for the first five degrees of equations in a single unknown To explain this in our modern symbols we suppose that the two solutions of the quadratic equation are 02 =++ qpxx α and β which means βα == xx Based on this logic the following thing must be true 0))(( =minusminus βα xx since only α andβ and no other values of x make this equation true This form of equation is just a rewritten form of the same equation If we multiply out those parentheses this rewritten equation turns to be Compared to the original equation the relationships between the solutions and the coefficients we obtain

0)(2 =++minus αββα xxqp =minus=+ αββα

(Derbyshire 2006) It is said that Viegravete discovered the solution formula called quadratic

10

formula today for general quadratic equations which is a

acbbx2

42

21minusplusmnminus

= for a general

quadratic equation (Olteanu 2007) 002 ne=++ acbxax Another French mathematician and philosopher who had strong influence in the history of algebra was Reneacute Descartes (1596-1650) His idea to use the Cartesian system of coordinates which was derived from his Latin name developed both algebra and geometry In his work La geacuteomeacutetrie written in 1637 Descartesndashthrough using numbers to indentify points in a Cartesian coordinatesndashconnected geometrical objects to algebraic numbers and made the classical geometry become analytical geometry He took up the plus and minus sign from the German Cossists of the previous century and also the square-root sign From Descartes the symbol of an unknown was represented by x which led to the modern system of literal symbolism (Derbyshire 2006) Descartes developed the idea of functions in Cartesian coordinates although the idea of function can be traced back to the Islamic mathematician Sharaf al-Din al-Tusi from Persia (Katz 2006) and Klaudius Ptolemaios about 2000 years ago (Olteanu 2007) Descartes declared that every curve in a Cartesian coordinate system has an equivalent equation which can represent the points on the curve or vice versa every equation containing x and y can be represented by a curve through its coordinate points However the word function was introduced by Gottfried Wilhelm Leibniz in 1693 and the definition of function was defined by Leonhard Euler in his work Introductio analysin infinitorium in 1748 Euler had also given the symbol of a denotation as a relying variable (Olteanu 2007) In the 18)(xf th century after the discovery of calculus by Newton and Leibniz algebra went into an area of analysis ndashthe study of limits infinite sequences and series functions derivatives and integrals (Derbyshire 2006) The purely abstract stage-algebra structure Since 17th century algebra has not anymore solely been about finding solutions for different degrees of equations mathematicians have started to integrate algebra with astronomy and physics Johann Kepler and Galileo Galilei were interested in curves and finding a mechanism for representing motion instead of quantity-like numbers But their arguments were not algebraic-symbolic but geometrical It was Fermat and Descartes who were regarded as the fathers of analytic geometry and who showed how to represent a curve described verbally through algebra analytic geometry and gave a mechanism for representing motion Newton in his Principia picked up on Fermat and Descartesrsquo representation and developed the calculus With the invention of the calculus mathematical problems were solved by curves not just points Algebra grew more and more to present paths of motion In order to be able to judge if the algebraic manipulations were correct axioms were formulated for arithmetic applied for algebraic manipulations In the late 18th century Lagrange introduced the idea of permutations into the search for solutions Therafter Galois developed methods for determining what conditions polynomial equations need to meet to be solvable New abstract algebraic concepts were found like ldquofieldrdquo and ldquogrouprdquo during the early 19th century In 1854 Ceyley gave an axiomatic definition of a group During the 1890s this definition entered textbooks along with the axiomatic definition of a field an idea which had roots in the work by Galois By the beginning of the 20th century algebra became less about finding solutions to equations and more about looking for common structures in many mathematical objects defined by sets of axioms (Katz 2006)

11

In short algebra development took 4000 years It took about 3500 years for mathematicians to start to use algebra symbols in systems Just almost 100 years ago algebra developed into a purely abstract science The long history may give both mathematics educators and students some explanations to why it is so difficult to teach and learn algebra

22 Three solving methods for quadratic equations Already in the previous part I have presented geometrically and rhetorically solving quadratic equations from a historical perspective In order to get more explicit understanding of quadratic equations quadratic formula and factorization from mathematics didactic point of view this part presents three common methods for solving quadratic equations 1) Completing the square 2) Factorization 3) Quadratic formula The reason to present these three methods is that they are the common topics for discussion in the articles searched in this area concerning mathematics education at upper-secondary school 1) Completing the square In two articles Solving quadratic equations by completing squares (Vinogradova 2007) and Geometric approaches to quadratic equations from other times and places (Allaire amp Bradley 2001) the authors advise mathematics teachers to use the ideas of ancient mathematician al-Khwarizmi and his method of completing quadratic equations by geometrical algebra and simplifying it in order to represent it to the students By doing so the teachers relate mathematics quantity to physical objects in a visual way Didactically teachers may start with a concrete example in which students can understand the content visually Later from the concrete example the teachers lead the students to another example expressed in algebraic symbols for example to study a rectangle whose area is )10( +xx and supposing this area is 39 this becomes 39)10( =+xx Four steps are followed to build a new square which is actually called completing the square x 10

Figure 6 x 5

Figure 7

x

x

5 5

12

A Begin with a rectangle of the area )10( +xx that is with the short side as x and the long side )10( +x The shaded area is 10bullx (Figure 6)

B Divide the shaded rectangle into two small shaded rectangles with the size 5x each and move them to each adjacent side of the xsup2 square (Figure 7) The total area is still

x10

)10(39 += xxC In Figure 7 a new square is shaped in the lower-right hand corner with the side 5 or

area 25 By adding this small square to the diagram the large square with an area of is ldquocompletedrdquo Therefore we can write 25)5(22 ++ xx 25)10(2539 ++=+ xx or

The large squarersquos area is now because the side has a length of Therefore

6425)5(22 =++ xx 2)5( +x5+x 38564)5( 2 =rArr=+rArr=+ xxx

After these three steps a solution to this quadratic equation is derived by completing the square (Allaire amp Bradley 2001) 2) Factorization Hoffman (1976) presents three approaches (A B and C) for factorizing quadratic expressions through observing and grouping He takes a second-degree polynomial as an example

273 2 ++ xx

a The inspection method 1) suggests factors of and 23x x3 x 2) 2 suggests factors of 1 and 2 or ndash1 and ndash2 3) Checking the linear term shows that the correct factorization is x7 )2)(13(273 2 ++=++ xxxx

This method according to Hoffman (1976) is based on the studentrsquos ability to simplify by inspecting the indicated product of a pair of linear expressions

b The method of decomposition of the linear term 1) Multiply and 2 to get 23x 26x2) Decompose into the sum of two terms whose product is x7 26x

This gives xxx += 67 3) Factorize by grouping 2)6(3 2 +++ xxx

2)6(3 2 +++ xxx = )

)2()63( 2 +++ xxx

= 2(1)2(3 +++ xxx= )13)(2( ++ xx

c Third method 1) Multiply 3 and 2 to get 6 2) Find two numbers whose product is 6 and whose sum is 7 The numbers are 6 and 1 3) 273 2 ++ xx

=3

)13)(63( ++ xx

= )13)(2( ++ xx

13

These three approaches can not actually be divided as three unrelated approaches especially the third method is seemingly unclear What Hoffman (1976) wants to demonstrate is that all the three approaches are based on observing the relationship among the coefficients and the constant of the polynomial Through finding the product and the sum of two factors the polynomial can be factorized in the product of two binomials In the authorrsquos opinion factorization is heavily dependent on skill in multiplication and using distributive law reversely He means that ldquostudents who do not have reasonably strong skills in multiplication should not be expected to develop strong skills in factorizationrdquo (p 55) Factorizing polynomials is not only for simplifying high-degree polynomials (for example the third or fourth-degree polynomials) but can also be applied for solving quadratic equations Supposing this polynomial forming a quadratic equation Solving this equation is actually to decompose the polynomial into a factoringrsquos form

0273 2 =++ xx0)13)(2( =++ xx

which is equivalent to the polynomial form The essential step to solve this equation is to follow the zero-factor propertyndashalso called null-factor law (nollprodukt in Swedish)ndashthat is for real numbers p and q p bull q = 0 if (and only if) p = 0 or q = 0 It means that either of the binomials on the left side of the equation has to be equal to zero in order to satisfy the

equivalent relation of this quadratic equation The solutions of two roots or 2minus=x31

minus=x

are obtained through making either 02 =+x or 013 =+x 3) Quadratic formula Solving quadratics equations by factorization is constrained within the simple quadratic equations over the whole integers and rational numbers as coefficients What happens when quadratic equations have coefficients that belong to an irrational domain or big quantity The quadratic formula is a solution to such a situation As mentioned previously historically the French mathematician Franccedilois Vieacutete discovered the quadratic formula

aacbbx

242 minusplusmnminus

= ) which is applicable for all kinds of quadratic equations In

Swedish textbooks this formula is written in PQ form by making for the equation

that is

0( nea

1=a

02 =++ qpxx qppx minus⎟⎠⎞

⎜⎝⎛plusmnminus=

2

22 with

abp = and

acq = In Oleatursquos study (2007)

studentsrsquo difficulties in using algebraic symbols no longer is the problem But on the other hand handling the parameters or coefficients in quadratic equations like and rewrite them in the equivalent form with p and q being real numbers becomes obstacles for the students when they study the algebra course at upper-secondary school

02 =++ cbxax02 =++ qpxx

The quadratic formula a

acbbx2

42 minusplusmnminus= has been regarded as the standard method

according to US mathematics teachers Obermeyer (1982) presents a way of how to derive this formula by completing a square through 11 steps (I will not present them here) At the same time an alternate method is also introduced according to the same principle All the methods mentioned so far are based on the square rules The standard method or PQ form might be regarded as an efficient and direct method but it may

222 2)( bababa +plusmn=plusmn

14

lead students to solve quadratic equations in a mechanical way besides having problems with the troublesome plusmn symbolism (Stover 1978) Both Stover (1978) and Olteanu (2007) have suggested that using a graph of quadratic functions to solve quadratic equations is another alternative method

23 Mathematical background of factorization Factorization in algebra structure Among the three solving methods presented above both the geometrical method and quadratic formula can be traced back to algebra history in section 21 What is the trace of factorization in algebra history Factorization was at the last stage in algebra history and belonged to algebra structure (Derbyshire 2006) In Swedish mathematics textbooks the chapter on algebra often includes polynomial factorization and quadratic equations But the factorization in the textbooks is not at the same abstract level as the factorization of algebra structure although they share the same axioms and there are certain connections between these two different levels of factorization How are polynomial and quadratic equations as well as factorization related to each other In order to find the theoretical background of factorizing polynomials this part constitutes a mathematical theoretical review by referring to two books concerning algebra structure (Vretblad 2000 Durbin 1992) and a research study (Bosseacute and Nandakumar 2005) A quadratic equation can be regarded as polynomial equation Solving quadratic equations has much to do with ldquounique factorization theoremrdquo The polynomial

is a second-degree polynomial A polynomial consists of coefficients in whole numbers rational numbers real numbers or complex numbers and variables (sometimes called unknowns) in different degrees as well as constants A polynomial

has symbols as coefficients which are real numbers If all are zero and so

)0(02 ne=++ acbxax

cbxax ++2

nnxaxaxaaxf ++++= )( 2

210 naaa 10

ka 1gek 0)( axf = is constant If all are zero then we say the is a zero polynomial If meaning that is not a zero polynomial and n is the biggest number and then we say that the polynomial is at degrees A zero polynomial has no degree at all According to the definition (Vredblad 2000) if is a polynomial the equation is called an algebraic equation or polynomial equation A solution or a root of this equation is a number

kaf 0nef f

0nena f nf

0)( =xfα so that 0)( =αf Such a number is even

called a zero point (nollstaumllle in Swedish) of the polynomial which means when f α=x the value of the polynomial is zero (p 161) Every polynomial whose degree is equal or greater than 1 can be written as a product of irreducible polynomial In this way through polynomial division a polynomial with n degrees can be expressed from a sum of terms into a product of its denominator and its quotient The process of changing the expressions is factorization for

instance 12

22

+=minusminusminus x

xxx The factoring form is )1)(2(22 minusminus=minusminus xxxx

Factorization of polynomials is analogue to factorization of whole numbers which is very much related to the concept of ring A ring consists of a set with two operations which are a sum and a product of two elements over a field Any polynomial from zero polynomial to n degree-polynomials can be written or operated through addition and multiplication of the

15

elements over a field F including whole numbers or integers rational numbers real numbers and complex numbers In a ring you can multiply any two elements but you canrsquot always divide Factorization is a way to deal with this fact Here are some examples In the ring of integers Z is an example of factorization in some polynomial rings for instance Q[x] the ring of polynomial in one variable with rational numbers as coefficients such as

7214 sdot=

)451)(2

51(8

52

251 2 +minus=minus+ xxxx is factorization Z[x] the ring of polynomial in one

variable with whole numbers as integers such as is factorization )1)(2(22 minusminus=minusminus xxxx The polynomial ring R[x] means the ring of polynomial in one variable with real numbers as coefficients Factorizing of such polynomials implies that the polynomial factors are the elements in the ring of R[x] for instance )2)(2(22 minus+=minus xxx The same thing is true for C[x] the ring of polynomial in one variable with complex numbers as coefficients Thus it is coefficients of polynomials that determine which ring a polynomial belongs to and if it is possible to be factorized in this ring For example number 6 can be factorized in two ways

in the ring of Z and 32times )51)(51( ii +minus in the ring of C (complex numbers) However there are cases in which divisions can not be done within the ring for example 52 or

)3()52( 2 ++minus xxx which means that no element in the ring can multiply with a denominator obtaining the numerator Therefore neither 5 nor can be factorized 522 +minus xx If a polynomial of degree at least one has no other divisors then it is considered to be irreducible or prime (Durbin 1992) Of every polynomial of degree at least one can be written as a product of irreducible polynomials according to unique factorization theorem each polynomial of degree at least one over a field F can be written as an element of F times a product of monic irreducible polynomials over F and except for the order in which these irreducible polynomials are written this can be done in only one way (p 221) Thus a factorable polynomial can be written into a product with another polynomial such as In other words the condition that )()()( xqaxxf minus= )( ax minus is a factor of a polynomial

in a variable x is only true if )(xf 0)( =af For example if and

6)( 2 minus+= xxxf0624)2( =minus+=f 0639)3( =minusminus=minusf the factors of are and )(xf )2( minusx )3( +x

The value of the polynomial is zero which is also expressed as the polynomial having a zero point (nollstaumllle in Swedish) when

)(xfα=x

Based on unique factorization theorem finding solutions of an algebraic equation is equivalent to finding the first-degree factors for a polynomial (Vredblad 2000) In this way solving a quadratic equation can be handled through finding the first-degree factors for the second-degree polynomial and there after finding the roots of the quadratic equation This is often called using factorization to solve quadratic equations Besides unique factorization theorem the connection of using factorization to solve a quadratic equation is that this method relates to prime numbers using distributive law backwards commutative law multiplication division and operating fractions As matter of fact it is about application of factorizing integers in polynomials at upper-secondary school level Since quadratic equations in the mathematics B course at Swedish upper-secondary school are limited to real numbers and often within the whole numbers (integers) many quadratic equations should be able to be solved by factorization according to my own

16

judgment Therefore using factorization for solving quadratic equations is very much related to arithmetic operations number concept and algebra structure It may offer students both mathematical conceptual understanding and operational skill For algebra beginners factorization can be traced back to studentsrsquo early years of mathematics education during which students do not only work with multiplication but also on factoring integers for example 56 = 8 middot 7 = 2 middot 2 middot 2 middot 7 = 4 middot 14 = 2 middot 28 In this example two operations are included multiplication and division Multiplication operation may help students to understand the multiplicative structure of the integers while factoring integers may become meaningful when it is related to divisibility divisors and prime numbers These two operational skills or competence are major techniques for simplifying fractions and finding common denominators Being able to understand and operate factoring integers has laid the essential grounds for algebra beginners to study factorization of polynomials Thus factorization reunites studentsrsquo early mathematics knowledge of factoring integers with later knowledge of algebra structure Factorability In a study on factorability by Bosseacute and Nandakumar (2005) the probability of factorability is reasoned through investigating the range of integers as coefficients of quadratic equations In the quadratic equation a b and c are randomly generated integers within a determined range r such as

002 ne=++ acbxaxyrx lele where x and y are integers When is a

perfect square or where

acb 42 minus

)4( 2 acb minus is an integer quadratics are factorable According to the studyrsquos data it shows that as the range for a b and c increases the probability of factorability of a quadratic with randomly selected integer values for a b and c decreases Bosseacute and Nandakumar confirm that as the range for coefficients expands to infinltltinfinminus r the probablility of factorability of a quadratic with randomly selected integer values for coefficients approaches zero The studyrsquos data collected from college algebra courses and textbooks demonstrates that about 15 of quadratics with integer coefficients [ ]1010minusisinr are factorable Thus about 85 of quadratics can not be factorized In spite of the limitation of factorability within the exercises concerning factoring practice chosen from 27 surveyed college algebra textbooks about 94 of the problems were factorable Within the textbooks 55 of the quadratics to be factored were within the range [-10 10] Even if many quadratic equations are factorable choosing the right factor pairs is time consuming and unnecessary for example

has to be factorized through choosing the right pairs of factors among nine pairs for 36 and 8 pairs of factors for 24 Comparing the three different methods for solving quadratic equations the study suggests that completing the square and quadratic formula are more effective informative and useful than factorization

0245936 2 =++ xx

231 Different kinds of factorizations for solving different quadratic equations How can we find the binomial factors of a quadratic equation or a second-degree polynomial without knowing one of the divisors as a binomial factor The answer is that we have to observe the relations among all coefficients and constants as well as the positive and negative signs before them under the condition of being factorable Most of the coefficients in quadratic equations or polynomials among exercises provided for practicing factoring are integers (here I borrow the same notation r as the one used by Bosseacute and Nandakumar) with

17

their range between ndash10 and 10 that is 10ler In this case it is possible for a student to make a judgment of whether or not a quadratic equation is factorable Five different kinds of factorization depending on five kinds of quadratic equations Factoring quadratic expressions can be divided into five different kinds depending on the type of quadratic equations The coefficients a b p q and constant c illustrated in the following quadratic equations are defined not to be equal to zero thus 000 nenene cba The first kind of quadratic equations is type 1 the obviously factorable type (Bosseacute and Nandakumar 2004) then the equation is factorized as

where the roots are

0002 nene=+ babxax

0)( =+ baxxabxx minus== 21 0

The second kind of quadratic equations is type 2 The factors of this second degree polynomial are so the roots to this type of quadratic

equations are

0)( 22 =minusbax))(()( 22 baxbaxbax minus+=minus

bax plusmn=21 This kind of factorization is based on the difference-of-squares

formula for example can be solved by factorizing into 049 2 =minusx 0)23)(23( =minus+ xx Thus

the roots are 32

32

21 =minus= xx This kind of factorization is also obviously factorable

The third kind of quadratic equations is type 3 This type of equations can be solved directly by factoring into two identical binomials

The double roots obtained from this

factorization are

02)( 22 =+plusmn babxax

0))(()(2)( 222 =plusmnplusmn=plusmn=+plusmn baxbaxbaxbabxax

abx plusmn=21 This method has utilized square rules which are quite easily

observed The forth kind of quadratic equations is type 4 In this equation the coefficient of the -term is 1 with p and q as positive or negative integers Type 4 equations can always be solved by the approaches of completing the square and quadratic formula However they can often be solved by factorization too since the quadratic expressions are factorable by using distributive laws reversely The roots are obtained through finding two factors for q At the same time these two factors become satisfying with the relation so that the sum of factors is equal to That way the two factors become two roots of a quadratic equation Denoting the two factors or two roots as the relationship between coefficient and constant is

02 =++ qpxx2x

qminus

21rr)( 2121 rrprrq +minus=bull= Quadratic equations can be written as the product of

two first-degree binomials or factoring form 0))(()( 212121

22 =minusminus=++minus=++ rxrxrrxrrxqpxx Through the factoring form the roots of this equation are obtained being The core of factorizing type 4 equations is seeking the connections between coefficients and roots which reveal the studentsrsquo basic ability of arithmetic multiplication and addition It is often more effective in this case to use factorization than to use quadratic formula or completing the square Let us have a look at the example In this equation we get

2211 rxrx ==

035122 =+minus xx

18

35)7)(5(12)7()5( =minusminus=minus=minus+minus= qp or 75)75( bull=bullminus= qp (here we can directly get roots) Factorizing the equation we obtain 0)7)(5( =minusminus xx and roots are The procedures of finding factors of the equation often occur in some studentsrsquo minds quickly without writing down the whole computing process but those students often have good skills in arithmetic operations (reference) Compared to using factorization for solving this equation application of quadratic formula seems ineffective A possible problem in this method is how to find the signs before the factors or how to judge if the factors in the parentheses are positive or negative integers In order to check if the solution is correct we can always multiply these two factored binomials by distributive law and see if the expanded polynomial is the same as the original quadratic equation

75 21 == xx

The fifth kind of quadratic equations is type 5 Still the methods of completing squares and quadratic formula can be applied for all type 5 equations but there are cases in which factorization can be used for solving this type of equations within a certain range of integers as mentioned before Sometimes procedures of factorizing such equations can be carried out in our minds quickly like type 4 equations but in many cases they can not As a matter of fact finding two binomial factors for a factorable quadratic equation or a factorable second-degree polynomial is a complicated process and requires systematic work In his article published in Mathematics in School Jackman (2005) demonstrates this systematic procedure via factorizing second-degree polynomials A second-degree polynomial can be written as where are all integers By multiplying out the polynomial becomes thus

0002 nene=++ bacbxax

))((2 srxqpxcbxax ++=++ srqpcbaqsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== Writing the factors of a and c on two lines p q times r s Finding the right pairs of factors for a and c has to be done by listing out the groups of products of different factors for example The factor pairs for 6 are (6 1) (1 6) (3 2) (2 3) and the factor pairs for (-4) are (4 -1) (-4 1) (2 -2) (-2 2) (1 -4) (-1 4) Writing them on two lines p 6 1 3 2 q 4 -4 2 -2 1 -1

456 2 minus+ xx

times r 1 6 2 3 s -1 1 -2 2 -4 4 Then comes calculating the values of ps and qr until the value of b = 5 is obtained The calculating procedures are divided into 4 groups and 6 steps in each group which means 4 times 6 steps in total Here I only give one grouprsquos procedure as an example 6 middot (-1) + 4 middot 1 = -2 6 middot 1 + 1middot (-4) = 2 6 middot (-2) + 1 middot 2 = -10 6 middot 2 + 1middot (-2) = 10 6 middot (-4) + 1 middot 1 = -23 6 middot 4 + 1 middot (-1) = 23 helliphellip After the 24ndashstep calculating procedures two computing procedures obtain the same correct result 3 (ndash1) + 4 2 = 5 (yes) 2 4 + (-1) 3 = 5 (yes) Are they the same From 3 (ndash1) + 4 2 = 5 the polynomial can be factorized into )12)(43( minus+ xx while from 2 4 + (ndash1) 3 = 5 the polynomial can be factorized into )43)(12( +minus xx According to communicative law these two binomial products are the same The shortcoming of this

19

method is that it is too ineffective and complicated The addition of matrixes is almost impossible and unbearable Nevertheless based on the same idea I suggest finding factors by guessing analyzing and checking Using the same example here we can find that one of the binomials must have a negative integer because of ndash4 which implies that 5 is the sum of a positive integer and a negative integer Then looking at a and c the factors of 6 can be (1 6) and (2 3) while the factors of 4 can be (1 4) and (2 2) when we regard them as absolute values We know that the factoring form of the polynomial will be like

and the connections of all coefficients are

456 2 minus+ xx

qsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== The coefficient 5=b is the difference of two products of factors for 6 and 4 First we try the biggest number of product such as 2446 =bull then 524 =minus the unknown is which is a prime number the only factors for 19 are (19 1) There is not any combination of products of factors for 6 and 4 satisfying 19 Therefore the multiplications of factors (1 6) and (1 4) are not included in this case So we move to factors (2 2) (2 3) and (1 4)

19 =

In the second step we try The result is negative it is obviously

4264466)23)(22( 22 minusminus=minus+minus=+minus xxxxxxx2 lt 5 which implies that the combination of (2 3) and (2 2) is not

considerable One pair of factors has to be removed and that is (2 2) which does not influence the coefficient of the -term Then we try The result this time is negative too but we can see that

2x 456)43)(12( 2 minusminus=minus+ xxxxb is 5 which guides us to interchanging the positions of

negative and positive signs in the parentheses The next attempt is and it is correct Seeking the correct factors for the polynomial

has taken four major operating steps by trying the combinations of factorsrsquo products and relating the sum of the products with the value of the x-term coefficient The strategy is to make use of distributive law reversely first and then multiply the product of every pair of parentheses in order to check if they are the right pair of polynomial factors Factorizing second-degree polynomials is the major step for solving quadratic equations After factoring polynomials into the product of two binomials as in the example given above the roots can be obtained by making each binomial equal to zero according to the theory of zero point (ldquonollstaumlllerdquo in Swedish) of a polynomial equation originating from unique factorization theorem

456)43)(12( 2 minus+=+minus xxxx456 2 minus+ xx

Factorization by using the technique of vertically and crosswise sutra Binomial expansion or multiplication of binomials by distributive law and factorization of the second-degree polynomials by using distributive law backwards have an extremely symmetrical effect and the results of two kinds of operations are equivalent It is crucial for the students to be able to perceive mathematics structure and be acquainted with arithmetic operations as well as to understand the relationship between quantities when they carry out factoring procedures The expansion of binomials by distributive law has been named as the method FOIL which stands for first outside inside and last according to multiplication orders of two binomials for example (Nataraj and Thomas 2006) Many students are used to this procedure Another technique to multiply two binomials is called vertically and crosswise sutra origin from the ancient Indian Vedic mathematics Vertically and crosswise sutra and FOIL are much the same in principle but it makes use of

824)4)(2( 2 +++=++ xxxxx

20

the power of visualization and improves facility with the expansion of binomials and the factorization of quadratic expressions In this method two binomials are written into two lines Using the same example the two lines are written as 824)4)(2( 2 +++=++ xxxxx

x + 2 x + 4 ______________ xsup2 4x + 2x 8

The order of multiplication is as shown below x + 2 x + 4

4 3 2 1 (the numbers 1 2 3 and 4 refer to steps) xsup2 + 2x + 4x + 8 Figure 3 The vertically and crosswise method of multiplication of binomials Step 2 and 3 can be interchangeable If we put this form into a line expanded according to the order above would be like

)4)(2( ++ xxxbullrarrbull 442 clockwise and xxx bullrarrbull2 in

counterclockwise direction According to sutra we can imagine drawing a thread starting from 2 then going through 4 and x clockwise and then draw the second thread starting from the same place 2 later going through x in the second bracket and finally x in the first bracket counterclockwise Learning expansion of binomials by the methods FOIL and Vertical and clockwise sutra is the basic process before students study factorization In this study Nataraj and Thomas (2006) have found that the students appear to do factorization using guess-and-check as well as decomposition or the vertical and clockwise approach Their finding is that afterwards the students performed better on algebra questions especially on the factorizations They conclude ldquohellipthe values of the method may lie in what it adds to the studentsrsquo overall algebraic conceptions and knowledge of mathematical structurerdquo (p 16) In general solving these five kinds of quadratic equations by factorization can be traced back to unique factorization theorem The important character in this method is that you can always check if factorization is correct or not through multiplying out the two pairs of a parenthesis and see if the expanded polynomial is the given one or not Using factorization is not only about method but also about fostering the studentsrsquo mathematical thinking The solving processes are more important than solutions

21

3 Review of related literature and research Besides the focus on mathematics content another focus in the background work for this research is about previous studies and research in the field of mathematics didactic concerning different methods for solving quadratic equations and the use of factorization In order to study the previous research and studies in the related field three scientific databases and a search engine have been used and 113 articles have been investigated Topics such as solving quadratic expressions and equations are often discussed as teaching ideas and strategies shared by mathematics educators in different countries from lower and upper-secondary schools to colleges according to the findings in the article search These didactic topics from mathematics teachersrsquo perspectives include for example using geometric approaches to solve quadratic equations (Allaire amp Bradley 2001) utilizing completing squares (Vinogradova 2007) factoring quadratics (Hoffman 1976 Cheng 1980) and using factorization completing the square graphical methods to solve quadratic equations (Macdonald 1986) Kostsopoulos (2007) links cognitive reasoning to pedagogical problems appearing in the mathematics classrooms in her article published in Australian Mathematics Teacher Based on her teaching experiences Kostsopoulos is aware of her pedagogical strategies lacking insight on dealing with studentsrsquo difficulties of multiplication and factorization of quadratics She assumes that ldquostudentsrsquo problems with factorization and with identifying varied representations of the same quadratic relationshiprdquo may be linked to the ways in which the brain constructs cognitive representations (p 19) Factoring of quadratics is the rewriting of polynomials as a product of polynomials It requires students to have both a conceptual understanding of multiplication of polynomials and the procedural knowledge to retrieve basic multiplication facts effectively Based on the theory about the order matters in the brainrsquos ability to retrieve number facts (Phenix amp Campbell 2001) Kostsopoulos conjectures that studentsrsquo ability to access the appropriate long-term semantic memory is limited when they are confused by mixing the order and varied forms of quadratics Although this article is not about a research study it seeks the reason why factorizing quadratics is difficult and therefore might provide math teachers with alternative explanations for similar pedagogical frustrations With Von Glasersfeldrsquos radical constructivism and Peter Gaumlrdenforsrsquo epistemic semantics theory as his theoretical frameworks Rauff (1994) suggests that belief-based teaching can be successful in teaching factoring According to Rauff constructivist learning theory takes the view that students construct their own beliefs and knowledge of mathematics Learning occurs when the students change their belief set Referring to belief set theory from Gaumlrdenfors (1988) Rauff (1994) claims that ldquoa belief set is expanded when a new belief is added to it and contracted when a proposition is no longer believed and is removed from the setrdquo (p 421) Expansion contraction and revision are three components to comprise basic mechanisms of changes of belief according to Rauff When examining and comparing studentsrsquo (ages 18-20) own definitions of factoring Rauff analyzes the errors of their definitions with constructivism and epistemic semantics as analyzing tools The result of this study shows that the approach to teaching factoring rely heavily upon what a student believes about factoring is quite productive in two dimensions First it reveals the source of nonstandard factoring Second it provides a starting point for modification of the studentrsquos underlying conceptions of factoring Rauff proposes that belief-based teaching can also be

22

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

bring a term from one side of an equation to the other gather up like terms for simplification and some elementary principles of expansion and factorization (Derbyshire 2006) Diophantus made his own literal symbolism with the use of special letter symbols for the unknown and its powers for subtraction and equality From Diophantus algebra history moved into another conceptual stage the equation-solving stage according to Katz (2006) In India quadratic formula appeared without any geometric support Brahmaggupta (598-665) was one of the first mathematicians who could systematically handle negative numbers and zero He gave a general solution to quadratic equations and realized that there were two roots for a quadratic equation It was possible that one of the roots was a negative number Baskharacharya (1114-1185) solved mathematics problems with the use of quadratic equations in his book Siddhanta Siromani (ldquoMathematical Pearlsrdquo) He presented an algorithm to reduce a quadratic equation to a first-degree equation (Olteanu 2007) It is commonly believed that the first true algebra text is the work on al-jabr and al-muqabala by Mohanmmad ibn Musa al-Khwarizmi (780-850) written in Baghdad around 825 (Katz 2006) The word algebra came from the title of this work The word al-jabr means restoration or reestablishment that is to eliminate negative terms through adding the same terms to both sides of equations The word of al-muqabalas means balance meaning to divide every term in a quadratic equation by the coefficient of the second degreersquos term (Olteanu 2007) The first part of his book is a manual for solving linear and quadratic equations Al-Khwarizimi classified equations into six types three of which were mixed quadratic equations For each type he presented an algorithm for its solution Five of the six types of equations were quadratic equations which can be expressed in modern form

cbxaxbxcaxcbxaxcaxbxax +==+=+== 22222 Here is an example of solving the equation ldquoTake the half of the number of the things that is five and multiply it by itself you obtain twenty-five Add this to thirty-nine you get sixty-four Take the square root or eight and subtract from it one half of the number of things which is five The result three is the thingrdquo (Kvasz 2006 p 292) Like Babylonian mathematicians al-Khwarizimirsquos algorithm is entirely verbal On the other hand al-Khwarizimi sometimes completed his algorithm by using geometrical explanations which can be translated as todayrsquos square-completing method Using the example of solving the quadratic equation the completed geometrical procedures are illustrated on next page in figures 3 4 and 5

39102 =+ xx

39102 =+ xx

8

x

xsup2

Figure 3 (Olteanu 2007 p 30) 5x2

25x

xsup2

25x

5x2

Figure 4 (Olteanu 2007 p 30) 254 254

39

254 254

Figure 5 (Olteanu 2007 p 30) According to Olteanursquos (2007) translation al-Khwarizimi started with a square whose sides are x and area is xsup2 (see Figure 3) Then he added four equal rectangles whose areas in total

was 10x along each side of the square that is xx sdotsdot=25410 Each rectanglersquos area is thus x

25

with its length x and its width 52 (see Figure 4) The sum of the big square and four rectangles was given which was 39 The equivalence relationship was Finally figure 4 was completed by adding four small equal squares which had an area the size of

39102 =+ xx

425

25

25

=sdot for each small square and the sum of those was 25 Through adding this sum to

both sides of the equation the area of the biggest square obtained in figure 5 was 64 The

equation translation is 425439

4254102 sdot+=sdot++ xx The side of the biggest square was 8 and

had its relation with other sides of different squares expressed in the first degree equation

9

25

258 ++= x Then x was 3 By ldquocut-and-pasterdquo geometry (Katz 2006 p 191)

al-Khwarizimi reduced the second degree of an equation to the first degree and thereafter solved it Unlike his Babylonian predecessors al-Khwarizimi always presented his problem abstractly rather than geometrically relating to lengths and widths The symbolic stage At this stage of algebra ldquoall numbers operations relationships are expressed through a set of easily recognized symbols and manipulations on the symbols take place according to well-understood rulesrdquo (Katz 2006 p 186) The ancient algebra and geometry had developed sophisticatedly in Egypt Persia Greece India and China After Medieval Islamic scholars gave us the word ldquoalgebrardquo Western Europe began the struggle for the development of algebra starting from some algebraists from Italy Italian mathematician Leonardo Pisano later known as Fibonacci traveled in the 12th and 13th centuries to Persia India and China When he returned to Italy he had wider knowledge of arithmetic and algebra His book Liber abbaci was the best math textbook since the end of Ancient world His book is credited with having introduced Indian numerals including zero to the West But his algebraic skills had been shown in two other works after this one With the introduction of printed books during the second half of the 15th century the development of algebra was sped up Several Italian mathematicians including Girolamo Cardano had figured out how to solve cubic and quadratic equations Algebra became purely abstract with the exception of an English mathematician named Robert Recorde who lived in the 16th century and created quadratic problems from real world experience (Derbyshire 2006) It was in France that algebra had developed into a well organized literal symbolism In his work In artem analyticem isagoge French mathematician Franςois Viegravete (1540-1603) in the late 16th century became the first mathematician to use letters representing numbers systematically and effectively (Derbyshire 2006) He made a range of letters available for many different quantities This was the beginning of modern literal symbolism Viegravetersquos unknown quantity was divided into two classes unknown quantities (meaning ldquothings soughtrdquo) denoted by A E I O U and Y while ldquothings givenrdquo was denoted by constants like B C Dhellip For example his A is our unknown x Viegravete was a pioneer in the study of equations His two papers on the theory of equations were published twelve years after his death In the second paper titled ldquoDe equationem emendationerdquo (ldquoOn the perfecting of equationsrdquo) Viegravete opened up the line of inquiry that led to the study of the symmetries of an equationrsquos solutions to Galois theory the theory of groups and all of modern algebra He found the relationship between the solutions of the equation and the coefficients for the first five degrees of equations in a single unknown To explain this in our modern symbols we suppose that the two solutions of the quadratic equation are 02 =++ qpxx α and β which means βα == xx Based on this logic the following thing must be true 0))(( =minusminus βα xx since only α andβ and no other values of x make this equation true This form of equation is just a rewritten form of the same equation If we multiply out those parentheses this rewritten equation turns to be Compared to the original equation the relationships between the solutions and the coefficients we obtain

0)(2 =++minus αββα xxqp =minus=+ αββα

(Derbyshire 2006) It is said that Viegravete discovered the solution formula called quadratic

10

formula today for general quadratic equations which is a

acbbx2

42

21minusplusmnminus

= for a general

quadratic equation (Olteanu 2007) 002 ne=++ acbxax Another French mathematician and philosopher who had strong influence in the history of algebra was Reneacute Descartes (1596-1650) His idea to use the Cartesian system of coordinates which was derived from his Latin name developed both algebra and geometry In his work La geacuteomeacutetrie written in 1637 Descartesndashthrough using numbers to indentify points in a Cartesian coordinatesndashconnected geometrical objects to algebraic numbers and made the classical geometry become analytical geometry He took up the plus and minus sign from the German Cossists of the previous century and also the square-root sign From Descartes the symbol of an unknown was represented by x which led to the modern system of literal symbolism (Derbyshire 2006) Descartes developed the idea of functions in Cartesian coordinates although the idea of function can be traced back to the Islamic mathematician Sharaf al-Din al-Tusi from Persia (Katz 2006) and Klaudius Ptolemaios about 2000 years ago (Olteanu 2007) Descartes declared that every curve in a Cartesian coordinate system has an equivalent equation which can represent the points on the curve or vice versa every equation containing x and y can be represented by a curve through its coordinate points However the word function was introduced by Gottfried Wilhelm Leibniz in 1693 and the definition of function was defined by Leonhard Euler in his work Introductio analysin infinitorium in 1748 Euler had also given the symbol of a denotation as a relying variable (Olteanu 2007) In the 18)(xf th century after the discovery of calculus by Newton and Leibniz algebra went into an area of analysis ndashthe study of limits infinite sequences and series functions derivatives and integrals (Derbyshire 2006) The purely abstract stage-algebra structure Since 17th century algebra has not anymore solely been about finding solutions for different degrees of equations mathematicians have started to integrate algebra with astronomy and physics Johann Kepler and Galileo Galilei were interested in curves and finding a mechanism for representing motion instead of quantity-like numbers But their arguments were not algebraic-symbolic but geometrical It was Fermat and Descartes who were regarded as the fathers of analytic geometry and who showed how to represent a curve described verbally through algebra analytic geometry and gave a mechanism for representing motion Newton in his Principia picked up on Fermat and Descartesrsquo representation and developed the calculus With the invention of the calculus mathematical problems were solved by curves not just points Algebra grew more and more to present paths of motion In order to be able to judge if the algebraic manipulations were correct axioms were formulated for arithmetic applied for algebraic manipulations In the late 18th century Lagrange introduced the idea of permutations into the search for solutions Therafter Galois developed methods for determining what conditions polynomial equations need to meet to be solvable New abstract algebraic concepts were found like ldquofieldrdquo and ldquogrouprdquo during the early 19th century In 1854 Ceyley gave an axiomatic definition of a group During the 1890s this definition entered textbooks along with the axiomatic definition of a field an idea which had roots in the work by Galois By the beginning of the 20th century algebra became less about finding solutions to equations and more about looking for common structures in many mathematical objects defined by sets of axioms (Katz 2006)

11

In short algebra development took 4000 years It took about 3500 years for mathematicians to start to use algebra symbols in systems Just almost 100 years ago algebra developed into a purely abstract science The long history may give both mathematics educators and students some explanations to why it is so difficult to teach and learn algebra

22 Three solving methods for quadratic equations Already in the previous part I have presented geometrically and rhetorically solving quadratic equations from a historical perspective In order to get more explicit understanding of quadratic equations quadratic formula and factorization from mathematics didactic point of view this part presents three common methods for solving quadratic equations 1) Completing the square 2) Factorization 3) Quadratic formula The reason to present these three methods is that they are the common topics for discussion in the articles searched in this area concerning mathematics education at upper-secondary school 1) Completing the square In two articles Solving quadratic equations by completing squares (Vinogradova 2007) and Geometric approaches to quadratic equations from other times and places (Allaire amp Bradley 2001) the authors advise mathematics teachers to use the ideas of ancient mathematician al-Khwarizmi and his method of completing quadratic equations by geometrical algebra and simplifying it in order to represent it to the students By doing so the teachers relate mathematics quantity to physical objects in a visual way Didactically teachers may start with a concrete example in which students can understand the content visually Later from the concrete example the teachers lead the students to another example expressed in algebraic symbols for example to study a rectangle whose area is )10( +xx and supposing this area is 39 this becomes 39)10( =+xx Four steps are followed to build a new square which is actually called completing the square x 10

Figure 6 x 5

Figure 7

x

x

5 5

12

A Begin with a rectangle of the area )10( +xx that is with the short side as x and the long side )10( +x The shaded area is 10bullx (Figure 6)

B Divide the shaded rectangle into two small shaded rectangles with the size 5x each and move them to each adjacent side of the xsup2 square (Figure 7) The total area is still

x10

)10(39 += xxC In Figure 7 a new square is shaped in the lower-right hand corner with the side 5 or

area 25 By adding this small square to the diagram the large square with an area of is ldquocompletedrdquo Therefore we can write 25)5(22 ++ xx 25)10(2539 ++=+ xx or

The large squarersquos area is now because the side has a length of Therefore

6425)5(22 =++ xx 2)5( +x5+x 38564)5( 2 =rArr=+rArr=+ xxx

After these three steps a solution to this quadratic equation is derived by completing the square (Allaire amp Bradley 2001) 2) Factorization Hoffman (1976) presents three approaches (A B and C) for factorizing quadratic expressions through observing and grouping He takes a second-degree polynomial as an example

273 2 ++ xx

a The inspection method 1) suggests factors of and 23x x3 x 2) 2 suggests factors of 1 and 2 or ndash1 and ndash2 3) Checking the linear term shows that the correct factorization is x7 )2)(13(273 2 ++=++ xxxx

This method according to Hoffman (1976) is based on the studentrsquos ability to simplify by inspecting the indicated product of a pair of linear expressions

b The method of decomposition of the linear term 1) Multiply and 2 to get 23x 26x2) Decompose into the sum of two terms whose product is x7 26x

This gives xxx += 67 3) Factorize by grouping 2)6(3 2 +++ xxx

2)6(3 2 +++ xxx = )

)2()63( 2 +++ xxx

= 2(1)2(3 +++ xxx= )13)(2( ++ xx

c Third method 1) Multiply 3 and 2 to get 6 2) Find two numbers whose product is 6 and whose sum is 7 The numbers are 6 and 1 3) 273 2 ++ xx

=3

)13)(63( ++ xx

= )13)(2( ++ xx

13

These three approaches can not actually be divided as three unrelated approaches especially the third method is seemingly unclear What Hoffman (1976) wants to demonstrate is that all the three approaches are based on observing the relationship among the coefficients and the constant of the polynomial Through finding the product and the sum of two factors the polynomial can be factorized in the product of two binomials In the authorrsquos opinion factorization is heavily dependent on skill in multiplication and using distributive law reversely He means that ldquostudents who do not have reasonably strong skills in multiplication should not be expected to develop strong skills in factorizationrdquo (p 55) Factorizing polynomials is not only for simplifying high-degree polynomials (for example the third or fourth-degree polynomials) but can also be applied for solving quadratic equations Supposing this polynomial forming a quadratic equation Solving this equation is actually to decompose the polynomial into a factoringrsquos form

0273 2 =++ xx0)13)(2( =++ xx

which is equivalent to the polynomial form The essential step to solve this equation is to follow the zero-factor propertyndashalso called null-factor law (nollprodukt in Swedish)ndashthat is for real numbers p and q p bull q = 0 if (and only if) p = 0 or q = 0 It means that either of the binomials on the left side of the equation has to be equal to zero in order to satisfy the

equivalent relation of this quadratic equation The solutions of two roots or 2minus=x31

minus=x

are obtained through making either 02 =+x or 013 =+x 3) Quadratic formula Solving quadratics equations by factorization is constrained within the simple quadratic equations over the whole integers and rational numbers as coefficients What happens when quadratic equations have coefficients that belong to an irrational domain or big quantity The quadratic formula is a solution to such a situation As mentioned previously historically the French mathematician Franccedilois Vieacutete discovered the quadratic formula

aacbbx

242 minusplusmnminus

= ) which is applicable for all kinds of quadratic equations In

Swedish textbooks this formula is written in PQ form by making for the equation

that is

0( nea

1=a

02 =++ qpxx qppx minus⎟⎠⎞

⎜⎝⎛plusmnminus=

2

22 with

abp = and

acq = In Oleatursquos study (2007)

studentsrsquo difficulties in using algebraic symbols no longer is the problem But on the other hand handling the parameters or coefficients in quadratic equations like and rewrite them in the equivalent form with p and q being real numbers becomes obstacles for the students when they study the algebra course at upper-secondary school

02 =++ cbxax02 =++ qpxx

The quadratic formula a

acbbx2

42 minusplusmnminus= has been regarded as the standard method

according to US mathematics teachers Obermeyer (1982) presents a way of how to derive this formula by completing a square through 11 steps (I will not present them here) At the same time an alternate method is also introduced according to the same principle All the methods mentioned so far are based on the square rules The standard method or PQ form might be regarded as an efficient and direct method but it may

222 2)( bababa +plusmn=plusmn

14

lead students to solve quadratic equations in a mechanical way besides having problems with the troublesome plusmn symbolism (Stover 1978) Both Stover (1978) and Olteanu (2007) have suggested that using a graph of quadratic functions to solve quadratic equations is another alternative method

23 Mathematical background of factorization Factorization in algebra structure Among the three solving methods presented above both the geometrical method and quadratic formula can be traced back to algebra history in section 21 What is the trace of factorization in algebra history Factorization was at the last stage in algebra history and belonged to algebra structure (Derbyshire 2006) In Swedish mathematics textbooks the chapter on algebra often includes polynomial factorization and quadratic equations But the factorization in the textbooks is not at the same abstract level as the factorization of algebra structure although they share the same axioms and there are certain connections between these two different levels of factorization How are polynomial and quadratic equations as well as factorization related to each other In order to find the theoretical background of factorizing polynomials this part constitutes a mathematical theoretical review by referring to two books concerning algebra structure (Vretblad 2000 Durbin 1992) and a research study (Bosseacute and Nandakumar 2005) A quadratic equation can be regarded as polynomial equation Solving quadratic equations has much to do with ldquounique factorization theoremrdquo The polynomial

is a second-degree polynomial A polynomial consists of coefficients in whole numbers rational numbers real numbers or complex numbers and variables (sometimes called unknowns) in different degrees as well as constants A polynomial

has symbols as coefficients which are real numbers If all are zero and so

)0(02 ne=++ acbxax

cbxax ++2

nnxaxaxaaxf ++++= )( 2

210 naaa 10

ka 1gek 0)( axf = is constant If all are zero then we say the is a zero polynomial If meaning that is not a zero polynomial and n is the biggest number and then we say that the polynomial is at degrees A zero polynomial has no degree at all According to the definition (Vredblad 2000) if is a polynomial the equation is called an algebraic equation or polynomial equation A solution or a root of this equation is a number

kaf 0nef f

0nena f nf

0)( =xfα so that 0)( =αf Such a number is even

called a zero point (nollstaumllle in Swedish) of the polynomial which means when f α=x the value of the polynomial is zero (p 161) Every polynomial whose degree is equal or greater than 1 can be written as a product of irreducible polynomial In this way through polynomial division a polynomial with n degrees can be expressed from a sum of terms into a product of its denominator and its quotient The process of changing the expressions is factorization for

instance 12

22

+=minusminusminus x

xxx The factoring form is )1)(2(22 minusminus=minusminus xxxx

Factorization of polynomials is analogue to factorization of whole numbers which is very much related to the concept of ring A ring consists of a set with two operations which are a sum and a product of two elements over a field Any polynomial from zero polynomial to n degree-polynomials can be written or operated through addition and multiplication of the

15

elements over a field F including whole numbers or integers rational numbers real numbers and complex numbers In a ring you can multiply any two elements but you canrsquot always divide Factorization is a way to deal with this fact Here are some examples In the ring of integers Z is an example of factorization in some polynomial rings for instance Q[x] the ring of polynomial in one variable with rational numbers as coefficients such as

7214 sdot=

)451)(2

51(8

52

251 2 +minus=minus+ xxxx is factorization Z[x] the ring of polynomial in one

variable with whole numbers as integers such as is factorization )1)(2(22 minusminus=minusminus xxxx The polynomial ring R[x] means the ring of polynomial in one variable with real numbers as coefficients Factorizing of such polynomials implies that the polynomial factors are the elements in the ring of R[x] for instance )2)(2(22 minus+=minus xxx The same thing is true for C[x] the ring of polynomial in one variable with complex numbers as coefficients Thus it is coefficients of polynomials that determine which ring a polynomial belongs to and if it is possible to be factorized in this ring For example number 6 can be factorized in two ways

in the ring of Z and 32times )51)(51( ii +minus in the ring of C (complex numbers) However there are cases in which divisions can not be done within the ring for example 52 or

)3()52( 2 ++minus xxx which means that no element in the ring can multiply with a denominator obtaining the numerator Therefore neither 5 nor can be factorized 522 +minus xx If a polynomial of degree at least one has no other divisors then it is considered to be irreducible or prime (Durbin 1992) Of every polynomial of degree at least one can be written as a product of irreducible polynomials according to unique factorization theorem each polynomial of degree at least one over a field F can be written as an element of F times a product of monic irreducible polynomials over F and except for the order in which these irreducible polynomials are written this can be done in only one way (p 221) Thus a factorable polynomial can be written into a product with another polynomial such as In other words the condition that )()()( xqaxxf minus= )( ax minus is a factor of a polynomial

in a variable x is only true if )(xf 0)( =af For example if and

6)( 2 minus+= xxxf0624)2( =minus+=f 0639)3( =minusminus=minusf the factors of are and )(xf )2( minusx )3( +x

The value of the polynomial is zero which is also expressed as the polynomial having a zero point (nollstaumllle in Swedish) when

)(xfα=x

Based on unique factorization theorem finding solutions of an algebraic equation is equivalent to finding the first-degree factors for a polynomial (Vredblad 2000) In this way solving a quadratic equation can be handled through finding the first-degree factors for the second-degree polynomial and there after finding the roots of the quadratic equation This is often called using factorization to solve quadratic equations Besides unique factorization theorem the connection of using factorization to solve a quadratic equation is that this method relates to prime numbers using distributive law backwards commutative law multiplication division and operating fractions As matter of fact it is about application of factorizing integers in polynomials at upper-secondary school level Since quadratic equations in the mathematics B course at Swedish upper-secondary school are limited to real numbers and often within the whole numbers (integers) many quadratic equations should be able to be solved by factorization according to my own

16

judgment Therefore using factorization for solving quadratic equations is very much related to arithmetic operations number concept and algebra structure It may offer students both mathematical conceptual understanding and operational skill For algebra beginners factorization can be traced back to studentsrsquo early years of mathematics education during which students do not only work with multiplication but also on factoring integers for example 56 = 8 middot 7 = 2 middot 2 middot 2 middot 7 = 4 middot 14 = 2 middot 28 In this example two operations are included multiplication and division Multiplication operation may help students to understand the multiplicative structure of the integers while factoring integers may become meaningful when it is related to divisibility divisors and prime numbers These two operational skills or competence are major techniques for simplifying fractions and finding common denominators Being able to understand and operate factoring integers has laid the essential grounds for algebra beginners to study factorization of polynomials Thus factorization reunites studentsrsquo early mathematics knowledge of factoring integers with later knowledge of algebra structure Factorability In a study on factorability by Bosseacute and Nandakumar (2005) the probability of factorability is reasoned through investigating the range of integers as coefficients of quadratic equations In the quadratic equation a b and c are randomly generated integers within a determined range r such as

002 ne=++ acbxaxyrx lele where x and y are integers When is a

perfect square or where

acb 42 minus

)4( 2 acb minus is an integer quadratics are factorable According to the studyrsquos data it shows that as the range for a b and c increases the probability of factorability of a quadratic with randomly selected integer values for a b and c decreases Bosseacute and Nandakumar confirm that as the range for coefficients expands to infinltltinfinminus r the probablility of factorability of a quadratic with randomly selected integer values for coefficients approaches zero The studyrsquos data collected from college algebra courses and textbooks demonstrates that about 15 of quadratics with integer coefficients [ ]1010minusisinr are factorable Thus about 85 of quadratics can not be factorized In spite of the limitation of factorability within the exercises concerning factoring practice chosen from 27 surveyed college algebra textbooks about 94 of the problems were factorable Within the textbooks 55 of the quadratics to be factored were within the range [-10 10] Even if many quadratic equations are factorable choosing the right factor pairs is time consuming and unnecessary for example

has to be factorized through choosing the right pairs of factors among nine pairs for 36 and 8 pairs of factors for 24 Comparing the three different methods for solving quadratic equations the study suggests that completing the square and quadratic formula are more effective informative and useful than factorization

0245936 2 =++ xx

231 Different kinds of factorizations for solving different quadratic equations How can we find the binomial factors of a quadratic equation or a second-degree polynomial without knowing one of the divisors as a binomial factor The answer is that we have to observe the relations among all coefficients and constants as well as the positive and negative signs before them under the condition of being factorable Most of the coefficients in quadratic equations or polynomials among exercises provided for practicing factoring are integers (here I borrow the same notation r as the one used by Bosseacute and Nandakumar) with

17

their range between ndash10 and 10 that is 10ler In this case it is possible for a student to make a judgment of whether or not a quadratic equation is factorable Five different kinds of factorization depending on five kinds of quadratic equations Factoring quadratic expressions can be divided into five different kinds depending on the type of quadratic equations The coefficients a b p q and constant c illustrated in the following quadratic equations are defined not to be equal to zero thus 000 nenene cba The first kind of quadratic equations is type 1 the obviously factorable type (Bosseacute and Nandakumar 2004) then the equation is factorized as

where the roots are

0002 nene=+ babxax

0)( =+ baxxabxx minus== 21 0

The second kind of quadratic equations is type 2 The factors of this second degree polynomial are so the roots to this type of quadratic

equations are

0)( 22 =minusbax))(()( 22 baxbaxbax minus+=minus

bax plusmn=21 This kind of factorization is based on the difference-of-squares

formula for example can be solved by factorizing into 049 2 =minusx 0)23)(23( =minus+ xx Thus

the roots are 32

32

21 =minus= xx This kind of factorization is also obviously factorable

The third kind of quadratic equations is type 3 This type of equations can be solved directly by factoring into two identical binomials

The double roots obtained from this

factorization are

02)( 22 =+plusmn babxax

0))(()(2)( 222 =plusmnplusmn=plusmn=+plusmn baxbaxbaxbabxax

abx plusmn=21 This method has utilized square rules which are quite easily

observed The forth kind of quadratic equations is type 4 In this equation the coefficient of the -term is 1 with p and q as positive or negative integers Type 4 equations can always be solved by the approaches of completing the square and quadratic formula However they can often be solved by factorization too since the quadratic expressions are factorable by using distributive laws reversely The roots are obtained through finding two factors for q At the same time these two factors become satisfying with the relation so that the sum of factors is equal to That way the two factors become two roots of a quadratic equation Denoting the two factors or two roots as the relationship between coefficient and constant is

02 =++ qpxx2x

qminus

21rr)( 2121 rrprrq +minus=bull= Quadratic equations can be written as the product of

two first-degree binomials or factoring form 0))(()( 212121

22 =minusminus=++minus=++ rxrxrrxrrxqpxx Through the factoring form the roots of this equation are obtained being The core of factorizing type 4 equations is seeking the connections between coefficients and roots which reveal the studentsrsquo basic ability of arithmetic multiplication and addition It is often more effective in this case to use factorization than to use quadratic formula or completing the square Let us have a look at the example In this equation we get

2211 rxrx ==

035122 =+minus xx

18

35)7)(5(12)7()5( =minusminus=minus=minus+minus= qp or 75)75( bull=bullminus= qp (here we can directly get roots) Factorizing the equation we obtain 0)7)(5( =minusminus xx and roots are The procedures of finding factors of the equation often occur in some studentsrsquo minds quickly without writing down the whole computing process but those students often have good skills in arithmetic operations (reference) Compared to using factorization for solving this equation application of quadratic formula seems ineffective A possible problem in this method is how to find the signs before the factors or how to judge if the factors in the parentheses are positive or negative integers In order to check if the solution is correct we can always multiply these two factored binomials by distributive law and see if the expanded polynomial is the same as the original quadratic equation

75 21 == xx

The fifth kind of quadratic equations is type 5 Still the methods of completing squares and quadratic formula can be applied for all type 5 equations but there are cases in which factorization can be used for solving this type of equations within a certain range of integers as mentioned before Sometimes procedures of factorizing such equations can be carried out in our minds quickly like type 4 equations but in many cases they can not As a matter of fact finding two binomial factors for a factorable quadratic equation or a factorable second-degree polynomial is a complicated process and requires systematic work In his article published in Mathematics in School Jackman (2005) demonstrates this systematic procedure via factorizing second-degree polynomials A second-degree polynomial can be written as where are all integers By multiplying out the polynomial becomes thus

0002 nene=++ bacbxax

))((2 srxqpxcbxax ++=++ srqpcbaqsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== Writing the factors of a and c on two lines p q times r s Finding the right pairs of factors for a and c has to be done by listing out the groups of products of different factors for example The factor pairs for 6 are (6 1) (1 6) (3 2) (2 3) and the factor pairs for (-4) are (4 -1) (-4 1) (2 -2) (-2 2) (1 -4) (-1 4) Writing them on two lines p 6 1 3 2 q 4 -4 2 -2 1 -1

456 2 minus+ xx

times r 1 6 2 3 s -1 1 -2 2 -4 4 Then comes calculating the values of ps and qr until the value of b = 5 is obtained The calculating procedures are divided into 4 groups and 6 steps in each group which means 4 times 6 steps in total Here I only give one grouprsquos procedure as an example 6 middot (-1) + 4 middot 1 = -2 6 middot 1 + 1middot (-4) = 2 6 middot (-2) + 1 middot 2 = -10 6 middot 2 + 1middot (-2) = 10 6 middot (-4) + 1 middot 1 = -23 6 middot 4 + 1 middot (-1) = 23 helliphellip After the 24ndashstep calculating procedures two computing procedures obtain the same correct result 3 (ndash1) + 4 2 = 5 (yes) 2 4 + (-1) 3 = 5 (yes) Are they the same From 3 (ndash1) + 4 2 = 5 the polynomial can be factorized into )12)(43( minus+ xx while from 2 4 + (ndash1) 3 = 5 the polynomial can be factorized into )43)(12( +minus xx According to communicative law these two binomial products are the same The shortcoming of this

19

method is that it is too ineffective and complicated The addition of matrixes is almost impossible and unbearable Nevertheless based on the same idea I suggest finding factors by guessing analyzing and checking Using the same example here we can find that one of the binomials must have a negative integer because of ndash4 which implies that 5 is the sum of a positive integer and a negative integer Then looking at a and c the factors of 6 can be (1 6) and (2 3) while the factors of 4 can be (1 4) and (2 2) when we regard them as absolute values We know that the factoring form of the polynomial will be like

and the connections of all coefficients are

456 2 minus+ xx

qsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== The coefficient 5=b is the difference of two products of factors for 6 and 4 First we try the biggest number of product such as 2446 =bull then 524 =minus the unknown is which is a prime number the only factors for 19 are (19 1) There is not any combination of products of factors for 6 and 4 satisfying 19 Therefore the multiplications of factors (1 6) and (1 4) are not included in this case So we move to factors (2 2) (2 3) and (1 4)

19 =

In the second step we try The result is negative it is obviously

4264466)23)(22( 22 minusminus=minus+minus=+minus xxxxxxx2 lt 5 which implies that the combination of (2 3) and (2 2) is not

considerable One pair of factors has to be removed and that is (2 2) which does not influence the coefficient of the -term Then we try The result this time is negative too but we can see that

2x 456)43)(12( 2 minusminus=minus+ xxxxb is 5 which guides us to interchanging the positions of

negative and positive signs in the parentheses The next attempt is and it is correct Seeking the correct factors for the polynomial

has taken four major operating steps by trying the combinations of factorsrsquo products and relating the sum of the products with the value of the x-term coefficient The strategy is to make use of distributive law reversely first and then multiply the product of every pair of parentheses in order to check if they are the right pair of polynomial factors Factorizing second-degree polynomials is the major step for solving quadratic equations After factoring polynomials into the product of two binomials as in the example given above the roots can be obtained by making each binomial equal to zero according to the theory of zero point (ldquonollstaumlllerdquo in Swedish) of a polynomial equation originating from unique factorization theorem

456)43)(12( 2 minus+=+minus xxxx456 2 minus+ xx

Factorization by using the technique of vertically and crosswise sutra Binomial expansion or multiplication of binomials by distributive law and factorization of the second-degree polynomials by using distributive law backwards have an extremely symmetrical effect and the results of two kinds of operations are equivalent It is crucial for the students to be able to perceive mathematics structure and be acquainted with arithmetic operations as well as to understand the relationship between quantities when they carry out factoring procedures The expansion of binomials by distributive law has been named as the method FOIL which stands for first outside inside and last according to multiplication orders of two binomials for example (Nataraj and Thomas 2006) Many students are used to this procedure Another technique to multiply two binomials is called vertically and crosswise sutra origin from the ancient Indian Vedic mathematics Vertically and crosswise sutra and FOIL are much the same in principle but it makes use of

824)4)(2( 2 +++=++ xxxxx

20

the power of visualization and improves facility with the expansion of binomials and the factorization of quadratic expressions In this method two binomials are written into two lines Using the same example the two lines are written as 824)4)(2( 2 +++=++ xxxxx

x + 2 x + 4 ______________ xsup2 4x + 2x 8

The order of multiplication is as shown below x + 2 x + 4

4 3 2 1 (the numbers 1 2 3 and 4 refer to steps) xsup2 + 2x + 4x + 8 Figure 3 The vertically and crosswise method of multiplication of binomials Step 2 and 3 can be interchangeable If we put this form into a line expanded according to the order above would be like

)4)(2( ++ xxxbullrarrbull 442 clockwise and xxx bullrarrbull2 in

counterclockwise direction According to sutra we can imagine drawing a thread starting from 2 then going through 4 and x clockwise and then draw the second thread starting from the same place 2 later going through x in the second bracket and finally x in the first bracket counterclockwise Learning expansion of binomials by the methods FOIL and Vertical and clockwise sutra is the basic process before students study factorization In this study Nataraj and Thomas (2006) have found that the students appear to do factorization using guess-and-check as well as decomposition or the vertical and clockwise approach Their finding is that afterwards the students performed better on algebra questions especially on the factorizations They conclude ldquohellipthe values of the method may lie in what it adds to the studentsrsquo overall algebraic conceptions and knowledge of mathematical structurerdquo (p 16) In general solving these five kinds of quadratic equations by factorization can be traced back to unique factorization theorem The important character in this method is that you can always check if factorization is correct or not through multiplying out the two pairs of a parenthesis and see if the expanded polynomial is the given one or not Using factorization is not only about method but also about fostering the studentsrsquo mathematical thinking The solving processes are more important than solutions

21

3 Review of related literature and research Besides the focus on mathematics content another focus in the background work for this research is about previous studies and research in the field of mathematics didactic concerning different methods for solving quadratic equations and the use of factorization In order to study the previous research and studies in the related field three scientific databases and a search engine have been used and 113 articles have been investigated Topics such as solving quadratic expressions and equations are often discussed as teaching ideas and strategies shared by mathematics educators in different countries from lower and upper-secondary schools to colleges according to the findings in the article search These didactic topics from mathematics teachersrsquo perspectives include for example using geometric approaches to solve quadratic equations (Allaire amp Bradley 2001) utilizing completing squares (Vinogradova 2007) factoring quadratics (Hoffman 1976 Cheng 1980) and using factorization completing the square graphical methods to solve quadratic equations (Macdonald 1986) Kostsopoulos (2007) links cognitive reasoning to pedagogical problems appearing in the mathematics classrooms in her article published in Australian Mathematics Teacher Based on her teaching experiences Kostsopoulos is aware of her pedagogical strategies lacking insight on dealing with studentsrsquo difficulties of multiplication and factorization of quadratics She assumes that ldquostudentsrsquo problems with factorization and with identifying varied representations of the same quadratic relationshiprdquo may be linked to the ways in which the brain constructs cognitive representations (p 19) Factoring of quadratics is the rewriting of polynomials as a product of polynomials It requires students to have both a conceptual understanding of multiplication of polynomials and the procedural knowledge to retrieve basic multiplication facts effectively Based on the theory about the order matters in the brainrsquos ability to retrieve number facts (Phenix amp Campbell 2001) Kostsopoulos conjectures that studentsrsquo ability to access the appropriate long-term semantic memory is limited when they are confused by mixing the order and varied forms of quadratics Although this article is not about a research study it seeks the reason why factorizing quadratics is difficult and therefore might provide math teachers with alternative explanations for similar pedagogical frustrations With Von Glasersfeldrsquos radical constructivism and Peter Gaumlrdenforsrsquo epistemic semantics theory as his theoretical frameworks Rauff (1994) suggests that belief-based teaching can be successful in teaching factoring According to Rauff constructivist learning theory takes the view that students construct their own beliefs and knowledge of mathematics Learning occurs when the students change their belief set Referring to belief set theory from Gaumlrdenfors (1988) Rauff (1994) claims that ldquoa belief set is expanded when a new belief is added to it and contracted when a proposition is no longer believed and is removed from the setrdquo (p 421) Expansion contraction and revision are three components to comprise basic mechanisms of changes of belief according to Rauff When examining and comparing studentsrsquo (ages 18-20) own definitions of factoring Rauff analyzes the errors of their definitions with constructivism and epistemic semantics as analyzing tools The result of this study shows that the approach to teaching factoring rely heavily upon what a student believes about factoring is quite productive in two dimensions First it reveals the source of nonstandard factoring Second it provides a starting point for modification of the studentrsquos underlying conceptions of factoring Rauff proposes that belief-based teaching can also be

22

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

x

xsup2

Figure 3 (Olteanu 2007 p 30) 5x2

25x

xsup2

25x

5x2

Figure 4 (Olteanu 2007 p 30) 254 254

39

254 254

Figure 5 (Olteanu 2007 p 30) According to Olteanursquos (2007) translation al-Khwarizimi started with a square whose sides are x and area is xsup2 (see Figure 3) Then he added four equal rectangles whose areas in total

was 10x along each side of the square that is xx sdotsdot=25410 Each rectanglersquos area is thus x

25

with its length x and its width 52 (see Figure 4) The sum of the big square and four rectangles was given which was 39 The equivalence relationship was Finally figure 4 was completed by adding four small equal squares which had an area the size of

39102 =+ xx

425

25

25

=sdot for each small square and the sum of those was 25 Through adding this sum to

both sides of the equation the area of the biggest square obtained in figure 5 was 64 The

equation translation is 425439

4254102 sdot+=sdot++ xx The side of the biggest square was 8 and

had its relation with other sides of different squares expressed in the first degree equation

9

25

258 ++= x Then x was 3 By ldquocut-and-pasterdquo geometry (Katz 2006 p 191)

al-Khwarizimi reduced the second degree of an equation to the first degree and thereafter solved it Unlike his Babylonian predecessors al-Khwarizimi always presented his problem abstractly rather than geometrically relating to lengths and widths The symbolic stage At this stage of algebra ldquoall numbers operations relationships are expressed through a set of easily recognized symbols and manipulations on the symbols take place according to well-understood rulesrdquo (Katz 2006 p 186) The ancient algebra and geometry had developed sophisticatedly in Egypt Persia Greece India and China After Medieval Islamic scholars gave us the word ldquoalgebrardquo Western Europe began the struggle for the development of algebra starting from some algebraists from Italy Italian mathematician Leonardo Pisano later known as Fibonacci traveled in the 12th and 13th centuries to Persia India and China When he returned to Italy he had wider knowledge of arithmetic and algebra His book Liber abbaci was the best math textbook since the end of Ancient world His book is credited with having introduced Indian numerals including zero to the West But his algebraic skills had been shown in two other works after this one With the introduction of printed books during the second half of the 15th century the development of algebra was sped up Several Italian mathematicians including Girolamo Cardano had figured out how to solve cubic and quadratic equations Algebra became purely abstract with the exception of an English mathematician named Robert Recorde who lived in the 16th century and created quadratic problems from real world experience (Derbyshire 2006) It was in France that algebra had developed into a well organized literal symbolism In his work In artem analyticem isagoge French mathematician Franςois Viegravete (1540-1603) in the late 16th century became the first mathematician to use letters representing numbers systematically and effectively (Derbyshire 2006) He made a range of letters available for many different quantities This was the beginning of modern literal symbolism Viegravetersquos unknown quantity was divided into two classes unknown quantities (meaning ldquothings soughtrdquo) denoted by A E I O U and Y while ldquothings givenrdquo was denoted by constants like B C Dhellip For example his A is our unknown x Viegravete was a pioneer in the study of equations His two papers on the theory of equations were published twelve years after his death In the second paper titled ldquoDe equationem emendationerdquo (ldquoOn the perfecting of equationsrdquo) Viegravete opened up the line of inquiry that led to the study of the symmetries of an equationrsquos solutions to Galois theory the theory of groups and all of modern algebra He found the relationship between the solutions of the equation and the coefficients for the first five degrees of equations in a single unknown To explain this in our modern symbols we suppose that the two solutions of the quadratic equation are 02 =++ qpxx α and β which means βα == xx Based on this logic the following thing must be true 0))(( =minusminus βα xx since only α andβ and no other values of x make this equation true This form of equation is just a rewritten form of the same equation If we multiply out those parentheses this rewritten equation turns to be Compared to the original equation the relationships between the solutions and the coefficients we obtain

0)(2 =++minus αββα xxqp =minus=+ αββα

(Derbyshire 2006) It is said that Viegravete discovered the solution formula called quadratic

10

formula today for general quadratic equations which is a

acbbx2

42

21minusplusmnminus

= for a general

quadratic equation (Olteanu 2007) 002 ne=++ acbxax Another French mathematician and philosopher who had strong influence in the history of algebra was Reneacute Descartes (1596-1650) His idea to use the Cartesian system of coordinates which was derived from his Latin name developed both algebra and geometry In his work La geacuteomeacutetrie written in 1637 Descartesndashthrough using numbers to indentify points in a Cartesian coordinatesndashconnected geometrical objects to algebraic numbers and made the classical geometry become analytical geometry He took up the plus and minus sign from the German Cossists of the previous century and also the square-root sign From Descartes the symbol of an unknown was represented by x which led to the modern system of literal symbolism (Derbyshire 2006) Descartes developed the idea of functions in Cartesian coordinates although the idea of function can be traced back to the Islamic mathematician Sharaf al-Din al-Tusi from Persia (Katz 2006) and Klaudius Ptolemaios about 2000 years ago (Olteanu 2007) Descartes declared that every curve in a Cartesian coordinate system has an equivalent equation which can represent the points on the curve or vice versa every equation containing x and y can be represented by a curve through its coordinate points However the word function was introduced by Gottfried Wilhelm Leibniz in 1693 and the definition of function was defined by Leonhard Euler in his work Introductio analysin infinitorium in 1748 Euler had also given the symbol of a denotation as a relying variable (Olteanu 2007) In the 18)(xf th century after the discovery of calculus by Newton and Leibniz algebra went into an area of analysis ndashthe study of limits infinite sequences and series functions derivatives and integrals (Derbyshire 2006) The purely abstract stage-algebra structure Since 17th century algebra has not anymore solely been about finding solutions for different degrees of equations mathematicians have started to integrate algebra with astronomy and physics Johann Kepler and Galileo Galilei were interested in curves and finding a mechanism for representing motion instead of quantity-like numbers But their arguments were not algebraic-symbolic but geometrical It was Fermat and Descartes who were regarded as the fathers of analytic geometry and who showed how to represent a curve described verbally through algebra analytic geometry and gave a mechanism for representing motion Newton in his Principia picked up on Fermat and Descartesrsquo representation and developed the calculus With the invention of the calculus mathematical problems were solved by curves not just points Algebra grew more and more to present paths of motion In order to be able to judge if the algebraic manipulations were correct axioms were formulated for arithmetic applied for algebraic manipulations In the late 18th century Lagrange introduced the idea of permutations into the search for solutions Therafter Galois developed methods for determining what conditions polynomial equations need to meet to be solvable New abstract algebraic concepts were found like ldquofieldrdquo and ldquogrouprdquo during the early 19th century In 1854 Ceyley gave an axiomatic definition of a group During the 1890s this definition entered textbooks along with the axiomatic definition of a field an idea which had roots in the work by Galois By the beginning of the 20th century algebra became less about finding solutions to equations and more about looking for common structures in many mathematical objects defined by sets of axioms (Katz 2006)

11

In short algebra development took 4000 years It took about 3500 years for mathematicians to start to use algebra symbols in systems Just almost 100 years ago algebra developed into a purely abstract science The long history may give both mathematics educators and students some explanations to why it is so difficult to teach and learn algebra

22 Three solving methods for quadratic equations Already in the previous part I have presented geometrically and rhetorically solving quadratic equations from a historical perspective In order to get more explicit understanding of quadratic equations quadratic formula and factorization from mathematics didactic point of view this part presents three common methods for solving quadratic equations 1) Completing the square 2) Factorization 3) Quadratic formula The reason to present these three methods is that they are the common topics for discussion in the articles searched in this area concerning mathematics education at upper-secondary school 1) Completing the square In two articles Solving quadratic equations by completing squares (Vinogradova 2007) and Geometric approaches to quadratic equations from other times and places (Allaire amp Bradley 2001) the authors advise mathematics teachers to use the ideas of ancient mathematician al-Khwarizmi and his method of completing quadratic equations by geometrical algebra and simplifying it in order to represent it to the students By doing so the teachers relate mathematics quantity to physical objects in a visual way Didactically teachers may start with a concrete example in which students can understand the content visually Later from the concrete example the teachers lead the students to another example expressed in algebraic symbols for example to study a rectangle whose area is )10( +xx and supposing this area is 39 this becomes 39)10( =+xx Four steps are followed to build a new square which is actually called completing the square x 10

Figure 6 x 5

Figure 7

x

x

5 5

12

A Begin with a rectangle of the area )10( +xx that is with the short side as x and the long side )10( +x The shaded area is 10bullx (Figure 6)

B Divide the shaded rectangle into two small shaded rectangles with the size 5x each and move them to each adjacent side of the xsup2 square (Figure 7) The total area is still

x10

)10(39 += xxC In Figure 7 a new square is shaped in the lower-right hand corner with the side 5 or

area 25 By adding this small square to the diagram the large square with an area of is ldquocompletedrdquo Therefore we can write 25)5(22 ++ xx 25)10(2539 ++=+ xx or

The large squarersquos area is now because the side has a length of Therefore

6425)5(22 =++ xx 2)5( +x5+x 38564)5( 2 =rArr=+rArr=+ xxx

After these three steps a solution to this quadratic equation is derived by completing the square (Allaire amp Bradley 2001) 2) Factorization Hoffman (1976) presents three approaches (A B and C) for factorizing quadratic expressions through observing and grouping He takes a second-degree polynomial as an example

273 2 ++ xx

a The inspection method 1) suggests factors of and 23x x3 x 2) 2 suggests factors of 1 and 2 or ndash1 and ndash2 3) Checking the linear term shows that the correct factorization is x7 )2)(13(273 2 ++=++ xxxx

This method according to Hoffman (1976) is based on the studentrsquos ability to simplify by inspecting the indicated product of a pair of linear expressions

b The method of decomposition of the linear term 1) Multiply and 2 to get 23x 26x2) Decompose into the sum of two terms whose product is x7 26x

This gives xxx += 67 3) Factorize by grouping 2)6(3 2 +++ xxx

2)6(3 2 +++ xxx = )

)2()63( 2 +++ xxx

= 2(1)2(3 +++ xxx= )13)(2( ++ xx

c Third method 1) Multiply 3 and 2 to get 6 2) Find two numbers whose product is 6 and whose sum is 7 The numbers are 6 and 1 3) 273 2 ++ xx

=3

)13)(63( ++ xx

= )13)(2( ++ xx

13

These three approaches can not actually be divided as three unrelated approaches especially the third method is seemingly unclear What Hoffman (1976) wants to demonstrate is that all the three approaches are based on observing the relationship among the coefficients and the constant of the polynomial Through finding the product and the sum of two factors the polynomial can be factorized in the product of two binomials In the authorrsquos opinion factorization is heavily dependent on skill in multiplication and using distributive law reversely He means that ldquostudents who do not have reasonably strong skills in multiplication should not be expected to develop strong skills in factorizationrdquo (p 55) Factorizing polynomials is not only for simplifying high-degree polynomials (for example the third or fourth-degree polynomials) but can also be applied for solving quadratic equations Supposing this polynomial forming a quadratic equation Solving this equation is actually to decompose the polynomial into a factoringrsquos form

0273 2 =++ xx0)13)(2( =++ xx

which is equivalent to the polynomial form The essential step to solve this equation is to follow the zero-factor propertyndashalso called null-factor law (nollprodukt in Swedish)ndashthat is for real numbers p and q p bull q = 0 if (and only if) p = 0 or q = 0 It means that either of the binomials on the left side of the equation has to be equal to zero in order to satisfy the

equivalent relation of this quadratic equation The solutions of two roots or 2minus=x31

minus=x

are obtained through making either 02 =+x or 013 =+x 3) Quadratic formula Solving quadratics equations by factorization is constrained within the simple quadratic equations over the whole integers and rational numbers as coefficients What happens when quadratic equations have coefficients that belong to an irrational domain or big quantity The quadratic formula is a solution to such a situation As mentioned previously historically the French mathematician Franccedilois Vieacutete discovered the quadratic formula

aacbbx

242 minusplusmnminus

= ) which is applicable for all kinds of quadratic equations In

Swedish textbooks this formula is written in PQ form by making for the equation

that is

0( nea

1=a

02 =++ qpxx qppx minus⎟⎠⎞

⎜⎝⎛plusmnminus=

2

22 with

abp = and

acq = In Oleatursquos study (2007)

studentsrsquo difficulties in using algebraic symbols no longer is the problem But on the other hand handling the parameters or coefficients in quadratic equations like and rewrite them in the equivalent form with p and q being real numbers becomes obstacles for the students when they study the algebra course at upper-secondary school

02 =++ cbxax02 =++ qpxx

The quadratic formula a

acbbx2

42 minusplusmnminus= has been regarded as the standard method

according to US mathematics teachers Obermeyer (1982) presents a way of how to derive this formula by completing a square through 11 steps (I will not present them here) At the same time an alternate method is also introduced according to the same principle All the methods mentioned so far are based on the square rules The standard method or PQ form might be regarded as an efficient and direct method but it may

222 2)( bababa +plusmn=plusmn

14

lead students to solve quadratic equations in a mechanical way besides having problems with the troublesome plusmn symbolism (Stover 1978) Both Stover (1978) and Olteanu (2007) have suggested that using a graph of quadratic functions to solve quadratic equations is another alternative method

23 Mathematical background of factorization Factorization in algebra structure Among the three solving methods presented above both the geometrical method and quadratic formula can be traced back to algebra history in section 21 What is the trace of factorization in algebra history Factorization was at the last stage in algebra history and belonged to algebra structure (Derbyshire 2006) In Swedish mathematics textbooks the chapter on algebra often includes polynomial factorization and quadratic equations But the factorization in the textbooks is not at the same abstract level as the factorization of algebra structure although they share the same axioms and there are certain connections between these two different levels of factorization How are polynomial and quadratic equations as well as factorization related to each other In order to find the theoretical background of factorizing polynomials this part constitutes a mathematical theoretical review by referring to two books concerning algebra structure (Vretblad 2000 Durbin 1992) and a research study (Bosseacute and Nandakumar 2005) A quadratic equation can be regarded as polynomial equation Solving quadratic equations has much to do with ldquounique factorization theoremrdquo The polynomial

is a second-degree polynomial A polynomial consists of coefficients in whole numbers rational numbers real numbers or complex numbers and variables (sometimes called unknowns) in different degrees as well as constants A polynomial

has symbols as coefficients which are real numbers If all are zero and so

)0(02 ne=++ acbxax

cbxax ++2

nnxaxaxaaxf ++++= )( 2

210 naaa 10

ka 1gek 0)( axf = is constant If all are zero then we say the is a zero polynomial If meaning that is not a zero polynomial and n is the biggest number and then we say that the polynomial is at degrees A zero polynomial has no degree at all According to the definition (Vredblad 2000) if is a polynomial the equation is called an algebraic equation or polynomial equation A solution or a root of this equation is a number

kaf 0nef f

0nena f nf

0)( =xfα so that 0)( =αf Such a number is even

called a zero point (nollstaumllle in Swedish) of the polynomial which means when f α=x the value of the polynomial is zero (p 161) Every polynomial whose degree is equal or greater than 1 can be written as a product of irreducible polynomial In this way through polynomial division a polynomial with n degrees can be expressed from a sum of terms into a product of its denominator and its quotient The process of changing the expressions is factorization for

instance 12

22

+=minusminusminus x

xxx The factoring form is )1)(2(22 minusminus=minusminus xxxx

Factorization of polynomials is analogue to factorization of whole numbers which is very much related to the concept of ring A ring consists of a set with two operations which are a sum and a product of two elements over a field Any polynomial from zero polynomial to n degree-polynomials can be written or operated through addition and multiplication of the

15

elements over a field F including whole numbers or integers rational numbers real numbers and complex numbers In a ring you can multiply any two elements but you canrsquot always divide Factorization is a way to deal with this fact Here are some examples In the ring of integers Z is an example of factorization in some polynomial rings for instance Q[x] the ring of polynomial in one variable with rational numbers as coefficients such as

7214 sdot=

)451)(2

51(8

52

251 2 +minus=minus+ xxxx is factorization Z[x] the ring of polynomial in one

variable with whole numbers as integers such as is factorization )1)(2(22 minusminus=minusminus xxxx The polynomial ring R[x] means the ring of polynomial in one variable with real numbers as coefficients Factorizing of such polynomials implies that the polynomial factors are the elements in the ring of R[x] for instance )2)(2(22 minus+=minus xxx The same thing is true for C[x] the ring of polynomial in one variable with complex numbers as coefficients Thus it is coefficients of polynomials that determine which ring a polynomial belongs to and if it is possible to be factorized in this ring For example number 6 can be factorized in two ways

in the ring of Z and 32times )51)(51( ii +minus in the ring of C (complex numbers) However there are cases in which divisions can not be done within the ring for example 52 or

)3()52( 2 ++minus xxx which means that no element in the ring can multiply with a denominator obtaining the numerator Therefore neither 5 nor can be factorized 522 +minus xx If a polynomial of degree at least one has no other divisors then it is considered to be irreducible or prime (Durbin 1992) Of every polynomial of degree at least one can be written as a product of irreducible polynomials according to unique factorization theorem each polynomial of degree at least one over a field F can be written as an element of F times a product of monic irreducible polynomials over F and except for the order in which these irreducible polynomials are written this can be done in only one way (p 221) Thus a factorable polynomial can be written into a product with another polynomial such as In other words the condition that )()()( xqaxxf minus= )( ax minus is a factor of a polynomial

in a variable x is only true if )(xf 0)( =af For example if and

6)( 2 minus+= xxxf0624)2( =minus+=f 0639)3( =minusminus=minusf the factors of are and )(xf )2( minusx )3( +x

The value of the polynomial is zero which is also expressed as the polynomial having a zero point (nollstaumllle in Swedish) when

)(xfα=x

Based on unique factorization theorem finding solutions of an algebraic equation is equivalent to finding the first-degree factors for a polynomial (Vredblad 2000) In this way solving a quadratic equation can be handled through finding the first-degree factors for the second-degree polynomial and there after finding the roots of the quadratic equation This is often called using factorization to solve quadratic equations Besides unique factorization theorem the connection of using factorization to solve a quadratic equation is that this method relates to prime numbers using distributive law backwards commutative law multiplication division and operating fractions As matter of fact it is about application of factorizing integers in polynomials at upper-secondary school level Since quadratic equations in the mathematics B course at Swedish upper-secondary school are limited to real numbers and often within the whole numbers (integers) many quadratic equations should be able to be solved by factorization according to my own

16

judgment Therefore using factorization for solving quadratic equations is very much related to arithmetic operations number concept and algebra structure It may offer students both mathematical conceptual understanding and operational skill For algebra beginners factorization can be traced back to studentsrsquo early years of mathematics education during which students do not only work with multiplication but also on factoring integers for example 56 = 8 middot 7 = 2 middot 2 middot 2 middot 7 = 4 middot 14 = 2 middot 28 In this example two operations are included multiplication and division Multiplication operation may help students to understand the multiplicative structure of the integers while factoring integers may become meaningful when it is related to divisibility divisors and prime numbers These two operational skills or competence are major techniques for simplifying fractions and finding common denominators Being able to understand and operate factoring integers has laid the essential grounds for algebra beginners to study factorization of polynomials Thus factorization reunites studentsrsquo early mathematics knowledge of factoring integers with later knowledge of algebra structure Factorability In a study on factorability by Bosseacute and Nandakumar (2005) the probability of factorability is reasoned through investigating the range of integers as coefficients of quadratic equations In the quadratic equation a b and c are randomly generated integers within a determined range r such as

002 ne=++ acbxaxyrx lele where x and y are integers When is a

perfect square or where

acb 42 minus

)4( 2 acb minus is an integer quadratics are factorable According to the studyrsquos data it shows that as the range for a b and c increases the probability of factorability of a quadratic with randomly selected integer values for a b and c decreases Bosseacute and Nandakumar confirm that as the range for coefficients expands to infinltltinfinminus r the probablility of factorability of a quadratic with randomly selected integer values for coefficients approaches zero The studyrsquos data collected from college algebra courses and textbooks demonstrates that about 15 of quadratics with integer coefficients [ ]1010minusisinr are factorable Thus about 85 of quadratics can not be factorized In spite of the limitation of factorability within the exercises concerning factoring practice chosen from 27 surveyed college algebra textbooks about 94 of the problems were factorable Within the textbooks 55 of the quadratics to be factored were within the range [-10 10] Even if many quadratic equations are factorable choosing the right factor pairs is time consuming and unnecessary for example

has to be factorized through choosing the right pairs of factors among nine pairs for 36 and 8 pairs of factors for 24 Comparing the three different methods for solving quadratic equations the study suggests that completing the square and quadratic formula are more effective informative and useful than factorization

0245936 2 =++ xx

231 Different kinds of factorizations for solving different quadratic equations How can we find the binomial factors of a quadratic equation or a second-degree polynomial without knowing one of the divisors as a binomial factor The answer is that we have to observe the relations among all coefficients and constants as well as the positive and negative signs before them under the condition of being factorable Most of the coefficients in quadratic equations or polynomials among exercises provided for practicing factoring are integers (here I borrow the same notation r as the one used by Bosseacute and Nandakumar) with

17

their range between ndash10 and 10 that is 10ler In this case it is possible for a student to make a judgment of whether or not a quadratic equation is factorable Five different kinds of factorization depending on five kinds of quadratic equations Factoring quadratic expressions can be divided into five different kinds depending on the type of quadratic equations The coefficients a b p q and constant c illustrated in the following quadratic equations are defined not to be equal to zero thus 000 nenene cba The first kind of quadratic equations is type 1 the obviously factorable type (Bosseacute and Nandakumar 2004) then the equation is factorized as

where the roots are

0002 nene=+ babxax

0)( =+ baxxabxx minus== 21 0

The second kind of quadratic equations is type 2 The factors of this second degree polynomial are so the roots to this type of quadratic

equations are

0)( 22 =minusbax))(()( 22 baxbaxbax minus+=minus

bax plusmn=21 This kind of factorization is based on the difference-of-squares

formula for example can be solved by factorizing into 049 2 =minusx 0)23)(23( =minus+ xx Thus

the roots are 32

32

21 =minus= xx This kind of factorization is also obviously factorable

The third kind of quadratic equations is type 3 This type of equations can be solved directly by factoring into two identical binomials

The double roots obtained from this

factorization are

02)( 22 =+plusmn babxax

0))(()(2)( 222 =plusmnplusmn=plusmn=+plusmn baxbaxbaxbabxax

abx plusmn=21 This method has utilized square rules which are quite easily

observed The forth kind of quadratic equations is type 4 In this equation the coefficient of the -term is 1 with p and q as positive or negative integers Type 4 equations can always be solved by the approaches of completing the square and quadratic formula However they can often be solved by factorization too since the quadratic expressions are factorable by using distributive laws reversely The roots are obtained through finding two factors for q At the same time these two factors become satisfying with the relation so that the sum of factors is equal to That way the two factors become two roots of a quadratic equation Denoting the two factors or two roots as the relationship between coefficient and constant is

02 =++ qpxx2x

qminus

21rr)( 2121 rrprrq +minus=bull= Quadratic equations can be written as the product of

two first-degree binomials or factoring form 0))(()( 212121

22 =minusminus=++minus=++ rxrxrrxrrxqpxx Through the factoring form the roots of this equation are obtained being The core of factorizing type 4 equations is seeking the connections between coefficients and roots which reveal the studentsrsquo basic ability of arithmetic multiplication and addition It is often more effective in this case to use factorization than to use quadratic formula or completing the square Let us have a look at the example In this equation we get

2211 rxrx ==

035122 =+minus xx

18

35)7)(5(12)7()5( =minusminus=minus=minus+minus= qp or 75)75( bull=bullminus= qp (here we can directly get roots) Factorizing the equation we obtain 0)7)(5( =minusminus xx and roots are The procedures of finding factors of the equation often occur in some studentsrsquo minds quickly without writing down the whole computing process but those students often have good skills in arithmetic operations (reference) Compared to using factorization for solving this equation application of quadratic formula seems ineffective A possible problem in this method is how to find the signs before the factors or how to judge if the factors in the parentheses are positive or negative integers In order to check if the solution is correct we can always multiply these two factored binomials by distributive law and see if the expanded polynomial is the same as the original quadratic equation

75 21 == xx

The fifth kind of quadratic equations is type 5 Still the methods of completing squares and quadratic formula can be applied for all type 5 equations but there are cases in which factorization can be used for solving this type of equations within a certain range of integers as mentioned before Sometimes procedures of factorizing such equations can be carried out in our minds quickly like type 4 equations but in many cases they can not As a matter of fact finding two binomial factors for a factorable quadratic equation or a factorable second-degree polynomial is a complicated process and requires systematic work In his article published in Mathematics in School Jackman (2005) demonstrates this systematic procedure via factorizing second-degree polynomials A second-degree polynomial can be written as where are all integers By multiplying out the polynomial becomes thus

0002 nene=++ bacbxax

))((2 srxqpxcbxax ++=++ srqpcbaqsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== Writing the factors of a and c on two lines p q times r s Finding the right pairs of factors for a and c has to be done by listing out the groups of products of different factors for example The factor pairs for 6 are (6 1) (1 6) (3 2) (2 3) and the factor pairs for (-4) are (4 -1) (-4 1) (2 -2) (-2 2) (1 -4) (-1 4) Writing them on two lines p 6 1 3 2 q 4 -4 2 -2 1 -1

456 2 minus+ xx

times r 1 6 2 3 s -1 1 -2 2 -4 4 Then comes calculating the values of ps and qr until the value of b = 5 is obtained The calculating procedures are divided into 4 groups and 6 steps in each group which means 4 times 6 steps in total Here I only give one grouprsquos procedure as an example 6 middot (-1) + 4 middot 1 = -2 6 middot 1 + 1middot (-4) = 2 6 middot (-2) + 1 middot 2 = -10 6 middot 2 + 1middot (-2) = 10 6 middot (-4) + 1 middot 1 = -23 6 middot 4 + 1 middot (-1) = 23 helliphellip After the 24ndashstep calculating procedures two computing procedures obtain the same correct result 3 (ndash1) + 4 2 = 5 (yes) 2 4 + (-1) 3 = 5 (yes) Are they the same From 3 (ndash1) + 4 2 = 5 the polynomial can be factorized into )12)(43( minus+ xx while from 2 4 + (ndash1) 3 = 5 the polynomial can be factorized into )43)(12( +minus xx According to communicative law these two binomial products are the same The shortcoming of this

19

method is that it is too ineffective and complicated The addition of matrixes is almost impossible and unbearable Nevertheless based on the same idea I suggest finding factors by guessing analyzing and checking Using the same example here we can find that one of the binomials must have a negative integer because of ndash4 which implies that 5 is the sum of a positive integer and a negative integer Then looking at a and c the factors of 6 can be (1 6) and (2 3) while the factors of 4 can be (1 4) and (2 2) when we regard them as absolute values We know that the factoring form of the polynomial will be like

and the connections of all coefficients are

456 2 minus+ xx

qsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== The coefficient 5=b is the difference of two products of factors for 6 and 4 First we try the biggest number of product such as 2446 =bull then 524 =minus the unknown is which is a prime number the only factors for 19 are (19 1) There is not any combination of products of factors for 6 and 4 satisfying 19 Therefore the multiplications of factors (1 6) and (1 4) are not included in this case So we move to factors (2 2) (2 3) and (1 4)

19 =

In the second step we try The result is negative it is obviously

4264466)23)(22( 22 minusminus=minus+minus=+minus xxxxxxx2 lt 5 which implies that the combination of (2 3) and (2 2) is not

considerable One pair of factors has to be removed and that is (2 2) which does not influence the coefficient of the -term Then we try The result this time is negative too but we can see that

2x 456)43)(12( 2 minusminus=minus+ xxxxb is 5 which guides us to interchanging the positions of

negative and positive signs in the parentheses The next attempt is and it is correct Seeking the correct factors for the polynomial

has taken four major operating steps by trying the combinations of factorsrsquo products and relating the sum of the products with the value of the x-term coefficient The strategy is to make use of distributive law reversely first and then multiply the product of every pair of parentheses in order to check if they are the right pair of polynomial factors Factorizing second-degree polynomials is the major step for solving quadratic equations After factoring polynomials into the product of two binomials as in the example given above the roots can be obtained by making each binomial equal to zero according to the theory of zero point (ldquonollstaumlllerdquo in Swedish) of a polynomial equation originating from unique factorization theorem

456)43)(12( 2 minus+=+minus xxxx456 2 minus+ xx

Factorization by using the technique of vertically and crosswise sutra Binomial expansion or multiplication of binomials by distributive law and factorization of the second-degree polynomials by using distributive law backwards have an extremely symmetrical effect and the results of two kinds of operations are equivalent It is crucial for the students to be able to perceive mathematics structure and be acquainted with arithmetic operations as well as to understand the relationship between quantities when they carry out factoring procedures The expansion of binomials by distributive law has been named as the method FOIL which stands for first outside inside and last according to multiplication orders of two binomials for example (Nataraj and Thomas 2006) Many students are used to this procedure Another technique to multiply two binomials is called vertically and crosswise sutra origin from the ancient Indian Vedic mathematics Vertically and crosswise sutra and FOIL are much the same in principle but it makes use of

824)4)(2( 2 +++=++ xxxxx

20

the power of visualization and improves facility with the expansion of binomials and the factorization of quadratic expressions In this method two binomials are written into two lines Using the same example the two lines are written as 824)4)(2( 2 +++=++ xxxxx

x + 2 x + 4 ______________ xsup2 4x + 2x 8

The order of multiplication is as shown below x + 2 x + 4

4 3 2 1 (the numbers 1 2 3 and 4 refer to steps) xsup2 + 2x + 4x + 8 Figure 3 The vertically and crosswise method of multiplication of binomials Step 2 and 3 can be interchangeable If we put this form into a line expanded according to the order above would be like

)4)(2( ++ xxxbullrarrbull 442 clockwise and xxx bullrarrbull2 in

counterclockwise direction According to sutra we can imagine drawing a thread starting from 2 then going through 4 and x clockwise and then draw the second thread starting from the same place 2 later going through x in the second bracket and finally x in the first bracket counterclockwise Learning expansion of binomials by the methods FOIL and Vertical and clockwise sutra is the basic process before students study factorization In this study Nataraj and Thomas (2006) have found that the students appear to do factorization using guess-and-check as well as decomposition or the vertical and clockwise approach Their finding is that afterwards the students performed better on algebra questions especially on the factorizations They conclude ldquohellipthe values of the method may lie in what it adds to the studentsrsquo overall algebraic conceptions and knowledge of mathematical structurerdquo (p 16) In general solving these five kinds of quadratic equations by factorization can be traced back to unique factorization theorem The important character in this method is that you can always check if factorization is correct or not through multiplying out the two pairs of a parenthesis and see if the expanded polynomial is the given one or not Using factorization is not only about method but also about fostering the studentsrsquo mathematical thinking The solving processes are more important than solutions

21

3 Review of related literature and research Besides the focus on mathematics content another focus in the background work for this research is about previous studies and research in the field of mathematics didactic concerning different methods for solving quadratic equations and the use of factorization In order to study the previous research and studies in the related field three scientific databases and a search engine have been used and 113 articles have been investigated Topics such as solving quadratic expressions and equations are often discussed as teaching ideas and strategies shared by mathematics educators in different countries from lower and upper-secondary schools to colleges according to the findings in the article search These didactic topics from mathematics teachersrsquo perspectives include for example using geometric approaches to solve quadratic equations (Allaire amp Bradley 2001) utilizing completing squares (Vinogradova 2007) factoring quadratics (Hoffman 1976 Cheng 1980) and using factorization completing the square graphical methods to solve quadratic equations (Macdonald 1986) Kostsopoulos (2007) links cognitive reasoning to pedagogical problems appearing in the mathematics classrooms in her article published in Australian Mathematics Teacher Based on her teaching experiences Kostsopoulos is aware of her pedagogical strategies lacking insight on dealing with studentsrsquo difficulties of multiplication and factorization of quadratics She assumes that ldquostudentsrsquo problems with factorization and with identifying varied representations of the same quadratic relationshiprdquo may be linked to the ways in which the brain constructs cognitive representations (p 19) Factoring of quadratics is the rewriting of polynomials as a product of polynomials It requires students to have both a conceptual understanding of multiplication of polynomials and the procedural knowledge to retrieve basic multiplication facts effectively Based on the theory about the order matters in the brainrsquos ability to retrieve number facts (Phenix amp Campbell 2001) Kostsopoulos conjectures that studentsrsquo ability to access the appropriate long-term semantic memory is limited when they are confused by mixing the order and varied forms of quadratics Although this article is not about a research study it seeks the reason why factorizing quadratics is difficult and therefore might provide math teachers with alternative explanations for similar pedagogical frustrations With Von Glasersfeldrsquos radical constructivism and Peter Gaumlrdenforsrsquo epistemic semantics theory as his theoretical frameworks Rauff (1994) suggests that belief-based teaching can be successful in teaching factoring According to Rauff constructivist learning theory takes the view that students construct their own beliefs and knowledge of mathematics Learning occurs when the students change their belief set Referring to belief set theory from Gaumlrdenfors (1988) Rauff (1994) claims that ldquoa belief set is expanded when a new belief is added to it and contracted when a proposition is no longer believed and is removed from the setrdquo (p 421) Expansion contraction and revision are three components to comprise basic mechanisms of changes of belief according to Rauff When examining and comparing studentsrsquo (ages 18-20) own definitions of factoring Rauff analyzes the errors of their definitions with constructivism and epistemic semantics as analyzing tools The result of this study shows that the approach to teaching factoring rely heavily upon what a student believes about factoring is quite productive in two dimensions First it reveals the source of nonstandard factoring Second it provides a starting point for modification of the studentrsquos underlying conceptions of factoring Rauff proposes that belief-based teaching can also be

22

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

25

258 ++= x Then x was 3 By ldquocut-and-pasterdquo geometry (Katz 2006 p 191)

al-Khwarizimi reduced the second degree of an equation to the first degree and thereafter solved it Unlike his Babylonian predecessors al-Khwarizimi always presented his problem abstractly rather than geometrically relating to lengths and widths The symbolic stage At this stage of algebra ldquoall numbers operations relationships are expressed through a set of easily recognized symbols and manipulations on the symbols take place according to well-understood rulesrdquo (Katz 2006 p 186) The ancient algebra and geometry had developed sophisticatedly in Egypt Persia Greece India and China After Medieval Islamic scholars gave us the word ldquoalgebrardquo Western Europe began the struggle for the development of algebra starting from some algebraists from Italy Italian mathematician Leonardo Pisano later known as Fibonacci traveled in the 12th and 13th centuries to Persia India and China When he returned to Italy he had wider knowledge of arithmetic and algebra His book Liber abbaci was the best math textbook since the end of Ancient world His book is credited with having introduced Indian numerals including zero to the West But his algebraic skills had been shown in two other works after this one With the introduction of printed books during the second half of the 15th century the development of algebra was sped up Several Italian mathematicians including Girolamo Cardano had figured out how to solve cubic and quadratic equations Algebra became purely abstract with the exception of an English mathematician named Robert Recorde who lived in the 16th century and created quadratic problems from real world experience (Derbyshire 2006) It was in France that algebra had developed into a well organized literal symbolism In his work In artem analyticem isagoge French mathematician Franςois Viegravete (1540-1603) in the late 16th century became the first mathematician to use letters representing numbers systematically and effectively (Derbyshire 2006) He made a range of letters available for many different quantities This was the beginning of modern literal symbolism Viegravetersquos unknown quantity was divided into two classes unknown quantities (meaning ldquothings soughtrdquo) denoted by A E I O U and Y while ldquothings givenrdquo was denoted by constants like B C Dhellip For example his A is our unknown x Viegravete was a pioneer in the study of equations His two papers on the theory of equations were published twelve years after his death In the second paper titled ldquoDe equationem emendationerdquo (ldquoOn the perfecting of equationsrdquo) Viegravete opened up the line of inquiry that led to the study of the symmetries of an equationrsquos solutions to Galois theory the theory of groups and all of modern algebra He found the relationship between the solutions of the equation and the coefficients for the first five degrees of equations in a single unknown To explain this in our modern symbols we suppose that the two solutions of the quadratic equation are 02 =++ qpxx α and β which means βα == xx Based on this logic the following thing must be true 0))(( =minusminus βα xx since only α andβ and no other values of x make this equation true This form of equation is just a rewritten form of the same equation If we multiply out those parentheses this rewritten equation turns to be Compared to the original equation the relationships between the solutions and the coefficients we obtain

0)(2 =++minus αββα xxqp =minus=+ αββα

(Derbyshire 2006) It is said that Viegravete discovered the solution formula called quadratic

10

formula today for general quadratic equations which is a

acbbx2

42

21minusplusmnminus

= for a general

quadratic equation (Olteanu 2007) 002 ne=++ acbxax Another French mathematician and philosopher who had strong influence in the history of algebra was Reneacute Descartes (1596-1650) His idea to use the Cartesian system of coordinates which was derived from his Latin name developed both algebra and geometry In his work La geacuteomeacutetrie written in 1637 Descartesndashthrough using numbers to indentify points in a Cartesian coordinatesndashconnected geometrical objects to algebraic numbers and made the classical geometry become analytical geometry He took up the plus and minus sign from the German Cossists of the previous century and also the square-root sign From Descartes the symbol of an unknown was represented by x which led to the modern system of literal symbolism (Derbyshire 2006) Descartes developed the idea of functions in Cartesian coordinates although the idea of function can be traced back to the Islamic mathematician Sharaf al-Din al-Tusi from Persia (Katz 2006) and Klaudius Ptolemaios about 2000 years ago (Olteanu 2007) Descartes declared that every curve in a Cartesian coordinate system has an equivalent equation which can represent the points on the curve or vice versa every equation containing x and y can be represented by a curve through its coordinate points However the word function was introduced by Gottfried Wilhelm Leibniz in 1693 and the definition of function was defined by Leonhard Euler in his work Introductio analysin infinitorium in 1748 Euler had also given the symbol of a denotation as a relying variable (Olteanu 2007) In the 18)(xf th century after the discovery of calculus by Newton and Leibniz algebra went into an area of analysis ndashthe study of limits infinite sequences and series functions derivatives and integrals (Derbyshire 2006) The purely abstract stage-algebra structure Since 17th century algebra has not anymore solely been about finding solutions for different degrees of equations mathematicians have started to integrate algebra with astronomy and physics Johann Kepler and Galileo Galilei were interested in curves and finding a mechanism for representing motion instead of quantity-like numbers But their arguments were not algebraic-symbolic but geometrical It was Fermat and Descartes who were regarded as the fathers of analytic geometry and who showed how to represent a curve described verbally through algebra analytic geometry and gave a mechanism for representing motion Newton in his Principia picked up on Fermat and Descartesrsquo representation and developed the calculus With the invention of the calculus mathematical problems were solved by curves not just points Algebra grew more and more to present paths of motion In order to be able to judge if the algebraic manipulations were correct axioms were formulated for arithmetic applied for algebraic manipulations In the late 18th century Lagrange introduced the idea of permutations into the search for solutions Therafter Galois developed methods for determining what conditions polynomial equations need to meet to be solvable New abstract algebraic concepts were found like ldquofieldrdquo and ldquogrouprdquo during the early 19th century In 1854 Ceyley gave an axiomatic definition of a group During the 1890s this definition entered textbooks along with the axiomatic definition of a field an idea which had roots in the work by Galois By the beginning of the 20th century algebra became less about finding solutions to equations and more about looking for common structures in many mathematical objects defined by sets of axioms (Katz 2006)

11

In short algebra development took 4000 years It took about 3500 years for mathematicians to start to use algebra symbols in systems Just almost 100 years ago algebra developed into a purely abstract science The long history may give both mathematics educators and students some explanations to why it is so difficult to teach and learn algebra

22 Three solving methods for quadratic equations Already in the previous part I have presented geometrically and rhetorically solving quadratic equations from a historical perspective In order to get more explicit understanding of quadratic equations quadratic formula and factorization from mathematics didactic point of view this part presents three common methods for solving quadratic equations 1) Completing the square 2) Factorization 3) Quadratic formula The reason to present these three methods is that they are the common topics for discussion in the articles searched in this area concerning mathematics education at upper-secondary school 1) Completing the square In two articles Solving quadratic equations by completing squares (Vinogradova 2007) and Geometric approaches to quadratic equations from other times and places (Allaire amp Bradley 2001) the authors advise mathematics teachers to use the ideas of ancient mathematician al-Khwarizmi and his method of completing quadratic equations by geometrical algebra and simplifying it in order to represent it to the students By doing so the teachers relate mathematics quantity to physical objects in a visual way Didactically teachers may start with a concrete example in which students can understand the content visually Later from the concrete example the teachers lead the students to another example expressed in algebraic symbols for example to study a rectangle whose area is )10( +xx and supposing this area is 39 this becomes 39)10( =+xx Four steps are followed to build a new square which is actually called completing the square x 10

Figure 6 x 5

Figure 7

x

x

5 5

12

A Begin with a rectangle of the area )10( +xx that is with the short side as x and the long side )10( +x The shaded area is 10bullx (Figure 6)

B Divide the shaded rectangle into two small shaded rectangles with the size 5x each and move them to each adjacent side of the xsup2 square (Figure 7) The total area is still

x10

)10(39 += xxC In Figure 7 a new square is shaped in the lower-right hand corner with the side 5 or

area 25 By adding this small square to the diagram the large square with an area of is ldquocompletedrdquo Therefore we can write 25)5(22 ++ xx 25)10(2539 ++=+ xx or

The large squarersquos area is now because the side has a length of Therefore

6425)5(22 =++ xx 2)5( +x5+x 38564)5( 2 =rArr=+rArr=+ xxx

After these three steps a solution to this quadratic equation is derived by completing the square (Allaire amp Bradley 2001) 2) Factorization Hoffman (1976) presents three approaches (A B and C) for factorizing quadratic expressions through observing and grouping He takes a second-degree polynomial as an example

273 2 ++ xx

a The inspection method 1) suggests factors of and 23x x3 x 2) 2 suggests factors of 1 and 2 or ndash1 and ndash2 3) Checking the linear term shows that the correct factorization is x7 )2)(13(273 2 ++=++ xxxx

This method according to Hoffman (1976) is based on the studentrsquos ability to simplify by inspecting the indicated product of a pair of linear expressions

b The method of decomposition of the linear term 1) Multiply and 2 to get 23x 26x2) Decompose into the sum of two terms whose product is x7 26x

This gives xxx += 67 3) Factorize by grouping 2)6(3 2 +++ xxx

2)6(3 2 +++ xxx = )

)2()63( 2 +++ xxx

= 2(1)2(3 +++ xxx= )13)(2( ++ xx

c Third method 1) Multiply 3 and 2 to get 6 2) Find two numbers whose product is 6 and whose sum is 7 The numbers are 6 and 1 3) 273 2 ++ xx

=3

)13)(63( ++ xx

= )13)(2( ++ xx

13

These three approaches can not actually be divided as three unrelated approaches especially the third method is seemingly unclear What Hoffman (1976) wants to demonstrate is that all the three approaches are based on observing the relationship among the coefficients and the constant of the polynomial Through finding the product and the sum of two factors the polynomial can be factorized in the product of two binomials In the authorrsquos opinion factorization is heavily dependent on skill in multiplication and using distributive law reversely He means that ldquostudents who do not have reasonably strong skills in multiplication should not be expected to develop strong skills in factorizationrdquo (p 55) Factorizing polynomials is not only for simplifying high-degree polynomials (for example the third or fourth-degree polynomials) but can also be applied for solving quadratic equations Supposing this polynomial forming a quadratic equation Solving this equation is actually to decompose the polynomial into a factoringrsquos form

0273 2 =++ xx0)13)(2( =++ xx

which is equivalent to the polynomial form The essential step to solve this equation is to follow the zero-factor propertyndashalso called null-factor law (nollprodukt in Swedish)ndashthat is for real numbers p and q p bull q = 0 if (and only if) p = 0 or q = 0 It means that either of the binomials on the left side of the equation has to be equal to zero in order to satisfy the

equivalent relation of this quadratic equation The solutions of two roots or 2minus=x31

minus=x

are obtained through making either 02 =+x or 013 =+x 3) Quadratic formula Solving quadratics equations by factorization is constrained within the simple quadratic equations over the whole integers and rational numbers as coefficients What happens when quadratic equations have coefficients that belong to an irrational domain or big quantity The quadratic formula is a solution to such a situation As mentioned previously historically the French mathematician Franccedilois Vieacutete discovered the quadratic formula

aacbbx

242 minusplusmnminus

= ) which is applicable for all kinds of quadratic equations In

Swedish textbooks this formula is written in PQ form by making for the equation

that is

0( nea

1=a

02 =++ qpxx qppx minus⎟⎠⎞

⎜⎝⎛plusmnminus=

2

22 with

abp = and

acq = In Oleatursquos study (2007)

studentsrsquo difficulties in using algebraic symbols no longer is the problem But on the other hand handling the parameters or coefficients in quadratic equations like and rewrite them in the equivalent form with p and q being real numbers becomes obstacles for the students when they study the algebra course at upper-secondary school

02 =++ cbxax02 =++ qpxx

The quadratic formula a

acbbx2

42 minusplusmnminus= has been regarded as the standard method

according to US mathematics teachers Obermeyer (1982) presents a way of how to derive this formula by completing a square through 11 steps (I will not present them here) At the same time an alternate method is also introduced according to the same principle All the methods mentioned so far are based on the square rules The standard method or PQ form might be regarded as an efficient and direct method but it may

222 2)( bababa +plusmn=plusmn

14

lead students to solve quadratic equations in a mechanical way besides having problems with the troublesome plusmn symbolism (Stover 1978) Both Stover (1978) and Olteanu (2007) have suggested that using a graph of quadratic functions to solve quadratic equations is another alternative method

23 Mathematical background of factorization Factorization in algebra structure Among the three solving methods presented above both the geometrical method and quadratic formula can be traced back to algebra history in section 21 What is the trace of factorization in algebra history Factorization was at the last stage in algebra history and belonged to algebra structure (Derbyshire 2006) In Swedish mathematics textbooks the chapter on algebra often includes polynomial factorization and quadratic equations But the factorization in the textbooks is not at the same abstract level as the factorization of algebra structure although they share the same axioms and there are certain connections between these two different levels of factorization How are polynomial and quadratic equations as well as factorization related to each other In order to find the theoretical background of factorizing polynomials this part constitutes a mathematical theoretical review by referring to two books concerning algebra structure (Vretblad 2000 Durbin 1992) and a research study (Bosseacute and Nandakumar 2005) A quadratic equation can be regarded as polynomial equation Solving quadratic equations has much to do with ldquounique factorization theoremrdquo The polynomial

is a second-degree polynomial A polynomial consists of coefficients in whole numbers rational numbers real numbers or complex numbers and variables (sometimes called unknowns) in different degrees as well as constants A polynomial

has symbols as coefficients which are real numbers If all are zero and so

)0(02 ne=++ acbxax

cbxax ++2

nnxaxaxaaxf ++++= )( 2

210 naaa 10

ka 1gek 0)( axf = is constant If all are zero then we say the is a zero polynomial If meaning that is not a zero polynomial and n is the biggest number and then we say that the polynomial is at degrees A zero polynomial has no degree at all According to the definition (Vredblad 2000) if is a polynomial the equation is called an algebraic equation or polynomial equation A solution or a root of this equation is a number

kaf 0nef f

0nena f nf

0)( =xfα so that 0)( =αf Such a number is even

called a zero point (nollstaumllle in Swedish) of the polynomial which means when f α=x the value of the polynomial is zero (p 161) Every polynomial whose degree is equal or greater than 1 can be written as a product of irreducible polynomial In this way through polynomial division a polynomial with n degrees can be expressed from a sum of terms into a product of its denominator and its quotient The process of changing the expressions is factorization for

instance 12

22

+=minusminusminus x

xxx The factoring form is )1)(2(22 minusminus=minusminus xxxx

Factorization of polynomials is analogue to factorization of whole numbers which is very much related to the concept of ring A ring consists of a set with two operations which are a sum and a product of two elements over a field Any polynomial from zero polynomial to n degree-polynomials can be written or operated through addition and multiplication of the

15

elements over a field F including whole numbers or integers rational numbers real numbers and complex numbers In a ring you can multiply any two elements but you canrsquot always divide Factorization is a way to deal with this fact Here are some examples In the ring of integers Z is an example of factorization in some polynomial rings for instance Q[x] the ring of polynomial in one variable with rational numbers as coefficients such as

7214 sdot=

)451)(2

51(8

52

251 2 +minus=minus+ xxxx is factorization Z[x] the ring of polynomial in one

variable with whole numbers as integers such as is factorization )1)(2(22 minusminus=minusminus xxxx The polynomial ring R[x] means the ring of polynomial in one variable with real numbers as coefficients Factorizing of such polynomials implies that the polynomial factors are the elements in the ring of R[x] for instance )2)(2(22 minus+=minus xxx The same thing is true for C[x] the ring of polynomial in one variable with complex numbers as coefficients Thus it is coefficients of polynomials that determine which ring a polynomial belongs to and if it is possible to be factorized in this ring For example number 6 can be factorized in two ways

in the ring of Z and 32times )51)(51( ii +minus in the ring of C (complex numbers) However there are cases in which divisions can not be done within the ring for example 52 or

)3()52( 2 ++minus xxx which means that no element in the ring can multiply with a denominator obtaining the numerator Therefore neither 5 nor can be factorized 522 +minus xx If a polynomial of degree at least one has no other divisors then it is considered to be irreducible or prime (Durbin 1992) Of every polynomial of degree at least one can be written as a product of irreducible polynomials according to unique factorization theorem each polynomial of degree at least one over a field F can be written as an element of F times a product of monic irreducible polynomials over F and except for the order in which these irreducible polynomials are written this can be done in only one way (p 221) Thus a factorable polynomial can be written into a product with another polynomial such as In other words the condition that )()()( xqaxxf minus= )( ax minus is a factor of a polynomial

in a variable x is only true if )(xf 0)( =af For example if and

6)( 2 minus+= xxxf0624)2( =minus+=f 0639)3( =minusminus=minusf the factors of are and )(xf )2( minusx )3( +x

The value of the polynomial is zero which is also expressed as the polynomial having a zero point (nollstaumllle in Swedish) when

)(xfα=x

Based on unique factorization theorem finding solutions of an algebraic equation is equivalent to finding the first-degree factors for a polynomial (Vredblad 2000) In this way solving a quadratic equation can be handled through finding the first-degree factors for the second-degree polynomial and there after finding the roots of the quadratic equation This is often called using factorization to solve quadratic equations Besides unique factorization theorem the connection of using factorization to solve a quadratic equation is that this method relates to prime numbers using distributive law backwards commutative law multiplication division and operating fractions As matter of fact it is about application of factorizing integers in polynomials at upper-secondary school level Since quadratic equations in the mathematics B course at Swedish upper-secondary school are limited to real numbers and often within the whole numbers (integers) many quadratic equations should be able to be solved by factorization according to my own

16

judgment Therefore using factorization for solving quadratic equations is very much related to arithmetic operations number concept and algebra structure It may offer students both mathematical conceptual understanding and operational skill For algebra beginners factorization can be traced back to studentsrsquo early years of mathematics education during which students do not only work with multiplication but also on factoring integers for example 56 = 8 middot 7 = 2 middot 2 middot 2 middot 7 = 4 middot 14 = 2 middot 28 In this example two operations are included multiplication and division Multiplication operation may help students to understand the multiplicative structure of the integers while factoring integers may become meaningful when it is related to divisibility divisors and prime numbers These two operational skills or competence are major techniques for simplifying fractions and finding common denominators Being able to understand and operate factoring integers has laid the essential grounds for algebra beginners to study factorization of polynomials Thus factorization reunites studentsrsquo early mathematics knowledge of factoring integers with later knowledge of algebra structure Factorability In a study on factorability by Bosseacute and Nandakumar (2005) the probability of factorability is reasoned through investigating the range of integers as coefficients of quadratic equations In the quadratic equation a b and c are randomly generated integers within a determined range r such as

002 ne=++ acbxaxyrx lele where x and y are integers When is a

perfect square or where

acb 42 minus

)4( 2 acb minus is an integer quadratics are factorable According to the studyrsquos data it shows that as the range for a b and c increases the probability of factorability of a quadratic with randomly selected integer values for a b and c decreases Bosseacute and Nandakumar confirm that as the range for coefficients expands to infinltltinfinminus r the probablility of factorability of a quadratic with randomly selected integer values for coefficients approaches zero The studyrsquos data collected from college algebra courses and textbooks demonstrates that about 15 of quadratics with integer coefficients [ ]1010minusisinr are factorable Thus about 85 of quadratics can not be factorized In spite of the limitation of factorability within the exercises concerning factoring practice chosen from 27 surveyed college algebra textbooks about 94 of the problems were factorable Within the textbooks 55 of the quadratics to be factored were within the range [-10 10] Even if many quadratic equations are factorable choosing the right factor pairs is time consuming and unnecessary for example

has to be factorized through choosing the right pairs of factors among nine pairs for 36 and 8 pairs of factors for 24 Comparing the three different methods for solving quadratic equations the study suggests that completing the square and quadratic formula are more effective informative and useful than factorization

0245936 2 =++ xx

231 Different kinds of factorizations for solving different quadratic equations How can we find the binomial factors of a quadratic equation or a second-degree polynomial without knowing one of the divisors as a binomial factor The answer is that we have to observe the relations among all coefficients and constants as well as the positive and negative signs before them under the condition of being factorable Most of the coefficients in quadratic equations or polynomials among exercises provided for practicing factoring are integers (here I borrow the same notation r as the one used by Bosseacute and Nandakumar) with

17

their range between ndash10 and 10 that is 10ler In this case it is possible for a student to make a judgment of whether or not a quadratic equation is factorable Five different kinds of factorization depending on five kinds of quadratic equations Factoring quadratic expressions can be divided into five different kinds depending on the type of quadratic equations The coefficients a b p q and constant c illustrated in the following quadratic equations are defined not to be equal to zero thus 000 nenene cba The first kind of quadratic equations is type 1 the obviously factorable type (Bosseacute and Nandakumar 2004) then the equation is factorized as

where the roots are

0002 nene=+ babxax

0)( =+ baxxabxx minus== 21 0

The second kind of quadratic equations is type 2 The factors of this second degree polynomial are so the roots to this type of quadratic

equations are

0)( 22 =minusbax))(()( 22 baxbaxbax minus+=minus

bax plusmn=21 This kind of factorization is based on the difference-of-squares

formula for example can be solved by factorizing into 049 2 =minusx 0)23)(23( =minus+ xx Thus

the roots are 32

32

21 =minus= xx This kind of factorization is also obviously factorable

The third kind of quadratic equations is type 3 This type of equations can be solved directly by factoring into two identical binomials

The double roots obtained from this

factorization are

02)( 22 =+plusmn babxax

0))(()(2)( 222 =plusmnplusmn=plusmn=+plusmn baxbaxbaxbabxax

abx plusmn=21 This method has utilized square rules which are quite easily

observed The forth kind of quadratic equations is type 4 In this equation the coefficient of the -term is 1 with p and q as positive or negative integers Type 4 equations can always be solved by the approaches of completing the square and quadratic formula However they can often be solved by factorization too since the quadratic expressions are factorable by using distributive laws reversely The roots are obtained through finding two factors for q At the same time these two factors become satisfying with the relation so that the sum of factors is equal to That way the two factors become two roots of a quadratic equation Denoting the two factors or two roots as the relationship between coefficient and constant is

02 =++ qpxx2x

qminus

21rr)( 2121 rrprrq +minus=bull= Quadratic equations can be written as the product of

two first-degree binomials or factoring form 0))(()( 212121

22 =minusminus=++minus=++ rxrxrrxrrxqpxx Through the factoring form the roots of this equation are obtained being The core of factorizing type 4 equations is seeking the connections between coefficients and roots which reveal the studentsrsquo basic ability of arithmetic multiplication and addition It is often more effective in this case to use factorization than to use quadratic formula or completing the square Let us have a look at the example In this equation we get

2211 rxrx ==

035122 =+minus xx

18

35)7)(5(12)7()5( =minusminus=minus=minus+minus= qp or 75)75( bull=bullminus= qp (here we can directly get roots) Factorizing the equation we obtain 0)7)(5( =minusminus xx and roots are The procedures of finding factors of the equation often occur in some studentsrsquo minds quickly without writing down the whole computing process but those students often have good skills in arithmetic operations (reference) Compared to using factorization for solving this equation application of quadratic formula seems ineffective A possible problem in this method is how to find the signs before the factors or how to judge if the factors in the parentheses are positive or negative integers In order to check if the solution is correct we can always multiply these two factored binomials by distributive law and see if the expanded polynomial is the same as the original quadratic equation

75 21 == xx

The fifth kind of quadratic equations is type 5 Still the methods of completing squares and quadratic formula can be applied for all type 5 equations but there are cases in which factorization can be used for solving this type of equations within a certain range of integers as mentioned before Sometimes procedures of factorizing such equations can be carried out in our minds quickly like type 4 equations but in many cases they can not As a matter of fact finding two binomial factors for a factorable quadratic equation or a factorable second-degree polynomial is a complicated process and requires systematic work In his article published in Mathematics in School Jackman (2005) demonstrates this systematic procedure via factorizing second-degree polynomials A second-degree polynomial can be written as where are all integers By multiplying out the polynomial becomes thus

0002 nene=++ bacbxax

))((2 srxqpxcbxax ++=++ srqpcbaqsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== Writing the factors of a and c on two lines p q times r s Finding the right pairs of factors for a and c has to be done by listing out the groups of products of different factors for example The factor pairs for 6 are (6 1) (1 6) (3 2) (2 3) and the factor pairs for (-4) are (4 -1) (-4 1) (2 -2) (-2 2) (1 -4) (-1 4) Writing them on two lines p 6 1 3 2 q 4 -4 2 -2 1 -1

456 2 minus+ xx

times r 1 6 2 3 s -1 1 -2 2 -4 4 Then comes calculating the values of ps and qr until the value of b = 5 is obtained The calculating procedures are divided into 4 groups and 6 steps in each group which means 4 times 6 steps in total Here I only give one grouprsquos procedure as an example 6 middot (-1) + 4 middot 1 = -2 6 middot 1 + 1middot (-4) = 2 6 middot (-2) + 1 middot 2 = -10 6 middot 2 + 1middot (-2) = 10 6 middot (-4) + 1 middot 1 = -23 6 middot 4 + 1 middot (-1) = 23 helliphellip After the 24ndashstep calculating procedures two computing procedures obtain the same correct result 3 (ndash1) + 4 2 = 5 (yes) 2 4 + (-1) 3 = 5 (yes) Are they the same From 3 (ndash1) + 4 2 = 5 the polynomial can be factorized into )12)(43( minus+ xx while from 2 4 + (ndash1) 3 = 5 the polynomial can be factorized into )43)(12( +minus xx According to communicative law these two binomial products are the same The shortcoming of this

19

method is that it is too ineffective and complicated The addition of matrixes is almost impossible and unbearable Nevertheless based on the same idea I suggest finding factors by guessing analyzing and checking Using the same example here we can find that one of the binomials must have a negative integer because of ndash4 which implies that 5 is the sum of a positive integer and a negative integer Then looking at a and c the factors of 6 can be (1 6) and (2 3) while the factors of 4 can be (1 4) and (2 2) when we regard them as absolute values We know that the factoring form of the polynomial will be like

and the connections of all coefficients are

456 2 minus+ xx

qsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== The coefficient 5=b is the difference of two products of factors for 6 and 4 First we try the biggest number of product such as 2446 =bull then 524 =minus the unknown is which is a prime number the only factors for 19 are (19 1) There is not any combination of products of factors for 6 and 4 satisfying 19 Therefore the multiplications of factors (1 6) and (1 4) are not included in this case So we move to factors (2 2) (2 3) and (1 4)

19 =

In the second step we try The result is negative it is obviously

4264466)23)(22( 22 minusminus=minus+minus=+minus xxxxxxx2 lt 5 which implies that the combination of (2 3) and (2 2) is not

considerable One pair of factors has to be removed and that is (2 2) which does not influence the coefficient of the -term Then we try The result this time is negative too but we can see that

2x 456)43)(12( 2 minusminus=minus+ xxxxb is 5 which guides us to interchanging the positions of

negative and positive signs in the parentheses The next attempt is and it is correct Seeking the correct factors for the polynomial

has taken four major operating steps by trying the combinations of factorsrsquo products and relating the sum of the products with the value of the x-term coefficient The strategy is to make use of distributive law reversely first and then multiply the product of every pair of parentheses in order to check if they are the right pair of polynomial factors Factorizing second-degree polynomials is the major step for solving quadratic equations After factoring polynomials into the product of two binomials as in the example given above the roots can be obtained by making each binomial equal to zero according to the theory of zero point (ldquonollstaumlllerdquo in Swedish) of a polynomial equation originating from unique factorization theorem

456)43)(12( 2 minus+=+minus xxxx456 2 minus+ xx

Factorization by using the technique of vertically and crosswise sutra Binomial expansion or multiplication of binomials by distributive law and factorization of the second-degree polynomials by using distributive law backwards have an extremely symmetrical effect and the results of two kinds of operations are equivalent It is crucial for the students to be able to perceive mathematics structure and be acquainted with arithmetic operations as well as to understand the relationship between quantities when they carry out factoring procedures The expansion of binomials by distributive law has been named as the method FOIL which stands for first outside inside and last according to multiplication orders of two binomials for example (Nataraj and Thomas 2006) Many students are used to this procedure Another technique to multiply two binomials is called vertically and crosswise sutra origin from the ancient Indian Vedic mathematics Vertically and crosswise sutra and FOIL are much the same in principle but it makes use of

824)4)(2( 2 +++=++ xxxxx

20

the power of visualization and improves facility with the expansion of binomials and the factorization of quadratic expressions In this method two binomials are written into two lines Using the same example the two lines are written as 824)4)(2( 2 +++=++ xxxxx

x + 2 x + 4 ______________ xsup2 4x + 2x 8

The order of multiplication is as shown below x + 2 x + 4

4 3 2 1 (the numbers 1 2 3 and 4 refer to steps) xsup2 + 2x + 4x + 8 Figure 3 The vertically and crosswise method of multiplication of binomials Step 2 and 3 can be interchangeable If we put this form into a line expanded according to the order above would be like

)4)(2( ++ xxxbullrarrbull 442 clockwise and xxx bullrarrbull2 in

counterclockwise direction According to sutra we can imagine drawing a thread starting from 2 then going through 4 and x clockwise and then draw the second thread starting from the same place 2 later going through x in the second bracket and finally x in the first bracket counterclockwise Learning expansion of binomials by the methods FOIL and Vertical and clockwise sutra is the basic process before students study factorization In this study Nataraj and Thomas (2006) have found that the students appear to do factorization using guess-and-check as well as decomposition or the vertical and clockwise approach Their finding is that afterwards the students performed better on algebra questions especially on the factorizations They conclude ldquohellipthe values of the method may lie in what it adds to the studentsrsquo overall algebraic conceptions and knowledge of mathematical structurerdquo (p 16) In general solving these five kinds of quadratic equations by factorization can be traced back to unique factorization theorem The important character in this method is that you can always check if factorization is correct or not through multiplying out the two pairs of a parenthesis and see if the expanded polynomial is the given one or not Using factorization is not only about method but also about fostering the studentsrsquo mathematical thinking The solving processes are more important than solutions

21

3 Review of related literature and research Besides the focus on mathematics content another focus in the background work for this research is about previous studies and research in the field of mathematics didactic concerning different methods for solving quadratic equations and the use of factorization In order to study the previous research and studies in the related field three scientific databases and a search engine have been used and 113 articles have been investigated Topics such as solving quadratic expressions and equations are often discussed as teaching ideas and strategies shared by mathematics educators in different countries from lower and upper-secondary schools to colleges according to the findings in the article search These didactic topics from mathematics teachersrsquo perspectives include for example using geometric approaches to solve quadratic equations (Allaire amp Bradley 2001) utilizing completing squares (Vinogradova 2007) factoring quadratics (Hoffman 1976 Cheng 1980) and using factorization completing the square graphical methods to solve quadratic equations (Macdonald 1986) Kostsopoulos (2007) links cognitive reasoning to pedagogical problems appearing in the mathematics classrooms in her article published in Australian Mathematics Teacher Based on her teaching experiences Kostsopoulos is aware of her pedagogical strategies lacking insight on dealing with studentsrsquo difficulties of multiplication and factorization of quadratics She assumes that ldquostudentsrsquo problems with factorization and with identifying varied representations of the same quadratic relationshiprdquo may be linked to the ways in which the brain constructs cognitive representations (p 19) Factoring of quadratics is the rewriting of polynomials as a product of polynomials It requires students to have both a conceptual understanding of multiplication of polynomials and the procedural knowledge to retrieve basic multiplication facts effectively Based on the theory about the order matters in the brainrsquos ability to retrieve number facts (Phenix amp Campbell 2001) Kostsopoulos conjectures that studentsrsquo ability to access the appropriate long-term semantic memory is limited when they are confused by mixing the order and varied forms of quadratics Although this article is not about a research study it seeks the reason why factorizing quadratics is difficult and therefore might provide math teachers with alternative explanations for similar pedagogical frustrations With Von Glasersfeldrsquos radical constructivism and Peter Gaumlrdenforsrsquo epistemic semantics theory as his theoretical frameworks Rauff (1994) suggests that belief-based teaching can be successful in teaching factoring According to Rauff constructivist learning theory takes the view that students construct their own beliefs and knowledge of mathematics Learning occurs when the students change their belief set Referring to belief set theory from Gaumlrdenfors (1988) Rauff (1994) claims that ldquoa belief set is expanded when a new belief is added to it and contracted when a proposition is no longer believed and is removed from the setrdquo (p 421) Expansion contraction and revision are three components to comprise basic mechanisms of changes of belief according to Rauff When examining and comparing studentsrsquo (ages 18-20) own definitions of factoring Rauff analyzes the errors of their definitions with constructivism and epistemic semantics as analyzing tools The result of this study shows that the approach to teaching factoring rely heavily upon what a student believes about factoring is quite productive in two dimensions First it reveals the source of nonstandard factoring Second it provides a starting point for modification of the studentrsquos underlying conceptions of factoring Rauff proposes that belief-based teaching can also be

22

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

formula today for general quadratic equations which is a

acbbx2

42

21minusplusmnminus

= for a general

quadratic equation (Olteanu 2007) 002 ne=++ acbxax Another French mathematician and philosopher who had strong influence in the history of algebra was Reneacute Descartes (1596-1650) His idea to use the Cartesian system of coordinates which was derived from his Latin name developed both algebra and geometry In his work La geacuteomeacutetrie written in 1637 Descartesndashthrough using numbers to indentify points in a Cartesian coordinatesndashconnected geometrical objects to algebraic numbers and made the classical geometry become analytical geometry He took up the plus and minus sign from the German Cossists of the previous century and also the square-root sign From Descartes the symbol of an unknown was represented by x which led to the modern system of literal symbolism (Derbyshire 2006) Descartes developed the idea of functions in Cartesian coordinates although the idea of function can be traced back to the Islamic mathematician Sharaf al-Din al-Tusi from Persia (Katz 2006) and Klaudius Ptolemaios about 2000 years ago (Olteanu 2007) Descartes declared that every curve in a Cartesian coordinate system has an equivalent equation which can represent the points on the curve or vice versa every equation containing x and y can be represented by a curve through its coordinate points However the word function was introduced by Gottfried Wilhelm Leibniz in 1693 and the definition of function was defined by Leonhard Euler in his work Introductio analysin infinitorium in 1748 Euler had also given the symbol of a denotation as a relying variable (Olteanu 2007) In the 18)(xf th century after the discovery of calculus by Newton and Leibniz algebra went into an area of analysis ndashthe study of limits infinite sequences and series functions derivatives and integrals (Derbyshire 2006) The purely abstract stage-algebra structure Since 17th century algebra has not anymore solely been about finding solutions for different degrees of equations mathematicians have started to integrate algebra with astronomy and physics Johann Kepler and Galileo Galilei were interested in curves and finding a mechanism for representing motion instead of quantity-like numbers But their arguments were not algebraic-symbolic but geometrical It was Fermat and Descartes who were regarded as the fathers of analytic geometry and who showed how to represent a curve described verbally through algebra analytic geometry and gave a mechanism for representing motion Newton in his Principia picked up on Fermat and Descartesrsquo representation and developed the calculus With the invention of the calculus mathematical problems were solved by curves not just points Algebra grew more and more to present paths of motion In order to be able to judge if the algebraic manipulations were correct axioms were formulated for arithmetic applied for algebraic manipulations In the late 18th century Lagrange introduced the idea of permutations into the search for solutions Therafter Galois developed methods for determining what conditions polynomial equations need to meet to be solvable New abstract algebraic concepts were found like ldquofieldrdquo and ldquogrouprdquo during the early 19th century In 1854 Ceyley gave an axiomatic definition of a group During the 1890s this definition entered textbooks along with the axiomatic definition of a field an idea which had roots in the work by Galois By the beginning of the 20th century algebra became less about finding solutions to equations and more about looking for common structures in many mathematical objects defined by sets of axioms (Katz 2006)

11

In short algebra development took 4000 years It took about 3500 years for mathematicians to start to use algebra symbols in systems Just almost 100 years ago algebra developed into a purely abstract science The long history may give both mathematics educators and students some explanations to why it is so difficult to teach and learn algebra

22 Three solving methods for quadratic equations Already in the previous part I have presented geometrically and rhetorically solving quadratic equations from a historical perspective In order to get more explicit understanding of quadratic equations quadratic formula and factorization from mathematics didactic point of view this part presents three common methods for solving quadratic equations 1) Completing the square 2) Factorization 3) Quadratic formula The reason to present these three methods is that they are the common topics for discussion in the articles searched in this area concerning mathematics education at upper-secondary school 1) Completing the square In two articles Solving quadratic equations by completing squares (Vinogradova 2007) and Geometric approaches to quadratic equations from other times and places (Allaire amp Bradley 2001) the authors advise mathematics teachers to use the ideas of ancient mathematician al-Khwarizmi and his method of completing quadratic equations by geometrical algebra and simplifying it in order to represent it to the students By doing so the teachers relate mathematics quantity to physical objects in a visual way Didactically teachers may start with a concrete example in which students can understand the content visually Later from the concrete example the teachers lead the students to another example expressed in algebraic symbols for example to study a rectangle whose area is )10( +xx and supposing this area is 39 this becomes 39)10( =+xx Four steps are followed to build a new square which is actually called completing the square x 10

Figure 6 x 5

Figure 7

x

x

5 5

12

A Begin with a rectangle of the area )10( +xx that is with the short side as x and the long side )10( +x The shaded area is 10bullx (Figure 6)

B Divide the shaded rectangle into two small shaded rectangles with the size 5x each and move them to each adjacent side of the xsup2 square (Figure 7) The total area is still

x10

)10(39 += xxC In Figure 7 a new square is shaped in the lower-right hand corner with the side 5 or

area 25 By adding this small square to the diagram the large square with an area of is ldquocompletedrdquo Therefore we can write 25)5(22 ++ xx 25)10(2539 ++=+ xx or

The large squarersquos area is now because the side has a length of Therefore

6425)5(22 =++ xx 2)5( +x5+x 38564)5( 2 =rArr=+rArr=+ xxx

After these three steps a solution to this quadratic equation is derived by completing the square (Allaire amp Bradley 2001) 2) Factorization Hoffman (1976) presents three approaches (A B and C) for factorizing quadratic expressions through observing and grouping He takes a second-degree polynomial as an example

273 2 ++ xx

a The inspection method 1) suggests factors of and 23x x3 x 2) 2 suggests factors of 1 and 2 or ndash1 and ndash2 3) Checking the linear term shows that the correct factorization is x7 )2)(13(273 2 ++=++ xxxx

This method according to Hoffman (1976) is based on the studentrsquos ability to simplify by inspecting the indicated product of a pair of linear expressions

b The method of decomposition of the linear term 1) Multiply and 2 to get 23x 26x2) Decompose into the sum of two terms whose product is x7 26x

This gives xxx += 67 3) Factorize by grouping 2)6(3 2 +++ xxx

2)6(3 2 +++ xxx = )

)2()63( 2 +++ xxx

= 2(1)2(3 +++ xxx= )13)(2( ++ xx

c Third method 1) Multiply 3 and 2 to get 6 2) Find two numbers whose product is 6 and whose sum is 7 The numbers are 6 and 1 3) 273 2 ++ xx

=3

)13)(63( ++ xx

= )13)(2( ++ xx

13

These three approaches can not actually be divided as three unrelated approaches especially the third method is seemingly unclear What Hoffman (1976) wants to demonstrate is that all the three approaches are based on observing the relationship among the coefficients and the constant of the polynomial Through finding the product and the sum of two factors the polynomial can be factorized in the product of two binomials In the authorrsquos opinion factorization is heavily dependent on skill in multiplication and using distributive law reversely He means that ldquostudents who do not have reasonably strong skills in multiplication should not be expected to develop strong skills in factorizationrdquo (p 55) Factorizing polynomials is not only for simplifying high-degree polynomials (for example the third or fourth-degree polynomials) but can also be applied for solving quadratic equations Supposing this polynomial forming a quadratic equation Solving this equation is actually to decompose the polynomial into a factoringrsquos form

0273 2 =++ xx0)13)(2( =++ xx

which is equivalent to the polynomial form The essential step to solve this equation is to follow the zero-factor propertyndashalso called null-factor law (nollprodukt in Swedish)ndashthat is for real numbers p and q p bull q = 0 if (and only if) p = 0 or q = 0 It means that either of the binomials on the left side of the equation has to be equal to zero in order to satisfy the

equivalent relation of this quadratic equation The solutions of two roots or 2minus=x31

minus=x

are obtained through making either 02 =+x or 013 =+x 3) Quadratic formula Solving quadratics equations by factorization is constrained within the simple quadratic equations over the whole integers and rational numbers as coefficients What happens when quadratic equations have coefficients that belong to an irrational domain or big quantity The quadratic formula is a solution to such a situation As mentioned previously historically the French mathematician Franccedilois Vieacutete discovered the quadratic formula

aacbbx

242 minusplusmnminus

= ) which is applicable for all kinds of quadratic equations In

Swedish textbooks this formula is written in PQ form by making for the equation

that is

0( nea

1=a

02 =++ qpxx qppx minus⎟⎠⎞

⎜⎝⎛plusmnminus=

2

22 with

abp = and

acq = In Oleatursquos study (2007)

studentsrsquo difficulties in using algebraic symbols no longer is the problem But on the other hand handling the parameters or coefficients in quadratic equations like and rewrite them in the equivalent form with p and q being real numbers becomes obstacles for the students when they study the algebra course at upper-secondary school

02 =++ cbxax02 =++ qpxx

The quadratic formula a

acbbx2

42 minusplusmnminus= has been regarded as the standard method

according to US mathematics teachers Obermeyer (1982) presents a way of how to derive this formula by completing a square through 11 steps (I will not present them here) At the same time an alternate method is also introduced according to the same principle All the methods mentioned so far are based on the square rules The standard method or PQ form might be regarded as an efficient and direct method but it may

222 2)( bababa +plusmn=plusmn

14

lead students to solve quadratic equations in a mechanical way besides having problems with the troublesome plusmn symbolism (Stover 1978) Both Stover (1978) and Olteanu (2007) have suggested that using a graph of quadratic functions to solve quadratic equations is another alternative method

23 Mathematical background of factorization Factorization in algebra structure Among the three solving methods presented above both the geometrical method and quadratic formula can be traced back to algebra history in section 21 What is the trace of factorization in algebra history Factorization was at the last stage in algebra history and belonged to algebra structure (Derbyshire 2006) In Swedish mathematics textbooks the chapter on algebra often includes polynomial factorization and quadratic equations But the factorization in the textbooks is not at the same abstract level as the factorization of algebra structure although they share the same axioms and there are certain connections between these two different levels of factorization How are polynomial and quadratic equations as well as factorization related to each other In order to find the theoretical background of factorizing polynomials this part constitutes a mathematical theoretical review by referring to two books concerning algebra structure (Vretblad 2000 Durbin 1992) and a research study (Bosseacute and Nandakumar 2005) A quadratic equation can be regarded as polynomial equation Solving quadratic equations has much to do with ldquounique factorization theoremrdquo The polynomial

is a second-degree polynomial A polynomial consists of coefficients in whole numbers rational numbers real numbers or complex numbers and variables (sometimes called unknowns) in different degrees as well as constants A polynomial

has symbols as coefficients which are real numbers If all are zero and so

)0(02 ne=++ acbxax

cbxax ++2

nnxaxaxaaxf ++++= )( 2

210 naaa 10

ka 1gek 0)( axf = is constant If all are zero then we say the is a zero polynomial If meaning that is not a zero polynomial and n is the biggest number and then we say that the polynomial is at degrees A zero polynomial has no degree at all According to the definition (Vredblad 2000) if is a polynomial the equation is called an algebraic equation or polynomial equation A solution or a root of this equation is a number

kaf 0nef f

0nena f nf

0)( =xfα so that 0)( =αf Such a number is even

called a zero point (nollstaumllle in Swedish) of the polynomial which means when f α=x the value of the polynomial is zero (p 161) Every polynomial whose degree is equal or greater than 1 can be written as a product of irreducible polynomial In this way through polynomial division a polynomial with n degrees can be expressed from a sum of terms into a product of its denominator and its quotient The process of changing the expressions is factorization for

instance 12

22

+=minusminusminus x

xxx The factoring form is )1)(2(22 minusminus=minusminus xxxx

Factorization of polynomials is analogue to factorization of whole numbers which is very much related to the concept of ring A ring consists of a set with two operations which are a sum and a product of two elements over a field Any polynomial from zero polynomial to n degree-polynomials can be written or operated through addition and multiplication of the

15

elements over a field F including whole numbers or integers rational numbers real numbers and complex numbers In a ring you can multiply any two elements but you canrsquot always divide Factorization is a way to deal with this fact Here are some examples In the ring of integers Z is an example of factorization in some polynomial rings for instance Q[x] the ring of polynomial in one variable with rational numbers as coefficients such as

7214 sdot=

)451)(2

51(8

52

251 2 +minus=minus+ xxxx is factorization Z[x] the ring of polynomial in one

variable with whole numbers as integers such as is factorization )1)(2(22 minusminus=minusminus xxxx The polynomial ring R[x] means the ring of polynomial in one variable with real numbers as coefficients Factorizing of such polynomials implies that the polynomial factors are the elements in the ring of R[x] for instance )2)(2(22 minus+=minus xxx The same thing is true for C[x] the ring of polynomial in one variable with complex numbers as coefficients Thus it is coefficients of polynomials that determine which ring a polynomial belongs to and if it is possible to be factorized in this ring For example number 6 can be factorized in two ways

in the ring of Z and 32times )51)(51( ii +minus in the ring of C (complex numbers) However there are cases in which divisions can not be done within the ring for example 52 or

)3()52( 2 ++minus xxx which means that no element in the ring can multiply with a denominator obtaining the numerator Therefore neither 5 nor can be factorized 522 +minus xx If a polynomial of degree at least one has no other divisors then it is considered to be irreducible or prime (Durbin 1992) Of every polynomial of degree at least one can be written as a product of irreducible polynomials according to unique factorization theorem each polynomial of degree at least one over a field F can be written as an element of F times a product of monic irreducible polynomials over F and except for the order in which these irreducible polynomials are written this can be done in only one way (p 221) Thus a factorable polynomial can be written into a product with another polynomial such as In other words the condition that )()()( xqaxxf minus= )( ax minus is a factor of a polynomial

in a variable x is only true if )(xf 0)( =af For example if and

6)( 2 minus+= xxxf0624)2( =minus+=f 0639)3( =minusminus=minusf the factors of are and )(xf )2( minusx )3( +x

The value of the polynomial is zero which is also expressed as the polynomial having a zero point (nollstaumllle in Swedish) when

)(xfα=x

Based on unique factorization theorem finding solutions of an algebraic equation is equivalent to finding the first-degree factors for a polynomial (Vredblad 2000) In this way solving a quadratic equation can be handled through finding the first-degree factors for the second-degree polynomial and there after finding the roots of the quadratic equation This is often called using factorization to solve quadratic equations Besides unique factorization theorem the connection of using factorization to solve a quadratic equation is that this method relates to prime numbers using distributive law backwards commutative law multiplication division and operating fractions As matter of fact it is about application of factorizing integers in polynomials at upper-secondary school level Since quadratic equations in the mathematics B course at Swedish upper-secondary school are limited to real numbers and often within the whole numbers (integers) many quadratic equations should be able to be solved by factorization according to my own

16

judgment Therefore using factorization for solving quadratic equations is very much related to arithmetic operations number concept and algebra structure It may offer students both mathematical conceptual understanding and operational skill For algebra beginners factorization can be traced back to studentsrsquo early years of mathematics education during which students do not only work with multiplication but also on factoring integers for example 56 = 8 middot 7 = 2 middot 2 middot 2 middot 7 = 4 middot 14 = 2 middot 28 In this example two operations are included multiplication and division Multiplication operation may help students to understand the multiplicative structure of the integers while factoring integers may become meaningful when it is related to divisibility divisors and prime numbers These two operational skills or competence are major techniques for simplifying fractions and finding common denominators Being able to understand and operate factoring integers has laid the essential grounds for algebra beginners to study factorization of polynomials Thus factorization reunites studentsrsquo early mathematics knowledge of factoring integers with later knowledge of algebra structure Factorability In a study on factorability by Bosseacute and Nandakumar (2005) the probability of factorability is reasoned through investigating the range of integers as coefficients of quadratic equations In the quadratic equation a b and c are randomly generated integers within a determined range r such as

002 ne=++ acbxaxyrx lele where x and y are integers When is a

perfect square or where

acb 42 minus

)4( 2 acb minus is an integer quadratics are factorable According to the studyrsquos data it shows that as the range for a b and c increases the probability of factorability of a quadratic with randomly selected integer values for a b and c decreases Bosseacute and Nandakumar confirm that as the range for coefficients expands to infinltltinfinminus r the probablility of factorability of a quadratic with randomly selected integer values for coefficients approaches zero The studyrsquos data collected from college algebra courses and textbooks demonstrates that about 15 of quadratics with integer coefficients [ ]1010minusisinr are factorable Thus about 85 of quadratics can not be factorized In spite of the limitation of factorability within the exercises concerning factoring practice chosen from 27 surveyed college algebra textbooks about 94 of the problems were factorable Within the textbooks 55 of the quadratics to be factored were within the range [-10 10] Even if many quadratic equations are factorable choosing the right factor pairs is time consuming and unnecessary for example

has to be factorized through choosing the right pairs of factors among nine pairs for 36 and 8 pairs of factors for 24 Comparing the three different methods for solving quadratic equations the study suggests that completing the square and quadratic formula are more effective informative and useful than factorization

0245936 2 =++ xx

231 Different kinds of factorizations for solving different quadratic equations How can we find the binomial factors of a quadratic equation or a second-degree polynomial without knowing one of the divisors as a binomial factor The answer is that we have to observe the relations among all coefficients and constants as well as the positive and negative signs before them under the condition of being factorable Most of the coefficients in quadratic equations or polynomials among exercises provided for practicing factoring are integers (here I borrow the same notation r as the one used by Bosseacute and Nandakumar) with

17

their range between ndash10 and 10 that is 10ler In this case it is possible for a student to make a judgment of whether or not a quadratic equation is factorable Five different kinds of factorization depending on five kinds of quadratic equations Factoring quadratic expressions can be divided into five different kinds depending on the type of quadratic equations The coefficients a b p q and constant c illustrated in the following quadratic equations are defined not to be equal to zero thus 000 nenene cba The first kind of quadratic equations is type 1 the obviously factorable type (Bosseacute and Nandakumar 2004) then the equation is factorized as

where the roots are

0002 nene=+ babxax

0)( =+ baxxabxx minus== 21 0

The second kind of quadratic equations is type 2 The factors of this second degree polynomial are so the roots to this type of quadratic

equations are

0)( 22 =minusbax))(()( 22 baxbaxbax minus+=minus

bax plusmn=21 This kind of factorization is based on the difference-of-squares

formula for example can be solved by factorizing into 049 2 =minusx 0)23)(23( =minus+ xx Thus

the roots are 32

32

21 =minus= xx This kind of factorization is also obviously factorable

The third kind of quadratic equations is type 3 This type of equations can be solved directly by factoring into two identical binomials

The double roots obtained from this

factorization are

02)( 22 =+plusmn babxax

0))(()(2)( 222 =plusmnplusmn=plusmn=+plusmn baxbaxbaxbabxax

abx plusmn=21 This method has utilized square rules which are quite easily

observed The forth kind of quadratic equations is type 4 In this equation the coefficient of the -term is 1 with p and q as positive or negative integers Type 4 equations can always be solved by the approaches of completing the square and quadratic formula However they can often be solved by factorization too since the quadratic expressions are factorable by using distributive laws reversely The roots are obtained through finding two factors for q At the same time these two factors become satisfying with the relation so that the sum of factors is equal to That way the two factors become two roots of a quadratic equation Denoting the two factors or two roots as the relationship between coefficient and constant is

02 =++ qpxx2x

qminus

21rr)( 2121 rrprrq +minus=bull= Quadratic equations can be written as the product of

two first-degree binomials or factoring form 0))(()( 212121

22 =minusminus=++minus=++ rxrxrrxrrxqpxx Through the factoring form the roots of this equation are obtained being The core of factorizing type 4 equations is seeking the connections between coefficients and roots which reveal the studentsrsquo basic ability of arithmetic multiplication and addition It is often more effective in this case to use factorization than to use quadratic formula or completing the square Let us have a look at the example In this equation we get

2211 rxrx ==

035122 =+minus xx

18

35)7)(5(12)7()5( =minusminus=minus=minus+minus= qp or 75)75( bull=bullminus= qp (here we can directly get roots) Factorizing the equation we obtain 0)7)(5( =minusminus xx and roots are The procedures of finding factors of the equation often occur in some studentsrsquo minds quickly without writing down the whole computing process but those students often have good skills in arithmetic operations (reference) Compared to using factorization for solving this equation application of quadratic formula seems ineffective A possible problem in this method is how to find the signs before the factors or how to judge if the factors in the parentheses are positive or negative integers In order to check if the solution is correct we can always multiply these two factored binomials by distributive law and see if the expanded polynomial is the same as the original quadratic equation

75 21 == xx

The fifth kind of quadratic equations is type 5 Still the methods of completing squares and quadratic formula can be applied for all type 5 equations but there are cases in which factorization can be used for solving this type of equations within a certain range of integers as mentioned before Sometimes procedures of factorizing such equations can be carried out in our minds quickly like type 4 equations but in many cases they can not As a matter of fact finding two binomial factors for a factorable quadratic equation or a factorable second-degree polynomial is a complicated process and requires systematic work In his article published in Mathematics in School Jackman (2005) demonstrates this systematic procedure via factorizing second-degree polynomials A second-degree polynomial can be written as where are all integers By multiplying out the polynomial becomes thus

0002 nene=++ bacbxax

))((2 srxqpxcbxax ++=++ srqpcbaqsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== Writing the factors of a and c on two lines p q times r s Finding the right pairs of factors for a and c has to be done by listing out the groups of products of different factors for example The factor pairs for 6 are (6 1) (1 6) (3 2) (2 3) and the factor pairs for (-4) are (4 -1) (-4 1) (2 -2) (-2 2) (1 -4) (-1 4) Writing them on two lines p 6 1 3 2 q 4 -4 2 -2 1 -1

456 2 minus+ xx

times r 1 6 2 3 s -1 1 -2 2 -4 4 Then comes calculating the values of ps and qr until the value of b = 5 is obtained The calculating procedures are divided into 4 groups and 6 steps in each group which means 4 times 6 steps in total Here I only give one grouprsquos procedure as an example 6 middot (-1) + 4 middot 1 = -2 6 middot 1 + 1middot (-4) = 2 6 middot (-2) + 1 middot 2 = -10 6 middot 2 + 1middot (-2) = 10 6 middot (-4) + 1 middot 1 = -23 6 middot 4 + 1 middot (-1) = 23 helliphellip After the 24ndashstep calculating procedures two computing procedures obtain the same correct result 3 (ndash1) + 4 2 = 5 (yes) 2 4 + (-1) 3 = 5 (yes) Are they the same From 3 (ndash1) + 4 2 = 5 the polynomial can be factorized into )12)(43( minus+ xx while from 2 4 + (ndash1) 3 = 5 the polynomial can be factorized into )43)(12( +minus xx According to communicative law these two binomial products are the same The shortcoming of this

19

method is that it is too ineffective and complicated The addition of matrixes is almost impossible and unbearable Nevertheless based on the same idea I suggest finding factors by guessing analyzing and checking Using the same example here we can find that one of the binomials must have a negative integer because of ndash4 which implies that 5 is the sum of a positive integer and a negative integer Then looking at a and c the factors of 6 can be (1 6) and (2 3) while the factors of 4 can be (1 4) and (2 2) when we regard them as absolute values We know that the factoring form of the polynomial will be like

and the connections of all coefficients are

456 2 minus+ xx

qsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== The coefficient 5=b is the difference of two products of factors for 6 and 4 First we try the biggest number of product such as 2446 =bull then 524 =minus the unknown is which is a prime number the only factors for 19 are (19 1) There is not any combination of products of factors for 6 and 4 satisfying 19 Therefore the multiplications of factors (1 6) and (1 4) are not included in this case So we move to factors (2 2) (2 3) and (1 4)

19 =

In the second step we try The result is negative it is obviously

4264466)23)(22( 22 minusminus=minus+minus=+minus xxxxxxx2 lt 5 which implies that the combination of (2 3) and (2 2) is not

considerable One pair of factors has to be removed and that is (2 2) which does not influence the coefficient of the -term Then we try The result this time is negative too but we can see that

2x 456)43)(12( 2 minusminus=minus+ xxxxb is 5 which guides us to interchanging the positions of

negative and positive signs in the parentheses The next attempt is and it is correct Seeking the correct factors for the polynomial

has taken four major operating steps by trying the combinations of factorsrsquo products and relating the sum of the products with the value of the x-term coefficient The strategy is to make use of distributive law reversely first and then multiply the product of every pair of parentheses in order to check if they are the right pair of polynomial factors Factorizing second-degree polynomials is the major step for solving quadratic equations After factoring polynomials into the product of two binomials as in the example given above the roots can be obtained by making each binomial equal to zero according to the theory of zero point (ldquonollstaumlllerdquo in Swedish) of a polynomial equation originating from unique factorization theorem

456)43)(12( 2 minus+=+minus xxxx456 2 minus+ xx

Factorization by using the technique of vertically and crosswise sutra Binomial expansion or multiplication of binomials by distributive law and factorization of the second-degree polynomials by using distributive law backwards have an extremely symmetrical effect and the results of two kinds of operations are equivalent It is crucial for the students to be able to perceive mathematics structure and be acquainted with arithmetic operations as well as to understand the relationship between quantities when they carry out factoring procedures The expansion of binomials by distributive law has been named as the method FOIL which stands for first outside inside and last according to multiplication orders of two binomials for example (Nataraj and Thomas 2006) Many students are used to this procedure Another technique to multiply two binomials is called vertically and crosswise sutra origin from the ancient Indian Vedic mathematics Vertically and crosswise sutra and FOIL are much the same in principle but it makes use of

824)4)(2( 2 +++=++ xxxxx

20

the power of visualization and improves facility with the expansion of binomials and the factorization of quadratic expressions In this method two binomials are written into two lines Using the same example the two lines are written as 824)4)(2( 2 +++=++ xxxxx

x + 2 x + 4 ______________ xsup2 4x + 2x 8

The order of multiplication is as shown below x + 2 x + 4

4 3 2 1 (the numbers 1 2 3 and 4 refer to steps) xsup2 + 2x + 4x + 8 Figure 3 The vertically and crosswise method of multiplication of binomials Step 2 and 3 can be interchangeable If we put this form into a line expanded according to the order above would be like

)4)(2( ++ xxxbullrarrbull 442 clockwise and xxx bullrarrbull2 in

counterclockwise direction According to sutra we can imagine drawing a thread starting from 2 then going through 4 and x clockwise and then draw the second thread starting from the same place 2 later going through x in the second bracket and finally x in the first bracket counterclockwise Learning expansion of binomials by the methods FOIL and Vertical and clockwise sutra is the basic process before students study factorization In this study Nataraj and Thomas (2006) have found that the students appear to do factorization using guess-and-check as well as decomposition or the vertical and clockwise approach Their finding is that afterwards the students performed better on algebra questions especially on the factorizations They conclude ldquohellipthe values of the method may lie in what it adds to the studentsrsquo overall algebraic conceptions and knowledge of mathematical structurerdquo (p 16) In general solving these five kinds of quadratic equations by factorization can be traced back to unique factorization theorem The important character in this method is that you can always check if factorization is correct or not through multiplying out the two pairs of a parenthesis and see if the expanded polynomial is the given one or not Using factorization is not only about method but also about fostering the studentsrsquo mathematical thinking The solving processes are more important than solutions

21

3 Review of related literature and research Besides the focus on mathematics content another focus in the background work for this research is about previous studies and research in the field of mathematics didactic concerning different methods for solving quadratic equations and the use of factorization In order to study the previous research and studies in the related field three scientific databases and a search engine have been used and 113 articles have been investigated Topics such as solving quadratic expressions and equations are often discussed as teaching ideas and strategies shared by mathematics educators in different countries from lower and upper-secondary schools to colleges according to the findings in the article search These didactic topics from mathematics teachersrsquo perspectives include for example using geometric approaches to solve quadratic equations (Allaire amp Bradley 2001) utilizing completing squares (Vinogradova 2007) factoring quadratics (Hoffman 1976 Cheng 1980) and using factorization completing the square graphical methods to solve quadratic equations (Macdonald 1986) Kostsopoulos (2007) links cognitive reasoning to pedagogical problems appearing in the mathematics classrooms in her article published in Australian Mathematics Teacher Based on her teaching experiences Kostsopoulos is aware of her pedagogical strategies lacking insight on dealing with studentsrsquo difficulties of multiplication and factorization of quadratics She assumes that ldquostudentsrsquo problems with factorization and with identifying varied representations of the same quadratic relationshiprdquo may be linked to the ways in which the brain constructs cognitive representations (p 19) Factoring of quadratics is the rewriting of polynomials as a product of polynomials It requires students to have both a conceptual understanding of multiplication of polynomials and the procedural knowledge to retrieve basic multiplication facts effectively Based on the theory about the order matters in the brainrsquos ability to retrieve number facts (Phenix amp Campbell 2001) Kostsopoulos conjectures that studentsrsquo ability to access the appropriate long-term semantic memory is limited when they are confused by mixing the order and varied forms of quadratics Although this article is not about a research study it seeks the reason why factorizing quadratics is difficult and therefore might provide math teachers with alternative explanations for similar pedagogical frustrations With Von Glasersfeldrsquos radical constructivism and Peter Gaumlrdenforsrsquo epistemic semantics theory as his theoretical frameworks Rauff (1994) suggests that belief-based teaching can be successful in teaching factoring According to Rauff constructivist learning theory takes the view that students construct their own beliefs and knowledge of mathematics Learning occurs when the students change their belief set Referring to belief set theory from Gaumlrdenfors (1988) Rauff (1994) claims that ldquoa belief set is expanded when a new belief is added to it and contracted when a proposition is no longer believed and is removed from the setrdquo (p 421) Expansion contraction and revision are three components to comprise basic mechanisms of changes of belief according to Rauff When examining and comparing studentsrsquo (ages 18-20) own definitions of factoring Rauff analyzes the errors of their definitions with constructivism and epistemic semantics as analyzing tools The result of this study shows that the approach to teaching factoring rely heavily upon what a student believes about factoring is quite productive in two dimensions First it reveals the source of nonstandard factoring Second it provides a starting point for modification of the studentrsquos underlying conceptions of factoring Rauff proposes that belief-based teaching can also be

22

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

In short algebra development took 4000 years It took about 3500 years for mathematicians to start to use algebra symbols in systems Just almost 100 years ago algebra developed into a purely abstract science The long history may give both mathematics educators and students some explanations to why it is so difficult to teach and learn algebra

22 Three solving methods for quadratic equations Already in the previous part I have presented geometrically and rhetorically solving quadratic equations from a historical perspective In order to get more explicit understanding of quadratic equations quadratic formula and factorization from mathematics didactic point of view this part presents three common methods for solving quadratic equations 1) Completing the square 2) Factorization 3) Quadratic formula The reason to present these three methods is that they are the common topics for discussion in the articles searched in this area concerning mathematics education at upper-secondary school 1) Completing the square In two articles Solving quadratic equations by completing squares (Vinogradova 2007) and Geometric approaches to quadratic equations from other times and places (Allaire amp Bradley 2001) the authors advise mathematics teachers to use the ideas of ancient mathematician al-Khwarizmi and his method of completing quadratic equations by geometrical algebra and simplifying it in order to represent it to the students By doing so the teachers relate mathematics quantity to physical objects in a visual way Didactically teachers may start with a concrete example in which students can understand the content visually Later from the concrete example the teachers lead the students to another example expressed in algebraic symbols for example to study a rectangle whose area is )10( +xx and supposing this area is 39 this becomes 39)10( =+xx Four steps are followed to build a new square which is actually called completing the square x 10

Figure 6 x 5

Figure 7

x

x

5 5

12

A Begin with a rectangle of the area )10( +xx that is with the short side as x and the long side )10( +x The shaded area is 10bullx (Figure 6)

B Divide the shaded rectangle into two small shaded rectangles with the size 5x each and move them to each adjacent side of the xsup2 square (Figure 7) The total area is still

x10

)10(39 += xxC In Figure 7 a new square is shaped in the lower-right hand corner with the side 5 or

area 25 By adding this small square to the diagram the large square with an area of is ldquocompletedrdquo Therefore we can write 25)5(22 ++ xx 25)10(2539 ++=+ xx or

The large squarersquos area is now because the side has a length of Therefore

6425)5(22 =++ xx 2)5( +x5+x 38564)5( 2 =rArr=+rArr=+ xxx

After these three steps a solution to this quadratic equation is derived by completing the square (Allaire amp Bradley 2001) 2) Factorization Hoffman (1976) presents three approaches (A B and C) for factorizing quadratic expressions through observing and grouping He takes a second-degree polynomial as an example

273 2 ++ xx

a The inspection method 1) suggests factors of and 23x x3 x 2) 2 suggests factors of 1 and 2 or ndash1 and ndash2 3) Checking the linear term shows that the correct factorization is x7 )2)(13(273 2 ++=++ xxxx

This method according to Hoffman (1976) is based on the studentrsquos ability to simplify by inspecting the indicated product of a pair of linear expressions

b The method of decomposition of the linear term 1) Multiply and 2 to get 23x 26x2) Decompose into the sum of two terms whose product is x7 26x

This gives xxx += 67 3) Factorize by grouping 2)6(3 2 +++ xxx

2)6(3 2 +++ xxx = )

)2()63( 2 +++ xxx

= 2(1)2(3 +++ xxx= )13)(2( ++ xx

c Third method 1) Multiply 3 and 2 to get 6 2) Find two numbers whose product is 6 and whose sum is 7 The numbers are 6 and 1 3) 273 2 ++ xx

=3

)13)(63( ++ xx

= )13)(2( ++ xx

13

These three approaches can not actually be divided as three unrelated approaches especially the third method is seemingly unclear What Hoffman (1976) wants to demonstrate is that all the three approaches are based on observing the relationship among the coefficients and the constant of the polynomial Through finding the product and the sum of two factors the polynomial can be factorized in the product of two binomials In the authorrsquos opinion factorization is heavily dependent on skill in multiplication and using distributive law reversely He means that ldquostudents who do not have reasonably strong skills in multiplication should not be expected to develop strong skills in factorizationrdquo (p 55) Factorizing polynomials is not only for simplifying high-degree polynomials (for example the third or fourth-degree polynomials) but can also be applied for solving quadratic equations Supposing this polynomial forming a quadratic equation Solving this equation is actually to decompose the polynomial into a factoringrsquos form

0273 2 =++ xx0)13)(2( =++ xx

which is equivalent to the polynomial form The essential step to solve this equation is to follow the zero-factor propertyndashalso called null-factor law (nollprodukt in Swedish)ndashthat is for real numbers p and q p bull q = 0 if (and only if) p = 0 or q = 0 It means that either of the binomials on the left side of the equation has to be equal to zero in order to satisfy the

equivalent relation of this quadratic equation The solutions of two roots or 2minus=x31

minus=x

are obtained through making either 02 =+x or 013 =+x 3) Quadratic formula Solving quadratics equations by factorization is constrained within the simple quadratic equations over the whole integers and rational numbers as coefficients What happens when quadratic equations have coefficients that belong to an irrational domain or big quantity The quadratic formula is a solution to such a situation As mentioned previously historically the French mathematician Franccedilois Vieacutete discovered the quadratic formula

aacbbx

242 minusplusmnminus

= ) which is applicable for all kinds of quadratic equations In

Swedish textbooks this formula is written in PQ form by making for the equation

that is

0( nea

1=a

02 =++ qpxx qppx minus⎟⎠⎞

⎜⎝⎛plusmnminus=

2

22 with

abp = and

acq = In Oleatursquos study (2007)

studentsrsquo difficulties in using algebraic symbols no longer is the problem But on the other hand handling the parameters or coefficients in quadratic equations like and rewrite them in the equivalent form with p and q being real numbers becomes obstacles for the students when they study the algebra course at upper-secondary school

02 =++ cbxax02 =++ qpxx

The quadratic formula a

acbbx2

42 minusplusmnminus= has been regarded as the standard method

according to US mathematics teachers Obermeyer (1982) presents a way of how to derive this formula by completing a square through 11 steps (I will not present them here) At the same time an alternate method is also introduced according to the same principle All the methods mentioned so far are based on the square rules The standard method or PQ form might be regarded as an efficient and direct method but it may

222 2)( bababa +plusmn=plusmn

14

lead students to solve quadratic equations in a mechanical way besides having problems with the troublesome plusmn symbolism (Stover 1978) Both Stover (1978) and Olteanu (2007) have suggested that using a graph of quadratic functions to solve quadratic equations is another alternative method

23 Mathematical background of factorization Factorization in algebra structure Among the three solving methods presented above both the geometrical method and quadratic formula can be traced back to algebra history in section 21 What is the trace of factorization in algebra history Factorization was at the last stage in algebra history and belonged to algebra structure (Derbyshire 2006) In Swedish mathematics textbooks the chapter on algebra often includes polynomial factorization and quadratic equations But the factorization in the textbooks is not at the same abstract level as the factorization of algebra structure although they share the same axioms and there are certain connections between these two different levels of factorization How are polynomial and quadratic equations as well as factorization related to each other In order to find the theoretical background of factorizing polynomials this part constitutes a mathematical theoretical review by referring to two books concerning algebra structure (Vretblad 2000 Durbin 1992) and a research study (Bosseacute and Nandakumar 2005) A quadratic equation can be regarded as polynomial equation Solving quadratic equations has much to do with ldquounique factorization theoremrdquo The polynomial

is a second-degree polynomial A polynomial consists of coefficients in whole numbers rational numbers real numbers or complex numbers and variables (sometimes called unknowns) in different degrees as well as constants A polynomial

has symbols as coefficients which are real numbers If all are zero and so

)0(02 ne=++ acbxax

cbxax ++2

nnxaxaxaaxf ++++= )( 2

210 naaa 10

ka 1gek 0)( axf = is constant If all are zero then we say the is a zero polynomial If meaning that is not a zero polynomial and n is the biggest number and then we say that the polynomial is at degrees A zero polynomial has no degree at all According to the definition (Vredblad 2000) if is a polynomial the equation is called an algebraic equation or polynomial equation A solution or a root of this equation is a number

kaf 0nef f

0nena f nf

0)( =xfα so that 0)( =αf Such a number is even

called a zero point (nollstaumllle in Swedish) of the polynomial which means when f α=x the value of the polynomial is zero (p 161) Every polynomial whose degree is equal or greater than 1 can be written as a product of irreducible polynomial In this way through polynomial division a polynomial with n degrees can be expressed from a sum of terms into a product of its denominator and its quotient The process of changing the expressions is factorization for

instance 12

22

+=minusminusminus x

xxx The factoring form is )1)(2(22 minusminus=minusminus xxxx

Factorization of polynomials is analogue to factorization of whole numbers which is very much related to the concept of ring A ring consists of a set with two operations which are a sum and a product of two elements over a field Any polynomial from zero polynomial to n degree-polynomials can be written or operated through addition and multiplication of the

15

elements over a field F including whole numbers or integers rational numbers real numbers and complex numbers In a ring you can multiply any two elements but you canrsquot always divide Factorization is a way to deal with this fact Here are some examples In the ring of integers Z is an example of factorization in some polynomial rings for instance Q[x] the ring of polynomial in one variable with rational numbers as coefficients such as

7214 sdot=

)451)(2

51(8

52

251 2 +minus=minus+ xxxx is factorization Z[x] the ring of polynomial in one

variable with whole numbers as integers such as is factorization )1)(2(22 minusminus=minusminus xxxx The polynomial ring R[x] means the ring of polynomial in one variable with real numbers as coefficients Factorizing of such polynomials implies that the polynomial factors are the elements in the ring of R[x] for instance )2)(2(22 minus+=minus xxx The same thing is true for C[x] the ring of polynomial in one variable with complex numbers as coefficients Thus it is coefficients of polynomials that determine which ring a polynomial belongs to and if it is possible to be factorized in this ring For example number 6 can be factorized in two ways

in the ring of Z and 32times )51)(51( ii +minus in the ring of C (complex numbers) However there are cases in which divisions can not be done within the ring for example 52 or

)3()52( 2 ++minus xxx which means that no element in the ring can multiply with a denominator obtaining the numerator Therefore neither 5 nor can be factorized 522 +minus xx If a polynomial of degree at least one has no other divisors then it is considered to be irreducible or prime (Durbin 1992) Of every polynomial of degree at least one can be written as a product of irreducible polynomials according to unique factorization theorem each polynomial of degree at least one over a field F can be written as an element of F times a product of monic irreducible polynomials over F and except for the order in which these irreducible polynomials are written this can be done in only one way (p 221) Thus a factorable polynomial can be written into a product with another polynomial such as In other words the condition that )()()( xqaxxf minus= )( ax minus is a factor of a polynomial

in a variable x is only true if )(xf 0)( =af For example if and

6)( 2 minus+= xxxf0624)2( =minus+=f 0639)3( =minusminus=minusf the factors of are and )(xf )2( minusx )3( +x

The value of the polynomial is zero which is also expressed as the polynomial having a zero point (nollstaumllle in Swedish) when

)(xfα=x

Based on unique factorization theorem finding solutions of an algebraic equation is equivalent to finding the first-degree factors for a polynomial (Vredblad 2000) In this way solving a quadratic equation can be handled through finding the first-degree factors for the second-degree polynomial and there after finding the roots of the quadratic equation This is often called using factorization to solve quadratic equations Besides unique factorization theorem the connection of using factorization to solve a quadratic equation is that this method relates to prime numbers using distributive law backwards commutative law multiplication division and operating fractions As matter of fact it is about application of factorizing integers in polynomials at upper-secondary school level Since quadratic equations in the mathematics B course at Swedish upper-secondary school are limited to real numbers and often within the whole numbers (integers) many quadratic equations should be able to be solved by factorization according to my own

16

judgment Therefore using factorization for solving quadratic equations is very much related to arithmetic operations number concept and algebra structure It may offer students both mathematical conceptual understanding and operational skill For algebra beginners factorization can be traced back to studentsrsquo early years of mathematics education during which students do not only work with multiplication but also on factoring integers for example 56 = 8 middot 7 = 2 middot 2 middot 2 middot 7 = 4 middot 14 = 2 middot 28 In this example two operations are included multiplication and division Multiplication operation may help students to understand the multiplicative structure of the integers while factoring integers may become meaningful when it is related to divisibility divisors and prime numbers These two operational skills or competence are major techniques for simplifying fractions and finding common denominators Being able to understand and operate factoring integers has laid the essential grounds for algebra beginners to study factorization of polynomials Thus factorization reunites studentsrsquo early mathematics knowledge of factoring integers with later knowledge of algebra structure Factorability In a study on factorability by Bosseacute and Nandakumar (2005) the probability of factorability is reasoned through investigating the range of integers as coefficients of quadratic equations In the quadratic equation a b and c are randomly generated integers within a determined range r such as

002 ne=++ acbxaxyrx lele where x and y are integers When is a

perfect square or where

acb 42 minus

)4( 2 acb minus is an integer quadratics are factorable According to the studyrsquos data it shows that as the range for a b and c increases the probability of factorability of a quadratic with randomly selected integer values for a b and c decreases Bosseacute and Nandakumar confirm that as the range for coefficients expands to infinltltinfinminus r the probablility of factorability of a quadratic with randomly selected integer values for coefficients approaches zero The studyrsquos data collected from college algebra courses and textbooks demonstrates that about 15 of quadratics with integer coefficients [ ]1010minusisinr are factorable Thus about 85 of quadratics can not be factorized In spite of the limitation of factorability within the exercises concerning factoring practice chosen from 27 surveyed college algebra textbooks about 94 of the problems were factorable Within the textbooks 55 of the quadratics to be factored were within the range [-10 10] Even if many quadratic equations are factorable choosing the right factor pairs is time consuming and unnecessary for example

has to be factorized through choosing the right pairs of factors among nine pairs for 36 and 8 pairs of factors for 24 Comparing the three different methods for solving quadratic equations the study suggests that completing the square and quadratic formula are more effective informative and useful than factorization

0245936 2 =++ xx

231 Different kinds of factorizations for solving different quadratic equations How can we find the binomial factors of a quadratic equation or a second-degree polynomial without knowing one of the divisors as a binomial factor The answer is that we have to observe the relations among all coefficients and constants as well as the positive and negative signs before them under the condition of being factorable Most of the coefficients in quadratic equations or polynomials among exercises provided for practicing factoring are integers (here I borrow the same notation r as the one used by Bosseacute and Nandakumar) with

17

their range between ndash10 and 10 that is 10ler In this case it is possible for a student to make a judgment of whether or not a quadratic equation is factorable Five different kinds of factorization depending on five kinds of quadratic equations Factoring quadratic expressions can be divided into five different kinds depending on the type of quadratic equations The coefficients a b p q and constant c illustrated in the following quadratic equations are defined not to be equal to zero thus 000 nenene cba The first kind of quadratic equations is type 1 the obviously factorable type (Bosseacute and Nandakumar 2004) then the equation is factorized as

where the roots are

0002 nene=+ babxax

0)( =+ baxxabxx minus== 21 0

The second kind of quadratic equations is type 2 The factors of this second degree polynomial are so the roots to this type of quadratic

equations are

0)( 22 =minusbax))(()( 22 baxbaxbax minus+=minus

bax plusmn=21 This kind of factorization is based on the difference-of-squares

formula for example can be solved by factorizing into 049 2 =minusx 0)23)(23( =minus+ xx Thus

the roots are 32

32

21 =minus= xx This kind of factorization is also obviously factorable

The third kind of quadratic equations is type 3 This type of equations can be solved directly by factoring into two identical binomials

The double roots obtained from this

factorization are

02)( 22 =+plusmn babxax

0))(()(2)( 222 =plusmnplusmn=plusmn=+plusmn baxbaxbaxbabxax

abx plusmn=21 This method has utilized square rules which are quite easily

observed The forth kind of quadratic equations is type 4 In this equation the coefficient of the -term is 1 with p and q as positive or negative integers Type 4 equations can always be solved by the approaches of completing the square and quadratic formula However they can often be solved by factorization too since the quadratic expressions are factorable by using distributive laws reversely The roots are obtained through finding two factors for q At the same time these two factors become satisfying with the relation so that the sum of factors is equal to That way the two factors become two roots of a quadratic equation Denoting the two factors or two roots as the relationship between coefficient and constant is

02 =++ qpxx2x

qminus

21rr)( 2121 rrprrq +minus=bull= Quadratic equations can be written as the product of

two first-degree binomials or factoring form 0))(()( 212121

22 =minusminus=++minus=++ rxrxrrxrrxqpxx Through the factoring form the roots of this equation are obtained being The core of factorizing type 4 equations is seeking the connections between coefficients and roots which reveal the studentsrsquo basic ability of arithmetic multiplication and addition It is often more effective in this case to use factorization than to use quadratic formula or completing the square Let us have a look at the example In this equation we get

2211 rxrx ==

035122 =+minus xx

18

35)7)(5(12)7()5( =minusminus=minus=minus+minus= qp or 75)75( bull=bullminus= qp (here we can directly get roots) Factorizing the equation we obtain 0)7)(5( =minusminus xx and roots are The procedures of finding factors of the equation often occur in some studentsrsquo minds quickly without writing down the whole computing process but those students often have good skills in arithmetic operations (reference) Compared to using factorization for solving this equation application of quadratic formula seems ineffective A possible problem in this method is how to find the signs before the factors or how to judge if the factors in the parentheses are positive or negative integers In order to check if the solution is correct we can always multiply these two factored binomials by distributive law and see if the expanded polynomial is the same as the original quadratic equation

75 21 == xx

The fifth kind of quadratic equations is type 5 Still the methods of completing squares and quadratic formula can be applied for all type 5 equations but there are cases in which factorization can be used for solving this type of equations within a certain range of integers as mentioned before Sometimes procedures of factorizing such equations can be carried out in our minds quickly like type 4 equations but in many cases they can not As a matter of fact finding two binomial factors for a factorable quadratic equation or a factorable second-degree polynomial is a complicated process and requires systematic work In his article published in Mathematics in School Jackman (2005) demonstrates this systematic procedure via factorizing second-degree polynomials A second-degree polynomial can be written as where are all integers By multiplying out the polynomial becomes thus

0002 nene=++ bacbxax

))((2 srxqpxcbxax ++=++ srqpcbaqsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== Writing the factors of a and c on two lines p q times r s Finding the right pairs of factors for a and c has to be done by listing out the groups of products of different factors for example The factor pairs for 6 are (6 1) (1 6) (3 2) (2 3) and the factor pairs for (-4) are (4 -1) (-4 1) (2 -2) (-2 2) (1 -4) (-1 4) Writing them on two lines p 6 1 3 2 q 4 -4 2 -2 1 -1

456 2 minus+ xx

times r 1 6 2 3 s -1 1 -2 2 -4 4 Then comes calculating the values of ps and qr until the value of b = 5 is obtained The calculating procedures are divided into 4 groups and 6 steps in each group which means 4 times 6 steps in total Here I only give one grouprsquos procedure as an example 6 middot (-1) + 4 middot 1 = -2 6 middot 1 + 1middot (-4) = 2 6 middot (-2) + 1 middot 2 = -10 6 middot 2 + 1middot (-2) = 10 6 middot (-4) + 1 middot 1 = -23 6 middot 4 + 1 middot (-1) = 23 helliphellip After the 24ndashstep calculating procedures two computing procedures obtain the same correct result 3 (ndash1) + 4 2 = 5 (yes) 2 4 + (-1) 3 = 5 (yes) Are they the same From 3 (ndash1) + 4 2 = 5 the polynomial can be factorized into )12)(43( minus+ xx while from 2 4 + (ndash1) 3 = 5 the polynomial can be factorized into )43)(12( +minus xx According to communicative law these two binomial products are the same The shortcoming of this

19

method is that it is too ineffective and complicated The addition of matrixes is almost impossible and unbearable Nevertheless based on the same idea I suggest finding factors by guessing analyzing and checking Using the same example here we can find that one of the binomials must have a negative integer because of ndash4 which implies that 5 is the sum of a positive integer and a negative integer Then looking at a and c the factors of 6 can be (1 6) and (2 3) while the factors of 4 can be (1 4) and (2 2) when we regard them as absolute values We know that the factoring form of the polynomial will be like

and the connections of all coefficients are

456 2 minus+ xx

qsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== The coefficient 5=b is the difference of two products of factors for 6 and 4 First we try the biggest number of product such as 2446 =bull then 524 =minus the unknown is which is a prime number the only factors for 19 are (19 1) There is not any combination of products of factors for 6 and 4 satisfying 19 Therefore the multiplications of factors (1 6) and (1 4) are not included in this case So we move to factors (2 2) (2 3) and (1 4)

19 =

In the second step we try The result is negative it is obviously

4264466)23)(22( 22 minusminus=minus+minus=+minus xxxxxxx2 lt 5 which implies that the combination of (2 3) and (2 2) is not

considerable One pair of factors has to be removed and that is (2 2) which does not influence the coefficient of the -term Then we try The result this time is negative too but we can see that

2x 456)43)(12( 2 minusminus=minus+ xxxxb is 5 which guides us to interchanging the positions of

negative and positive signs in the parentheses The next attempt is and it is correct Seeking the correct factors for the polynomial

has taken four major operating steps by trying the combinations of factorsrsquo products and relating the sum of the products with the value of the x-term coefficient The strategy is to make use of distributive law reversely first and then multiply the product of every pair of parentheses in order to check if they are the right pair of polynomial factors Factorizing second-degree polynomials is the major step for solving quadratic equations After factoring polynomials into the product of two binomials as in the example given above the roots can be obtained by making each binomial equal to zero according to the theory of zero point (ldquonollstaumlllerdquo in Swedish) of a polynomial equation originating from unique factorization theorem

456)43)(12( 2 minus+=+minus xxxx456 2 minus+ xx

Factorization by using the technique of vertically and crosswise sutra Binomial expansion or multiplication of binomials by distributive law and factorization of the second-degree polynomials by using distributive law backwards have an extremely symmetrical effect and the results of two kinds of operations are equivalent It is crucial for the students to be able to perceive mathematics structure and be acquainted with arithmetic operations as well as to understand the relationship between quantities when they carry out factoring procedures The expansion of binomials by distributive law has been named as the method FOIL which stands for first outside inside and last according to multiplication orders of two binomials for example (Nataraj and Thomas 2006) Many students are used to this procedure Another technique to multiply two binomials is called vertically and crosswise sutra origin from the ancient Indian Vedic mathematics Vertically and crosswise sutra and FOIL are much the same in principle but it makes use of

824)4)(2( 2 +++=++ xxxxx

20

the power of visualization and improves facility with the expansion of binomials and the factorization of quadratic expressions In this method two binomials are written into two lines Using the same example the two lines are written as 824)4)(2( 2 +++=++ xxxxx

x + 2 x + 4 ______________ xsup2 4x + 2x 8

The order of multiplication is as shown below x + 2 x + 4

4 3 2 1 (the numbers 1 2 3 and 4 refer to steps) xsup2 + 2x + 4x + 8 Figure 3 The vertically and crosswise method of multiplication of binomials Step 2 and 3 can be interchangeable If we put this form into a line expanded according to the order above would be like

)4)(2( ++ xxxbullrarrbull 442 clockwise and xxx bullrarrbull2 in

counterclockwise direction According to sutra we can imagine drawing a thread starting from 2 then going through 4 and x clockwise and then draw the second thread starting from the same place 2 later going through x in the second bracket and finally x in the first bracket counterclockwise Learning expansion of binomials by the methods FOIL and Vertical and clockwise sutra is the basic process before students study factorization In this study Nataraj and Thomas (2006) have found that the students appear to do factorization using guess-and-check as well as decomposition or the vertical and clockwise approach Their finding is that afterwards the students performed better on algebra questions especially on the factorizations They conclude ldquohellipthe values of the method may lie in what it adds to the studentsrsquo overall algebraic conceptions and knowledge of mathematical structurerdquo (p 16) In general solving these five kinds of quadratic equations by factorization can be traced back to unique factorization theorem The important character in this method is that you can always check if factorization is correct or not through multiplying out the two pairs of a parenthesis and see if the expanded polynomial is the given one or not Using factorization is not only about method but also about fostering the studentsrsquo mathematical thinking The solving processes are more important than solutions

21

3 Review of related literature and research Besides the focus on mathematics content another focus in the background work for this research is about previous studies and research in the field of mathematics didactic concerning different methods for solving quadratic equations and the use of factorization In order to study the previous research and studies in the related field three scientific databases and a search engine have been used and 113 articles have been investigated Topics such as solving quadratic expressions and equations are often discussed as teaching ideas and strategies shared by mathematics educators in different countries from lower and upper-secondary schools to colleges according to the findings in the article search These didactic topics from mathematics teachersrsquo perspectives include for example using geometric approaches to solve quadratic equations (Allaire amp Bradley 2001) utilizing completing squares (Vinogradova 2007) factoring quadratics (Hoffman 1976 Cheng 1980) and using factorization completing the square graphical methods to solve quadratic equations (Macdonald 1986) Kostsopoulos (2007) links cognitive reasoning to pedagogical problems appearing in the mathematics classrooms in her article published in Australian Mathematics Teacher Based on her teaching experiences Kostsopoulos is aware of her pedagogical strategies lacking insight on dealing with studentsrsquo difficulties of multiplication and factorization of quadratics She assumes that ldquostudentsrsquo problems with factorization and with identifying varied representations of the same quadratic relationshiprdquo may be linked to the ways in which the brain constructs cognitive representations (p 19) Factoring of quadratics is the rewriting of polynomials as a product of polynomials It requires students to have both a conceptual understanding of multiplication of polynomials and the procedural knowledge to retrieve basic multiplication facts effectively Based on the theory about the order matters in the brainrsquos ability to retrieve number facts (Phenix amp Campbell 2001) Kostsopoulos conjectures that studentsrsquo ability to access the appropriate long-term semantic memory is limited when they are confused by mixing the order and varied forms of quadratics Although this article is not about a research study it seeks the reason why factorizing quadratics is difficult and therefore might provide math teachers with alternative explanations for similar pedagogical frustrations With Von Glasersfeldrsquos radical constructivism and Peter Gaumlrdenforsrsquo epistemic semantics theory as his theoretical frameworks Rauff (1994) suggests that belief-based teaching can be successful in teaching factoring According to Rauff constructivist learning theory takes the view that students construct their own beliefs and knowledge of mathematics Learning occurs when the students change their belief set Referring to belief set theory from Gaumlrdenfors (1988) Rauff (1994) claims that ldquoa belief set is expanded when a new belief is added to it and contracted when a proposition is no longer believed and is removed from the setrdquo (p 421) Expansion contraction and revision are three components to comprise basic mechanisms of changes of belief according to Rauff When examining and comparing studentsrsquo (ages 18-20) own definitions of factoring Rauff analyzes the errors of their definitions with constructivism and epistemic semantics as analyzing tools The result of this study shows that the approach to teaching factoring rely heavily upon what a student believes about factoring is quite productive in two dimensions First it reveals the source of nonstandard factoring Second it provides a starting point for modification of the studentrsquos underlying conceptions of factoring Rauff proposes that belief-based teaching can also be

22

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

A Begin with a rectangle of the area )10( +xx that is with the short side as x and the long side )10( +x The shaded area is 10bullx (Figure 6)

B Divide the shaded rectangle into two small shaded rectangles with the size 5x each and move them to each adjacent side of the xsup2 square (Figure 7) The total area is still

x10

)10(39 += xxC In Figure 7 a new square is shaped in the lower-right hand corner with the side 5 or

area 25 By adding this small square to the diagram the large square with an area of is ldquocompletedrdquo Therefore we can write 25)5(22 ++ xx 25)10(2539 ++=+ xx or

The large squarersquos area is now because the side has a length of Therefore

6425)5(22 =++ xx 2)5( +x5+x 38564)5( 2 =rArr=+rArr=+ xxx

After these three steps a solution to this quadratic equation is derived by completing the square (Allaire amp Bradley 2001) 2) Factorization Hoffman (1976) presents three approaches (A B and C) for factorizing quadratic expressions through observing and grouping He takes a second-degree polynomial as an example

273 2 ++ xx

a The inspection method 1) suggests factors of and 23x x3 x 2) 2 suggests factors of 1 and 2 or ndash1 and ndash2 3) Checking the linear term shows that the correct factorization is x7 )2)(13(273 2 ++=++ xxxx

This method according to Hoffman (1976) is based on the studentrsquos ability to simplify by inspecting the indicated product of a pair of linear expressions

b The method of decomposition of the linear term 1) Multiply and 2 to get 23x 26x2) Decompose into the sum of two terms whose product is x7 26x

This gives xxx += 67 3) Factorize by grouping 2)6(3 2 +++ xxx

2)6(3 2 +++ xxx = )

)2()63( 2 +++ xxx

= 2(1)2(3 +++ xxx= )13)(2( ++ xx

c Third method 1) Multiply 3 and 2 to get 6 2) Find two numbers whose product is 6 and whose sum is 7 The numbers are 6 and 1 3) 273 2 ++ xx

=3

)13)(63( ++ xx

= )13)(2( ++ xx

13

These three approaches can not actually be divided as three unrelated approaches especially the third method is seemingly unclear What Hoffman (1976) wants to demonstrate is that all the three approaches are based on observing the relationship among the coefficients and the constant of the polynomial Through finding the product and the sum of two factors the polynomial can be factorized in the product of two binomials In the authorrsquos opinion factorization is heavily dependent on skill in multiplication and using distributive law reversely He means that ldquostudents who do not have reasonably strong skills in multiplication should not be expected to develop strong skills in factorizationrdquo (p 55) Factorizing polynomials is not only for simplifying high-degree polynomials (for example the third or fourth-degree polynomials) but can also be applied for solving quadratic equations Supposing this polynomial forming a quadratic equation Solving this equation is actually to decompose the polynomial into a factoringrsquos form

0273 2 =++ xx0)13)(2( =++ xx

which is equivalent to the polynomial form The essential step to solve this equation is to follow the zero-factor propertyndashalso called null-factor law (nollprodukt in Swedish)ndashthat is for real numbers p and q p bull q = 0 if (and only if) p = 0 or q = 0 It means that either of the binomials on the left side of the equation has to be equal to zero in order to satisfy the

equivalent relation of this quadratic equation The solutions of two roots or 2minus=x31

minus=x

are obtained through making either 02 =+x or 013 =+x 3) Quadratic formula Solving quadratics equations by factorization is constrained within the simple quadratic equations over the whole integers and rational numbers as coefficients What happens when quadratic equations have coefficients that belong to an irrational domain or big quantity The quadratic formula is a solution to such a situation As mentioned previously historically the French mathematician Franccedilois Vieacutete discovered the quadratic formula

aacbbx

242 minusplusmnminus

= ) which is applicable for all kinds of quadratic equations In

Swedish textbooks this formula is written in PQ form by making for the equation

that is

0( nea

1=a

02 =++ qpxx qppx minus⎟⎠⎞

⎜⎝⎛plusmnminus=

2

22 with

abp = and

acq = In Oleatursquos study (2007)

studentsrsquo difficulties in using algebraic symbols no longer is the problem But on the other hand handling the parameters or coefficients in quadratic equations like and rewrite them in the equivalent form with p and q being real numbers becomes obstacles for the students when they study the algebra course at upper-secondary school

02 =++ cbxax02 =++ qpxx

The quadratic formula a

acbbx2

42 minusplusmnminus= has been regarded as the standard method

according to US mathematics teachers Obermeyer (1982) presents a way of how to derive this formula by completing a square through 11 steps (I will not present them here) At the same time an alternate method is also introduced according to the same principle All the methods mentioned so far are based on the square rules The standard method or PQ form might be regarded as an efficient and direct method but it may

222 2)( bababa +plusmn=plusmn

14

lead students to solve quadratic equations in a mechanical way besides having problems with the troublesome plusmn symbolism (Stover 1978) Both Stover (1978) and Olteanu (2007) have suggested that using a graph of quadratic functions to solve quadratic equations is another alternative method

23 Mathematical background of factorization Factorization in algebra structure Among the three solving methods presented above both the geometrical method and quadratic formula can be traced back to algebra history in section 21 What is the trace of factorization in algebra history Factorization was at the last stage in algebra history and belonged to algebra structure (Derbyshire 2006) In Swedish mathematics textbooks the chapter on algebra often includes polynomial factorization and quadratic equations But the factorization in the textbooks is not at the same abstract level as the factorization of algebra structure although they share the same axioms and there are certain connections between these two different levels of factorization How are polynomial and quadratic equations as well as factorization related to each other In order to find the theoretical background of factorizing polynomials this part constitutes a mathematical theoretical review by referring to two books concerning algebra structure (Vretblad 2000 Durbin 1992) and a research study (Bosseacute and Nandakumar 2005) A quadratic equation can be regarded as polynomial equation Solving quadratic equations has much to do with ldquounique factorization theoremrdquo The polynomial

is a second-degree polynomial A polynomial consists of coefficients in whole numbers rational numbers real numbers or complex numbers and variables (sometimes called unknowns) in different degrees as well as constants A polynomial

has symbols as coefficients which are real numbers If all are zero and so

)0(02 ne=++ acbxax

cbxax ++2

nnxaxaxaaxf ++++= )( 2

210 naaa 10

ka 1gek 0)( axf = is constant If all are zero then we say the is a zero polynomial If meaning that is not a zero polynomial and n is the biggest number and then we say that the polynomial is at degrees A zero polynomial has no degree at all According to the definition (Vredblad 2000) if is a polynomial the equation is called an algebraic equation or polynomial equation A solution or a root of this equation is a number

kaf 0nef f

0nena f nf

0)( =xfα so that 0)( =αf Such a number is even

called a zero point (nollstaumllle in Swedish) of the polynomial which means when f α=x the value of the polynomial is zero (p 161) Every polynomial whose degree is equal or greater than 1 can be written as a product of irreducible polynomial In this way through polynomial division a polynomial with n degrees can be expressed from a sum of terms into a product of its denominator and its quotient The process of changing the expressions is factorization for

instance 12

22

+=minusminusminus x

xxx The factoring form is )1)(2(22 minusminus=minusminus xxxx

Factorization of polynomials is analogue to factorization of whole numbers which is very much related to the concept of ring A ring consists of a set with two operations which are a sum and a product of two elements over a field Any polynomial from zero polynomial to n degree-polynomials can be written or operated through addition and multiplication of the

15

elements over a field F including whole numbers or integers rational numbers real numbers and complex numbers In a ring you can multiply any two elements but you canrsquot always divide Factorization is a way to deal with this fact Here are some examples In the ring of integers Z is an example of factorization in some polynomial rings for instance Q[x] the ring of polynomial in one variable with rational numbers as coefficients such as

7214 sdot=

)451)(2

51(8

52

251 2 +minus=minus+ xxxx is factorization Z[x] the ring of polynomial in one

variable with whole numbers as integers such as is factorization )1)(2(22 minusminus=minusminus xxxx The polynomial ring R[x] means the ring of polynomial in one variable with real numbers as coefficients Factorizing of such polynomials implies that the polynomial factors are the elements in the ring of R[x] for instance )2)(2(22 minus+=minus xxx The same thing is true for C[x] the ring of polynomial in one variable with complex numbers as coefficients Thus it is coefficients of polynomials that determine which ring a polynomial belongs to and if it is possible to be factorized in this ring For example number 6 can be factorized in two ways

in the ring of Z and 32times )51)(51( ii +minus in the ring of C (complex numbers) However there are cases in which divisions can not be done within the ring for example 52 or

)3()52( 2 ++minus xxx which means that no element in the ring can multiply with a denominator obtaining the numerator Therefore neither 5 nor can be factorized 522 +minus xx If a polynomial of degree at least one has no other divisors then it is considered to be irreducible or prime (Durbin 1992) Of every polynomial of degree at least one can be written as a product of irreducible polynomials according to unique factorization theorem each polynomial of degree at least one over a field F can be written as an element of F times a product of monic irreducible polynomials over F and except for the order in which these irreducible polynomials are written this can be done in only one way (p 221) Thus a factorable polynomial can be written into a product with another polynomial such as In other words the condition that )()()( xqaxxf minus= )( ax minus is a factor of a polynomial

in a variable x is only true if )(xf 0)( =af For example if and

6)( 2 minus+= xxxf0624)2( =minus+=f 0639)3( =minusminus=minusf the factors of are and )(xf )2( minusx )3( +x

The value of the polynomial is zero which is also expressed as the polynomial having a zero point (nollstaumllle in Swedish) when

)(xfα=x

Based on unique factorization theorem finding solutions of an algebraic equation is equivalent to finding the first-degree factors for a polynomial (Vredblad 2000) In this way solving a quadratic equation can be handled through finding the first-degree factors for the second-degree polynomial and there after finding the roots of the quadratic equation This is often called using factorization to solve quadratic equations Besides unique factorization theorem the connection of using factorization to solve a quadratic equation is that this method relates to prime numbers using distributive law backwards commutative law multiplication division and operating fractions As matter of fact it is about application of factorizing integers in polynomials at upper-secondary school level Since quadratic equations in the mathematics B course at Swedish upper-secondary school are limited to real numbers and often within the whole numbers (integers) many quadratic equations should be able to be solved by factorization according to my own

16

judgment Therefore using factorization for solving quadratic equations is very much related to arithmetic operations number concept and algebra structure It may offer students both mathematical conceptual understanding and operational skill For algebra beginners factorization can be traced back to studentsrsquo early years of mathematics education during which students do not only work with multiplication but also on factoring integers for example 56 = 8 middot 7 = 2 middot 2 middot 2 middot 7 = 4 middot 14 = 2 middot 28 In this example two operations are included multiplication and division Multiplication operation may help students to understand the multiplicative structure of the integers while factoring integers may become meaningful when it is related to divisibility divisors and prime numbers These two operational skills or competence are major techniques for simplifying fractions and finding common denominators Being able to understand and operate factoring integers has laid the essential grounds for algebra beginners to study factorization of polynomials Thus factorization reunites studentsrsquo early mathematics knowledge of factoring integers with later knowledge of algebra structure Factorability In a study on factorability by Bosseacute and Nandakumar (2005) the probability of factorability is reasoned through investigating the range of integers as coefficients of quadratic equations In the quadratic equation a b and c are randomly generated integers within a determined range r such as

002 ne=++ acbxaxyrx lele where x and y are integers When is a

perfect square or where

acb 42 minus

)4( 2 acb minus is an integer quadratics are factorable According to the studyrsquos data it shows that as the range for a b and c increases the probability of factorability of a quadratic with randomly selected integer values for a b and c decreases Bosseacute and Nandakumar confirm that as the range for coefficients expands to infinltltinfinminus r the probablility of factorability of a quadratic with randomly selected integer values for coefficients approaches zero The studyrsquos data collected from college algebra courses and textbooks demonstrates that about 15 of quadratics with integer coefficients [ ]1010minusisinr are factorable Thus about 85 of quadratics can not be factorized In spite of the limitation of factorability within the exercises concerning factoring practice chosen from 27 surveyed college algebra textbooks about 94 of the problems were factorable Within the textbooks 55 of the quadratics to be factored were within the range [-10 10] Even if many quadratic equations are factorable choosing the right factor pairs is time consuming and unnecessary for example

has to be factorized through choosing the right pairs of factors among nine pairs for 36 and 8 pairs of factors for 24 Comparing the three different methods for solving quadratic equations the study suggests that completing the square and quadratic formula are more effective informative and useful than factorization

0245936 2 =++ xx

231 Different kinds of factorizations for solving different quadratic equations How can we find the binomial factors of a quadratic equation or a second-degree polynomial without knowing one of the divisors as a binomial factor The answer is that we have to observe the relations among all coefficients and constants as well as the positive and negative signs before them under the condition of being factorable Most of the coefficients in quadratic equations or polynomials among exercises provided for practicing factoring are integers (here I borrow the same notation r as the one used by Bosseacute and Nandakumar) with

17

their range between ndash10 and 10 that is 10ler In this case it is possible for a student to make a judgment of whether or not a quadratic equation is factorable Five different kinds of factorization depending on five kinds of quadratic equations Factoring quadratic expressions can be divided into five different kinds depending on the type of quadratic equations The coefficients a b p q and constant c illustrated in the following quadratic equations are defined not to be equal to zero thus 000 nenene cba The first kind of quadratic equations is type 1 the obviously factorable type (Bosseacute and Nandakumar 2004) then the equation is factorized as

where the roots are

0002 nene=+ babxax

0)( =+ baxxabxx minus== 21 0

The second kind of quadratic equations is type 2 The factors of this second degree polynomial are so the roots to this type of quadratic

equations are

0)( 22 =minusbax))(()( 22 baxbaxbax minus+=minus

bax plusmn=21 This kind of factorization is based on the difference-of-squares

formula for example can be solved by factorizing into 049 2 =minusx 0)23)(23( =minus+ xx Thus

the roots are 32

32

21 =minus= xx This kind of factorization is also obviously factorable

The third kind of quadratic equations is type 3 This type of equations can be solved directly by factoring into two identical binomials

The double roots obtained from this

factorization are

02)( 22 =+plusmn babxax

0))(()(2)( 222 =plusmnplusmn=plusmn=+plusmn baxbaxbaxbabxax

abx plusmn=21 This method has utilized square rules which are quite easily

observed The forth kind of quadratic equations is type 4 In this equation the coefficient of the -term is 1 with p and q as positive or negative integers Type 4 equations can always be solved by the approaches of completing the square and quadratic formula However they can often be solved by factorization too since the quadratic expressions are factorable by using distributive laws reversely The roots are obtained through finding two factors for q At the same time these two factors become satisfying with the relation so that the sum of factors is equal to That way the two factors become two roots of a quadratic equation Denoting the two factors or two roots as the relationship between coefficient and constant is

02 =++ qpxx2x

qminus

21rr)( 2121 rrprrq +minus=bull= Quadratic equations can be written as the product of

two first-degree binomials or factoring form 0))(()( 212121

22 =minusminus=++minus=++ rxrxrrxrrxqpxx Through the factoring form the roots of this equation are obtained being The core of factorizing type 4 equations is seeking the connections between coefficients and roots which reveal the studentsrsquo basic ability of arithmetic multiplication and addition It is often more effective in this case to use factorization than to use quadratic formula or completing the square Let us have a look at the example In this equation we get

2211 rxrx ==

035122 =+minus xx

18

35)7)(5(12)7()5( =minusminus=minus=minus+minus= qp or 75)75( bull=bullminus= qp (here we can directly get roots) Factorizing the equation we obtain 0)7)(5( =minusminus xx and roots are The procedures of finding factors of the equation often occur in some studentsrsquo minds quickly without writing down the whole computing process but those students often have good skills in arithmetic operations (reference) Compared to using factorization for solving this equation application of quadratic formula seems ineffective A possible problem in this method is how to find the signs before the factors or how to judge if the factors in the parentheses are positive or negative integers In order to check if the solution is correct we can always multiply these two factored binomials by distributive law and see if the expanded polynomial is the same as the original quadratic equation

75 21 == xx

The fifth kind of quadratic equations is type 5 Still the methods of completing squares and quadratic formula can be applied for all type 5 equations but there are cases in which factorization can be used for solving this type of equations within a certain range of integers as mentioned before Sometimes procedures of factorizing such equations can be carried out in our minds quickly like type 4 equations but in many cases they can not As a matter of fact finding two binomial factors for a factorable quadratic equation or a factorable second-degree polynomial is a complicated process and requires systematic work In his article published in Mathematics in School Jackman (2005) demonstrates this systematic procedure via factorizing second-degree polynomials A second-degree polynomial can be written as where are all integers By multiplying out the polynomial becomes thus

0002 nene=++ bacbxax

))((2 srxqpxcbxax ++=++ srqpcbaqsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== Writing the factors of a and c on two lines p q times r s Finding the right pairs of factors for a and c has to be done by listing out the groups of products of different factors for example The factor pairs for 6 are (6 1) (1 6) (3 2) (2 3) and the factor pairs for (-4) are (4 -1) (-4 1) (2 -2) (-2 2) (1 -4) (-1 4) Writing them on two lines p 6 1 3 2 q 4 -4 2 -2 1 -1

456 2 minus+ xx

times r 1 6 2 3 s -1 1 -2 2 -4 4 Then comes calculating the values of ps and qr until the value of b = 5 is obtained The calculating procedures are divided into 4 groups and 6 steps in each group which means 4 times 6 steps in total Here I only give one grouprsquos procedure as an example 6 middot (-1) + 4 middot 1 = -2 6 middot 1 + 1middot (-4) = 2 6 middot (-2) + 1 middot 2 = -10 6 middot 2 + 1middot (-2) = 10 6 middot (-4) + 1 middot 1 = -23 6 middot 4 + 1 middot (-1) = 23 helliphellip After the 24ndashstep calculating procedures two computing procedures obtain the same correct result 3 (ndash1) + 4 2 = 5 (yes) 2 4 + (-1) 3 = 5 (yes) Are they the same From 3 (ndash1) + 4 2 = 5 the polynomial can be factorized into )12)(43( minus+ xx while from 2 4 + (ndash1) 3 = 5 the polynomial can be factorized into )43)(12( +minus xx According to communicative law these two binomial products are the same The shortcoming of this

19

method is that it is too ineffective and complicated The addition of matrixes is almost impossible and unbearable Nevertheless based on the same idea I suggest finding factors by guessing analyzing and checking Using the same example here we can find that one of the binomials must have a negative integer because of ndash4 which implies that 5 is the sum of a positive integer and a negative integer Then looking at a and c the factors of 6 can be (1 6) and (2 3) while the factors of 4 can be (1 4) and (2 2) when we regard them as absolute values We know that the factoring form of the polynomial will be like

and the connections of all coefficients are

456 2 minus+ xx

qsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== The coefficient 5=b is the difference of two products of factors for 6 and 4 First we try the biggest number of product such as 2446 =bull then 524 =minus the unknown is which is a prime number the only factors for 19 are (19 1) There is not any combination of products of factors for 6 and 4 satisfying 19 Therefore the multiplications of factors (1 6) and (1 4) are not included in this case So we move to factors (2 2) (2 3) and (1 4)

19 =

In the second step we try The result is negative it is obviously

4264466)23)(22( 22 minusminus=minus+minus=+minus xxxxxxx2 lt 5 which implies that the combination of (2 3) and (2 2) is not

considerable One pair of factors has to be removed and that is (2 2) which does not influence the coefficient of the -term Then we try The result this time is negative too but we can see that

2x 456)43)(12( 2 minusminus=minus+ xxxxb is 5 which guides us to interchanging the positions of

negative and positive signs in the parentheses The next attempt is and it is correct Seeking the correct factors for the polynomial

has taken four major operating steps by trying the combinations of factorsrsquo products and relating the sum of the products with the value of the x-term coefficient The strategy is to make use of distributive law reversely first and then multiply the product of every pair of parentheses in order to check if they are the right pair of polynomial factors Factorizing second-degree polynomials is the major step for solving quadratic equations After factoring polynomials into the product of two binomials as in the example given above the roots can be obtained by making each binomial equal to zero according to the theory of zero point (ldquonollstaumlllerdquo in Swedish) of a polynomial equation originating from unique factorization theorem

456)43)(12( 2 minus+=+minus xxxx456 2 minus+ xx

Factorization by using the technique of vertically and crosswise sutra Binomial expansion or multiplication of binomials by distributive law and factorization of the second-degree polynomials by using distributive law backwards have an extremely symmetrical effect and the results of two kinds of operations are equivalent It is crucial for the students to be able to perceive mathematics structure and be acquainted with arithmetic operations as well as to understand the relationship between quantities when they carry out factoring procedures The expansion of binomials by distributive law has been named as the method FOIL which stands for first outside inside and last according to multiplication orders of two binomials for example (Nataraj and Thomas 2006) Many students are used to this procedure Another technique to multiply two binomials is called vertically and crosswise sutra origin from the ancient Indian Vedic mathematics Vertically and crosswise sutra and FOIL are much the same in principle but it makes use of

824)4)(2( 2 +++=++ xxxxx

20

the power of visualization and improves facility with the expansion of binomials and the factorization of quadratic expressions In this method two binomials are written into two lines Using the same example the two lines are written as 824)4)(2( 2 +++=++ xxxxx

x + 2 x + 4 ______________ xsup2 4x + 2x 8

The order of multiplication is as shown below x + 2 x + 4

4 3 2 1 (the numbers 1 2 3 and 4 refer to steps) xsup2 + 2x + 4x + 8 Figure 3 The vertically and crosswise method of multiplication of binomials Step 2 and 3 can be interchangeable If we put this form into a line expanded according to the order above would be like

)4)(2( ++ xxxbullrarrbull 442 clockwise and xxx bullrarrbull2 in

counterclockwise direction According to sutra we can imagine drawing a thread starting from 2 then going through 4 and x clockwise and then draw the second thread starting from the same place 2 later going through x in the second bracket and finally x in the first bracket counterclockwise Learning expansion of binomials by the methods FOIL and Vertical and clockwise sutra is the basic process before students study factorization In this study Nataraj and Thomas (2006) have found that the students appear to do factorization using guess-and-check as well as decomposition or the vertical and clockwise approach Their finding is that afterwards the students performed better on algebra questions especially on the factorizations They conclude ldquohellipthe values of the method may lie in what it adds to the studentsrsquo overall algebraic conceptions and knowledge of mathematical structurerdquo (p 16) In general solving these five kinds of quadratic equations by factorization can be traced back to unique factorization theorem The important character in this method is that you can always check if factorization is correct or not through multiplying out the two pairs of a parenthesis and see if the expanded polynomial is the given one or not Using factorization is not only about method but also about fostering the studentsrsquo mathematical thinking The solving processes are more important than solutions

21

3 Review of related literature and research Besides the focus on mathematics content another focus in the background work for this research is about previous studies and research in the field of mathematics didactic concerning different methods for solving quadratic equations and the use of factorization In order to study the previous research and studies in the related field three scientific databases and a search engine have been used and 113 articles have been investigated Topics such as solving quadratic expressions and equations are often discussed as teaching ideas and strategies shared by mathematics educators in different countries from lower and upper-secondary schools to colleges according to the findings in the article search These didactic topics from mathematics teachersrsquo perspectives include for example using geometric approaches to solve quadratic equations (Allaire amp Bradley 2001) utilizing completing squares (Vinogradova 2007) factoring quadratics (Hoffman 1976 Cheng 1980) and using factorization completing the square graphical methods to solve quadratic equations (Macdonald 1986) Kostsopoulos (2007) links cognitive reasoning to pedagogical problems appearing in the mathematics classrooms in her article published in Australian Mathematics Teacher Based on her teaching experiences Kostsopoulos is aware of her pedagogical strategies lacking insight on dealing with studentsrsquo difficulties of multiplication and factorization of quadratics She assumes that ldquostudentsrsquo problems with factorization and with identifying varied representations of the same quadratic relationshiprdquo may be linked to the ways in which the brain constructs cognitive representations (p 19) Factoring of quadratics is the rewriting of polynomials as a product of polynomials It requires students to have both a conceptual understanding of multiplication of polynomials and the procedural knowledge to retrieve basic multiplication facts effectively Based on the theory about the order matters in the brainrsquos ability to retrieve number facts (Phenix amp Campbell 2001) Kostsopoulos conjectures that studentsrsquo ability to access the appropriate long-term semantic memory is limited when they are confused by mixing the order and varied forms of quadratics Although this article is not about a research study it seeks the reason why factorizing quadratics is difficult and therefore might provide math teachers with alternative explanations for similar pedagogical frustrations With Von Glasersfeldrsquos radical constructivism and Peter Gaumlrdenforsrsquo epistemic semantics theory as his theoretical frameworks Rauff (1994) suggests that belief-based teaching can be successful in teaching factoring According to Rauff constructivist learning theory takes the view that students construct their own beliefs and knowledge of mathematics Learning occurs when the students change their belief set Referring to belief set theory from Gaumlrdenfors (1988) Rauff (1994) claims that ldquoa belief set is expanded when a new belief is added to it and contracted when a proposition is no longer believed and is removed from the setrdquo (p 421) Expansion contraction and revision are three components to comprise basic mechanisms of changes of belief according to Rauff When examining and comparing studentsrsquo (ages 18-20) own definitions of factoring Rauff analyzes the errors of their definitions with constructivism and epistemic semantics as analyzing tools The result of this study shows that the approach to teaching factoring rely heavily upon what a student believes about factoring is quite productive in two dimensions First it reveals the source of nonstandard factoring Second it provides a starting point for modification of the studentrsquos underlying conceptions of factoring Rauff proposes that belief-based teaching can also be

22

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

These three approaches can not actually be divided as three unrelated approaches especially the third method is seemingly unclear What Hoffman (1976) wants to demonstrate is that all the three approaches are based on observing the relationship among the coefficients and the constant of the polynomial Through finding the product and the sum of two factors the polynomial can be factorized in the product of two binomials In the authorrsquos opinion factorization is heavily dependent on skill in multiplication and using distributive law reversely He means that ldquostudents who do not have reasonably strong skills in multiplication should not be expected to develop strong skills in factorizationrdquo (p 55) Factorizing polynomials is not only for simplifying high-degree polynomials (for example the third or fourth-degree polynomials) but can also be applied for solving quadratic equations Supposing this polynomial forming a quadratic equation Solving this equation is actually to decompose the polynomial into a factoringrsquos form

0273 2 =++ xx0)13)(2( =++ xx

which is equivalent to the polynomial form The essential step to solve this equation is to follow the zero-factor propertyndashalso called null-factor law (nollprodukt in Swedish)ndashthat is for real numbers p and q p bull q = 0 if (and only if) p = 0 or q = 0 It means that either of the binomials on the left side of the equation has to be equal to zero in order to satisfy the

equivalent relation of this quadratic equation The solutions of two roots or 2minus=x31

minus=x

are obtained through making either 02 =+x or 013 =+x 3) Quadratic formula Solving quadratics equations by factorization is constrained within the simple quadratic equations over the whole integers and rational numbers as coefficients What happens when quadratic equations have coefficients that belong to an irrational domain or big quantity The quadratic formula is a solution to such a situation As mentioned previously historically the French mathematician Franccedilois Vieacutete discovered the quadratic formula

aacbbx

242 minusplusmnminus

= ) which is applicable for all kinds of quadratic equations In

Swedish textbooks this formula is written in PQ form by making for the equation

that is

0( nea

1=a

02 =++ qpxx qppx minus⎟⎠⎞

⎜⎝⎛plusmnminus=

2

22 with

abp = and

acq = In Oleatursquos study (2007)

studentsrsquo difficulties in using algebraic symbols no longer is the problem But on the other hand handling the parameters or coefficients in quadratic equations like and rewrite them in the equivalent form with p and q being real numbers becomes obstacles for the students when they study the algebra course at upper-secondary school

02 =++ cbxax02 =++ qpxx

The quadratic formula a

acbbx2

42 minusplusmnminus= has been regarded as the standard method

according to US mathematics teachers Obermeyer (1982) presents a way of how to derive this formula by completing a square through 11 steps (I will not present them here) At the same time an alternate method is also introduced according to the same principle All the methods mentioned so far are based on the square rules The standard method or PQ form might be regarded as an efficient and direct method but it may

222 2)( bababa +plusmn=plusmn

14

lead students to solve quadratic equations in a mechanical way besides having problems with the troublesome plusmn symbolism (Stover 1978) Both Stover (1978) and Olteanu (2007) have suggested that using a graph of quadratic functions to solve quadratic equations is another alternative method

23 Mathematical background of factorization Factorization in algebra structure Among the three solving methods presented above both the geometrical method and quadratic formula can be traced back to algebra history in section 21 What is the trace of factorization in algebra history Factorization was at the last stage in algebra history and belonged to algebra structure (Derbyshire 2006) In Swedish mathematics textbooks the chapter on algebra often includes polynomial factorization and quadratic equations But the factorization in the textbooks is not at the same abstract level as the factorization of algebra structure although they share the same axioms and there are certain connections between these two different levels of factorization How are polynomial and quadratic equations as well as factorization related to each other In order to find the theoretical background of factorizing polynomials this part constitutes a mathematical theoretical review by referring to two books concerning algebra structure (Vretblad 2000 Durbin 1992) and a research study (Bosseacute and Nandakumar 2005) A quadratic equation can be regarded as polynomial equation Solving quadratic equations has much to do with ldquounique factorization theoremrdquo The polynomial

is a second-degree polynomial A polynomial consists of coefficients in whole numbers rational numbers real numbers or complex numbers and variables (sometimes called unknowns) in different degrees as well as constants A polynomial

has symbols as coefficients which are real numbers If all are zero and so

)0(02 ne=++ acbxax

cbxax ++2

nnxaxaxaaxf ++++= )( 2

210 naaa 10

ka 1gek 0)( axf = is constant If all are zero then we say the is a zero polynomial If meaning that is not a zero polynomial and n is the biggest number and then we say that the polynomial is at degrees A zero polynomial has no degree at all According to the definition (Vredblad 2000) if is a polynomial the equation is called an algebraic equation or polynomial equation A solution or a root of this equation is a number

kaf 0nef f

0nena f nf

0)( =xfα so that 0)( =αf Such a number is even

called a zero point (nollstaumllle in Swedish) of the polynomial which means when f α=x the value of the polynomial is zero (p 161) Every polynomial whose degree is equal or greater than 1 can be written as a product of irreducible polynomial In this way through polynomial division a polynomial with n degrees can be expressed from a sum of terms into a product of its denominator and its quotient The process of changing the expressions is factorization for

instance 12

22

+=minusminusminus x

xxx The factoring form is )1)(2(22 minusminus=minusminus xxxx

Factorization of polynomials is analogue to factorization of whole numbers which is very much related to the concept of ring A ring consists of a set with two operations which are a sum and a product of two elements over a field Any polynomial from zero polynomial to n degree-polynomials can be written or operated through addition and multiplication of the

15

elements over a field F including whole numbers or integers rational numbers real numbers and complex numbers In a ring you can multiply any two elements but you canrsquot always divide Factorization is a way to deal with this fact Here are some examples In the ring of integers Z is an example of factorization in some polynomial rings for instance Q[x] the ring of polynomial in one variable with rational numbers as coefficients such as

7214 sdot=

)451)(2

51(8

52

251 2 +minus=minus+ xxxx is factorization Z[x] the ring of polynomial in one

variable with whole numbers as integers such as is factorization )1)(2(22 minusminus=minusminus xxxx The polynomial ring R[x] means the ring of polynomial in one variable with real numbers as coefficients Factorizing of such polynomials implies that the polynomial factors are the elements in the ring of R[x] for instance )2)(2(22 minus+=minus xxx The same thing is true for C[x] the ring of polynomial in one variable with complex numbers as coefficients Thus it is coefficients of polynomials that determine which ring a polynomial belongs to and if it is possible to be factorized in this ring For example number 6 can be factorized in two ways

in the ring of Z and 32times )51)(51( ii +minus in the ring of C (complex numbers) However there are cases in which divisions can not be done within the ring for example 52 or

)3()52( 2 ++minus xxx which means that no element in the ring can multiply with a denominator obtaining the numerator Therefore neither 5 nor can be factorized 522 +minus xx If a polynomial of degree at least one has no other divisors then it is considered to be irreducible or prime (Durbin 1992) Of every polynomial of degree at least one can be written as a product of irreducible polynomials according to unique factorization theorem each polynomial of degree at least one over a field F can be written as an element of F times a product of monic irreducible polynomials over F and except for the order in which these irreducible polynomials are written this can be done in only one way (p 221) Thus a factorable polynomial can be written into a product with another polynomial such as In other words the condition that )()()( xqaxxf minus= )( ax minus is a factor of a polynomial

in a variable x is only true if )(xf 0)( =af For example if and

6)( 2 minus+= xxxf0624)2( =minus+=f 0639)3( =minusminus=minusf the factors of are and )(xf )2( minusx )3( +x

The value of the polynomial is zero which is also expressed as the polynomial having a zero point (nollstaumllle in Swedish) when

)(xfα=x

Based on unique factorization theorem finding solutions of an algebraic equation is equivalent to finding the first-degree factors for a polynomial (Vredblad 2000) In this way solving a quadratic equation can be handled through finding the first-degree factors for the second-degree polynomial and there after finding the roots of the quadratic equation This is often called using factorization to solve quadratic equations Besides unique factorization theorem the connection of using factorization to solve a quadratic equation is that this method relates to prime numbers using distributive law backwards commutative law multiplication division and operating fractions As matter of fact it is about application of factorizing integers in polynomials at upper-secondary school level Since quadratic equations in the mathematics B course at Swedish upper-secondary school are limited to real numbers and often within the whole numbers (integers) many quadratic equations should be able to be solved by factorization according to my own

16

judgment Therefore using factorization for solving quadratic equations is very much related to arithmetic operations number concept and algebra structure It may offer students both mathematical conceptual understanding and operational skill For algebra beginners factorization can be traced back to studentsrsquo early years of mathematics education during which students do not only work with multiplication but also on factoring integers for example 56 = 8 middot 7 = 2 middot 2 middot 2 middot 7 = 4 middot 14 = 2 middot 28 In this example two operations are included multiplication and division Multiplication operation may help students to understand the multiplicative structure of the integers while factoring integers may become meaningful when it is related to divisibility divisors and prime numbers These two operational skills or competence are major techniques for simplifying fractions and finding common denominators Being able to understand and operate factoring integers has laid the essential grounds for algebra beginners to study factorization of polynomials Thus factorization reunites studentsrsquo early mathematics knowledge of factoring integers with later knowledge of algebra structure Factorability In a study on factorability by Bosseacute and Nandakumar (2005) the probability of factorability is reasoned through investigating the range of integers as coefficients of quadratic equations In the quadratic equation a b and c are randomly generated integers within a determined range r such as

002 ne=++ acbxaxyrx lele where x and y are integers When is a

perfect square or where

acb 42 minus

)4( 2 acb minus is an integer quadratics are factorable According to the studyrsquos data it shows that as the range for a b and c increases the probability of factorability of a quadratic with randomly selected integer values for a b and c decreases Bosseacute and Nandakumar confirm that as the range for coefficients expands to infinltltinfinminus r the probablility of factorability of a quadratic with randomly selected integer values for coefficients approaches zero The studyrsquos data collected from college algebra courses and textbooks demonstrates that about 15 of quadratics with integer coefficients [ ]1010minusisinr are factorable Thus about 85 of quadratics can not be factorized In spite of the limitation of factorability within the exercises concerning factoring practice chosen from 27 surveyed college algebra textbooks about 94 of the problems were factorable Within the textbooks 55 of the quadratics to be factored were within the range [-10 10] Even if many quadratic equations are factorable choosing the right factor pairs is time consuming and unnecessary for example

has to be factorized through choosing the right pairs of factors among nine pairs for 36 and 8 pairs of factors for 24 Comparing the three different methods for solving quadratic equations the study suggests that completing the square and quadratic formula are more effective informative and useful than factorization

0245936 2 =++ xx

231 Different kinds of factorizations for solving different quadratic equations How can we find the binomial factors of a quadratic equation or a second-degree polynomial without knowing one of the divisors as a binomial factor The answer is that we have to observe the relations among all coefficients and constants as well as the positive and negative signs before them under the condition of being factorable Most of the coefficients in quadratic equations or polynomials among exercises provided for practicing factoring are integers (here I borrow the same notation r as the one used by Bosseacute and Nandakumar) with

17

their range between ndash10 and 10 that is 10ler In this case it is possible for a student to make a judgment of whether or not a quadratic equation is factorable Five different kinds of factorization depending on five kinds of quadratic equations Factoring quadratic expressions can be divided into five different kinds depending on the type of quadratic equations The coefficients a b p q and constant c illustrated in the following quadratic equations are defined not to be equal to zero thus 000 nenene cba The first kind of quadratic equations is type 1 the obviously factorable type (Bosseacute and Nandakumar 2004) then the equation is factorized as

where the roots are

0002 nene=+ babxax

0)( =+ baxxabxx minus== 21 0

The second kind of quadratic equations is type 2 The factors of this second degree polynomial are so the roots to this type of quadratic

equations are

0)( 22 =minusbax))(()( 22 baxbaxbax minus+=minus

bax plusmn=21 This kind of factorization is based on the difference-of-squares

formula for example can be solved by factorizing into 049 2 =minusx 0)23)(23( =minus+ xx Thus

the roots are 32

32

21 =minus= xx This kind of factorization is also obviously factorable

The third kind of quadratic equations is type 3 This type of equations can be solved directly by factoring into two identical binomials

The double roots obtained from this

factorization are

02)( 22 =+plusmn babxax

0))(()(2)( 222 =plusmnplusmn=plusmn=+plusmn baxbaxbaxbabxax

abx plusmn=21 This method has utilized square rules which are quite easily

observed The forth kind of quadratic equations is type 4 In this equation the coefficient of the -term is 1 with p and q as positive or negative integers Type 4 equations can always be solved by the approaches of completing the square and quadratic formula However they can often be solved by factorization too since the quadratic expressions are factorable by using distributive laws reversely The roots are obtained through finding two factors for q At the same time these two factors become satisfying with the relation so that the sum of factors is equal to That way the two factors become two roots of a quadratic equation Denoting the two factors or two roots as the relationship between coefficient and constant is

02 =++ qpxx2x

qminus

21rr)( 2121 rrprrq +minus=bull= Quadratic equations can be written as the product of

two first-degree binomials or factoring form 0))(()( 212121

22 =minusminus=++minus=++ rxrxrrxrrxqpxx Through the factoring form the roots of this equation are obtained being The core of factorizing type 4 equations is seeking the connections between coefficients and roots which reveal the studentsrsquo basic ability of arithmetic multiplication and addition It is often more effective in this case to use factorization than to use quadratic formula or completing the square Let us have a look at the example In this equation we get

2211 rxrx ==

035122 =+minus xx

18

35)7)(5(12)7()5( =minusminus=minus=minus+minus= qp or 75)75( bull=bullminus= qp (here we can directly get roots) Factorizing the equation we obtain 0)7)(5( =minusminus xx and roots are The procedures of finding factors of the equation often occur in some studentsrsquo minds quickly without writing down the whole computing process but those students often have good skills in arithmetic operations (reference) Compared to using factorization for solving this equation application of quadratic formula seems ineffective A possible problem in this method is how to find the signs before the factors or how to judge if the factors in the parentheses are positive or negative integers In order to check if the solution is correct we can always multiply these two factored binomials by distributive law and see if the expanded polynomial is the same as the original quadratic equation

75 21 == xx

The fifth kind of quadratic equations is type 5 Still the methods of completing squares and quadratic formula can be applied for all type 5 equations but there are cases in which factorization can be used for solving this type of equations within a certain range of integers as mentioned before Sometimes procedures of factorizing such equations can be carried out in our minds quickly like type 4 equations but in many cases they can not As a matter of fact finding two binomial factors for a factorable quadratic equation or a factorable second-degree polynomial is a complicated process and requires systematic work In his article published in Mathematics in School Jackman (2005) demonstrates this systematic procedure via factorizing second-degree polynomials A second-degree polynomial can be written as where are all integers By multiplying out the polynomial becomes thus

0002 nene=++ bacbxax

))((2 srxqpxcbxax ++=++ srqpcbaqsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== Writing the factors of a and c on two lines p q times r s Finding the right pairs of factors for a and c has to be done by listing out the groups of products of different factors for example The factor pairs for 6 are (6 1) (1 6) (3 2) (2 3) and the factor pairs for (-4) are (4 -1) (-4 1) (2 -2) (-2 2) (1 -4) (-1 4) Writing them on two lines p 6 1 3 2 q 4 -4 2 -2 1 -1

456 2 minus+ xx

times r 1 6 2 3 s -1 1 -2 2 -4 4 Then comes calculating the values of ps and qr until the value of b = 5 is obtained The calculating procedures are divided into 4 groups and 6 steps in each group which means 4 times 6 steps in total Here I only give one grouprsquos procedure as an example 6 middot (-1) + 4 middot 1 = -2 6 middot 1 + 1middot (-4) = 2 6 middot (-2) + 1 middot 2 = -10 6 middot 2 + 1middot (-2) = 10 6 middot (-4) + 1 middot 1 = -23 6 middot 4 + 1 middot (-1) = 23 helliphellip After the 24ndashstep calculating procedures two computing procedures obtain the same correct result 3 (ndash1) + 4 2 = 5 (yes) 2 4 + (-1) 3 = 5 (yes) Are they the same From 3 (ndash1) + 4 2 = 5 the polynomial can be factorized into )12)(43( minus+ xx while from 2 4 + (ndash1) 3 = 5 the polynomial can be factorized into )43)(12( +minus xx According to communicative law these two binomial products are the same The shortcoming of this

19

method is that it is too ineffective and complicated The addition of matrixes is almost impossible and unbearable Nevertheless based on the same idea I suggest finding factors by guessing analyzing and checking Using the same example here we can find that one of the binomials must have a negative integer because of ndash4 which implies that 5 is the sum of a positive integer and a negative integer Then looking at a and c the factors of 6 can be (1 6) and (2 3) while the factors of 4 can be (1 4) and (2 2) when we regard them as absolute values We know that the factoring form of the polynomial will be like

and the connections of all coefficients are

456 2 minus+ xx

qsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== The coefficient 5=b is the difference of two products of factors for 6 and 4 First we try the biggest number of product such as 2446 =bull then 524 =minus the unknown is which is a prime number the only factors for 19 are (19 1) There is not any combination of products of factors for 6 and 4 satisfying 19 Therefore the multiplications of factors (1 6) and (1 4) are not included in this case So we move to factors (2 2) (2 3) and (1 4)

19 =

In the second step we try The result is negative it is obviously

4264466)23)(22( 22 minusminus=minus+minus=+minus xxxxxxx2 lt 5 which implies that the combination of (2 3) and (2 2) is not

considerable One pair of factors has to be removed and that is (2 2) which does not influence the coefficient of the -term Then we try The result this time is negative too but we can see that

2x 456)43)(12( 2 minusminus=minus+ xxxxb is 5 which guides us to interchanging the positions of

negative and positive signs in the parentheses The next attempt is and it is correct Seeking the correct factors for the polynomial

has taken four major operating steps by trying the combinations of factorsrsquo products and relating the sum of the products with the value of the x-term coefficient The strategy is to make use of distributive law reversely first and then multiply the product of every pair of parentheses in order to check if they are the right pair of polynomial factors Factorizing second-degree polynomials is the major step for solving quadratic equations After factoring polynomials into the product of two binomials as in the example given above the roots can be obtained by making each binomial equal to zero according to the theory of zero point (ldquonollstaumlllerdquo in Swedish) of a polynomial equation originating from unique factorization theorem

456)43)(12( 2 minus+=+minus xxxx456 2 minus+ xx

Factorization by using the technique of vertically and crosswise sutra Binomial expansion or multiplication of binomials by distributive law and factorization of the second-degree polynomials by using distributive law backwards have an extremely symmetrical effect and the results of two kinds of operations are equivalent It is crucial for the students to be able to perceive mathematics structure and be acquainted with arithmetic operations as well as to understand the relationship between quantities when they carry out factoring procedures The expansion of binomials by distributive law has been named as the method FOIL which stands for first outside inside and last according to multiplication orders of two binomials for example (Nataraj and Thomas 2006) Many students are used to this procedure Another technique to multiply two binomials is called vertically and crosswise sutra origin from the ancient Indian Vedic mathematics Vertically and crosswise sutra and FOIL are much the same in principle but it makes use of

824)4)(2( 2 +++=++ xxxxx

20

the power of visualization and improves facility with the expansion of binomials and the factorization of quadratic expressions In this method two binomials are written into two lines Using the same example the two lines are written as 824)4)(2( 2 +++=++ xxxxx

x + 2 x + 4 ______________ xsup2 4x + 2x 8

The order of multiplication is as shown below x + 2 x + 4

4 3 2 1 (the numbers 1 2 3 and 4 refer to steps) xsup2 + 2x + 4x + 8 Figure 3 The vertically and crosswise method of multiplication of binomials Step 2 and 3 can be interchangeable If we put this form into a line expanded according to the order above would be like

)4)(2( ++ xxxbullrarrbull 442 clockwise and xxx bullrarrbull2 in

counterclockwise direction According to sutra we can imagine drawing a thread starting from 2 then going through 4 and x clockwise and then draw the second thread starting from the same place 2 later going through x in the second bracket and finally x in the first bracket counterclockwise Learning expansion of binomials by the methods FOIL and Vertical and clockwise sutra is the basic process before students study factorization In this study Nataraj and Thomas (2006) have found that the students appear to do factorization using guess-and-check as well as decomposition or the vertical and clockwise approach Their finding is that afterwards the students performed better on algebra questions especially on the factorizations They conclude ldquohellipthe values of the method may lie in what it adds to the studentsrsquo overall algebraic conceptions and knowledge of mathematical structurerdquo (p 16) In general solving these five kinds of quadratic equations by factorization can be traced back to unique factorization theorem The important character in this method is that you can always check if factorization is correct or not through multiplying out the two pairs of a parenthesis and see if the expanded polynomial is the given one or not Using factorization is not only about method but also about fostering the studentsrsquo mathematical thinking The solving processes are more important than solutions

21

3 Review of related literature and research Besides the focus on mathematics content another focus in the background work for this research is about previous studies and research in the field of mathematics didactic concerning different methods for solving quadratic equations and the use of factorization In order to study the previous research and studies in the related field three scientific databases and a search engine have been used and 113 articles have been investigated Topics such as solving quadratic expressions and equations are often discussed as teaching ideas and strategies shared by mathematics educators in different countries from lower and upper-secondary schools to colleges according to the findings in the article search These didactic topics from mathematics teachersrsquo perspectives include for example using geometric approaches to solve quadratic equations (Allaire amp Bradley 2001) utilizing completing squares (Vinogradova 2007) factoring quadratics (Hoffman 1976 Cheng 1980) and using factorization completing the square graphical methods to solve quadratic equations (Macdonald 1986) Kostsopoulos (2007) links cognitive reasoning to pedagogical problems appearing in the mathematics classrooms in her article published in Australian Mathematics Teacher Based on her teaching experiences Kostsopoulos is aware of her pedagogical strategies lacking insight on dealing with studentsrsquo difficulties of multiplication and factorization of quadratics She assumes that ldquostudentsrsquo problems with factorization and with identifying varied representations of the same quadratic relationshiprdquo may be linked to the ways in which the brain constructs cognitive representations (p 19) Factoring of quadratics is the rewriting of polynomials as a product of polynomials It requires students to have both a conceptual understanding of multiplication of polynomials and the procedural knowledge to retrieve basic multiplication facts effectively Based on the theory about the order matters in the brainrsquos ability to retrieve number facts (Phenix amp Campbell 2001) Kostsopoulos conjectures that studentsrsquo ability to access the appropriate long-term semantic memory is limited when they are confused by mixing the order and varied forms of quadratics Although this article is not about a research study it seeks the reason why factorizing quadratics is difficult and therefore might provide math teachers with alternative explanations for similar pedagogical frustrations With Von Glasersfeldrsquos radical constructivism and Peter Gaumlrdenforsrsquo epistemic semantics theory as his theoretical frameworks Rauff (1994) suggests that belief-based teaching can be successful in teaching factoring According to Rauff constructivist learning theory takes the view that students construct their own beliefs and knowledge of mathematics Learning occurs when the students change their belief set Referring to belief set theory from Gaumlrdenfors (1988) Rauff (1994) claims that ldquoa belief set is expanded when a new belief is added to it and contracted when a proposition is no longer believed and is removed from the setrdquo (p 421) Expansion contraction and revision are three components to comprise basic mechanisms of changes of belief according to Rauff When examining and comparing studentsrsquo (ages 18-20) own definitions of factoring Rauff analyzes the errors of their definitions with constructivism and epistemic semantics as analyzing tools The result of this study shows that the approach to teaching factoring rely heavily upon what a student believes about factoring is quite productive in two dimensions First it reveals the source of nonstandard factoring Second it provides a starting point for modification of the studentrsquos underlying conceptions of factoring Rauff proposes that belief-based teaching can also be

22

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

lead students to solve quadratic equations in a mechanical way besides having problems with the troublesome plusmn symbolism (Stover 1978) Both Stover (1978) and Olteanu (2007) have suggested that using a graph of quadratic functions to solve quadratic equations is another alternative method

23 Mathematical background of factorization Factorization in algebra structure Among the three solving methods presented above both the geometrical method and quadratic formula can be traced back to algebra history in section 21 What is the trace of factorization in algebra history Factorization was at the last stage in algebra history and belonged to algebra structure (Derbyshire 2006) In Swedish mathematics textbooks the chapter on algebra often includes polynomial factorization and quadratic equations But the factorization in the textbooks is not at the same abstract level as the factorization of algebra structure although they share the same axioms and there are certain connections between these two different levels of factorization How are polynomial and quadratic equations as well as factorization related to each other In order to find the theoretical background of factorizing polynomials this part constitutes a mathematical theoretical review by referring to two books concerning algebra structure (Vretblad 2000 Durbin 1992) and a research study (Bosseacute and Nandakumar 2005) A quadratic equation can be regarded as polynomial equation Solving quadratic equations has much to do with ldquounique factorization theoremrdquo The polynomial

is a second-degree polynomial A polynomial consists of coefficients in whole numbers rational numbers real numbers or complex numbers and variables (sometimes called unknowns) in different degrees as well as constants A polynomial

has symbols as coefficients which are real numbers If all are zero and so

)0(02 ne=++ acbxax

cbxax ++2

nnxaxaxaaxf ++++= )( 2

210 naaa 10

ka 1gek 0)( axf = is constant If all are zero then we say the is a zero polynomial If meaning that is not a zero polynomial and n is the biggest number and then we say that the polynomial is at degrees A zero polynomial has no degree at all According to the definition (Vredblad 2000) if is a polynomial the equation is called an algebraic equation or polynomial equation A solution or a root of this equation is a number

kaf 0nef f

0nena f nf

0)( =xfα so that 0)( =αf Such a number is even

called a zero point (nollstaumllle in Swedish) of the polynomial which means when f α=x the value of the polynomial is zero (p 161) Every polynomial whose degree is equal or greater than 1 can be written as a product of irreducible polynomial In this way through polynomial division a polynomial with n degrees can be expressed from a sum of terms into a product of its denominator and its quotient The process of changing the expressions is factorization for

instance 12

22

+=minusminusminus x

xxx The factoring form is )1)(2(22 minusminus=minusminus xxxx

Factorization of polynomials is analogue to factorization of whole numbers which is very much related to the concept of ring A ring consists of a set with two operations which are a sum and a product of two elements over a field Any polynomial from zero polynomial to n degree-polynomials can be written or operated through addition and multiplication of the

15

elements over a field F including whole numbers or integers rational numbers real numbers and complex numbers In a ring you can multiply any two elements but you canrsquot always divide Factorization is a way to deal with this fact Here are some examples In the ring of integers Z is an example of factorization in some polynomial rings for instance Q[x] the ring of polynomial in one variable with rational numbers as coefficients such as

7214 sdot=

)451)(2

51(8

52

251 2 +minus=minus+ xxxx is factorization Z[x] the ring of polynomial in one

variable with whole numbers as integers such as is factorization )1)(2(22 minusminus=minusminus xxxx The polynomial ring R[x] means the ring of polynomial in one variable with real numbers as coefficients Factorizing of such polynomials implies that the polynomial factors are the elements in the ring of R[x] for instance )2)(2(22 minus+=minus xxx The same thing is true for C[x] the ring of polynomial in one variable with complex numbers as coefficients Thus it is coefficients of polynomials that determine which ring a polynomial belongs to and if it is possible to be factorized in this ring For example number 6 can be factorized in two ways

in the ring of Z and 32times )51)(51( ii +minus in the ring of C (complex numbers) However there are cases in which divisions can not be done within the ring for example 52 or

)3()52( 2 ++minus xxx which means that no element in the ring can multiply with a denominator obtaining the numerator Therefore neither 5 nor can be factorized 522 +minus xx If a polynomial of degree at least one has no other divisors then it is considered to be irreducible or prime (Durbin 1992) Of every polynomial of degree at least one can be written as a product of irreducible polynomials according to unique factorization theorem each polynomial of degree at least one over a field F can be written as an element of F times a product of monic irreducible polynomials over F and except for the order in which these irreducible polynomials are written this can be done in only one way (p 221) Thus a factorable polynomial can be written into a product with another polynomial such as In other words the condition that )()()( xqaxxf minus= )( ax minus is a factor of a polynomial

in a variable x is only true if )(xf 0)( =af For example if and

6)( 2 minus+= xxxf0624)2( =minus+=f 0639)3( =minusminus=minusf the factors of are and )(xf )2( minusx )3( +x

The value of the polynomial is zero which is also expressed as the polynomial having a zero point (nollstaumllle in Swedish) when

)(xfα=x

Based on unique factorization theorem finding solutions of an algebraic equation is equivalent to finding the first-degree factors for a polynomial (Vredblad 2000) In this way solving a quadratic equation can be handled through finding the first-degree factors for the second-degree polynomial and there after finding the roots of the quadratic equation This is often called using factorization to solve quadratic equations Besides unique factorization theorem the connection of using factorization to solve a quadratic equation is that this method relates to prime numbers using distributive law backwards commutative law multiplication division and operating fractions As matter of fact it is about application of factorizing integers in polynomials at upper-secondary school level Since quadratic equations in the mathematics B course at Swedish upper-secondary school are limited to real numbers and often within the whole numbers (integers) many quadratic equations should be able to be solved by factorization according to my own

16

judgment Therefore using factorization for solving quadratic equations is very much related to arithmetic operations number concept and algebra structure It may offer students both mathematical conceptual understanding and operational skill For algebra beginners factorization can be traced back to studentsrsquo early years of mathematics education during which students do not only work with multiplication but also on factoring integers for example 56 = 8 middot 7 = 2 middot 2 middot 2 middot 7 = 4 middot 14 = 2 middot 28 In this example two operations are included multiplication and division Multiplication operation may help students to understand the multiplicative structure of the integers while factoring integers may become meaningful when it is related to divisibility divisors and prime numbers These two operational skills or competence are major techniques for simplifying fractions and finding common denominators Being able to understand and operate factoring integers has laid the essential grounds for algebra beginners to study factorization of polynomials Thus factorization reunites studentsrsquo early mathematics knowledge of factoring integers with later knowledge of algebra structure Factorability In a study on factorability by Bosseacute and Nandakumar (2005) the probability of factorability is reasoned through investigating the range of integers as coefficients of quadratic equations In the quadratic equation a b and c are randomly generated integers within a determined range r such as

002 ne=++ acbxaxyrx lele where x and y are integers When is a

perfect square or where

acb 42 minus

)4( 2 acb minus is an integer quadratics are factorable According to the studyrsquos data it shows that as the range for a b and c increases the probability of factorability of a quadratic with randomly selected integer values for a b and c decreases Bosseacute and Nandakumar confirm that as the range for coefficients expands to infinltltinfinminus r the probablility of factorability of a quadratic with randomly selected integer values for coefficients approaches zero The studyrsquos data collected from college algebra courses and textbooks demonstrates that about 15 of quadratics with integer coefficients [ ]1010minusisinr are factorable Thus about 85 of quadratics can not be factorized In spite of the limitation of factorability within the exercises concerning factoring practice chosen from 27 surveyed college algebra textbooks about 94 of the problems were factorable Within the textbooks 55 of the quadratics to be factored were within the range [-10 10] Even if many quadratic equations are factorable choosing the right factor pairs is time consuming and unnecessary for example

has to be factorized through choosing the right pairs of factors among nine pairs for 36 and 8 pairs of factors for 24 Comparing the three different methods for solving quadratic equations the study suggests that completing the square and quadratic formula are more effective informative and useful than factorization

0245936 2 =++ xx

231 Different kinds of factorizations for solving different quadratic equations How can we find the binomial factors of a quadratic equation or a second-degree polynomial without knowing one of the divisors as a binomial factor The answer is that we have to observe the relations among all coefficients and constants as well as the positive and negative signs before them under the condition of being factorable Most of the coefficients in quadratic equations or polynomials among exercises provided for practicing factoring are integers (here I borrow the same notation r as the one used by Bosseacute and Nandakumar) with

17

their range between ndash10 and 10 that is 10ler In this case it is possible for a student to make a judgment of whether or not a quadratic equation is factorable Five different kinds of factorization depending on five kinds of quadratic equations Factoring quadratic expressions can be divided into five different kinds depending on the type of quadratic equations The coefficients a b p q and constant c illustrated in the following quadratic equations are defined not to be equal to zero thus 000 nenene cba The first kind of quadratic equations is type 1 the obviously factorable type (Bosseacute and Nandakumar 2004) then the equation is factorized as

where the roots are

0002 nene=+ babxax

0)( =+ baxxabxx minus== 21 0

The second kind of quadratic equations is type 2 The factors of this second degree polynomial are so the roots to this type of quadratic

equations are

0)( 22 =minusbax))(()( 22 baxbaxbax minus+=minus

bax plusmn=21 This kind of factorization is based on the difference-of-squares

formula for example can be solved by factorizing into 049 2 =minusx 0)23)(23( =minus+ xx Thus

the roots are 32

32

21 =minus= xx This kind of factorization is also obviously factorable

The third kind of quadratic equations is type 3 This type of equations can be solved directly by factoring into two identical binomials

The double roots obtained from this

factorization are

02)( 22 =+plusmn babxax

0))(()(2)( 222 =plusmnplusmn=plusmn=+plusmn baxbaxbaxbabxax

abx plusmn=21 This method has utilized square rules which are quite easily

observed The forth kind of quadratic equations is type 4 In this equation the coefficient of the -term is 1 with p and q as positive or negative integers Type 4 equations can always be solved by the approaches of completing the square and quadratic formula However they can often be solved by factorization too since the quadratic expressions are factorable by using distributive laws reversely The roots are obtained through finding two factors for q At the same time these two factors become satisfying with the relation so that the sum of factors is equal to That way the two factors become two roots of a quadratic equation Denoting the two factors or two roots as the relationship between coefficient and constant is

02 =++ qpxx2x

qminus

21rr)( 2121 rrprrq +minus=bull= Quadratic equations can be written as the product of

two first-degree binomials or factoring form 0))(()( 212121

22 =minusminus=++minus=++ rxrxrrxrrxqpxx Through the factoring form the roots of this equation are obtained being The core of factorizing type 4 equations is seeking the connections between coefficients and roots which reveal the studentsrsquo basic ability of arithmetic multiplication and addition It is often more effective in this case to use factorization than to use quadratic formula or completing the square Let us have a look at the example In this equation we get

2211 rxrx ==

035122 =+minus xx

18

35)7)(5(12)7()5( =minusminus=minus=minus+minus= qp or 75)75( bull=bullminus= qp (here we can directly get roots) Factorizing the equation we obtain 0)7)(5( =minusminus xx and roots are The procedures of finding factors of the equation often occur in some studentsrsquo minds quickly without writing down the whole computing process but those students often have good skills in arithmetic operations (reference) Compared to using factorization for solving this equation application of quadratic formula seems ineffective A possible problem in this method is how to find the signs before the factors or how to judge if the factors in the parentheses are positive or negative integers In order to check if the solution is correct we can always multiply these two factored binomials by distributive law and see if the expanded polynomial is the same as the original quadratic equation

75 21 == xx

The fifth kind of quadratic equations is type 5 Still the methods of completing squares and quadratic formula can be applied for all type 5 equations but there are cases in which factorization can be used for solving this type of equations within a certain range of integers as mentioned before Sometimes procedures of factorizing such equations can be carried out in our minds quickly like type 4 equations but in many cases they can not As a matter of fact finding two binomial factors for a factorable quadratic equation or a factorable second-degree polynomial is a complicated process and requires systematic work In his article published in Mathematics in School Jackman (2005) demonstrates this systematic procedure via factorizing second-degree polynomials A second-degree polynomial can be written as where are all integers By multiplying out the polynomial becomes thus

0002 nene=++ bacbxax

))((2 srxqpxcbxax ++=++ srqpcbaqsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== Writing the factors of a and c on two lines p q times r s Finding the right pairs of factors for a and c has to be done by listing out the groups of products of different factors for example The factor pairs for 6 are (6 1) (1 6) (3 2) (2 3) and the factor pairs for (-4) are (4 -1) (-4 1) (2 -2) (-2 2) (1 -4) (-1 4) Writing them on two lines p 6 1 3 2 q 4 -4 2 -2 1 -1

456 2 minus+ xx

times r 1 6 2 3 s -1 1 -2 2 -4 4 Then comes calculating the values of ps and qr until the value of b = 5 is obtained The calculating procedures are divided into 4 groups and 6 steps in each group which means 4 times 6 steps in total Here I only give one grouprsquos procedure as an example 6 middot (-1) + 4 middot 1 = -2 6 middot 1 + 1middot (-4) = 2 6 middot (-2) + 1 middot 2 = -10 6 middot 2 + 1middot (-2) = 10 6 middot (-4) + 1 middot 1 = -23 6 middot 4 + 1 middot (-1) = 23 helliphellip After the 24ndashstep calculating procedures two computing procedures obtain the same correct result 3 (ndash1) + 4 2 = 5 (yes) 2 4 + (-1) 3 = 5 (yes) Are they the same From 3 (ndash1) + 4 2 = 5 the polynomial can be factorized into )12)(43( minus+ xx while from 2 4 + (ndash1) 3 = 5 the polynomial can be factorized into )43)(12( +minus xx According to communicative law these two binomial products are the same The shortcoming of this

19

method is that it is too ineffective and complicated The addition of matrixes is almost impossible and unbearable Nevertheless based on the same idea I suggest finding factors by guessing analyzing and checking Using the same example here we can find that one of the binomials must have a negative integer because of ndash4 which implies that 5 is the sum of a positive integer and a negative integer Then looking at a and c the factors of 6 can be (1 6) and (2 3) while the factors of 4 can be (1 4) and (2 2) when we regard them as absolute values We know that the factoring form of the polynomial will be like

and the connections of all coefficients are

456 2 minus+ xx

qsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== The coefficient 5=b is the difference of two products of factors for 6 and 4 First we try the biggest number of product such as 2446 =bull then 524 =minus the unknown is which is a prime number the only factors for 19 are (19 1) There is not any combination of products of factors for 6 and 4 satisfying 19 Therefore the multiplications of factors (1 6) and (1 4) are not included in this case So we move to factors (2 2) (2 3) and (1 4)

19 =

In the second step we try The result is negative it is obviously

4264466)23)(22( 22 minusminus=minus+minus=+minus xxxxxxx2 lt 5 which implies that the combination of (2 3) and (2 2) is not

considerable One pair of factors has to be removed and that is (2 2) which does not influence the coefficient of the -term Then we try The result this time is negative too but we can see that

2x 456)43)(12( 2 minusminus=minus+ xxxxb is 5 which guides us to interchanging the positions of

negative and positive signs in the parentheses The next attempt is and it is correct Seeking the correct factors for the polynomial

has taken four major operating steps by trying the combinations of factorsrsquo products and relating the sum of the products with the value of the x-term coefficient The strategy is to make use of distributive law reversely first and then multiply the product of every pair of parentheses in order to check if they are the right pair of polynomial factors Factorizing second-degree polynomials is the major step for solving quadratic equations After factoring polynomials into the product of two binomials as in the example given above the roots can be obtained by making each binomial equal to zero according to the theory of zero point (ldquonollstaumlllerdquo in Swedish) of a polynomial equation originating from unique factorization theorem

456)43)(12( 2 minus+=+minus xxxx456 2 minus+ xx

Factorization by using the technique of vertically and crosswise sutra Binomial expansion or multiplication of binomials by distributive law and factorization of the second-degree polynomials by using distributive law backwards have an extremely symmetrical effect and the results of two kinds of operations are equivalent It is crucial for the students to be able to perceive mathematics structure and be acquainted with arithmetic operations as well as to understand the relationship between quantities when they carry out factoring procedures The expansion of binomials by distributive law has been named as the method FOIL which stands for first outside inside and last according to multiplication orders of two binomials for example (Nataraj and Thomas 2006) Many students are used to this procedure Another technique to multiply two binomials is called vertically and crosswise sutra origin from the ancient Indian Vedic mathematics Vertically and crosswise sutra and FOIL are much the same in principle but it makes use of

824)4)(2( 2 +++=++ xxxxx

20

the power of visualization and improves facility with the expansion of binomials and the factorization of quadratic expressions In this method two binomials are written into two lines Using the same example the two lines are written as 824)4)(2( 2 +++=++ xxxxx

x + 2 x + 4 ______________ xsup2 4x + 2x 8

The order of multiplication is as shown below x + 2 x + 4

4 3 2 1 (the numbers 1 2 3 and 4 refer to steps) xsup2 + 2x + 4x + 8 Figure 3 The vertically and crosswise method of multiplication of binomials Step 2 and 3 can be interchangeable If we put this form into a line expanded according to the order above would be like

)4)(2( ++ xxxbullrarrbull 442 clockwise and xxx bullrarrbull2 in

counterclockwise direction According to sutra we can imagine drawing a thread starting from 2 then going through 4 and x clockwise and then draw the second thread starting from the same place 2 later going through x in the second bracket and finally x in the first bracket counterclockwise Learning expansion of binomials by the methods FOIL and Vertical and clockwise sutra is the basic process before students study factorization In this study Nataraj and Thomas (2006) have found that the students appear to do factorization using guess-and-check as well as decomposition or the vertical and clockwise approach Their finding is that afterwards the students performed better on algebra questions especially on the factorizations They conclude ldquohellipthe values of the method may lie in what it adds to the studentsrsquo overall algebraic conceptions and knowledge of mathematical structurerdquo (p 16) In general solving these five kinds of quadratic equations by factorization can be traced back to unique factorization theorem The important character in this method is that you can always check if factorization is correct or not through multiplying out the two pairs of a parenthesis and see if the expanded polynomial is the given one or not Using factorization is not only about method but also about fostering the studentsrsquo mathematical thinking The solving processes are more important than solutions

21

3 Review of related literature and research Besides the focus on mathematics content another focus in the background work for this research is about previous studies and research in the field of mathematics didactic concerning different methods for solving quadratic equations and the use of factorization In order to study the previous research and studies in the related field three scientific databases and a search engine have been used and 113 articles have been investigated Topics such as solving quadratic expressions and equations are often discussed as teaching ideas and strategies shared by mathematics educators in different countries from lower and upper-secondary schools to colleges according to the findings in the article search These didactic topics from mathematics teachersrsquo perspectives include for example using geometric approaches to solve quadratic equations (Allaire amp Bradley 2001) utilizing completing squares (Vinogradova 2007) factoring quadratics (Hoffman 1976 Cheng 1980) and using factorization completing the square graphical methods to solve quadratic equations (Macdonald 1986) Kostsopoulos (2007) links cognitive reasoning to pedagogical problems appearing in the mathematics classrooms in her article published in Australian Mathematics Teacher Based on her teaching experiences Kostsopoulos is aware of her pedagogical strategies lacking insight on dealing with studentsrsquo difficulties of multiplication and factorization of quadratics She assumes that ldquostudentsrsquo problems with factorization and with identifying varied representations of the same quadratic relationshiprdquo may be linked to the ways in which the brain constructs cognitive representations (p 19) Factoring of quadratics is the rewriting of polynomials as a product of polynomials It requires students to have both a conceptual understanding of multiplication of polynomials and the procedural knowledge to retrieve basic multiplication facts effectively Based on the theory about the order matters in the brainrsquos ability to retrieve number facts (Phenix amp Campbell 2001) Kostsopoulos conjectures that studentsrsquo ability to access the appropriate long-term semantic memory is limited when they are confused by mixing the order and varied forms of quadratics Although this article is not about a research study it seeks the reason why factorizing quadratics is difficult and therefore might provide math teachers with alternative explanations for similar pedagogical frustrations With Von Glasersfeldrsquos radical constructivism and Peter Gaumlrdenforsrsquo epistemic semantics theory as his theoretical frameworks Rauff (1994) suggests that belief-based teaching can be successful in teaching factoring According to Rauff constructivist learning theory takes the view that students construct their own beliefs and knowledge of mathematics Learning occurs when the students change their belief set Referring to belief set theory from Gaumlrdenfors (1988) Rauff (1994) claims that ldquoa belief set is expanded when a new belief is added to it and contracted when a proposition is no longer believed and is removed from the setrdquo (p 421) Expansion contraction and revision are three components to comprise basic mechanisms of changes of belief according to Rauff When examining and comparing studentsrsquo (ages 18-20) own definitions of factoring Rauff analyzes the errors of their definitions with constructivism and epistemic semantics as analyzing tools The result of this study shows that the approach to teaching factoring rely heavily upon what a student believes about factoring is quite productive in two dimensions First it reveals the source of nonstandard factoring Second it provides a starting point for modification of the studentrsquos underlying conceptions of factoring Rauff proposes that belief-based teaching can also be

22

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

elements over a field F including whole numbers or integers rational numbers real numbers and complex numbers In a ring you can multiply any two elements but you canrsquot always divide Factorization is a way to deal with this fact Here are some examples In the ring of integers Z is an example of factorization in some polynomial rings for instance Q[x] the ring of polynomial in one variable with rational numbers as coefficients such as

7214 sdot=

)451)(2

51(8

52

251 2 +minus=minus+ xxxx is factorization Z[x] the ring of polynomial in one

variable with whole numbers as integers such as is factorization )1)(2(22 minusminus=minusminus xxxx The polynomial ring R[x] means the ring of polynomial in one variable with real numbers as coefficients Factorizing of such polynomials implies that the polynomial factors are the elements in the ring of R[x] for instance )2)(2(22 minus+=minus xxx The same thing is true for C[x] the ring of polynomial in one variable with complex numbers as coefficients Thus it is coefficients of polynomials that determine which ring a polynomial belongs to and if it is possible to be factorized in this ring For example number 6 can be factorized in two ways

in the ring of Z and 32times )51)(51( ii +minus in the ring of C (complex numbers) However there are cases in which divisions can not be done within the ring for example 52 or

)3()52( 2 ++minus xxx which means that no element in the ring can multiply with a denominator obtaining the numerator Therefore neither 5 nor can be factorized 522 +minus xx If a polynomial of degree at least one has no other divisors then it is considered to be irreducible or prime (Durbin 1992) Of every polynomial of degree at least one can be written as a product of irreducible polynomials according to unique factorization theorem each polynomial of degree at least one over a field F can be written as an element of F times a product of monic irreducible polynomials over F and except for the order in which these irreducible polynomials are written this can be done in only one way (p 221) Thus a factorable polynomial can be written into a product with another polynomial such as In other words the condition that )()()( xqaxxf minus= )( ax minus is a factor of a polynomial

in a variable x is only true if )(xf 0)( =af For example if and

6)( 2 minus+= xxxf0624)2( =minus+=f 0639)3( =minusminus=minusf the factors of are and )(xf )2( minusx )3( +x

The value of the polynomial is zero which is also expressed as the polynomial having a zero point (nollstaumllle in Swedish) when

)(xfα=x

Based on unique factorization theorem finding solutions of an algebraic equation is equivalent to finding the first-degree factors for a polynomial (Vredblad 2000) In this way solving a quadratic equation can be handled through finding the first-degree factors for the second-degree polynomial and there after finding the roots of the quadratic equation This is often called using factorization to solve quadratic equations Besides unique factorization theorem the connection of using factorization to solve a quadratic equation is that this method relates to prime numbers using distributive law backwards commutative law multiplication division and operating fractions As matter of fact it is about application of factorizing integers in polynomials at upper-secondary school level Since quadratic equations in the mathematics B course at Swedish upper-secondary school are limited to real numbers and often within the whole numbers (integers) many quadratic equations should be able to be solved by factorization according to my own

16

judgment Therefore using factorization for solving quadratic equations is very much related to arithmetic operations number concept and algebra structure It may offer students both mathematical conceptual understanding and operational skill For algebra beginners factorization can be traced back to studentsrsquo early years of mathematics education during which students do not only work with multiplication but also on factoring integers for example 56 = 8 middot 7 = 2 middot 2 middot 2 middot 7 = 4 middot 14 = 2 middot 28 In this example two operations are included multiplication and division Multiplication operation may help students to understand the multiplicative structure of the integers while factoring integers may become meaningful when it is related to divisibility divisors and prime numbers These two operational skills or competence are major techniques for simplifying fractions and finding common denominators Being able to understand and operate factoring integers has laid the essential grounds for algebra beginners to study factorization of polynomials Thus factorization reunites studentsrsquo early mathematics knowledge of factoring integers with later knowledge of algebra structure Factorability In a study on factorability by Bosseacute and Nandakumar (2005) the probability of factorability is reasoned through investigating the range of integers as coefficients of quadratic equations In the quadratic equation a b and c are randomly generated integers within a determined range r such as

002 ne=++ acbxaxyrx lele where x and y are integers When is a

perfect square or where

acb 42 minus

)4( 2 acb minus is an integer quadratics are factorable According to the studyrsquos data it shows that as the range for a b and c increases the probability of factorability of a quadratic with randomly selected integer values for a b and c decreases Bosseacute and Nandakumar confirm that as the range for coefficients expands to infinltltinfinminus r the probablility of factorability of a quadratic with randomly selected integer values for coefficients approaches zero The studyrsquos data collected from college algebra courses and textbooks demonstrates that about 15 of quadratics with integer coefficients [ ]1010minusisinr are factorable Thus about 85 of quadratics can not be factorized In spite of the limitation of factorability within the exercises concerning factoring practice chosen from 27 surveyed college algebra textbooks about 94 of the problems were factorable Within the textbooks 55 of the quadratics to be factored were within the range [-10 10] Even if many quadratic equations are factorable choosing the right factor pairs is time consuming and unnecessary for example

has to be factorized through choosing the right pairs of factors among nine pairs for 36 and 8 pairs of factors for 24 Comparing the three different methods for solving quadratic equations the study suggests that completing the square and quadratic formula are more effective informative and useful than factorization

0245936 2 =++ xx

231 Different kinds of factorizations for solving different quadratic equations How can we find the binomial factors of a quadratic equation or a second-degree polynomial without knowing one of the divisors as a binomial factor The answer is that we have to observe the relations among all coefficients and constants as well as the positive and negative signs before them under the condition of being factorable Most of the coefficients in quadratic equations or polynomials among exercises provided for practicing factoring are integers (here I borrow the same notation r as the one used by Bosseacute and Nandakumar) with

17

their range between ndash10 and 10 that is 10ler In this case it is possible for a student to make a judgment of whether or not a quadratic equation is factorable Five different kinds of factorization depending on five kinds of quadratic equations Factoring quadratic expressions can be divided into five different kinds depending on the type of quadratic equations The coefficients a b p q and constant c illustrated in the following quadratic equations are defined not to be equal to zero thus 000 nenene cba The first kind of quadratic equations is type 1 the obviously factorable type (Bosseacute and Nandakumar 2004) then the equation is factorized as

where the roots are

0002 nene=+ babxax

0)( =+ baxxabxx minus== 21 0

The second kind of quadratic equations is type 2 The factors of this second degree polynomial are so the roots to this type of quadratic

equations are

0)( 22 =minusbax))(()( 22 baxbaxbax minus+=minus

bax plusmn=21 This kind of factorization is based on the difference-of-squares

formula for example can be solved by factorizing into 049 2 =minusx 0)23)(23( =minus+ xx Thus

the roots are 32

32

21 =minus= xx This kind of factorization is also obviously factorable

The third kind of quadratic equations is type 3 This type of equations can be solved directly by factoring into two identical binomials

The double roots obtained from this

factorization are

02)( 22 =+plusmn babxax

0))(()(2)( 222 =plusmnplusmn=plusmn=+plusmn baxbaxbaxbabxax

abx plusmn=21 This method has utilized square rules which are quite easily

observed The forth kind of quadratic equations is type 4 In this equation the coefficient of the -term is 1 with p and q as positive or negative integers Type 4 equations can always be solved by the approaches of completing the square and quadratic formula However they can often be solved by factorization too since the quadratic expressions are factorable by using distributive laws reversely The roots are obtained through finding two factors for q At the same time these two factors become satisfying with the relation so that the sum of factors is equal to That way the two factors become two roots of a quadratic equation Denoting the two factors or two roots as the relationship between coefficient and constant is

02 =++ qpxx2x

qminus

21rr)( 2121 rrprrq +minus=bull= Quadratic equations can be written as the product of

two first-degree binomials or factoring form 0))(()( 212121

22 =minusminus=++minus=++ rxrxrrxrrxqpxx Through the factoring form the roots of this equation are obtained being The core of factorizing type 4 equations is seeking the connections between coefficients and roots which reveal the studentsrsquo basic ability of arithmetic multiplication and addition It is often more effective in this case to use factorization than to use quadratic formula or completing the square Let us have a look at the example In this equation we get

2211 rxrx ==

035122 =+minus xx

18

35)7)(5(12)7()5( =minusminus=minus=minus+minus= qp or 75)75( bull=bullminus= qp (here we can directly get roots) Factorizing the equation we obtain 0)7)(5( =minusminus xx and roots are The procedures of finding factors of the equation often occur in some studentsrsquo minds quickly without writing down the whole computing process but those students often have good skills in arithmetic operations (reference) Compared to using factorization for solving this equation application of quadratic formula seems ineffective A possible problem in this method is how to find the signs before the factors or how to judge if the factors in the parentheses are positive or negative integers In order to check if the solution is correct we can always multiply these two factored binomials by distributive law and see if the expanded polynomial is the same as the original quadratic equation

75 21 == xx

The fifth kind of quadratic equations is type 5 Still the methods of completing squares and quadratic formula can be applied for all type 5 equations but there are cases in which factorization can be used for solving this type of equations within a certain range of integers as mentioned before Sometimes procedures of factorizing such equations can be carried out in our minds quickly like type 4 equations but in many cases they can not As a matter of fact finding two binomial factors for a factorable quadratic equation or a factorable second-degree polynomial is a complicated process and requires systematic work In his article published in Mathematics in School Jackman (2005) demonstrates this systematic procedure via factorizing second-degree polynomials A second-degree polynomial can be written as where are all integers By multiplying out the polynomial becomes thus

0002 nene=++ bacbxax

))((2 srxqpxcbxax ++=++ srqpcbaqsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== Writing the factors of a and c on two lines p q times r s Finding the right pairs of factors for a and c has to be done by listing out the groups of products of different factors for example The factor pairs for 6 are (6 1) (1 6) (3 2) (2 3) and the factor pairs for (-4) are (4 -1) (-4 1) (2 -2) (-2 2) (1 -4) (-1 4) Writing them on two lines p 6 1 3 2 q 4 -4 2 -2 1 -1

456 2 minus+ xx

times r 1 6 2 3 s -1 1 -2 2 -4 4 Then comes calculating the values of ps and qr until the value of b = 5 is obtained The calculating procedures are divided into 4 groups and 6 steps in each group which means 4 times 6 steps in total Here I only give one grouprsquos procedure as an example 6 middot (-1) + 4 middot 1 = -2 6 middot 1 + 1middot (-4) = 2 6 middot (-2) + 1 middot 2 = -10 6 middot 2 + 1middot (-2) = 10 6 middot (-4) + 1 middot 1 = -23 6 middot 4 + 1 middot (-1) = 23 helliphellip After the 24ndashstep calculating procedures two computing procedures obtain the same correct result 3 (ndash1) + 4 2 = 5 (yes) 2 4 + (-1) 3 = 5 (yes) Are they the same From 3 (ndash1) + 4 2 = 5 the polynomial can be factorized into )12)(43( minus+ xx while from 2 4 + (ndash1) 3 = 5 the polynomial can be factorized into )43)(12( +minus xx According to communicative law these two binomial products are the same The shortcoming of this

19

method is that it is too ineffective and complicated The addition of matrixes is almost impossible and unbearable Nevertheless based on the same idea I suggest finding factors by guessing analyzing and checking Using the same example here we can find that one of the binomials must have a negative integer because of ndash4 which implies that 5 is the sum of a positive integer and a negative integer Then looking at a and c the factors of 6 can be (1 6) and (2 3) while the factors of 4 can be (1 4) and (2 2) when we regard them as absolute values We know that the factoring form of the polynomial will be like

and the connections of all coefficients are

456 2 minus+ xx

qsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== The coefficient 5=b is the difference of two products of factors for 6 and 4 First we try the biggest number of product such as 2446 =bull then 524 =minus the unknown is which is a prime number the only factors for 19 are (19 1) There is not any combination of products of factors for 6 and 4 satisfying 19 Therefore the multiplications of factors (1 6) and (1 4) are not included in this case So we move to factors (2 2) (2 3) and (1 4)

19 =

In the second step we try The result is negative it is obviously

4264466)23)(22( 22 minusminus=minus+minus=+minus xxxxxxx2 lt 5 which implies that the combination of (2 3) and (2 2) is not

considerable One pair of factors has to be removed and that is (2 2) which does not influence the coefficient of the -term Then we try The result this time is negative too but we can see that

2x 456)43)(12( 2 minusminus=minus+ xxxxb is 5 which guides us to interchanging the positions of

negative and positive signs in the parentheses The next attempt is and it is correct Seeking the correct factors for the polynomial

has taken four major operating steps by trying the combinations of factorsrsquo products and relating the sum of the products with the value of the x-term coefficient The strategy is to make use of distributive law reversely first and then multiply the product of every pair of parentheses in order to check if they are the right pair of polynomial factors Factorizing second-degree polynomials is the major step for solving quadratic equations After factoring polynomials into the product of two binomials as in the example given above the roots can be obtained by making each binomial equal to zero according to the theory of zero point (ldquonollstaumlllerdquo in Swedish) of a polynomial equation originating from unique factorization theorem

456)43)(12( 2 minus+=+minus xxxx456 2 minus+ xx

Factorization by using the technique of vertically and crosswise sutra Binomial expansion or multiplication of binomials by distributive law and factorization of the second-degree polynomials by using distributive law backwards have an extremely symmetrical effect and the results of two kinds of operations are equivalent It is crucial for the students to be able to perceive mathematics structure and be acquainted with arithmetic operations as well as to understand the relationship between quantities when they carry out factoring procedures The expansion of binomials by distributive law has been named as the method FOIL which stands for first outside inside and last according to multiplication orders of two binomials for example (Nataraj and Thomas 2006) Many students are used to this procedure Another technique to multiply two binomials is called vertically and crosswise sutra origin from the ancient Indian Vedic mathematics Vertically and crosswise sutra and FOIL are much the same in principle but it makes use of

824)4)(2( 2 +++=++ xxxxx

20

the power of visualization and improves facility with the expansion of binomials and the factorization of quadratic expressions In this method two binomials are written into two lines Using the same example the two lines are written as 824)4)(2( 2 +++=++ xxxxx

x + 2 x + 4 ______________ xsup2 4x + 2x 8

The order of multiplication is as shown below x + 2 x + 4

4 3 2 1 (the numbers 1 2 3 and 4 refer to steps) xsup2 + 2x + 4x + 8 Figure 3 The vertically and crosswise method of multiplication of binomials Step 2 and 3 can be interchangeable If we put this form into a line expanded according to the order above would be like

)4)(2( ++ xxxbullrarrbull 442 clockwise and xxx bullrarrbull2 in

counterclockwise direction According to sutra we can imagine drawing a thread starting from 2 then going through 4 and x clockwise and then draw the second thread starting from the same place 2 later going through x in the second bracket and finally x in the first bracket counterclockwise Learning expansion of binomials by the methods FOIL and Vertical and clockwise sutra is the basic process before students study factorization In this study Nataraj and Thomas (2006) have found that the students appear to do factorization using guess-and-check as well as decomposition or the vertical and clockwise approach Their finding is that afterwards the students performed better on algebra questions especially on the factorizations They conclude ldquohellipthe values of the method may lie in what it adds to the studentsrsquo overall algebraic conceptions and knowledge of mathematical structurerdquo (p 16) In general solving these five kinds of quadratic equations by factorization can be traced back to unique factorization theorem The important character in this method is that you can always check if factorization is correct or not through multiplying out the two pairs of a parenthesis and see if the expanded polynomial is the given one or not Using factorization is not only about method but also about fostering the studentsrsquo mathematical thinking The solving processes are more important than solutions

21

3 Review of related literature and research Besides the focus on mathematics content another focus in the background work for this research is about previous studies and research in the field of mathematics didactic concerning different methods for solving quadratic equations and the use of factorization In order to study the previous research and studies in the related field three scientific databases and a search engine have been used and 113 articles have been investigated Topics such as solving quadratic expressions and equations are often discussed as teaching ideas and strategies shared by mathematics educators in different countries from lower and upper-secondary schools to colleges according to the findings in the article search These didactic topics from mathematics teachersrsquo perspectives include for example using geometric approaches to solve quadratic equations (Allaire amp Bradley 2001) utilizing completing squares (Vinogradova 2007) factoring quadratics (Hoffman 1976 Cheng 1980) and using factorization completing the square graphical methods to solve quadratic equations (Macdonald 1986) Kostsopoulos (2007) links cognitive reasoning to pedagogical problems appearing in the mathematics classrooms in her article published in Australian Mathematics Teacher Based on her teaching experiences Kostsopoulos is aware of her pedagogical strategies lacking insight on dealing with studentsrsquo difficulties of multiplication and factorization of quadratics She assumes that ldquostudentsrsquo problems with factorization and with identifying varied representations of the same quadratic relationshiprdquo may be linked to the ways in which the brain constructs cognitive representations (p 19) Factoring of quadratics is the rewriting of polynomials as a product of polynomials It requires students to have both a conceptual understanding of multiplication of polynomials and the procedural knowledge to retrieve basic multiplication facts effectively Based on the theory about the order matters in the brainrsquos ability to retrieve number facts (Phenix amp Campbell 2001) Kostsopoulos conjectures that studentsrsquo ability to access the appropriate long-term semantic memory is limited when they are confused by mixing the order and varied forms of quadratics Although this article is not about a research study it seeks the reason why factorizing quadratics is difficult and therefore might provide math teachers with alternative explanations for similar pedagogical frustrations With Von Glasersfeldrsquos radical constructivism and Peter Gaumlrdenforsrsquo epistemic semantics theory as his theoretical frameworks Rauff (1994) suggests that belief-based teaching can be successful in teaching factoring According to Rauff constructivist learning theory takes the view that students construct their own beliefs and knowledge of mathematics Learning occurs when the students change their belief set Referring to belief set theory from Gaumlrdenfors (1988) Rauff (1994) claims that ldquoa belief set is expanded when a new belief is added to it and contracted when a proposition is no longer believed and is removed from the setrdquo (p 421) Expansion contraction and revision are three components to comprise basic mechanisms of changes of belief according to Rauff When examining and comparing studentsrsquo (ages 18-20) own definitions of factoring Rauff analyzes the errors of their definitions with constructivism and epistemic semantics as analyzing tools The result of this study shows that the approach to teaching factoring rely heavily upon what a student believes about factoring is quite productive in two dimensions First it reveals the source of nonstandard factoring Second it provides a starting point for modification of the studentrsquos underlying conceptions of factoring Rauff proposes that belief-based teaching can also be

22

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

judgment Therefore using factorization for solving quadratic equations is very much related to arithmetic operations number concept and algebra structure It may offer students both mathematical conceptual understanding and operational skill For algebra beginners factorization can be traced back to studentsrsquo early years of mathematics education during which students do not only work with multiplication but also on factoring integers for example 56 = 8 middot 7 = 2 middot 2 middot 2 middot 7 = 4 middot 14 = 2 middot 28 In this example two operations are included multiplication and division Multiplication operation may help students to understand the multiplicative structure of the integers while factoring integers may become meaningful when it is related to divisibility divisors and prime numbers These two operational skills or competence are major techniques for simplifying fractions and finding common denominators Being able to understand and operate factoring integers has laid the essential grounds for algebra beginners to study factorization of polynomials Thus factorization reunites studentsrsquo early mathematics knowledge of factoring integers with later knowledge of algebra structure Factorability In a study on factorability by Bosseacute and Nandakumar (2005) the probability of factorability is reasoned through investigating the range of integers as coefficients of quadratic equations In the quadratic equation a b and c are randomly generated integers within a determined range r such as

002 ne=++ acbxaxyrx lele where x and y are integers When is a

perfect square or where

acb 42 minus

)4( 2 acb minus is an integer quadratics are factorable According to the studyrsquos data it shows that as the range for a b and c increases the probability of factorability of a quadratic with randomly selected integer values for a b and c decreases Bosseacute and Nandakumar confirm that as the range for coefficients expands to infinltltinfinminus r the probablility of factorability of a quadratic with randomly selected integer values for coefficients approaches zero The studyrsquos data collected from college algebra courses and textbooks demonstrates that about 15 of quadratics with integer coefficients [ ]1010minusisinr are factorable Thus about 85 of quadratics can not be factorized In spite of the limitation of factorability within the exercises concerning factoring practice chosen from 27 surveyed college algebra textbooks about 94 of the problems were factorable Within the textbooks 55 of the quadratics to be factored were within the range [-10 10] Even if many quadratic equations are factorable choosing the right factor pairs is time consuming and unnecessary for example

has to be factorized through choosing the right pairs of factors among nine pairs for 36 and 8 pairs of factors for 24 Comparing the three different methods for solving quadratic equations the study suggests that completing the square and quadratic formula are more effective informative and useful than factorization

0245936 2 =++ xx

231 Different kinds of factorizations for solving different quadratic equations How can we find the binomial factors of a quadratic equation or a second-degree polynomial without knowing one of the divisors as a binomial factor The answer is that we have to observe the relations among all coefficients and constants as well as the positive and negative signs before them under the condition of being factorable Most of the coefficients in quadratic equations or polynomials among exercises provided for practicing factoring are integers (here I borrow the same notation r as the one used by Bosseacute and Nandakumar) with

17

their range between ndash10 and 10 that is 10ler In this case it is possible for a student to make a judgment of whether or not a quadratic equation is factorable Five different kinds of factorization depending on five kinds of quadratic equations Factoring quadratic expressions can be divided into five different kinds depending on the type of quadratic equations The coefficients a b p q and constant c illustrated in the following quadratic equations are defined not to be equal to zero thus 000 nenene cba The first kind of quadratic equations is type 1 the obviously factorable type (Bosseacute and Nandakumar 2004) then the equation is factorized as

where the roots are

0002 nene=+ babxax

0)( =+ baxxabxx minus== 21 0

The second kind of quadratic equations is type 2 The factors of this second degree polynomial are so the roots to this type of quadratic

equations are

0)( 22 =minusbax))(()( 22 baxbaxbax minus+=minus

bax plusmn=21 This kind of factorization is based on the difference-of-squares

formula for example can be solved by factorizing into 049 2 =minusx 0)23)(23( =minus+ xx Thus

the roots are 32

32

21 =minus= xx This kind of factorization is also obviously factorable

The third kind of quadratic equations is type 3 This type of equations can be solved directly by factoring into two identical binomials

The double roots obtained from this

factorization are

02)( 22 =+plusmn babxax

0))(()(2)( 222 =plusmnplusmn=plusmn=+plusmn baxbaxbaxbabxax

abx plusmn=21 This method has utilized square rules which are quite easily

observed The forth kind of quadratic equations is type 4 In this equation the coefficient of the -term is 1 with p and q as positive or negative integers Type 4 equations can always be solved by the approaches of completing the square and quadratic formula However they can often be solved by factorization too since the quadratic expressions are factorable by using distributive laws reversely The roots are obtained through finding two factors for q At the same time these two factors become satisfying with the relation so that the sum of factors is equal to That way the two factors become two roots of a quadratic equation Denoting the two factors or two roots as the relationship between coefficient and constant is

02 =++ qpxx2x

qminus

21rr)( 2121 rrprrq +minus=bull= Quadratic equations can be written as the product of

two first-degree binomials or factoring form 0))(()( 212121

22 =minusminus=++minus=++ rxrxrrxrrxqpxx Through the factoring form the roots of this equation are obtained being The core of factorizing type 4 equations is seeking the connections between coefficients and roots which reveal the studentsrsquo basic ability of arithmetic multiplication and addition It is often more effective in this case to use factorization than to use quadratic formula or completing the square Let us have a look at the example In this equation we get

2211 rxrx ==

035122 =+minus xx

18

35)7)(5(12)7()5( =minusminus=minus=minus+minus= qp or 75)75( bull=bullminus= qp (here we can directly get roots) Factorizing the equation we obtain 0)7)(5( =minusminus xx and roots are The procedures of finding factors of the equation often occur in some studentsrsquo minds quickly without writing down the whole computing process but those students often have good skills in arithmetic operations (reference) Compared to using factorization for solving this equation application of quadratic formula seems ineffective A possible problem in this method is how to find the signs before the factors or how to judge if the factors in the parentheses are positive or negative integers In order to check if the solution is correct we can always multiply these two factored binomials by distributive law and see if the expanded polynomial is the same as the original quadratic equation

75 21 == xx

The fifth kind of quadratic equations is type 5 Still the methods of completing squares and quadratic formula can be applied for all type 5 equations but there are cases in which factorization can be used for solving this type of equations within a certain range of integers as mentioned before Sometimes procedures of factorizing such equations can be carried out in our minds quickly like type 4 equations but in many cases they can not As a matter of fact finding two binomial factors for a factorable quadratic equation or a factorable second-degree polynomial is a complicated process and requires systematic work In his article published in Mathematics in School Jackman (2005) demonstrates this systematic procedure via factorizing second-degree polynomials A second-degree polynomial can be written as where are all integers By multiplying out the polynomial becomes thus

0002 nene=++ bacbxax

))((2 srxqpxcbxax ++=++ srqpcbaqsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== Writing the factors of a and c on two lines p q times r s Finding the right pairs of factors for a and c has to be done by listing out the groups of products of different factors for example The factor pairs for 6 are (6 1) (1 6) (3 2) (2 3) and the factor pairs for (-4) are (4 -1) (-4 1) (2 -2) (-2 2) (1 -4) (-1 4) Writing them on two lines p 6 1 3 2 q 4 -4 2 -2 1 -1

456 2 minus+ xx

times r 1 6 2 3 s -1 1 -2 2 -4 4 Then comes calculating the values of ps and qr until the value of b = 5 is obtained The calculating procedures are divided into 4 groups and 6 steps in each group which means 4 times 6 steps in total Here I only give one grouprsquos procedure as an example 6 middot (-1) + 4 middot 1 = -2 6 middot 1 + 1middot (-4) = 2 6 middot (-2) + 1 middot 2 = -10 6 middot 2 + 1middot (-2) = 10 6 middot (-4) + 1 middot 1 = -23 6 middot 4 + 1 middot (-1) = 23 helliphellip After the 24ndashstep calculating procedures two computing procedures obtain the same correct result 3 (ndash1) + 4 2 = 5 (yes) 2 4 + (-1) 3 = 5 (yes) Are they the same From 3 (ndash1) + 4 2 = 5 the polynomial can be factorized into )12)(43( minus+ xx while from 2 4 + (ndash1) 3 = 5 the polynomial can be factorized into )43)(12( +minus xx According to communicative law these two binomial products are the same The shortcoming of this

19

method is that it is too ineffective and complicated The addition of matrixes is almost impossible and unbearable Nevertheless based on the same idea I suggest finding factors by guessing analyzing and checking Using the same example here we can find that one of the binomials must have a negative integer because of ndash4 which implies that 5 is the sum of a positive integer and a negative integer Then looking at a and c the factors of 6 can be (1 6) and (2 3) while the factors of 4 can be (1 4) and (2 2) when we regard them as absolute values We know that the factoring form of the polynomial will be like

and the connections of all coefficients are

456 2 minus+ xx

qsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== The coefficient 5=b is the difference of two products of factors for 6 and 4 First we try the biggest number of product such as 2446 =bull then 524 =minus the unknown is which is a prime number the only factors for 19 are (19 1) There is not any combination of products of factors for 6 and 4 satisfying 19 Therefore the multiplications of factors (1 6) and (1 4) are not included in this case So we move to factors (2 2) (2 3) and (1 4)

19 =

In the second step we try The result is negative it is obviously

4264466)23)(22( 22 minusminus=minus+minus=+minus xxxxxxx2 lt 5 which implies that the combination of (2 3) and (2 2) is not

considerable One pair of factors has to be removed and that is (2 2) which does not influence the coefficient of the -term Then we try The result this time is negative too but we can see that

2x 456)43)(12( 2 minusminus=minus+ xxxxb is 5 which guides us to interchanging the positions of

negative and positive signs in the parentheses The next attempt is and it is correct Seeking the correct factors for the polynomial

has taken four major operating steps by trying the combinations of factorsrsquo products and relating the sum of the products with the value of the x-term coefficient The strategy is to make use of distributive law reversely first and then multiply the product of every pair of parentheses in order to check if they are the right pair of polynomial factors Factorizing second-degree polynomials is the major step for solving quadratic equations After factoring polynomials into the product of two binomials as in the example given above the roots can be obtained by making each binomial equal to zero according to the theory of zero point (ldquonollstaumlllerdquo in Swedish) of a polynomial equation originating from unique factorization theorem

456)43)(12( 2 minus+=+minus xxxx456 2 minus+ xx

Factorization by using the technique of vertically and crosswise sutra Binomial expansion or multiplication of binomials by distributive law and factorization of the second-degree polynomials by using distributive law backwards have an extremely symmetrical effect and the results of two kinds of operations are equivalent It is crucial for the students to be able to perceive mathematics structure and be acquainted with arithmetic operations as well as to understand the relationship between quantities when they carry out factoring procedures The expansion of binomials by distributive law has been named as the method FOIL which stands for first outside inside and last according to multiplication orders of two binomials for example (Nataraj and Thomas 2006) Many students are used to this procedure Another technique to multiply two binomials is called vertically and crosswise sutra origin from the ancient Indian Vedic mathematics Vertically and crosswise sutra and FOIL are much the same in principle but it makes use of

824)4)(2( 2 +++=++ xxxxx

20

the power of visualization and improves facility with the expansion of binomials and the factorization of quadratic expressions In this method two binomials are written into two lines Using the same example the two lines are written as 824)4)(2( 2 +++=++ xxxxx

x + 2 x + 4 ______________ xsup2 4x + 2x 8

The order of multiplication is as shown below x + 2 x + 4

4 3 2 1 (the numbers 1 2 3 and 4 refer to steps) xsup2 + 2x + 4x + 8 Figure 3 The vertically and crosswise method of multiplication of binomials Step 2 and 3 can be interchangeable If we put this form into a line expanded according to the order above would be like

)4)(2( ++ xxxbullrarrbull 442 clockwise and xxx bullrarrbull2 in

counterclockwise direction According to sutra we can imagine drawing a thread starting from 2 then going through 4 and x clockwise and then draw the second thread starting from the same place 2 later going through x in the second bracket and finally x in the first bracket counterclockwise Learning expansion of binomials by the methods FOIL and Vertical and clockwise sutra is the basic process before students study factorization In this study Nataraj and Thomas (2006) have found that the students appear to do factorization using guess-and-check as well as decomposition or the vertical and clockwise approach Their finding is that afterwards the students performed better on algebra questions especially on the factorizations They conclude ldquohellipthe values of the method may lie in what it adds to the studentsrsquo overall algebraic conceptions and knowledge of mathematical structurerdquo (p 16) In general solving these five kinds of quadratic equations by factorization can be traced back to unique factorization theorem The important character in this method is that you can always check if factorization is correct or not through multiplying out the two pairs of a parenthesis and see if the expanded polynomial is the given one or not Using factorization is not only about method but also about fostering the studentsrsquo mathematical thinking The solving processes are more important than solutions

21

3 Review of related literature and research Besides the focus on mathematics content another focus in the background work for this research is about previous studies and research in the field of mathematics didactic concerning different methods for solving quadratic equations and the use of factorization In order to study the previous research and studies in the related field three scientific databases and a search engine have been used and 113 articles have been investigated Topics such as solving quadratic expressions and equations are often discussed as teaching ideas and strategies shared by mathematics educators in different countries from lower and upper-secondary schools to colleges according to the findings in the article search These didactic topics from mathematics teachersrsquo perspectives include for example using geometric approaches to solve quadratic equations (Allaire amp Bradley 2001) utilizing completing squares (Vinogradova 2007) factoring quadratics (Hoffman 1976 Cheng 1980) and using factorization completing the square graphical methods to solve quadratic equations (Macdonald 1986) Kostsopoulos (2007) links cognitive reasoning to pedagogical problems appearing in the mathematics classrooms in her article published in Australian Mathematics Teacher Based on her teaching experiences Kostsopoulos is aware of her pedagogical strategies lacking insight on dealing with studentsrsquo difficulties of multiplication and factorization of quadratics She assumes that ldquostudentsrsquo problems with factorization and with identifying varied representations of the same quadratic relationshiprdquo may be linked to the ways in which the brain constructs cognitive representations (p 19) Factoring of quadratics is the rewriting of polynomials as a product of polynomials It requires students to have both a conceptual understanding of multiplication of polynomials and the procedural knowledge to retrieve basic multiplication facts effectively Based on the theory about the order matters in the brainrsquos ability to retrieve number facts (Phenix amp Campbell 2001) Kostsopoulos conjectures that studentsrsquo ability to access the appropriate long-term semantic memory is limited when they are confused by mixing the order and varied forms of quadratics Although this article is not about a research study it seeks the reason why factorizing quadratics is difficult and therefore might provide math teachers with alternative explanations for similar pedagogical frustrations With Von Glasersfeldrsquos radical constructivism and Peter Gaumlrdenforsrsquo epistemic semantics theory as his theoretical frameworks Rauff (1994) suggests that belief-based teaching can be successful in teaching factoring According to Rauff constructivist learning theory takes the view that students construct their own beliefs and knowledge of mathematics Learning occurs when the students change their belief set Referring to belief set theory from Gaumlrdenfors (1988) Rauff (1994) claims that ldquoa belief set is expanded when a new belief is added to it and contracted when a proposition is no longer believed and is removed from the setrdquo (p 421) Expansion contraction and revision are three components to comprise basic mechanisms of changes of belief according to Rauff When examining and comparing studentsrsquo (ages 18-20) own definitions of factoring Rauff analyzes the errors of their definitions with constructivism and epistemic semantics as analyzing tools The result of this study shows that the approach to teaching factoring rely heavily upon what a student believes about factoring is quite productive in two dimensions First it reveals the source of nonstandard factoring Second it provides a starting point for modification of the studentrsquos underlying conceptions of factoring Rauff proposes that belief-based teaching can also be

22

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

their range between ndash10 and 10 that is 10ler In this case it is possible for a student to make a judgment of whether or not a quadratic equation is factorable Five different kinds of factorization depending on five kinds of quadratic equations Factoring quadratic expressions can be divided into five different kinds depending on the type of quadratic equations The coefficients a b p q and constant c illustrated in the following quadratic equations are defined not to be equal to zero thus 000 nenene cba The first kind of quadratic equations is type 1 the obviously factorable type (Bosseacute and Nandakumar 2004) then the equation is factorized as

where the roots are

0002 nene=+ babxax

0)( =+ baxxabxx minus== 21 0

The second kind of quadratic equations is type 2 The factors of this second degree polynomial are so the roots to this type of quadratic

equations are

0)( 22 =minusbax))(()( 22 baxbaxbax minus+=minus

bax plusmn=21 This kind of factorization is based on the difference-of-squares

formula for example can be solved by factorizing into 049 2 =minusx 0)23)(23( =minus+ xx Thus

the roots are 32

32

21 =minus= xx This kind of factorization is also obviously factorable

The third kind of quadratic equations is type 3 This type of equations can be solved directly by factoring into two identical binomials

The double roots obtained from this

factorization are

02)( 22 =+plusmn babxax

0))(()(2)( 222 =plusmnplusmn=plusmn=+plusmn baxbaxbaxbabxax

abx plusmn=21 This method has utilized square rules which are quite easily

observed The forth kind of quadratic equations is type 4 In this equation the coefficient of the -term is 1 with p and q as positive or negative integers Type 4 equations can always be solved by the approaches of completing the square and quadratic formula However they can often be solved by factorization too since the quadratic expressions are factorable by using distributive laws reversely The roots are obtained through finding two factors for q At the same time these two factors become satisfying with the relation so that the sum of factors is equal to That way the two factors become two roots of a quadratic equation Denoting the two factors or two roots as the relationship between coefficient and constant is

02 =++ qpxx2x

qminus

21rr)( 2121 rrprrq +minus=bull= Quadratic equations can be written as the product of

two first-degree binomials or factoring form 0))(()( 212121

22 =minusminus=++minus=++ rxrxrrxrrxqpxx Through the factoring form the roots of this equation are obtained being The core of factorizing type 4 equations is seeking the connections between coefficients and roots which reveal the studentsrsquo basic ability of arithmetic multiplication and addition It is often more effective in this case to use factorization than to use quadratic formula or completing the square Let us have a look at the example In this equation we get

2211 rxrx ==

035122 =+minus xx

18

35)7)(5(12)7()5( =minusminus=minus=minus+minus= qp or 75)75( bull=bullminus= qp (here we can directly get roots) Factorizing the equation we obtain 0)7)(5( =minusminus xx and roots are The procedures of finding factors of the equation often occur in some studentsrsquo minds quickly without writing down the whole computing process but those students often have good skills in arithmetic operations (reference) Compared to using factorization for solving this equation application of quadratic formula seems ineffective A possible problem in this method is how to find the signs before the factors or how to judge if the factors in the parentheses are positive or negative integers In order to check if the solution is correct we can always multiply these two factored binomials by distributive law and see if the expanded polynomial is the same as the original quadratic equation

75 21 == xx

The fifth kind of quadratic equations is type 5 Still the methods of completing squares and quadratic formula can be applied for all type 5 equations but there are cases in which factorization can be used for solving this type of equations within a certain range of integers as mentioned before Sometimes procedures of factorizing such equations can be carried out in our minds quickly like type 4 equations but in many cases they can not As a matter of fact finding two binomial factors for a factorable quadratic equation or a factorable second-degree polynomial is a complicated process and requires systematic work In his article published in Mathematics in School Jackman (2005) demonstrates this systematic procedure via factorizing second-degree polynomials A second-degree polynomial can be written as where are all integers By multiplying out the polynomial becomes thus

0002 nene=++ bacbxax

))((2 srxqpxcbxax ++=++ srqpcbaqsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== Writing the factors of a and c on two lines p q times r s Finding the right pairs of factors for a and c has to be done by listing out the groups of products of different factors for example The factor pairs for 6 are (6 1) (1 6) (3 2) (2 3) and the factor pairs for (-4) are (4 -1) (-4 1) (2 -2) (-2 2) (1 -4) (-1 4) Writing them on two lines p 6 1 3 2 q 4 -4 2 -2 1 -1

456 2 minus+ xx

times r 1 6 2 3 s -1 1 -2 2 -4 4 Then comes calculating the values of ps and qr until the value of b = 5 is obtained The calculating procedures are divided into 4 groups and 6 steps in each group which means 4 times 6 steps in total Here I only give one grouprsquos procedure as an example 6 middot (-1) + 4 middot 1 = -2 6 middot 1 + 1middot (-4) = 2 6 middot (-2) + 1 middot 2 = -10 6 middot 2 + 1middot (-2) = 10 6 middot (-4) + 1 middot 1 = -23 6 middot 4 + 1 middot (-1) = 23 helliphellip After the 24ndashstep calculating procedures two computing procedures obtain the same correct result 3 (ndash1) + 4 2 = 5 (yes) 2 4 + (-1) 3 = 5 (yes) Are they the same From 3 (ndash1) + 4 2 = 5 the polynomial can be factorized into )12)(43( minus+ xx while from 2 4 + (ndash1) 3 = 5 the polynomial can be factorized into )43)(12( +minus xx According to communicative law these two binomial products are the same The shortcoming of this

19

method is that it is too ineffective and complicated The addition of matrixes is almost impossible and unbearable Nevertheless based on the same idea I suggest finding factors by guessing analyzing and checking Using the same example here we can find that one of the binomials must have a negative integer because of ndash4 which implies that 5 is the sum of a positive integer and a negative integer Then looking at a and c the factors of 6 can be (1 6) and (2 3) while the factors of 4 can be (1 4) and (2 2) when we regard them as absolute values We know that the factoring form of the polynomial will be like

and the connections of all coefficients are

456 2 minus+ xx

qsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== The coefficient 5=b is the difference of two products of factors for 6 and 4 First we try the biggest number of product such as 2446 =bull then 524 =minus the unknown is which is a prime number the only factors for 19 are (19 1) There is not any combination of products of factors for 6 and 4 satisfying 19 Therefore the multiplications of factors (1 6) and (1 4) are not included in this case So we move to factors (2 2) (2 3) and (1 4)

19 =

In the second step we try The result is negative it is obviously

4264466)23)(22( 22 minusminus=minus+minus=+minus xxxxxxx2 lt 5 which implies that the combination of (2 3) and (2 2) is not

considerable One pair of factors has to be removed and that is (2 2) which does not influence the coefficient of the -term Then we try The result this time is negative too but we can see that

2x 456)43)(12( 2 minusminus=minus+ xxxxb is 5 which guides us to interchanging the positions of

negative and positive signs in the parentheses The next attempt is and it is correct Seeking the correct factors for the polynomial

has taken four major operating steps by trying the combinations of factorsrsquo products and relating the sum of the products with the value of the x-term coefficient The strategy is to make use of distributive law reversely first and then multiply the product of every pair of parentheses in order to check if they are the right pair of polynomial factors Factorizing second-degree polynomials is the major step for solving quadratic equations After factoring polynomials into the product of two binomials as in the example given above the roots can be obtained by making each binomial equal to zero according to the theory of zero point (ldquonollstaumlllerdquo in Swedish) of a polynomial equation originating from unique factorization theorem

456)43)(12( 2 minus+=+minus xxxx456 2 minus+ xx

Factorization by using the technique of vertically and crosswise sutra Binomial expansion or multiplication of binomials by distributive law and factorization of the second-degree polynomials by using distributive law backwards have an extremely symmetrical effect and the results of two kinds of operations are equivalent It is crucial for the students to be able to perceive mathematics structure and be acquainted with arithmetic operations as well as to understand the relationship between quantities when they carry out factoring procedures The expansion of binomials by distributive law has been named as the method FOIL which stands for first outside inside and last according to multiplication orders of two binomials for example (Nataraj and Thomas 2006) Many students are used to this procedure Another technique to multiply two binomials is called vertically and crosswise sutra origin from the ancient Indian Vedic mathematics Vertically and crosswise sutra and FOIL are much the same in principle but it makes use of

824)4)(2( 2 +++=++ xxxxx

20

the power of visualization and improves facility with the expansion of binomials and the factorization of quadratic expressions In this method two binomials are written into two lines Using the same example the two lines are written as 824)4)(2( 2 +++=++ xxxxx

x + 2 x + 4 ______________ xsup2 4x + 2x 8

The order of multiplication is as shown below x + 2 x + 4

4 3 2 1 (the numbers 1 2 3 and 4 refer to steps) xsup2 + 2x + 4x + 8 Figure 3 The vertically and crosswise method of multiplication of binomials Step 2 and 3 can be interchangeable If we put this form into a line expanded according to the order above would be like

)4)(2( ++ xxxbullrarrbull 442 clockwise and xxx bullrarrbull2 in

counterclockwise direction According to sutra we can imagine drawing a thread starting from 2 then going through 4 and x clockwise and then draw the second thread starting from the same place 2 later going through x in the second bracket and finally x in the first bracket counterclockwise Learning expansion of binomials by the methods FOIL and Vertical and clockwise sutra is the basic process before students study factorization In this study Nataraj and Thomas (2006) have found that the students appear to do factorization using guess-and-check as well as decomposition or the vertical and clockwise approach Their finding is that afterwards the students performed better on algebra questions especially on the factorizations They conclude ldquohellipthe values of the method may lie in what it adds to the studentsrsquo overall algebraic conceptions and knowledge of mathematical structurerdquo (p 16) In general solving these five kinds of quadratic equations by factorization can be traced back to unique factorization theorem The important character in this method is that you can always check if factorization is correct or not through multiplying out the two pairs of a parenthesis and see if the expanded polynomial is the given one or not Using factorization is not only about method but also about fostering the studentsrsquo mathematical thinking The solving processes are more important than solutions

21

3 Review of related literature and research Besides the focus on mathematics content another focus in the background work for this research is about previous studies and research in the field of mathematics didactic concerning different methods for solving quadratic equations and the use of factorization In order to study the previous research and studies in the related field three scientific databases and a search engine have been used and 113 articles have been investigated Topics such as solving quadratic expressions and equations are often discussed as teaching ideas and strategies shared by mathematics educators in different countries from lower and upper-secondary schools to colleges according to the findings in the article search These didactic topics from mathematics teachersrsquo perspectives include for example using geometric approaches to solve quadratic equations (Allaire amp Bradley 2001) utilizing completing squares (Vinogradova 2007) factoring quadratics (Hoffman 1976 Cheng 1980) and using factorization completing the square graphical methods to solve quadratic equations (Macdonald 1986) Kostsopoulos (2007) links cognitive reasoning to pedagogical problems appearing in the mathematics classrooms in her article published in Australian Mathematics Teacher Based on her teaching experiences Kostsopoulos is aware of her pedagogical strategies lacking insight on dealing with studentsrsquo difficulties of multiplication and factorization of quadratics She assumes that ldquostudentsrsquo problems with factorization and with identifying varied representations of the same quadratic relationshiprdquo may be linked to the ways in which the brain constructs cognitive representations (p 19) Factoring of quadratics is the rewriting of polynomials as a product of polynomials It requires students to have both a conceptual understanding of multiplication of polynomials and the procedural knowledge to retrieve basic multiplication facts effectively Based on the theory about the order matters in the brainrsquos ability to retrieve number facts (Phenix amp Campbell 2001) Kostsopoulos conjectures that studentsrsquo ability to access the appropriate long-term semantic memory is limited when they are confused by mixing the order and varied forms of quadratics Although this article is not about a research study it seeks the reason why factorizing quadratics is difficult and therefore might provide math teachers with alternative explanations for similar pedagogical frustrations With Von Glasersfeldrsquos radical constructivism and Peter Gaumlrdenforsrsquo epistemic semantics theory as his theoretical frameworks Rauff (1994) suggests that belief-based teaching can be successful in teaching factoring According to Rauff constructivist learning theory takes the view that students construct their own beliefs and knowledge of mathematics Learning occurs when the students change their belief set Referring to belief set theory from Gaumlrdenfors (1988) Rauff (1994) claims that ldquoa belief set is expanded when a new belief is added to it and contracted when a proposition is no longer believed and is removed from the setrdquo (p 421) Expansion contraction and revision are three components to comprise basic mechanisms of changes of belief according to Rauff When examining and comparing studentsrsquo (ages 18-20) own definitions of factoring Rauff analyzes the errors of their definitions with constructivism and epistemic semantics as analyzing tools The result of this study shows that the approach to teaching factoring rely heavily upon what a student believes about factoring is quite productive in two dimensions First it reveals the source of nonstandard factoring Second it provides a starting point for modification of the studentrsquos underlying conceptions of factoring Rauff proposes that belief-based teaching can also be

22

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

35)7)(5(12)7()5( =minusminus=minus=minus+minus= qp or 75)75( bull=bullminus= qp (here we can directly get roots) Factorizing the equation we obtain 0)7)(5( =minusminus xx and roots are The procedures of finding factors of the equation often occur in some studentsrsquo minds quickly without writing down the whole computing process but those students often have good skills in arithmetic operations (reference) Compared to using factorization for solving this equation application of quadratic formula seems ineffective A possible problem in this method is how to find the signs before the factors or how to judge if the factors in the parentheses are positive or negative integers In order to check if the solution is correct we can always multiply these two factored binomials by distributive law and see if the expanded polynomial is the same as the original quadratic equation

75 21 == xx

The fifth kind of quadratic equations is type 5 Still the methods of completing squares and quadratic formula can be applied for all type 5 equations but there are cases in which factorization can be used for solving this type of equations within a certain range of integers as mentioned before Sometimes procedures of factorizing such equations can be carried out in our minds quickly like type 4 equations but in many cases they can not As a matter of fact finding two binomial factors for a factorable quadratic equation or a factorable second-degree polynomial is a complicated process and requires systematic work In his article published in Mathematics in School Jackman (2005) demonstrates this systematic procedure via factorizing second-degree polynomials A second-degree polynomial can be written as where are all integers By multiplying out the polynomial becomes thus

0002 nene=++ bacbxax

))((2 srxqpxcbxax ++=++ srqpcbaqsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== Writing the factors of a and c on two lines p q times r s Finding the right pairs of factors for a and c has to be done by listing out the groups of products of different factors for example The factor pairs for 6 are (6 1) (1 6) (3 2) (2 3) and the factor pairs for (-4) are (4 -1) (-4 1) (2 -2) (-2 2) (1 -4) (-1 4) Writing them on two lines p 6 1 3 2 q 4 -4 2 -2 1 -1

456 2 minus+ xx

times r 1 6 2 3 s -1 1 -2 2 -4 4 Then comes calculating the values of ps and qr until the value of b = 5 is obtained The calculating procedures are divided into 4 groups and 6 steps in each group which means 4 times 6 steps in total Here I only give one grouprsquos procedure as an example 6 middot (-1) + 4 middot 1 = -2 6 middot 1 + 1middot (-4) = 2 6 middot (-2) + 1 middot 2 = -10 6 middot 2 + 1middot (-2) = 10 6 middot (-4) + 1 middot 1 = -23 6 middot 4 + 1 middot (-1) = 23 helliphellip After the 24ndashstep calculating procedures two computing procedures obtain the same correct result 3 (ndash1) + 4 2 = 5 (yes) 2 4 + (-1) 3 = 5 (yes) Are they the same From 3 (ndash1) + 4 2 = 5 the polynomial can be factorized into )12)(43( minus+ xx while from 2 4 + (ndash1) 3 = 5 the polynomial can be factorized into )43)(12( +minus xx According to communicative law these two binomial products are the same The shortcoming of this

19

method is that it is too ineffective and complicated The addition of matrixes is almost impossible and unbearable Nevertheless based on the same idea I suggest finding factors by guessing analyzing and checking Using the same example here we can find that one of the binomials must have a negative integer because of ndash4 which implies that 5 is the sum of a positive integer and a negative integer Then looking at a and c the factors of 6 can be (1 6) and (2 3) while the factors of 4 can be (1 4) and (2 2) when we regard them as absolute values We know that the factoring form of the polynomial will be like

and the connections of all coefficients are

456 2 minus+ xx

qsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== The coefficient 5=b is the difference of two products of factors for 6 and 4 First we try the biggest number of product such as 2446 =bull then 524 =minus the unknown is which is a prime number the only factors for 19 are (19 1) There is not any combination of products of factors for 6 and 4 satisfying 19 Therefore the multiplications of factors (1 6) and (1 4) are not included in this case So we move to factors (2 2) (2 3) and (1 4)

19 =

In the second step we try The result is negative it is obviously

4264466)23)(22( 22 minusminus=minus+minus=+minus xxxxxxx2 lt 5 which implies that the combination of (2 3) and (2 2) is not

considerable One pair of factors has to be removed and that is (2 2) which does not influence the coefficient of the -term Then we try The result this time is negative too but we can see that

2x 456)43)(12( 2 minusminus=minus+ xxxxb is 5 which guides us to interchanging the positions of

negative and positive signs in the parentheses The next attempt is and it is correct Seeking the correct factors for the polynomial

has taken four major operating steps by trying the combinations of factorsrsquo products and relating the sum of the products with the value of the x-term coefficient The strategy is to make use of distributive law reversely first and then multiply the product of every pair of parentheses in order to check if they are the right pair of polynomial factors Factorizing second-degree polynomials is the major step for solving quadratic equations After factoring polynomials into the product of two binomials as in the example given above the roots can be obtained by making each binomial equal to zero according to the theory of zero point (ldquonollstaumlllerdquo in Swedish) of a polynomial equation originating from unique factorization theorem

456)43)(12( 2 minus+=+minus xxxx456 2 minus+ xx

Factorization by using the technique of vertically and crosswise sutra Binomial expansion or multiplication of binomials by distributive law and factorization of the second-degree polynomials by using distributive law backwards have an extremely symmetrical effect and the results of two kinds of operations are equivalent It is crucial for the students to be able to perceive mathematics structure and be acquainted with arithmetic operations as well as to understand the relationship between quantities when they carry out factoring procedures The expansion of binomials by distributive law has been named as the method FOIL which stands for first outside inside and last according to multiplication orders of two binomials for example (Nataraj and Thomas 2006) Many students are used to this procedure Another technique to multiply two binomials is called vertically and crosswise sutra origin from the ancient Indian Vedic mathematics Vertically and crosswise sutra and FOIL are much the same in principle but it makes use of

824)4)(2( 2 +++=++ xxxxx

20

the power of visualization and improves facility with the expansion of binomials and the factorization of quadratic expressions In this method two binomials are written into two lines Using the same example the two lines are written as 824)4)(2( 2 +++=++ xxxxx

x + 2 x + 4 ______________ xsup2 4x + 2x 8

The order of multiplication is as shown below x + 2 x + 4

4 3 2 1 (the numbers 1 2 3 and 4 refer to steps) xsup2 + 2x + 4x + 8 Figure 3 The vertically and crosswise method of multiplication of binomials Step 2 and 3 can be interchangeable If we put this form into a line expanded according to the order above would be like

)4)(2( ++ xxxbullrarrbull 442 clockwise and xxx bullrarrbull2 in

counterclockwise direction According to sutra we can imagine drawing a thread starting from 2 then going through 4 and x clockwise and then draw the second thread starting from the same place 2 later going through x in the second bracket and finally x in the first bracket counterclockwise Learning expansion of binomials by the methods FOIL and Vertical and clockwise sutra is the basic process before students study factorization In this study Nataraj and Thomas (2006) have found that the students appear to do factorization using guess-and-check as well as decomposition or the vertical and clockwise approach Their finding is that afterwards the students performed better on algebra questions especially on the factorizations They conclude ldquohellipthe values of the method may lie in what it adds to the studentsrsquo overall algebraic conceptions and knowledge of mathematical structurerdquo (p 16) In general solving these five kinds of quadratic equations by factorization can be traced back to unique factorization theorem The important character in this method is that you can always check if factorization is correct or not through multiplying out the two pairs of a parenthesis and see if the expanded polynomial is the given one or not Using factorization is not only about method but also about fostering the studentsrsquo mathematical thinking The solving processes are more important than solutions

21

3 Review of related literature and research Besides the focus on mathematics content another focus in the background work for this research is about previous studies and research in the field of mathematics didactic concerning different methods for solving quadratic equations and the use of factorization In order to study the previous research and studies in the related field three scientific databases and a search engine have been used and 113 articles have been investigated Topics such as solving quadratic expressions and equations are often discussed as teaching ideas and strategies shared by mathematics educators in different countries from lower and upper-secondary schools to colleges according to the findings in the article search These didactic topics from mathematics teachersrsquo perspectives include for example using geometric approaches to solve quadratic equations (Allaire amp Bradley 2001) utilizing completing squares (Vinogradova 2007) factoring quadratics (Hoffman 1976 Cheng 1980) and using factorization completing the square graphical methods to solve quadratic equations (Macdonald 1986) Kostsopoulos (2007) links cognitive reasoning to pedagogical problems appearing in the mathematics classrooms in her article published in Australian Mathematics Teacher Based on her teaching experiences Kostsopoulos is aware of her pedagogical strategies lacking insight on dealing with studentsrsquo difficulties of multiplication and factorization of quadratics She assumes that ldquostudentsrsquo problems with factorization and with identifying varied representations of the same quadratic relationshiprdquo may be linked to the ways in which the brain constructs cognitive representations (p 19) Factoring of quadratics is the rewriting of polynomials as a product of polynomials It requires students to have both a conceptual understanding of multiplication of polynomials and the procedural knowledge to retrieve basic multiplication facts effectively Based on the theory about the order matters in the brainrsquos ability to retrieve number facts (Phenix amp Campbell 2001) Kostsopoulos conjectures that studentsrsquo ability to access the appropriate long-term semantic memory is limited when they are confused by mixing the order and varied forms of quadratics Although this article is not about a research study it seeks the reason why factorizing quadratics is difficult and therefore might provide math teachers with alternative explanations for similar pedagogical frustrations With Von Glasersfeldrsquos radical constructivism and Peter Gaumlrdenforsrsquo epistemic semantics theory as his theoretical frameworks Rauff (1994) suggests that belief-based teaching can be successful in teaching factoring According to Rauff constructivist learning theory takes the view that students construct their own beliefs and knowledge of mathematics Learning occurs when the students change their belief set Referring to belief set theory from Gaumlrdenfors (1988) Rauff (1994) claims that ldquoa belief set is expanded when a new belief is added to it and contracted when a proposition is no longer believed and is removed from the setrdquo (p 421) Expansion contraction and revision are three components to comprise basic mechanisms of changes of belief according to Rauff When examining and comparing studentsrsquo (ages 18-20) own definitions of factoring Rauff analyzes the errors of their definitions with constructivism and epistemic semantics as analyzing tools The result of this study shows that the approach to teaching factoring rely heavily upon what a student believes about factoring is quite productive in two dimensions First it reveals the source of nonstandard factoring Second it provides a starting point for modification of the studentrsquos underlying conceptions of factoring Rauff proposes that belief-based teaching can also be

22

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

method is that it is too ineffective and complicated The addition of matrixes is almost impossible and unbearable Nevertheless based on the same idea I suggest finding factors by guessing analyzing and checking Using the same example here we can find that one of the binomials must have a negative integer because of ndash4 which implies that 5 is the sum of a positive integer and a negative integer Then looking at a and c the factors of 6 can be (1 6) and (2 3) while the factors of 4 can be (1 4) and (2 2) when we regard them as absolute values We know that the factoring form of the polynomial will be like

and the connections of all coefficients are

456 2 minus+ xx

qsxqrpsprxsrxqpx +++=++ )())(( 2

qscqrpsbpra =+== The coefficient 5=b is the difference of two products of factors for 6 and 4 First we try the biggest number of product such as 2446 =bull then 524 =minus the unknown is which is a prime number the only factors for 19 are (19 1) There is not any combination of products of factors for 6 and 4 satisfying 19 Therefore the multiplications of factors (1 6) and (1 4) are not included in this case So we move to factors (2 2) (2 3) and (1 4)

19 =

In the second step we try The result is negative it is obviously

4264466)23)(22( 22 minusminus=minus+minus=+minus xxxxxxx2 lt 5 which implies that the combination of (2 3) and (2 2) is not

considerable One pair of factors has to be removed and that is (2 2) which does not influence the coefficient of the -term Then we try The result this time is negative too but we can see that

2x 456)43)(12( 2 minusminus=minus+ xxxxb is 5 which guides us to interchanging the positions of

negative and positive signs in the parentheses The next attempt is and it is correct Seeking the correct factors for the polynomial

has taken four major operating steps by trying the combinations of factorsrsquo products and relating the sum of the products with the value of the x-term coefficient The strategy is to make use of distributive law reversely first and then multiply the product of every pair of parentheses in order to check if they are the right pair of polynomial factors Factorizing second-degree polynomials is the major step for solving quadratic equations After factoring polynomials into the product of two binomials as in the example given above the roots can be obtained by making each binomial equal to zero according to the theory of zero point (ldquonollstaumlllerdquo in Swedish) of a polynomial equation originating from unique factorization theorem

456)43)(12( 2 minus+=+minus xxxx456 2 minus+ xx

Factorization by using the technique of vertically and crosswise sutra Binomial expansion or multiplication of binomials by distributive law and factorization of the second-degree polynomials by using distributive law backwards have an extremely symmetrical effect and the results of two kinds of operations are equivalent It is crucial for the students to be able to perceive mathematics structure and be acquainted with arithmetic operations as well as to understand the relationship between quantities when they carry out factoring procedures The expansion of binomials by distributive law has been named as the method FOIL which stands for first outside inside and last according to multiplication orders of two binomials for example (Nataraj and Thomas 2006) Many students are used to this procedure Another technique to multiply two binomials is called vertically and crosswise sutra origin from the ancient Indian Vedic mathematics Vertically and crosswise sutra and FOIL are much the same in principle but it makes use of

824)4)(2( 2 +++=++ xxxxx

20

the power of visualization and improves facility with the expansion of binomials and the factorization of quadratic expressions In this method two binomials are written into two lines Using the same example the two lines are written as 824)4)(2( 2 +++=++ xxxxx

x + 2 x + 4 ______________ xsup2 4x + 2x 8

The order of multiplication is as shown below x + 2 x + 4

4 3 2 1 (the numbers 1 2 3 and 4 refer to steps) xsup2 + 2x + 4x + 8 Figure 3 The vertically and crosswise method of multiplication of binomials Step 2 and 3 can be interchangeable If we put this form into a line expanded according to the order above would be like

)4)(2( ++ xxxbullrarrbull 442 clockwise and xxx bullrarrbull2 in

counterclockwise direction According to sutra we can imagine drawing a thread starting from 2 then going through 4 and x clockwise and then draw the second thread starting from the same place 2 later going through x in the second bracket and finally x in the first bracket counterclockwise Learning expansion of binomials by the methods FOIL and Vertical and clockwise sutra is the basic process before students study factorization In this study Nataraj and Thomas (2006) have found that the students appear to do factorization using guess-and-check as well as decomposition or the vertical and clockwise approach Their finding is that afterwards the students performed better on algebra questions especially on the factorizations They conclude ldquohellipthe values of the method may lie in what it adds to the studentsrsquo overall algebraic conceptions and knowledge of mathematical structurerdquo (p 16) In general solving these five kinds of quadratic equations by factorization can be traced back to unique factorization theorem The important character in this method is that you can always check if factorization is correct or not through multiplying out the two pairs of a parenthesis and see if the expanded polynomial is the given one or not Using factorization is not only about method but also about fostering the studentsrsquo mathematical thinking The solving processes are more important than solutions

21

3 Review of related literature and research Besides the focus on mathematics content another focus in the background work for this research is about previous studies and research in the field of mathematics didactic concerning different methods for solving quadratic equations and the use of factorization In order to study the previous research and studies in the related field three scientific databases and a search engine have been used and 113 articles have been investigated Topics such as solving quadratic expressions and equations are often discussed as teaching ideas and strategies shared by mathematics educators in different countries from lower and upper-secondary schools to colleges according to the findings in the article search These didactic topics from mathematics teachersrsquo perspectives include for example using geometric approaches to solve quadratic equations (Allaire amp Bradley 2001) utilizing completing squares (Vinogradova 2007) factoring quadratics (Hoffman 1976 Cheng 1980) and using factorization completing the square graphical methods to solve quadratic equations (Macdonald 1986) Kostsopoulos (2007) links cognitive reasoning to pedagogical problems appearing in the mathematics classrooms in her article published in Australian Mathematics Teacher Based on her teaching experiences Kostsopoulos is aware of her pedagogical strategies lacking insight on dealing with studentsrsquo difficulties of multiplication and factorization of quadratics She assumes that ldquostudentsrsquo problems with factorization and with identifying varied representations of the same quadratic relationshiprdquo may be linked to the ways in which the brain constructs cognitive representations (p 19) Factoring of quadratics is the rewriting of polynomials as a product of polynomials It requires students to have both a conceptual understanding of multiplication of polynomials and the procedural knowledge to retrieve basic multiplication facts effectively Based on the theory about the order matters in the brainrsquos ability to retrieve number facts (Phenix amp Campbell 2001) Kostsopoulos conjectures that studentsrsquo ability to access the appropriate long-term semantic memory is limited when they are confused by mixing the order and varied forms of quadratics Although this article is not about a research study it seeks the reason why factorizing quadratics is difficult and therefore might provide math teachers with alternative explanations for similar pedagogical frustrations With Von Glasersfeldrsquos radical constructivism and Peter Gaumlrdenforsrsquo epistemic semantics theory as his theoretical frameworks Rauff (1994) suggests that belief-based teaching can be successful in teaching factoring According to Rauff constructivist learning theory takes the view that students construct their own beliefs and knowledge of mathematics Learning occurs when the students change their belief set Referring to belief set theory from Gaumlrdenfors (1988) Rauff (1994) claims that ldquoa belief set is expanded when a new belief is added to it and contracted when a proposition is no longer believed and is removed from the setrdquo (p 421) Expansion contraction and revision are three components to comprise basic mechanisms of changes of belief according to Rauff When examining and comparing studentsrsquo (ages 18-20) own definitions of factoring Rauff analyzes the errors of their definitions with constructivism and epistemic semantics as analyzing tools The result of this study shows that the approach to teaching factoring rely heavily upon what a student believes about factoring is quite productive in two dimensions First it reveals the source of nonstandard factoring Second it provides a starting point for modification of the studentrsquos underlying conceptions of factoring Rauff proposes that belief-based teaching can also be

22

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

the power of visualization and improves facility with the expansion of binomials and the factorization of quadratic expressions In this method two binomials are written into two lines Using the same example the two lines are written as 824)4)(2( 2 +++=++ xxxxx

x + 2 x + 4 ______________ xsup2 4x + 2x 8

The order of multiplication is as shown below x + 2 x + 4

4 3 2 1 (the numbers 1 2 3 and 4 refer to steps) xsup2 + 2x + 4x + 8 Figure 3 The vertically and crosswise method of multiplication of binomials Step 2 and 3 can be interchangeable If we put this form into a line expanded according to the order above would be like

)4)(2( ++ xxxbullrarrbull 442 clockwise and xxx bullrarrbull2 in

counterclockwise direction According to sutra we can imagine drawing a thread starting from 2 then going through 4 and x clockwise and then draw the second thread starting from the same place 2 later going through x in the second bracket and finally x in the first bracket counterclockwise Learning expansion of binomials by the methods FOIL and Vertical and clockwise sutra is the basic process before students study factorization In this study Nataraj and Thomas (2006) have found that the students appear to do factorization using guess-and-check as well as decomposition or the vertical and clockwise approach Their finding is that afterwards the students performed better on algebra questions especially on the factorizations They conclude ldquohellipthe values of the method may lie in what it adds to the studentsrsquo overall algebraic conceptions and knowledge of mathematical structurerdquo (p 16) In general solving these five kinds of quadratic equations by factorization can be traced back to unique factorization theorem The important character in this method is that you can always check if factorization is correct or not through multiplying out the two pairs of a parenthesis and see if the expanded polynomial is the given one or not Using factorization is not only about method but also about fostering the studentsrsquo mathematical thinking The solving processes are more important than solutions

21

3 Review of related literature and research Besides the focus on mathematics content another focus in the background work for this research is about previous studies and research in the field of mathematics didactic concerning different methods for solving quadratic equations and the use of factorization In order to study the previous research and studies in the related field three scientific databases and a search engine have been used and 113 articles have been investigated Topics such as solving quadratic expressions and equations are often discussed as teaching ideas and strategies shared by mathematics educators in different countries from lower and upper-secondary schools to colleges according to the findings in the article search These didactic topics from mathematics teachersrsquo perspectives include for example using geometric approaches to solve quadratic equations (Allaire amp Bradley 2001) utilizing completing squares (Vinogradova 2007) factoring quadratics (Hoffman 1976 Cheng 1980) and using factorization completing the square graphical methods to solve quadratic equations (Macdonald 1986) Kostsopoulos (2007) links cognitive reasoning to pedagogical problems appearing in the mathematics classrooms in her article published in Australian Mathematics Teacher Based on her teaching experiences Kostsopoulos is aware of her pedagogical strategies lacking insight on dealing with studentsrsquo difficulties of multiplication and factorization of quadratics She assumes that ldquostudentsrsquo problems with factorization and with identifying varied representations of the same quadratic relationshiprdquo may be linked to the ways in which the brain constructs cognitive representations (p 19) Factoring of quadratics is the rewriting of polynomials as a product of polynomials It requires students to have both a conceptual understanding of multiplication of polynomials and the procedural knowledge to retrieve basic multiplication facts effectively Based on the theory about the order matters in the brainrsquos ability to retrieve number facts (Phenix amp Campbell 2001) Kostsopoulos conjectures that studentsrsquo ability to access the appropriate long-term semantic memory is limited when they are confused by mixing the order and varied forms of quadratics Although this article is not about a research study it seeks the reason why factorizing quadratics is difficult and therefore might provide math teachers with alternative explanations for similar pedagogical frustrations With Von Glasersfeldrsquos radical constructivism and Peter Gaumlrdenforsrsquo epistemic semantics theory as his theoretical frameworks Rauff (1994) suggests that belief-based teaching can be successful in teaching factoring According to Rauff constructivist learning theory takes the view that students construct their own beliefs and knowledge of mathematics Learning occurs when the students change their belief set Referring to belief set theory from Gaumlrdenfors (1988) Rauff (1994) claims that ldquoa belief set is expanded when a new belief is added to it and contracted when a proposition is no longer believed and is removed from the setrdquo (p 421) Expansion contraction and revision are three components to comprise basic mechanisms of changes of belief according to Rauff When examining and comparing studentsrsquo (ages 18-20) own definitions of factoring Rauff analyzes the errors of their definitions with constructivism and epistemic semantics as analyzing tools The result of this study shows that the approach to teaching factoring rely heavily upon what a student believes about factoring is quite productive in two dimensions First it reveals the source of nonstandard factoring Second it provides a starting point for modification of the studentrsquos underlying conceptions of factoring Rauff proposes that belief-based teaching can also be

22

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

3 Review of related literature and research Besides the focus on mathematics content another focus in the background work for this research is about previous studies and research in the field of mathematics didactic concerning different methods for solving quadratic equations and the use of factorization In order to study the previous research and studies in the related field three scientific databases and a search engine have been used and 113 articles have been investigated Topics such as solving quadratic expressions and equations are often discussed as teaching ideas and strategies shared by mathematics educators in different countries from lower and upper-secondary schools to colleges according to the findings in the article search These didactic topics from mathematics teachersrsquo perspectives include for example using geometric approaches to solve quadratic equations (Allaire amp Bradley 2001) utilizing completing squares (Vinogradova 2007) factoring quadratics (Hoffman 1976 Cheng 1980) and using factorization completing the square graphical methods to solve quadratic equations (Macdonald 1986) Kostsopoulos (2007) links cognitive reasoning to pedagogical problems appearing in the mathematics classrooms in her article published in Australian Mathematics Teacher Based on her teaching experiences Kostsopoulos is aware of her pedagogical strategies lacking insight on dealing with studentsrsquo difficulties of multiplication and factorization of quadratics She assumes that ldquostudentsrsquo problems with factorization and with identifying varied representations of the same quadratic relationshiprdquo may be linked to the ways in which the brain constructs cognitive representations (p 19) Factoring of quadratics is the rewriting of polynomials as a product of polynomials It requires students to have both a conceptual understanding of multiplication of polynomials and the procedural knowledge to retrieve basic multiplication facts effectively Based on the theory about the order matters in the brainrsquos ability to retrieve number facts (Phenix amp Campbell 2001) Kostsopoulos conjectures that studentsrsquo ability to access the appropriate long-term semantic memory is limited when they are confused by mixing the order and varied forms of quadratics Although this article is not about a research study it seeks the reason why factorizing quadratics is difficult and therefore might provide math teachers with alternative explanations for similar pedagogical frustrations With Von Glasersfeldrsquos radical constructivism and Peter Gaumlrdenforsrsquo epistemic semantics theory as his theoretical frameworks Rauff (1994) suggests that belief-based teaching can be successful in teaching factoring According to Rauff constructivist learning theory takes the view that students construct their own beliefs and knowledge of mathematics Learning occurs when the students change their belief set Referring to belief set theory from Gaumlrdenfors (1988) Rauff (1994) claims that ldquoa belief set is expanded when a new belief is added to it and contracted when a proposition is no longer believed and is removed from the setrdquo (p 421) Expansion contraction and revision are three components to comprise basic mechanisms of changes of belief according to Rauff When examining and comparing studentsrsquo (ages 18-20) own definitions of factoring Rauff analyzes the errors of their definitions with constructivism and epistemic semantics as analyzing tools The result of this study shows that the approach to teaching factoring rely heavily upon what a student believes about factoring is quite productive in two dimensions First it reveals the source of nonstandard factoring Second it provides a starting point for modification of the studentrsquos underlying conceptions of factoring Rauff proposes that belief-based teaching can also be

22

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

applied to teaching other aspects of algebra such as polynomial multiplication In his study however he has not investigated the source in relation to subject matter from textbooks An ambiguity of his theoretical frameworks is how true knowledge in mathematics is if mathematics can be constructed by anybody What I mean is that teachers have to know what is right or wrong in mathematics principles Vaiyavutjamai and Clements (2006) have carried out a study involving 231 students from two government secondary schools in Thailand Their aim of the study is to investigate how traditional lessons on quadratic equations in particular when teaching grade 9 students to solve quadratic equations by factorization (and application of the null-factor law) by completing the square and by the quadratic formula influence studentsrsquo understanding of quadratic equations The study consists of 18 lessons on 50 minutes each with observations recorded on audiotape pre- and post-teaching tests as well as 18 interviews for exploring studentsrsquo understanding of quadratic equations and their unknowns In the literature review the study points out the lack of research on the learning of quadratic equations in associated understanding variables in quadratic equations ldquoStudent thinking in such contexts appeared to be dominated by a need to achieve procedural mastery and usually there was no guarantee that relational understanding was achievedrdquo (p 49) With Skemprsquos instrumental understanding and relational understanding as their theoretical framework Vaiyavutjamai and Clements (2006) aim at finding if a traditional teaching approach does improve studentsrsquo relational understanding of quadratic equations The study points out misconceptions regarding variables as obstacles for the students in understanding quadratic equations The students have difficulties in discerning

and 01582 =+minus xx 0)5)(3( =minusminus xx These two equations are actually equivalent and have the same structure the factoring form is just another form of the quadratic equation The students do not think that x in 0)5)(3( =minusminus xx represents different variables They do not really understand the null-factor law Many students obtained correct solutions but had serious misconceptions about what quadratic equations actually are from a mathematical point of view The study reveals that the traditional teaching approach may improve studentsrsquo rote-learned knowledge and performance skills but do not help their relational understanding of quadratic equations The study suggests teaching quadratic equations within the teaching of functions by the use of modern technology like graphic calculators To a great extend Vaiyavutjamai and Clementsrsquo study relates to my research though they have their focus on studentsrsquo understanding of variables Within traditional teaching pedagogy an experimental study from a Chinese middle school has given evidence to effective self-learning of factorization without lectures but instead with chosen examples and problems (Zhu amp Simon 1987) Another similar study of exploring the result of teaching a Vedic method for learning factorization has been done at a secondary school in New Zealand which has been reviewed in Section 231 The result shows an improvement of the studentsrsquo performance in factorization of quadratic expressions and expansion of binomials as well as conceptual understanding of quadratic expressions (Nataraj amp Thomas 2006) Not all studies support the use of factorization for solving quadratic equations Bosseacute and Nandakumar (2005) arguendashthrough investigations of college courses and 27 college algebra textbooksndashthat the employment of factorization is not efficient compared to utilizing the quadratic formula or completing the square They have found that only 15 of quadratics

23

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

with integer coefficients at range of [ndash10 10] among problems and examples from the textbooks are factorable The probability of factorability of a quadratic with randomly selected integer coefficients from the textbooks was small The study criticizes the practice of factoring prior to other methods according to the conventional curriculum based on NCTM recommendations in US schools ldquoentertains a false dichotomy pitting mathematics against pedagogyrdquo (p 147) By demonstrating the strengths of using the techniques of completing the square and the quadratic formula as well as solving quadratic functions in graphs Bosseacute and Nandakumar declare that these methods are useful for all kinds of quadratic equations and more efficient and informative than factorization which ldquois only appropriate for quadratic equations with rational rootsrdquo (p 151)

4 Research issues From the previous review one learns that there are different methods for solving quadratic equations in mathematics teaching at the upper secondary-school level There are arguments both for and against teaching factorization How mathematics textbooks and other teaching material handle this algebraic topic is a matter of mathematics pedagogy and nature of mathematics In this research I want to explore how these different methods or approaches of solving quadratic equations are presented and sequenced in mathematics textbooks Furthermore I want to investigate how these different methods might be related to each other The research questions thus are 1 What are mathematical and pedagogical rationales in the presentations of algebra contents

concerning the different methods for solving quadratic equations in Swedish mathematics textbooks and other teaching material for mathematics course B at upper-secondary school

2 Is there any factorization potentiality among tasks on solving quadratic equations in the mathematics textbooks investigated If so what do they look like

3 How do mathematics teachers use mathematics textbooks and other teaching material concerning the teaching of solving quadratic equations and factorization

In order to find the answer to research questions the research includes two studies one is a study of mathematics textbooks and the other of teaching materials The second study focuses on how mathematics teachers use textbooks as a resource when teaching and the study will be carried out through interviewing mathematics teachers

5 Theoretical frameworks

51 Pedagogical Content Knowledge Teaching should consider the transformation of a specific content knowledge into forms suitable for studentsrsquo learning How a teacher transforms the content knowledge in a subject includes many factors teacherrsquos understanding of subject which means the teacherrsquos own subject knowledge the sources of teacher knowledge such as textbooks subject literature teaching material and so on teacherrsquos knowledge of students and their learning teacherrsquos knowledge of curricular organization of subject material etc All these factors and the organization of them in a teacherrsquos mind are referred to as a teacherrsquos content knowledge (Shulman 1986) Subject matter content knowledge pedagogical content knowledge and curricular knowledge are three categories of content knowledge A teacher understanding subject matter content knowledge requires his or her understanding the facts or concepts of

24

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

the subject contents and the structures of the subject ldquoGoing beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teachingrdquo is a teacherrsquos Pedagogical Content Knowledge (Shulman 1986 p 9) In other words PCK relates to a teacherrsquos ways of representing and formulating the subject in order to make it comprehensible to others and teacherrsquos understanding of what makes the learning of specific topics easy or difficult Agreeing with Cochran et al (1993) Emanuelsson (2001) claims that PCK defined by Shulman has an unclear epistemology Although Cochran et al consequently reformulate PCKnowledge into PCKnowing to make it clearer through drawing upon constructivist epistemology it does not solve the problem of epistemology according to Emanuelsson (2001) Instead he means that taking into account studentsrsquo possibilities to learn in a better way can be a solution of rephrasing PCK originated by Shulman

52 Procedural and conceptual knowledge Investigating different methods for solving quadratic equations is not merely about procedures or rules but also about exploring what these methods can offer to students when it comes to perceiving algebra structure and enriching conceptual connections The character of generalization in algebraic rules makes it both easy and difficult for the students It is simple in a way because a rule can be used for many different situations and is generalized The students can usually solve analogical problems by memorizing the rule and the description of the problem mechanically Anyhow the result will be correct The students seem to be right but they do not understand why Unfortunately many problems in textbooks are formulated simply and students can easily recognize them therefore it is not hard to find rules which fit into the situation On the other hand algebra is difficult because students can solve problems or operate equations with symbols without really understanding what those letters (or symbols) mean What a student needs is not only procedural knowledge but also conceptual knowledge Conceptual misunderstanding in algebra is common among students like seeing an unknown as a special number or ignoring variables (Booth 1984) It is probably the reason why some students do not realize that there are actually two roots or results in a quadratic equation if they take a variable for just one special number Here it is actually the problem of lacking conceptual knowledge of variables J Hiebert (1986) generalizes knowledge into two kinds procedural and conceptual knowledge The conceptual knowledge is often regarded as the knowledge of concepts and principles compared to the procedural knowledge about the knowledge of rules and procedures (Star 2005) Conceptual knowledge is rich in relationships (Hiebert amp Lefevre 1986) It is not an isolated piece of information instead it is part of a network On the other hand procedural knowledge is defined as a sequence of actions for solving mathematical problems It is knowledge of rules and procedures (Hiebert amp Carpenter 2007) In school mathematics many teachers ldquobehave as if mathematics is a subject whose use for students in the end is as a set of procedures for solving problemsrdquo Teachers expect the students to learn the skills of solving particular kinds of problems (Stigler amp Hiebert 1999 p 89) There are arguments on which kind of knowledge that is more important and how conceptual and procedural knowledge interact Teachers are still wondering whether to be concerned with conceptual relationships or procedural proficiency first ldquoWell-rehearsed procedures capture a kind of mathematical power because they exploit the consistency and patterns in mathematical systems and guide the seemingly effortless solution of routine problemsrdquo

25

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

(Hiebert amp Carpenter 2007 p 78) It is about action and doing things in my opinion To understand an algebraic concept or definition it is not enough to grasp procedural knowledge This is probably the reason of mechanical learning Algebraic conceptual knowledge is required for the students in order to understand the meaning of algebra The relationships between conceptual and procedural knowledge depend on the connections learners construct between their representations Procedures in mathematics always depend on principles represented conceptually Sometimes the relationship between them is so close that they become difficult to distinguish (Hiebert amp Carpenter 2007) The deep procedural and conceptual knowledge is obtained when they are completely integrated The ability to understand a concept increases with the rich connections between its conceptual and procedural knowledge (Baroody A Feil Y amp Johnson A (2007)

It is commonly believed that the conceptual knowledge precedes procedural knowledge In mathematics the procedural knowledge is often related to steps rules practice and memory with goals of efficiency and accuracy (Hiebert amp Handa 2004) According to Hiebert and Handa (2004) the dichotomy between procedural and conceptual is not the case in Hong Kong classrooms Memorization in Hong Kong classrooms is not rote memory but memorization with mathematical connections and meanings The distinction between procedural and conceptual is not so clear and absolute all the time the interplay between them is actually very complex and intertwined Therefore Hiebert and Handa declare the need of new terms that can capture the activities including the interaction of both conceptual and procedural knowledge Algebra uses symbols as its language In 1989 L R Booth describes the meaning of algebraic structure and the letter-symbolic form as Our ability to manipulate algebraic symbols successfully requires that we first understand the structural properties of mathematical operations and relations which distinguish allowable transformations from those that are not These structural properties constitute the semantic aspects of algebrahellip The essential feature of algebraic representation and symbol manipulation then is that it should proceed from an understanding of the semantics or referential meanings that underlie it (Kieran 2007 p 711)

According to Kieranrsquos interpretation of Boothrsquos words above the symbols become transparent when a learner can ldquoseerdquo abstract ideas hidden behind the symbols (Kieran 2007) This way algebra becomes meaningful In fact to make algebra meaningful is a matter of the relation between algebraic conceptual knowledge and procedural knowledge Besides the theory of conceptual and procedural knowledge another important theoretical framework is about relational and instrumental understanding formed by Richard R Skemp and used in his model of analysis According to Skemp instrumental understanding means knowing the rules and having the ability to use them but lacking the knowledge of why (Chueng 1980) Relational understanding means knowing both what to do and why (Skemp 1976)

26

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

6 Mathematics textbooks analyzing frameworks Some related research on textbooks analyses has provided the important criteria as analyzing tools for this research I will discuss the analyzing frameworks used by Braumlndstroumlm (2005) Pepin and Haggarty (2001) and Van Dormolen (1986) below

61 A brief review on previous research regarding analyzing frameworks Braumlndstroumlm (2005) has analyzed the levels of difficulty for differentiated tasks in mathematics textbooks used in Swedish lower-secondary school from grade 7 to 9 In her analysis Braumlndstroumlm uses two perspectives to analyze the study of the thought process of a student when he solves a task One perspective of her analyzing framework is based on the revised version of Bloomrsquos taxonomy by Anderson (2001) Bloomrsquos taxonomy was originally ldquocreated to represent the intended outcome of the educational process and categories the studentsrsquo behaviorrdquo (Braumlndstroumlm p 27) Braumlndstroumlm (2005) has referring to Krathwohl (2001) stated that Bloomrsquos taxonomy has been regarded as a framework for classifying what teachers expect students to learn as a result of instruction According to her interpretation of Bloomrsquos taxonomy (1956) Braumlndstroumlm mentions three domains of educational activities identified in the taxonomy cognitive affective and psychomotor with focus on the cognitive domain The cognitive domain refers to the cognitive demonstrated by knowledge recall and intellectual skills (eg understanding ideas and applying knowledge) There are six hierarchical categories in this domain beginning with simple behavior and building to the most complex knowledge comprehension application analysis synthesis and evaluation A student performing at a higher level demonstrates a more complex level of cognitive thinking (Braumlndstroumlm 2005) However according to Braumlndstroumlm here referring to other researchers different expressions of criticism about using taxonomy arise during its application such as difficulty in interpreting the categories and the independence of content from process and categories isolated from any context Thus the revised version of Bloomrsquos taxonomy by Anderson (2001) is applied to Braumlndstroumlmrsquos research In this revised version the categories are renamed as remembering understanding applying analyzing evaluating and creating These categories indicate a studentrsquos thinking at different levels from the lowest level of remembering to the highest level of creating Braumlndstroumlm uses a framework by Smith and Stein (1998) as another aspect for her framework to study the thought process of a student when solving a task In this framework four aspects are included memorization procedures with connections to concepts or meaning procedures without connections to concepts or meaning doing mathematics Through the combination of the two frameworks above Braumlndstroumlms own framework for analyzing what is demanded by a student when he or she solves a mathematics task includes four perspectives pictures (none decorative functional) operations (one operation or more than one operation) processes (remembering understanding applying analyzing evaluating and creating) demands (memorization connections no connections and doing mathematics) (Braumlndstroumlm 2005) Pepin et al (2001) have carried out a study of textbook analyses and mathematics teachersrsquo use of textbooks in school contexts They have analyzed textbooks used in English French and German mathematics classrooms at lower-secondary level In their article (Pepin et al 2001) they generalize four areas concerning analyses of mathematics textbooks in terms of

27

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

their content and structure the mathematical intentions of textbooks pedagogical intentions of textbooks sociological contexts of textbooks and the culture traditions represented in textbooks The mathematics intentions of textbooks are divided into three areas ldquowhat kind of mathematics that is represented in the textbooks beliefs about the nature of mathematics that are implicit in textbooks and the presentation of mathematical knowledgerdquo (p 3) What school mathematics is becomes discussed in the article when it is related to the topic of the nature of mathematics Referring to La Transposition Didactique by Y Chevellard Pepin et al (2001) write that mathematics in school has been regarded as a special version of mathematics or taught knowledge distinguishing it from the real scientific knowledge They consider it important to investigate the intended views of the nature of mathematics projected in textbooks In their analysis presentation of the mathematical knowledge in textbooks Pepin et al (2001) have consulted analyzing aspects by Van Dormolen (1986) and by Schmidt et al (1996 1997) Referring to Van Dormolen (1986) an analyst might look for the following aspects when analyzing mathematics textbooks theoretical aspect (theorems definitions axioms) an algorithmic aspect (explicitly how to dohellip) a logical aspect (rules about how we are and are not allowed to handle theory) a methodological aspect (how to dohellip more heuristically for example how to use mathematical induction) a communicative aspect (conventions or how to write down an argument for example) Schmidt et al keep their focus on the understanding of the content in terms of its topic complexity (which topics when what is emphasized with what conceptual demands) developmental complexity (ways of sequencing and developing topics across lessons and across the whole curriculum) cognitive complexity (the pedagogical intention for the topic ie what you want the students to do as a result of having learned the topic) (Pepin et al 2001 p 4)Van Dormolenrsquos classification seems to explore the nature of mathematics from a mathematics textbook analysis and represent a teaching perspective while Schmidt et al put the focus of their classification on a learning perspective Van Dormolenrsquos classification has its starting point in what mathematics is from my own point of view Van Dormolen (1986) generalizes two contrasting extremist views One is from the formalistic point of view ldquoIn the formalistic view mathematics is a set of concepts rules theorems and structures This is a culture heritage that we must pass on partly at schoolrdquo (p 143) In order to be able to reason in mathematics the students must acquire mathematics knowledge through first learning certain skills such as algorithmic skills and deductive reasoning ability Plausible and intuitive reasoning do not belong among genuine mathematical activities and serve only as preliminary stages towards real knowledge Authors agreeing with this view may write texts that tightly guide students towards knowledge of certain concepts and rules like instructions Such books may contain only definitions theorems proofs and exercises Another extreme point of view is the opposite of the formalistic view Mathematics consists of the students engaging in activities like generalizing classifying formalizing ordering abstracting exploring patterns and so on Intuitive reasoning is to be considered as a useful mathematical activity and regarded as genuine mathematics Students are supposed to discover patterns rules structures New ideas are encouraged The formalistic view is indirectly interpreted as reception learning and the opposing view as discovery learning A third view is also mentioned by Van Dormolen being a mixture of both views Mathematics can be seen as a human activity on the one hand and cultural heritage in the formal knowledge on the other hand and this view leads to a kind of learning called guided discovery What kind of view a textbook author agrees on concerning mathematics will influence his or her writing of mathematics textbooks

28

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

62 What analyzing criteria from previous frameworks might be useful for this research In this research mathematics textbooks analyses include analyzing content structures in eight mathematics textbooks for Swedish upper-secondary school and detailed analyses of three chosen textbooks among those eight ones I will investigate how the mathematics content concerning quadratic equations and their solving methods as well as related tasks are presented in the mathematics textbooks from the three perspectives mentioned in the research aim The result will be used as the basic material for further interviews with mathematics teachers In analyzing I will combine the aspects taken from Van Dormolenrsquos classification and the one by Schmidt et al as well as Braumlndstroumlmrsquos categories of analyzing mathematics contents and tasks Classifications of Van Dormolen and Schmidt et al will be chosen and applied when analyzing instructional texts (also called theoretical parts) and Braumlndstroumlmrsquos categories for exercises after instructions will be chosen The following aspects can be considered global theoretical algorithmic communicative and cognitive complexity Global perspective means consistency considering first whether a student has been prepared in the past for the new topic or mathematical content in the text second whether the new content in the text serves a purpose for future learning (Van Dormolen 1986) This criterion will be used specially for analyzing the tables of contents in the eight textbooks Cognitive complexity is used for examining what an author expects a learner to do as a result of having learned the topic or the new skill recognizing recalling rules or concepts performing routine procedures explaining reasoning and creating The analyzing aspects consist of two parts one for analyzing theoretical contents in the textbooks the other for analyzing exercises and activities These two groups of criteria will be presented in detail in Section 71 and Section 72

7 Methods and some results of the analysis The analyses of the textbooks will be an attempt to explore the mathematical and pedagogical rationale in the presentations of algebra contents related to solving quadratic equations and factorization in mathematics textbooks for the mathematics course B Four analyses have almost been completed so far They include one on the tables of contents and content structures in the eight mathematics textbooks and three detailed analyses of three of those eight textbooks The eight mathematics textbooks are all used in the course Mathematics B at Swedish upper-secondary school They are Matematik 3000 Kurs B (Bjoumlrk et al 2001) for the SP and ES programs Matematik 4000 Kurs B [Green book] (Alfredsson et al 2008) the difficulty level of this book is medium Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) the difficulty level of this book is the highest Matematik fraringn A till E Kurs B (Holmstroumlm et al 2001) Matematik 2000 B Nya Delta Matematik kurs A och B (Bjoumlrup et al 2000) Δ NTa+b Kurs A och B (Wallin et al 2000) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) The three textbooks for detailed analyses are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B [Blue book] (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) These three textbooks make

29

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

up a series of mathematics books which appear in different colored volumes according to their varied levels of difficulty Four of the eight textbooks were randomly chosen from the library collections while the other four textbooksndashamong which the three selected for detailed analyses can be foundndashwere chosen based on information via e-mail from three publishers Gleerup Utbildning AB Natur amp Kultur and Studentlitteratur The information indicates that Matematik 4000 Matematik 3000 Exponent B Matematik foumlr gymnasieskolan Matematik fraringn A till E Kurs B were the best-selling textbooks on the Swedish market in 2008

71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B The analysis of the eight textbooks concerning the tables of contents and content structures has a mathematical and pedagogical intention with the textbooks as its focus Since the investigation has a PCK perspective I have combined the classification aspects from Van Dormolen and Schmidt et al (Pepin et al 2001) mentioned in Chapter 6 for analyzing the theoretical texts When analyzing mathematics exercises or problems and activities from the textbooks I will use some of Braumlndstroumlmrsquos (2005) criteria as a tool for my analysis but part of the analysis of mathematics tasks has not been completed yet The following aspects have been considered regarding the theoretical parts in the textbooks

1 Content structure and consistency this aspect will be investigated both in a chapter and a whole book Which topic is the focus of a chapter and how does it relate to other kinds of content in the same book Has it been mentioned before this chapter Does it appear later in the book

2 Mathematical theoretical aspects what theorems definitions and rules are described in the text before the part with exercises

3 Algorithmic aspects how are a set of rules explained explicitly for solving a particular problem

4 Representativeness of given examples to which extent does a given example represent a relating theory or rule

5 Number of methods how many methods for solving quadratic equations have been presented in a textbook

6 Language use in which way are these theorems definitions rules explained and illustrated for readersstudents formally in mathematical language or pedagogically in combination with everyday language

The table on the following page is an example of the result of investigating the tables of contents and content structure in the eight textbooks concerning criteria aspects 1 2 and 5 stated above

30

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

Content structure Mathematical theoretical aspects Textbooks

N=8 Preceding related contents The concept of functions linear

function and equation systems linear inequality

6

Mathematical concepts and rules before the introduction of quadratic equations

Polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple)

7

Solving methods for simple quadratic equations axsup2 + c = 0 or axsup2 + bx = 0 (a ne 0 b ne 0)

Using null-factor law square root method

8

Solving methods for complete quadratic equations xsup2 + px + q = 0

Using completing the square quadratic formula

8

Connections (how are quadratic equations presented in relation to other kinds of content)

Relating to quadratic functions in a graphical presentation or connecting with geometry

2 graphical 4 geometrical 2 no connection

After the chapter on quadratic equations

6 Quadratic functions with cases of interceptions with x-axis

Table 1 Three analyzing aspects from the eight textbooks The content of quadratic equations belongs to the chapter on algebra in all of the books except one The definitions of function of linear function and of equation systems provide students with pre-knowledge of quadratic equations Pedagogical intention can be seen as necessary steps to learning quadratic equations through the presentations related to the concept of polynomials distributive law or product of two binomials the difference-of-squares formula square rules factorization (simple) before learning quadratic equations After all these moments the students are prepared to meet quadratic equations in both simple forms and complete forms as well as their solving methods Even later in the chapter on quadratic functions applying quadratic formula for finding the symmetry line and intercepting coordinates in a quadratic function is emphasized If we look at the historical development of algebra presented earlier solving quadratic equations had developed in the stages from geometrical paralleling with rhetorical to symbolical and graphical stage Quadratic formula appeared after the late stage of the symbolical one Four out of these eight textbooks have geometrical illustrations for the method of completing squares in the beginning and then later present quadratic formula The graphical solving method is presented in the chapter on function after the geometrical presentation and quadratic formula in six out of the eight books Thus the order of the presentation of algebra contents in six textbooks has reflected the development in the historical stages The content structure regarding algebra contents and solving quadratic equations in the textbooks is consistent and related to each other before and after the part on quadratic equations In spite of this consistency it is short of descriptions of the relations among these different algebra contents for example why should polynomial be presented first in the chapter on algebra Why must factorization or distributive law be presented before the part on quadratic equations Is the method ldquoNull-product methodrdquondashalso called null-factor

31

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

lawndashthe same as factorization These theoretical contents are introduced without much of context and connections The presentation of different kinds of quadratic equations in examples is restrained mostly within simple and complete quadratics with ldquo1rdquo as coefficient of -term like in all the books The detailed presentation of solving standard-form quadratic equations like

is absent in six books though there are many tasks of this kind in the books The presentation of the transformation from the standard-form quadratic equations to complete quadratic equations is absent in all eight books although I think that this is an essential step in order to help students understand how a quadratic formula is derived from solving standard quadratic equations The varied form of quadratic formula is especially useful in solving quadratic functions

2x 02 =++ qpxx

002 ne=++ acbxax

Every book presents four different solving skills null-factor law or factorization and square-root method for simple quadratic equations completing the square and quadratic formula for complete quadratic equations All the books put emphasis on completing the square and quadratic formula Factorization takes little space among these skills related to the null-factor law Factorization in these books is applied only for solving simple quadratic equations in the form of axsup2 + bx = 0 (a ne 0 b ne 0) or the expanded form derived from the difference-of-squares formula Using factorization to solve the standard quadratic equations is absent for example On the other hand the use of factorization for solving standard quadratic equations has been popularly discussed and taught by many mathematics educators internationally Is it too difficult or complicated to present it in textbooks because of its mathematics complexity as Bosseacute and Nandakumar (2005) have argued Or is it too early to learn this kind of factorization in view of the studentsrsquo cognitive understanding Or at last is it probably unnecessary to use this old technique because of the employment of computer programs and calculators

)2)(32(62 2 minusminus=minus+ xxxx

72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail I have chosen three of the eight textbooks to analyze in detail These three textbooks have sold well on the market in Sweden recently and they are Matematik 3000 Kurs B (Bjoumlrk et al 2001) Matematik 4000 Kurs B (Blue book) (Alfredsson et al 2008) and Exponent B Matematik foumlr gymnasieskolan (Gennow et al 2007) as mentioned before Unlike analyzing the content structure of the eight mathematics textbooks the analysis of the three textbooks is done by describing and evaluating the contents in detail and exercises page by page in every part of the related chapters in these three textbooks This is carried out with the use of the theoretical criteria presented above in six points and the criteria for analyzing exercises based on Braumlndstroumlm (2005) The aim to analyze exercise parts in the three textbooks is to investigate first how many methods that can possibly be used for solving a quadratic equation second the potentials of utilizing factorization among all the exercises and third mathematics complexity at different levels among these exercises The criteria or analyzing tool used for analyzing mathematics exercises or tasks are partly based on the ones from Braumlndstroumlm (2005) They are

1) Which of the four solving methods is one supposed to use for solving a quadratic equation according to the exercise instruction

32

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

2) Which method is most effective among factorization square root methodcompleting squares and quadratic formula

3) How many solving methods can actually be used without considering the instruction for solving quadratic equations

4) Can a complete or standard quadratic equation be solved by factorization or not Factorability in short

5) Which domains of coefficients in a quadratic equation belong to for example integers (if it is bigger than 10) rational numbers real numbers or irrational numbers

6) How many operational steps are needed for solving one task 7) How many conceptual connections can be found in each task 8) Does a task require mathematics generalization 9) Does a task require a student to create hisher own mathematics problem

Every exercise may have four to six separate tasks or may consist of different subproblems Therefore I categorize an exercise with a few separate tasks into small independent tasks If an exercise with more than two subproblems is undividable in a mathematical sense I regard it as one independent task in a whole Tasks are included in an exercise An example of analyzing tasks is illustrated below in Table 2

Tasks

0562 =+minus xx

0562 =++ xx

01242 =minus+ xx

33

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

0762 =minus+ xx

010155 2 =+minus xx

096444 2 =minus+minus xx

070242 2 =++ xx

080505 2 =+minus xx

0288 2 =+minus zz

39)10( =+AA

34

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

102 2 =x

014 2 =minus xx

0122 =minusminus xx

510 2 =minus yy

024142 =+minus xx

35

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

05562 =minusminus xx

0432 =minusminus xx

0652 =minusminus xx

0132 =+minus xx

0552 =minus+ xx

05260 2 =minusminus xx

36

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

0904031 2 =minusminus xx

26543 nn minusminus=

69445823 22 ++=++ zzzz

25)1)(1()4( 2 =+minus+minus xxx

3)1()12( 2 ++=minus xxx

42 minus=x

37

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

092 =minusx

162 2 =x

043 2 =+x

032122 =+minus xx

38

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

02562 =+minus xx

034162 2 =++ xx

081182 =+minus xx

39

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

abxbax 6)23(2 =minus+

73221 2 =++ nn ( xn =2 )

Table 2 The analysis of exercises on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) The table above is a general analysis on all the tasks (36) given on page 103 in the textbook Matematik 3000 B (Bjoumlrk et al 2001) according to the four methods that have been presented in this chapter SQRCSQ for the square-root method and CSQ for completing squares (I regard it as a submethod belonging to the square-root method) F for the null-factor law (factorization) PQ for the solving formula (the quadratic formula) In fact a graphical approach through finding the coordinates of parabolas cutting the x-axis can be used for all quadratic equations regarded as special cases of quadratic functions Thus I will not include it in the analysis of quadratic equations In the table columns P is the initial of the word ldquopossiblerdquo which means that the method listed in the column can also be used and T for textbook means that the textbook requires the method in this column to be used for example can be solved by all three methods but the textbook requires that only the PQ formula is used Thus I have marked the columns with P P and T Open means that methods are chosen by a student himself E means that using this method is effective M for method shows how many methods that can be applied for this task The column labeled Roots shows which number set the roots of each quadratic equation belongs to

0562 =+minus xx

Result of the analysis of the exercises in Table 2 SQR denoted for the square-root method and CSQ for completing the square are two different methods However I put them in one column because it is easy to compare these two methods within the same space in addition to how unclear the definitions of these two techniques are presented in the textbook Here I generalize all equations above into 1 or ( ) cax plusmn=2 02 =plusmn cax 0nea2 ( ) 02 =plusmn bxax 0nea3 (02 =plusmnplusmn cbxax 0nea ) 4 ( ) 02 =plusmnplusmn cbxx 1=a In total 25 tasks out of 36 (about 69 of the tasks) can be solved by all four methods 7 tasks out of all 36 (19) are not factorable but can be solved by the other three methods 4 tasks out of all 36 can not be solved at all on this level since they have to be solved in the complex-number system which is not dealt with in the B course There are 14 tasks for which one is

40

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

supposed to apply the quadratic formula called PQ formula and 4 tasks with completing the square There are 18 open tasks meaning that the students have to choose the solving methods by themselves There is not any instruction about using factorization included for any exercise even though there are 25 out of 36 tasks that can be solved by factorization Only 7 out of 36 tasks are not factorable and all their roots are irrational numbers The roots of factorable quadratic equations are mostly integers and 4 rational numbers They belong to real number system (domain) Obviously all of the quadratic equations from these exercises can be solved by both the method of completing the square and the PQ formula There are 4 out of 36 tasks which are solved by the square-root method (SQR) compared to 27 that one is to solve by using the method completing the square These four quadratic equations belong to the first type listed above which shows that the x term of every quadratic equation is missing Half of the tasks (18 out of 36) are type four quadratic equations The coefficient of x square is one One quadratic equation is type two There are 13 out of all 36 tasks which are type three quadratic equations The most common ones are type four and three It is worth noticing that 12 tasks can be solved effectively by factorizing directly upon inspection and all of them are type four quadratic equations The other 13 equations denoted ldquoIndirect Frdquo are type three equations In order to factorize this kind of equations the students have to consider the relations among coefficients of xsup2 term and x term as well as the constant All coefficients are integers except three rational numbers Among the coefficients 0 lt a lt10 The most effective method for solving the complete and standard form quadratic equations in this table in my opinion is factorization since it requires fewer operational steps at the same time as it trains studentsrsquo structure sense but factorization for standard quadratic equations is absent in Matematik 3000 B The most inefficient method is the quadratic formula since the algorithm procedures are unnecessarily long on the other hand the method completing the square and quadratic formula are the techniques that guarantee success with solving quadratic equations How much have I done in this research work Finally the analysis of the overall content structure and the tables of contents concerning algebra in the eight mathematics textbooks has been carried out according to the three criteria reported in Section 71 The theoretical contentsrsquo analyses of the three textbooks have been carried out Analyses of exercises in the three books have not been completed The first analyzing text is about the textbook Matematik 3000 B the second is Matematik 4000 B and the third is Exponent B The contents arrangement in every related chapter of each textbook is very similar presented in the following pattern a short introduction rarr theoretical expositions rarr examples with answers for helping to understand theory rarr exercises at three levels rarr historical backgrounds rarr algebra and its application in the real world as well as this kind of exercises rarr more exercises like discovery activities rarr summary of the whole chapter rarr mixed exercises with previous knowledge and newly learned skills rarr the most challenging tasks Each of the three books has the same algebra contents simplifying polynomials or computing polynomials the difference-of-squares formula and square rules factorization simple

41

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

quadratic equations the null-factor law also called the null-product equations complete quadratic equations and quadratic formula Discussion on my further research work Since the purpose of analyzing the mathematics textbooks is to answer the first two research issues on the mathematical and pedagogical rationales of relating algebra contents and potentials of factorization I might need to analyze a few more books among these eight alternatives as a comparison Based on the result of the analysis of the textbooks I will design interview questions for the next study of interviewing mathematics teachers from upper-secondary schools and universities aiming at finding how mathematics textbooks are used by teachers concerning teaching quadratic equations These two studies will be used for the licentiate thesis For my continuing research after the licentiate thesis there might be a study on teaching the solving of quadratic equations based on classroom observations

References Alfredsson L Brolin H Erixon P Heikne H amp Ristamaumlki A (2008) Matematik 4000

Kurs B (Blaring bok) Stockholm Natur och Kultur Alfredsson L Brolin H Erixon amp P Heikne H (2008) Matematik 4000 Kurs B

(Groumlnbok) Stockholm Natur och Kultur Allaire P R amp Bradley R E (2001) Geometric approaches to quadratic equations from

other times and places Mathematics Teacher 94 (4) 308-319 Baroody A Feil Y amp Johnson A (2007) An alternative reconceptualization of procedural and conceptual knowledge Journal for Research in Mathematics Education 38(2) 115-131 Bergsten C (2002) Faces of Swedish research in mathematics education In C Bergsten G

Dahland amp B Grevholm (red) Research and action in the mathematics classroom Proceedings of the 2nd Swedish mathematics Education Research Seminar (21-36) Linkoumlping Svensk Foumlrening foumlr Matematikdidaktisk Forskning

Bjoumlrk L E Brolin H amp Ekstig K (1995) Matematik 2000 Kurs B Stockholm Natur och

Kultur Bjoumlrk L E Borg K Brolin H Ekstig K Heikne H amp Larsson K (2001) Matematik 3000 Kurs B (Groumln bok) Stockholm Natur och Kultur Bjoumlrup K Koumlrner S Oscarsson E amp Sandhall Aring (2000) Nya delta matematik kurs A och

B Malmouml Gleerups Booth L R (1984) Algebra Childrenrsquos strategies and errors Windsor UK NFER-Nelson Booth L R (1989) A question of structure In S Wagner amp C Kieran (Eds) Research issues

in the learning and teaching of algebra (Research agenda for mathematics

42

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

education 4 57-59) Reston VA National Council of Teachers of Mathematics

Bosseacute M J amp Nandakumar N R (2005) The factorability of quadratics motivation for

more techniques (section A) Teaching Mathematics and its Applications 24 (4) 143-153

Braumlndstroumlm A (2005) Differentiated Tasks in Mathematics Textbooks An Analysis of the Levels of Difficulty Licentiate thesis Lulearing University of Technology

Cheung Y L (1980) Learning ideas for mathematics teacher education Journal of Science and Mathematics Education in S E Asia 3 (2) 12-19

Derbyshire J (2006) Unknown quantity Washington D C Joseph Henry Press Durbin J R (1992) Modern algebra an introduction New York Chichester Brisbane

Toronto and Singapore John Wiley amp Sons Inc Emanuelsson J (2001) En fraringga om fraringgor Hur laumlrares fraringgor i klassrummet goumlr det

moumljligt att faring reda paring elevernas saumltt att foumlrstaring det som undervisningen behandlar i matematik or naturvetenskap Goumlteborg Acta Universitatis Gothoburgensis

Gennow S Gustafsson I-M amp Silborn B (2007) Exponent B Matematik foumlr

gymnasieskolan (Roumld bok) Malmouml Gleerups Holmstroumlm M Smedhamre E amp Liber AB (2001) Matematik fraringn A till E gymnasiets

matematik kurs B Stockholm Liber AB Hiebert J and Carpenter T P (2007) Learning and teaching with understanding In F K

Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 65-97 Charlotte NC Information Age Pub

Hiebert J and Lefevre P (1986) Conceptual and procedural knowledge in mathematics

An introductory analysis In J Hiebert (Ed) Conceptual and procedural knowledge The case of mathematics 1-27 Hillsdale NJ Lawrence Erlbaum

Hiebert J amp Handa Y (2004) A modest proposal for reconceptualising the activity of learning mathematical procedures The American Educational Research Association [Paper from conference] San Diego April 2004 pp 1-13

Hoffman N (1976) Factorisation of quadratics Mathematics teaching 76 54-55 Haumlggstroumlm J Skolans algebra ndash Varfoumlr saring svaringrt (2006) Proceedings of Matematik bygger

broar Malmouml Malmouml Houmlgskola

Jackman T (2005) Another method of factorizing quadratics Mathematics in School 34(3) 20-21

43

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

Johansson M (2006) Teaching mathematics with textbooks a classroom and curricular perspective Doctorial thesis Lulearing University of Technology

Katz V J (2007) Stages in the history of algebra with implications for teaching Educational Studies in Mathematics 66 185-201

Kieran C (2007) Learning and teaching of algebra at the middle school through college levels building meaning for symbols and their manipulation In F K Lester Jr (Ed) Second handbook of research on mathematics teaching and learning a project of the National Council of Teachers of Mathematics 2 707-762 Charlotte NC Information Age Pub

Kotsopoulos D (2007) Unravelling student challenges with quadratics a cognitive approach

Australian Mathematics Teacher 63 (2) 19-24 Kvasz L (2006) The History of Algebra and the Development of the Form of its Language

Philosophia mathematica (Series III Volume 14 Number 3 October 2006) Oxford Oxford University Press

MacDonald T H (1986) Problems in presenting quadratics as a unifying topic The

Australian Mathematics Teachers 42(3) 20-22

Nataraj M S amp Thomas M O J (2006) Expansion of binomials and factorisation of quadratic expressions exploring a Vedic method Australian Senior Mathematics Journal 20 (2) 8-17

Obermeyer D D (1982) Another look at the quadratic formula Mathematics Teacher 75 (2) 146-152

Olteanu C (2007) rdquoVad skull x kunna varardquo andragradsekvation och andragradsfunktion som objekt foumlr laumlrande [rdquoWhat could x berdquo second degree equation and qudratic function as objects of learning] Institutionen foumlr beteendevetenskap Houmlgskolan i Kristianstad

Pepin B amp Haggarty L (2001) Mathematics textbooks and their use in English French and

German classrooms a way to understand teaching and learning cultures Zentralblatt fuumlr Didaktik der Mathematik 33 (5) pp 158-175

Rauff J V (1994) Constructivism factoring and beliefs School Science and Mathematics

94 (8) 421-426 Shulman L S (1986) Those who understand Knowledge growth in teaching Educational

Research 15 (2) 4-14 Skemp R R (1976) Relational understanding and instrumental understanding Mathematics

Teaching 77 20-26

Skolverket (2000) Naturvetenskapsprogrammet Gy 2000 Programmaringl kursplaner betygskriterier och kommentarer Stockholm Fritzes

44

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References

Star J (2005) Reconceptualizing procedural knowledge Journal for research in Mathematics Education 36 404-411

Stover D W (1978) Teaching quadratic problem solving Mathematics Teacher 71 (1) 13-16

Stigler J W amp Hiebert J (1999) The Teaching gap Best ideas from the Worldrsquos Teachers

for Improving Education in the Classroom New York The Free Press 89 Vaiyavutjamai P amp Clements M A (2006) Effects of Classroom Instruction on Studentsrsquo

Understanding of Quadratic Equations Mathematics Education Research Journal 18(1) 47-77

Van Dormolen (1986) Textual analysis In Christiansen B Howson A G and Otte M (Eds)

Perspectives on Mathematics Education Dordrecht D Reidel Vinogradova N (2007) Solving quadratic equations by completing squares Mathematics

Teaching in the Middle School 12 (7) 403-405 Vretblad A (2000) Algebra och geometri Kristianstad Gleerups Wallin H Lithner J Wiklund S Jacobsson S amp Liber AB (2000) Δ NT a+b

gymnasiematematik foumlr naturvetenskaps- och teknikprogrammen kurs A och B Stockholm Liber AB

Zhu X amp Simon H A (1987) Learning mathematics from examples and by doing

Cognition and Instruction 4 (3) 137-166

45

• Alternative Approaches of Solving Quadratic Equations in Mathematics Teaching
• • An Empirical Study of Mathematics Textbooks and Teaching Material for Swedish Upper-Secondary School
  • • 1 AIM AND BACKGROUND
    • 11 Introduction
    • • 12 Why is it about algebra
      • 13 The choice of which mathematics textbooks to investigate
      • 14 Research Aim
      • The aim of this research is to explore the rationale of Pedagogical Content Knowledge in the presentations of algebra contents on the topic of quadratic equations in Swedish mathematics textbooks and other teaching material for the B course in mathematics at upper-secondary school Three perspectives will be considered algebra historical perspective mathematics as a scientific discipline and pedagogical perspective Research questions will be put forward in Chapter 4 after the mathematics background introductions and previous research review in Chapter 2 and Chapter 3
      • • 2 Mathematics background in the field of algebra
        • • 21 Algebra history and its development
          • 22 Three solving methods for quadratic equations
          • 23 Mathematical background of factorization
          • • 231 Different kinds of factorizations for solving different quadratic equations
            • • 3 Review of related literature and research
              • 4 Research issues
              • 5 Theoretical frameworks
              • • 51 Pedagogical Content Knowledge
                • 52 Procedural and conceptual knowledge
                • • 6 Mathematics textbooks analyzing frameworks
                  • • 61 A brief review on previous research regarding analyzing frameworks
                    • • 7 Methods and some results of the analysis
                      • • 71 Investigating the tables of contents and content structures in eight mathematics textbooks for Mathematics course B
                        • 72 Analyzing three of the eight mathematics textbooks for Mathematics course B in detail
                        • • References