structure sense in algebraic expressions and equations of

Journal of Educational and Human Research Development 6:140-154 (2018)Southern Leyte State University, Sogod, Southern Leyte, Philippines

Structure Sense in Algebraic Expressions andEquations of Groups of Students

Joseph G. TabanEdelyn A. Cadorna

University of Northern PhilippinesVigan City, Ilocos Sur, Philippines

Abstract

High school Algebra is a critical subject that bridges students’ ability to reaching a higher levelof mathematical knowledge, attitudes, and skills. This study determined the “structure sense”in algebraic expressions and equations of groups of students. Using in-depth analysis on thewritten outputs of the Grades 8, 9, and 10 students, the study explored the algebraic structureof their solutions. The students portrayed different ways of solving Algebraic expressions andequations. Results show that those who have adequate knowledge and skills were successfulin carrying out the works while those who are not familiar with the structural properties of thetasks found difficulty in solving, which yielded an ambiguous solution, which led to incorrectanswers. The students lacked the conceptual understanding of the given problem, and theyhave poor skills in manipulating expressions, calculation mistakes, and technical errors. Theyeven displayed different methods of solving algebraic expressions and equations, from simple,to more detailed, and to more complicated but understandable solutions. Recognition of the“structure sense” of every term in algebraic expressions and equations are necessary forstudents to perform appropriate manipulations.

Keywords: Algebraic expressions and equations; Qualitative; Structure sense

Introduction

In high school Algebra, students find difficultywith basic concepts particularly in applyingalgebraic techniques in contexts, which aredifferent from what they experienced.

According to Novotna and Hoch (2008),there are excellent Mathematics performers injunior high school who are poor in algebraicmanipulations. Students who succeed well inGrade 10 algebra show disappointing resultslater on because of algebra. Students founddifficulty in formulating equations when solvingword problems, and questions which requirethem to manipulate algebraic expressionscorrectly; demonstrated weak arithmetic skills;and made errors of an arithmetical naturecausing them to make algebraic errors too

(Matzin & Shahrill, 2015). They also finddifficulty applying algebraic techniques theyhave learned earlier.

Jupri, Drijvers and Panhuizen (2014)identified types of difficulties in initial algebrasuch as applying arithmetical operationsin numerical and algebraic expressions,understanding the notion of variable,understanding algebraic expressions,understanding the meanings of the equalsign, and mathematization. This difficulty maybe due to lack of structure sense.” (Novotna &Hoch, 2008).

Structure sense is associated with thedifficulties of students when learning theconcepts of algebra. (Hoch & Dreyfus,2010). As stated in the paper of Drijversand Gravemeijer (2014), and as described by

*Correspondence: [email protected] ISSN 2545-9732

Taban and Cadorna JEHRD Vol.6, 2018

Hoch and Dreyfus (2010), structure sense isused to analyze students’ use of previouslylearned algebraic techniques. A studentdisplay structure sense (SS) if s/he can: 1)recognize a familiar structure in its simplestform, 2) deal with a compound term as a singleentity and through an appropriate substitutionrecognize a familiar structure in a morecomplex form, and 3) choose appropriatemanipulations to make the best use of thestructure. The substitution principle whichstates that if a variable or parameter isreplaced by a compound term (product orsum), or if a parameter replaces a compoundterm, the structure remains the same, anessential feature of a structure sense.

Therefore, there is a need for studentsto master structure sense because this isconsidered as an important collection ofabilities to deal with algebra. In the contextsof high school algebra, it includes the abilityto see an algebraic expression or sentence asan entity, recognize an algebraic expression orsentence as a previously met structure, dividean entity into sub-structures, recognize mutualconnections between structures, recognizewhich manipulations is possible to perform,and recognize which manipulations is useful toperform. (Hoch & Dreyfus; Novotna & Hoch; inJupri & Sispiyati, 2017, p.1).

A prerequisite for students to masterstructure sense in algebra is to determinetheir proficiency in basic algebraic conceptslike algebraic expressions and equationsfocusing on how the students do the tasks.An analysis of the students’ solutions willenable one to see their difficulty in applyingalgebraic techniques. Studies in mathematicseducation focus on analyzing the result of atest that describes students’ performance andbehavior in a specific area of mathematics.Researches had shown several difficultiesof students in dealing with algebra. Bushand Karp (2013) cited students’ difficultieswith proportional reasoning, fragmentedunderstanding of fractions concepts, andincorrect understanding of equality. Lamon(2012) also stated students’ confusion about

ratios when written as fractions.There are limited studies that explain how

students solve problems regarding structuresense, and these are foreign studies (Novotna& Hoch, 2008; Hock & Dreyfus, 2010; Hoch,2003; Jupri & Sispiyati, 2017; Bush & Karp,2013; and Lamon, 2012). One method ofdetermining the students’ use of structuresense is to investigate how students solveproblems particularly tasks that deal withalgebraic expressions and equations.

This study aimed to investigate the structuresense in algebraic expressions and equationsof different groups of students (Grades 8, 9,and 10).

Specifically, it sought to determine thealgebraic task/s where the students displayedstructure sense, to explore the structure ofthe solutions of students in different algebraictasks; and to see the pattern of the structuresense of the students by grade level.

Framework of the StudyIn mathematics education, structure is seenas a broad view analysis of the way in whichan entity is made up of its parts. Theanalysis describes the systems of connectionsor relationships between the parts (Hoch &Dreyfus, 2010). Hoch and Dreyfus claimedthat any algebraic expression or sentencerepresents an algebraic structure. Theexternal appearance or shape reveals, or ifnecessary can be transformed to reveal aninternal order. The relationships between thequantities and operations that are parts of thestructure determine the internal order (Hoch &Dreyfus, 2010).

Novotna and Hoch (2008) cited Linchevskiand Livneh who pointed out that studentshave difficulty with algebraic structure, i.e.,students cannot distinguish structural featuresof equations. Many students commit a lot oferros in solving algebraic tasks.

Identifying and correcting errors areessential aspects of the constructivist theory ofdevelopmental learning. These errors are dueto incorrect procedures and misconceptions

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or inadequate understanding.Studies hadshown that the primary cause of errorin basic mathematics among students isthe lack of understanding of the basicrules. The students experienced problemsin representing and translating problemstatements; relations between percentages,decimals, and fractions; and dealing withnegative numbers. They used the methodof guessing extensively in solving wordedproblems. Non-mathematics causes of errorsinclude carelessness, haphazard reasoning,lack or non-use of metacognitive skills,creativity, poor memory, and test anxiety(Cruz, Duavis, & Colmo, 2015).Moreover,mathematics textbooks tend to focus on theprocedures and not so much on symbol senseand structure sense. This reality may be thereason for the students’ tendency to focuson routine procedures as a consequence ofdidactical contract in the textbooks (Stiphout,Drijvers, & Gravemeijer, 2013).

Reaching a higher level of conceptualunderstanding is inherently not an easytask because it involves a shift of thinking.How to attain this higher level is a corefundamental concern of the mathematicseducation research community. The call fromeducators and politicians for more attentionto routine and procedural skills will not solvestudents’ problems because the problemswith more difficult items do not primarily stemfrom a lack of procedural skills, but more froma lack of conceptual understanding (Stiphout,Drijvers, & Gravemeijer, 2013).

From the literature search, there areonly very few studies on structure sense,particularly on algebraic expressions andequations. Error analysis was conducted, butthe focus is not on structure sense. In thelocality, not one has ever done a study onstructure sense.

Methodology

Research Design

This study employed a combination ofquantitative and qualitative methods ofresearch and were used to explore thestructural properties of students’ solutions toalgebraic tasks. The study made an in-depthanalysis of the written outputs of the students.

Population and Sample

The respondents of the study included all thestudents in one of the two classes in each ofthe three grade levels (Grade 8, 9, & 10) from astate university-laboratory school in Region 1,Philippines. The classes were selected usingthe method of randomization. Table 1 showsthe distribution of the respondents.

For the interview part of the study, the studyconsidered nine volunteer students (three foreach grade level).

Table 1. Distribution of the respondentsGrade Level N

Grade 10 39Grade 9 43Grade 8 38Total 120

Instrument

The study used a test in Algebra to determinethe structure sense of students in algebra(randomly selecting one section per gradelevel). The test focused on algebraic skillstaught in the lower secondary levels (Grade7-9) formulated by the DepEd under theK-12 curriculum. The 10- item problem testcovered only expanding brackets, simplifyingexpressions, distribution, solving linearequations, factorization, solving rationalequations, evaluating algebraic expressions,rational expressions, derivation of formula,and solving radical equations. The testrequired the students to give a detailedsolution to the problems. The test was content

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validated by three mathematics teachers andpilot tested to determine its reliability. The testhas a reliability index of 0.87.

Data Gathering Procerue

Before the collection of data, the researchersasked permission from the Principal of thelaboratory school to allow the administrationof the proficiency test to the students. Thestudents in the three grade levels answeredthe same test within one hour. Moreover, aone-on-one interview with volunteer studentswas also done to elicit their reasoning abouttheir solutions. The unstructured interviewwas utilized. This method helped theresearchers to give a clearer picture of thestudents’ answers.

Ethical Consideration

The researchers observed research ethics inthe conduct of the study. Before the testadministration, the data collectors informedthe respondents about the nature and purposeof the activity, the plans for using the testsand the protocols observed to protect theiranonymity. Since the study included minors,the students were asked to fill up assentform which was signed by their parents.The researchers conducted the study out ofpersonal interest but for the improvement ofmathematics teaching and learning. Datagathered were all kept confidential especiallyin reporting the students’ solutions in thebody of the research. There were no risksassociated with the conduct of the study. Onlythe students who volunteered to be part of thestudy were requested to answer the test.

Analysis of Data

Data in this study were the written outputs ofthe students which underwent scrutiny by theresearchers. The researchers both analyzedseparately the outputs of the students.Comparison of analyses was done. In casesof contradicting cases, the researchers

discussed and agreed. Frequency andpercentage were used to determine theproportion of students who were able todisplay structure sense in doing algebraicexpressions and equations.

Results and Discussion

The main purpose of the study was to explorethe structure sense in algebraic expressionsand equations of three groups of students.

Algebraic Tasks Where StudentsDisplayed Structure Sense

Table 2 shows the proportion of studentswho were able to demonstrate their ability torecognize the structure of each test item inAlgebra.

Table 2 shows that out of 120 students, thehighest proportion of students that displayedstructure sense on expanding brackets (88or 73.33%) and distribution (87 or 72.5%).More than 50% of the total number ofstudents also obtained correct answers onsimplifying expressions (70.83%), solvinglinear equations (66.67%), evaluating analgebraic expression (59.17%), and solving aradical equation (58.33%).

On the other hand, the least of the studentsdisplayed structure sense on making a rationalexpression undefined. They struggled indetermining the value of x that makes thegiven rational expression undefined. Many ofthem did not solve this item because of theunfamiliarity of the tasks. Less than 50% of thestudents obtained correct answers on derivinga formula (33.33%), solving a rational equation(37.50%), and factorization (47.50%).

By grade level, more grade 8 students hadcorrect solutions and answers on distribution(71.05%); grade 9 students on both expandingbrackets (72.09%) and distribution (72.09%);and grade 10 students on expanding bracket(84.62%). All the three groups of studentshad the lowest correct answers on makingexpressions undefined.

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Table 2. The proportion of students who obtained correct solutions and answersLeanring Areas Grade8

(n1=38)Grade9(n2=43)

Grade10(n3=39)

Overall(n=120)

Expanding brackets 24 (63.16%) 31 (72.09%) 33 (84.62%) 88 (73.33%)Simplifying expressions 25 (65.80%) 29 (67.44%) 31 (79.49%) 85 (70.83%)Distribution 27 (71.05%) 31 (72.09%) 29 (74.36%) 87 (72.50%)Solving linear equations 23 (60.53%) 29 (67.44%) 28 (71.79%) 80 (66.67%)Factorization 13 (34.21%) 21 (48.84%) 23 (58.97%) 57 (47.50%)Solving a rational equation 8 (21.05%) 18 (41.86%) 19 (48.72%) 45 (37.50%)Evaluating an algebraicexpression

17 (44.74%) 27 (62.79%) 30 (76.92%) 71 (59.17%)

Making rational expressionundefined

8 (21.05%) 10 (23.26%) 13 (33.33%) 31 (25.83%)

Deriving a formula 9 (23.68%) 14 (32.56%) 17 (43.59%) 40 (33.33%)Solving a radical equation 16 (42.1%) 24 (55.81%) 30 (76.92%) 70 (58.33%)

Student’s Structure Sense inAlgebraic Expressions andEquations

To determine the structure sense of thestudents in solving algebraic expressionsand equations, the researchers performed aqualitative analysis of the written outputs ofthe students in the different algebraic tasks.For each item, at least one common selectedoutput from each grade level are presentedand analyzed.

Item 1. On expanding brackets

The first item of the test required the studentsto do expansion by multiplying a monomialand binomial expressions. More than one-halfof the students were able to get the correctanswer. However, some students gaveincorrect solutions/answers. Figures 1a–1dare selected samples of students’ solutions inthe given tasks.

The students have commonly appliedthe distributive property of multiplication,multiplying the monomial to each term in thebinomial. The students can give an instantanswer to the problem but, as per direction,they showed a clear solution on how to arriveat a final answer.

The grade 8 (G8) student carefully multipliedterm by term to come up with a product.The grade 9 (G9) student used arrows in

multiplying (3p) to (4p+q). A similar solutionwas carried out by a grade 10 (G10) student,but another grade 10 simply gave the correctfinal answer.

The students were able to see the structureof the item which involves multiplying amonomial with a binomial using distributiveproperty of multiplication as supported by theresponse of a student:

It is easy to obtain the answer because I onlyhave to multiply 3p and (4p + q). Since the twoterms in the product 12p2 + 3pq are not similar,it is already the final answer.

The students recognized the need tomultiply terms but failed to give the correctanswer as shown in fig. 1e and 1f.

The student whose solution appears inFigure 1f was wondering about his solution.He said, “I multiplied 3p and (4p + q) sir, butwhy was my answer is wrong?”. The reactionof the student depicts that he was surprisedabout committing a mistake. Hence, the errorwas due to carelessness and not because oflack of knowledge.

Item 2. Simplifying an algebraicexpression

Simplifying algebraic expressions is done bycombining like terms (expressions with acommon variable raised to the same power).Figures 2a, 2b, and 2c show the tasks ofselected students displaying different methods

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Figure 1. Student’s solutions on expanding brackets

of simplifying the given algebraic expression.The grade 8 student wrote similar terms

next to each other and marked them as termsthat can be combined and then arrived ata correct answer. Meanwhile, the grade 9student showed another way of combiningsimilar terms by writing these on top of theother than adding the terms to simplify thealgebraic expression. Lastly, the grade 10student combined similar terms by groupingthem inside a parenthesis.

When a student was asked about hisanswer, he replied: I learned that in simplifyingalgebraic expressions, I could only add orsubtract similar terms. 5x3 and -4x3 aresimilar, 3y and -9y are similar.

The common mistakes of students are dueto their inability to perform the proper operation(Fig. 2d-2f).

The grade 8 student recognized the needto combine similar terms but disregardedthe negative signs of 4x3 and 9y (Fig.2d).Meanwhile, the grade 9 student tried to groupsimilar terms by using a parenthesis but hemistakenly grouped (y2 and 2) as similar terms(Fig.2e). He also forgot to put separators

between the groups, then multiplied the termsin each group. Lastly, the grade 10 studenthad his focus on the exponents (Fig.2f). Hesimplified the algebraic expression by gettingthe powers of the coefficient of each term.

Item 3. Distribution

Distribution is a more challenging task ofexpansion which requires the use of thedistributive property of multiplication. The finalanswer is obtained by combining like terms.The given task has the structure in the formof (x+a)(x+b)=x2+(a+b)x+ab.

The following figures are solutions ofstudents taken from each grade level:

The grade 8 student wrote an explicitsolution by applying the FOIL method bymultiplying the First terms together, followedby Outer terms, Inner terms, and Last terms.He obtained the final answer by combiningsimilar terms. The grade 9 student also usedthe same method but with the use of arrowsto track which pairs of the term he shouldmultiply. A unique solution is seen in thework of a grade 10 student (Fig.3c). He alsoused the FOIL method. However, he portrayed

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Figure 2. Student’s solutions on simplifying an algebraic expression

Figure 3. Student’s solutions on distribution

mastery of the solution by verifying the answerusing the synthetic division method. Thestudent used the product (a2+10a+16) as thedividend and one of the factors (a+2) as adivisor. By using this method, he showed that

he must obtain a quotient (a+8) equal to theremaining factor.

From these outputs, the studentsrecognized that the structure of this expressionrequires identifying a pair of terms “to be

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multiplied” with the aid of the FOIL method.Moreover, they need to combine “similarterms” from these products.

The selected works of students prove thecommon difficulties experienced by studentswhen dealing with distribution (please refer toFigures 3d-3f).

In Figure 3d, the student was unable tosee the algebraic structure of the expressionwhich requires multiplying each term in the firstbinomial to each of the terms in the secondbinomial. In Figure 3e, the student multipliedthe terms in each binomial. In Figure 3f,the first step of the solution was correct, butcommitted errors in the succeeding steps.

Item 4. Solving linear equations

In solving linear equations, the students findfor the value of the variable which satisfies anequation.

The grade 8 student provided a detailedsolution by combining similar terms on bothsides of the equation. Then, he solvedfor x by dividing -3 to both sides of theequation. The grade 9 student immediatelyadded similar terms and obtained the finalanswer by immediately dividing the right sideof the equation by 3 giving:

x =−8

3

The solution of the grade 10 student is notdifferent from those of the grades 8 and 9students. However, he found it necessary toconvert the fraction as a mixed number and toexpress its decimal value.

The students manifested their masteryon the tasks by explaining their solutionssystematically. One student narrated asfollows:

This is a linear equation in which I solvedfor the value of x by simply combining similarterms in the opposite sides of the equalsymbol. Then I divided both sides of theequation by – 3 to solve for the final value ofx which is equal to -8/3.

On the other hand, some students haddifficulty in performing the task (refer toFigures 4d-4f).

The grade 8 student did not know what itmeans to solve a linear equation. He triedcombining similar terms but did not give avalue of x which satisfied the equation. Thegrade 9 and 10 students got the same butincorrect answer due to incorrect arithmeticoperation.

Item 5. Simplifying rational expressions

In simplifying rational expressions, thestudents must recall the different factoringtechniques like factoring with a commonfactor, factoring difference of two squares,factoring quadratic trinomials and others.This task requires the use of the law ofcancellation after showing common factors inthe numerator and denominator of a rationalexpression. The students should recognizethe form (pq/p) as equivalent to (q). Figures5a-5c are sample works of selected students.

The grade 8 student cancelled 9 in bothnumerator and denominator and divided x2by 3x which yielded an answer of x/3. Thisoutput represents almost the works of allthe grade 8 students. A grade 9 studentdivided the first and the second terms together.These common mistakes of the studentswere verified in the explanations given by thestudents as follows:

I crossed out the same numbers that arefound in the numerator and denominator. Iobtained 3x−1-1 by dividing the first terms andsecond terms together that are found in thefraction.

On the other hand, a grade 10 studentshowed the proper way of simplifying a rationalexpression. He applied two methods offactoring. He factored out the common factorin the numerator and factored as a product of asum and difference in the denominator. Then,he cancelled (x+3), a common factor presentin both numerator and denominator.

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Figure 4. Student’s solutions on solving linear equation

Figure 5. Student’s solutions on simplifying rational expression

Item 6. Solving rational equations

Rational equations contain the unknownusually found in the denominator of one sideof the equation. Solving this kind of equationrequires the application of the differentproperties of equality to solve the value of theunknown variable that satisfies an equation.

The grade 8 student committed an error atthe start of his solution. He thought that thestructure of the left side of the equation is inthe form of:

a+ b

c=

a

c+

b

c

As a result, the student divided 10 by 2 whichgives 5. He, therefore got 4x-5 for the left side

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Figure 6. Student’s solutions on solving rational expression

of the equation. The student properly carriedout his next solution, but his mistake from thestart led to an incorrect answer. He explainedhis solution as follows:

I cannot divide 10 by 4x, but I can divide 10by -2. So, I got 4x-5. Then I solve the resultingequation. My final answer is x is equal to 6/4.

The grade 9 student was able to recognizethe structure of the equation by starting withcross-multiplication as a strategy. He mighthave obtained the correct answer if he wrotethe expression (4x-2) instead of (4x+2).

The grade 10 student was also familiar withthe task by starting with cross-multiplication.However, the student committed a mistakeby using the correct sign of (4x) aftertransposition.

An analysis of the outputs revealed thaterrors committed by students were dueto carelessness and incorrect arithmeticoperation.

Item 7. Evaluating an algebraicexpression

This task is done by substitution using valuesof the variables in an algebraic expression.Most of the students had evaluated thealgebraic expression correctly. Please refer to

students’ solutions in Figures 7a-7c.However, there were still students who failed

to evaluate the algebraic expression properlyas shown in Figures 7d-7f.

The grade 8 student considered the firstterm (a2b) and the second term (ab) bothequal to (-12) after substituting the values of(a=-1) and (b=2) in the algebraic expression.The grade 9 student committed a mistakein writing the correct signs of the first andthird terms. Meanwhile, the grade 10 studentmissed getting the cube of the first term.

Item 8. Determining the value of x whichmakes a rational expression undefined

This task is related to finding the domain ofa function. A rational expression is in theform: When Q is equal to zero, the rationalexpression becomes undefined. Hence, thestudents must understand that not all numberscan be used to substitute a variable in arational expression. The students’ task wasto identify value/s that make/s the expressionundefined.

The grade 8 student lacked the conceptualunderstanding because he was looking for anequation to solve for the value of x. The grade9 student knew that rational expression is

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Figure 7. Student’s solutions on evaluating an algebraic expression

Figure 8. Student’s solutions on determining the value of x which makes arational expression undefined

undefined if the denominator is equal to zero.By the additive inverse property, he looked fora value of x that will make 3x=9 so that whenadded to –9 or subtracted from 9, the resultis zero. The grade 10 student presented inhis solution to the structure of this task. Hefurther verified his answer by substitution andfound that when x=3, the rational expression is

undefined.

Item 9. Deriving a formula

Deriving a formula requires the students towrite an equation of a variable as a functionof another variable. Students need thismathematical skill to avoid memorizing lots offormula because they only need to manipulate

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Figure 9. Student’s solutions on deriving formula

the original equation or formula. Below aresamples of students’ outputs from grades 8 to10.

The figures show that the students areaware that deriving a formula requiresmanipulating the equation to obtain anotherformula for the desired variable. The grade8 student came up with a formula for q.He committed an error in his first step byeliminating 4 in the denominator of the rightside of the equation. In Figures 9b and 9c, thesolutions were accurately carried out leadingto correct final answers, although the twosolutions were not the same in form. The workof the grade 10 student, as compared to thegrades 8 and 9 students, is further simplifiedby writing the expression in the form :

a+ b

c=

a

c+

b

c

Item 10. Solving radical equations

The last item deals with solving a radicalequation which is somewhat more complexin its structure because of the radical symbolused. Many students did not work on thistask maybe because of the square root symbolwhich put difficulty in solving the equation.However, some students managed to do thetask successfully. The students recognized

that in solving this kind of equation, they firstneed to eliminate the radical symbol by gettingthe square of both sides of the equation.

When a grade 8 student was asked how heobtained his answer (Fig.10a), he replied,

I don’t know how to solve a radical equationbecause my teacher in Math did not teach usyet in solving radical equation, but I am surethat x must be 1 so that the number inside theradical equation becomes 4, and the squareroot of 4 is equal to 2.

Though this task is not included as a mathtopic taught to grade 8 students, he was ableto look for the value of x that satisfies theequation.

The solutions of the grade 9 and grade 10students were similar. Here are quotes fromthe students, respectively:

I should start by removing the radicalsymbol. Then I solve the resulting equation.I squared both sides of the equation. So, I gotx+3=4. Next, I solve for x by placing 3 to theright side. That’s why I got x=1 as my finalanswer.

Both students needed to apply algebraicmanipulation. The students recognized thatin simplifying the equation, they need totransform the radical equation into a linearequation by getting the squares of both sidesof the equation.

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Figure 10. Student’s solutions on solving radical equations

On the other hand, some students gotincorrect answers because they havepoor procedural skills to perform properoperations(Fig. 10d & 10e).

During the interview, a student stated:

”I remove the square root symbol so thatI can solve for x.”

The student was right in his conceptualunderstanding about solving a radicalequation. However, he failed to apply theproper procedure in removing the radicalsymbol.

Furthermore, a student also disregardedthe square root symbol (Fig.10e). An alarmingpart of his solution was when he added theterms of the radicand which are not similar.The solution manifests that some students donot recognize the need to simplify the radicalequation by removing the radical symbol bygetting its square and not just by disregardingit.

Structure Sense of Students byGrade Level

Based on the data presented in Table 2 andfrom the solutions of the students, the studentsshown an increasing mastery and portrayedstructure sense in utilizing previously learnedalgebraic techniques to solve each task. Thestudents in the higher grade level had a betterunderstanding of Algebra and structure sensein algebraic expressions and equations thanthose in the lower grade levels. Many ofthe students in grade 8 did not attempt tosolve the last six items of the test. Moreover,the students displayed different methods ofsolving algebraic expressions and equations.The grade 8 students had simpler and shortersolutions than the grade 9 and 10. The grade9 students gave more detailed solutionsthan the grade 8 students. Meanwhile, thegrade 10 students gave clear and moreunderstandable solutions and were morecareful in simplifying final answers.

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ConclusionIt is necessary to equip the students withthe needed algebraic knowledge and skillsespecially at the secondary level becausethese are necessary for higher or advancedmathematics. The ability of the student tosee the structure of an algebraic expressionor equation with an understanding to dothe manipulations contributes to success insolving problems in algebra.

Many of the students portrayed difficultyin working with algebraic expressions andequations. These difficulties are due toinability to see the algebraic structures of thetasks, inadequate conceptual understandingabout the given problem, poor skills inmanipulating expressions, and technicalerrors like carelessness in copying term/sthroughout the process of solving the problem.

The inability of some students to see the“structure sense” of an algebraic expressionor equation yields to an ambiguous solutionwhich led to an incorrect answer. Thus,it is necessary for students to recognizewith ease the “structure sense” of everyterm in an expression to be able to performthe appropriate manipulations. Most of thestudents who did not use structure sensewere not successful in carrying out thealgebraic tasks. Those who have adequateknowledge and skills got the answer quicklyand accurately. These groups of students arethose who have displayed structure sense inalgebraic tasks.

Mathematics teachers should, therefore,provide students with adequate learningopportunities that will help them buildconfidence when dealing with mathematicaltasks and become more proficient in Algebra.They should give immediate feedback tostudents’ solutions to solve the difficulties ofthe students and correct their misconceptions.The teaching of Algebra in high school shouldemphasize structure sense.

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