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Word Problems: A Framework for Understanding, Analysis and Teaching

Jarmila Novotná, Leo Rogers

1. Word problems

Throughout the history of mathematics, mathematical problems serve to carry information, to practice techniques, to teach, and to diagnose the acquisition of skills. We may be presented with a diagram, or some data, but until we know what to do with this, it means little. We need some words, some description, some challenge, in order to know how to proceed further. In the development of mathematics, semi-literate cultures needed to know certain arithmetical techniques in order to function in their economy, and the earliest way of transmitting this information was orally. Over time, oral transmission developed these sets of instructions into stories or puzzles, and so the tradition of the Word Problem was born.

Word Problems then, can be regarded as linguistic descriptions of problem situations where questions are raised, and the answer obtained by the application of mathematical operations to numerical or logical data available in the problem statement. Typically, word problems take the form of brief texts describing the essentials of some situation where some quantities or relations are explicitly given and others are not, and where the solver is required to give an answer to a specific question by using the quantities given in the text and the mathematical relationships inferred between those quantities.

“Most life situations are described in words. Word problems constitute one of the few school mathematics domains which require mathematisation of situations described in words and the transformation of a mathematical solution back to the context of the problem.” (Novotná, 2000).

However, in some cases, in order just to practice the particular mathematical techniques, the problems are so decontextualised, as to be virtually meaningless in a real ‘real-life’ situation.

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For example: Word problem: To make a blouse, mother needs 1.5 m of 140 cm wide cloth. How much of 90 cm wide cloth does she need?

This can be solved easily by ensuring the data are all in the same units, setting up an expression about equivalent areas, and performing a simple transformation. However, if anyone has experience of working with cloth, or cutting patterns, they will see how far this is removed from reality.

Verbally stated numerical problems are not considered as word problems: Solve the quadratic equation x2 + 3x – 7 = 0.

This is just an instruction – do this!. The mathematical relationship given implies certain ways of working, and clearly the solver has to know how to proceed.

In tackling word problems as with any other problems, the help the teacher gives is most important.

“arithmetic requires an immediate search for a solution, on the contrary algebra postpones the search for a solution and begins with a formal transpositioning from the domain of natural language to a specific system of representation.” (ELTMAPS ArAl Project p.10)

However, in searching for a solution, the representation system is very important. The choice of a representation system should be open – pupils can be supported and encouraged to build their own representations; lists, pictures, diagrams, which can be examined for a helpful structure and which may be used if they are found to be helpful and developable. It is important to encourage the careful transposition from the natural language in which problems are posed into the formal arithmetic or algebraic language where the relations between the data are established.

If we are aware of the various aspects of understanding and solving word problems this should help us to diagnose the obstacles that pupils face when trying to solve them. When we know more about how understanding increases during the stages of the solution process, we

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may be able to prepare more effective instructional programmes for word problems.

The following notes are intended as a guide planning lessons, and as an ‘aide memoire’ for the various possible aspects of the process of problem solving and the analysis of pupils’ work. 2. Understanding and word problems

Understanding is a very complex process. Much of understanding relies on cultural contexts, and problems which are expressed almost entirely in words can be a barrier to some people. Different kinds of understanding have been proposed to see if we can identify what happens when an act of understanding takes place. For example:

Instrumental understanding is the ability to apply an appropriate remembered rule to the solution of a problem without knowing why the rule works.

Structural (relational) understanding is the ability to deduce specific rules or procedures from more general mathematical relationships.

Intuitive understanding is the ability to solve a problem without prior analysis of the problem.

Formal understanding is the ability to connect mathematical symbolism and notation with relevant mathematical ideas and to combine these ideas into chains of logical reasoning. (Herscovics & Bergeron, 1983)

The claim is that we use Instrumental understanding most of the time. We remember algorithms and apply them to problems. However, when the algorithm does not work, we get stuck - unless we have some Relational understanding of the situation which helps us to build a new solution process from the perceived relations between the data and the mathematics we already know. With Intuitive understanding, sometimes we are able to see a solution immediately without being aware of any internal thinking process, while we have Formal understanding when we are able to represent the problem in a mathematical language.

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3. Representations

The use of representations is strongly influenced by a teacher's demands and pupils' habits. The representation can have different forms, from the detailed rewriting of the assignment to a more clearly organised form (word representation) on one hand and a considerably shortened record such as a draft (graphic representation) on the other. The choice of an appropriate representation can help to get insight into the problem structure during the solving process.

Functions of representations

to re-present: Its purpose is to create an adequate mental picture. This is another way of presenting the problem containing the data, conditions and unknowns. It can range from a nearly realistic to a completely schematic drawing, or a mathematical statement.

to organise: Its purpose is to bring order into a solver's already existing mental picture and knowledge in order to connect them together.

to interpret: Its purpose is to facilitate the understanding of what are the unknowns of the problem and to reveal the relationships among the facts given. It helps to eliminate the formation of erroneous mental pictures.

to transform: Its purpose is to influence the solver's information processing by changing the used reference language to another one that is more suitable for the respective solver and to help systematically to recall helpful information stored in his/her memory.

4. The grasping process

The first stage in the solving process is to understand the information given in the text of its assignment. We will call this process grasping process, and it is intimately linked with the solvers ability to make an adequate representation of the problem.

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Basic stages of the grasping process:

a) identifying separate pieces of information during the reading of the problem,

b) determining what the question was asking, c) searching for a unifying view, d) looking for and finding relationships relevant to the solving process, e) getting an overall insight (finding how all the pieces of information

are mutually connected).

We can propose three components characterising the levels of a student’s understanding of word problem assignment: 1. reaching the grasping process stages (levels of understanding of the

assignment are related to the successfully finished stages of the grasping process; the stages are not necessarily expressed in a student’s solution in an explicit way, they may also be hidden in implicit steps)

2. how many times the solver refers back to the assignment (a scalar component)

3. quality of the grasping process (terminology related to the terminology used in (Hejný, 1995)): grasping with understanding if a particular stage of the grasping

process results in understanding what was searched for incomplete understanding if the solver only grasps a part of the

assigned information prothetic grasping if no understanding occurs in a particular

stage of the grasping process 5. Solving strategies and their relation to understanding the problem

The level of the solver’s understanding of the word problem structure strongly influences their solving strategy.

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Items for the characterisation of the chosen solving strategy: (i) if the solution was found accidentally or after gaining an insight into the problem structure, (ii) if the solution was based on the identification of key words/word groups in the text or on the insight into the problem structure. Stages of word problem solving process (Novotná, 1997a) encoding stage (grasping the assignment), transformation stage (transfer to the language of mathematics) calculation stage (mathematical solution of the problem), storage stage (transfer of mathematical results back into the context). Terminology (Novotná, 1999) Coding of word problem assignment is the transformation of the

word problem text into a suitable system (reference language) in which data, conditions and unknowns can be recorded in a more clearly organised and/or more economical form.

The reference language contains basic symbols and rules for legend creation.

The result of this process is a representation. The legend constructed in a pictorial form is called a graphic

representation. 6. Difficulties with solving and analysis of children’s work

“A problem is not truly solved unless the learner understands what he has done and knows his actions were appropriate.” “Given the proper understanding of mathematical concepts and procedures, students would be better able to apply their knowledge in novel situations.” (Brownell, 1928)

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Students’ solving strategies and their relation to students’ understanding of the solved word problems

The level of the solver’s understanding of the word problem structure strongly influences his/her solving strategy.

Items for the characterisation of the chosen solving strategy: (i) if the solution was found accidentally or after gaining an insight

into the problem structure (ii) if the solution was based on the identification of key words/word

groups in the text or on the insight into the problem structure

(i) Solution found accidentally Ota (boy, 15 years old) Problem: Ota and Pavel each had some money but Ota had 10 CZK more than Pavel. Pavel managed to double his amount of money and Ota got 20 CZK more. They now found that both of them had the same amount. How many crowns did each of them have at the beginning?

I ‘ gessed’.

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(i) Solution found after systematic trials

Hana (girl, 11 years old)

Problem: Petr, David and Jirka play marbles. They have 198 marbles altogether. Petr has 6 times more marbles than David and Jirka has 2 times more marbles than David. How many marbles has each boy got?

(ii) Identification of key words in the assignment

Filip (boy, 12 years old)

Problem: Petr, David and Jirka play marbles. They have 198 marbles altogether. Petr has 6 times more marbles than David and 3 times more marbles than Jirka. How many marbles has each boy got?

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(ii) Understanding of the problem structure

Jana (girl, 12 years old)

Problem: Petr, David and Jirka play marbles. They have 198 marbles altogether. Petr has 6 times more marbles than David and Jirka has 2 times more marbles than David. How many marbles has each boy got?

Classification of solving strategies

The main classification of students' strategies is based on their division into arithmetical and algebraic ones. But some students use such strategies which do not fit any of these types and sometimes it is even impossible to determine the strategy.

Arithmetical strategies The student solves the problem using only arithmetical means. The student grasps the structure of the problem. (Solutions in which

estimations, visualization etc. are used also belong to this subgroup.)

Algebraic strategies Solutions where one or more equations are used.

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(Jitka, girl, 12 years old) – Arithmetical strategy Problem: The total fee which Mr. Novák’s daughters, Pavla and Marie, received was 181 CZK. Marie had 37 CZK more than Pavla. How many crowns did each daughter receive?

(Zdeněk, boy, 14 years old) – Algebraic strategy

Problem: A buffet sells three different dishes – pizzas, hamburgers and langoses1). In one day 288 dishes of hamburgers and langoses were sold altogether. Four times more pizzas than hamburgers and seven times more langoses than hamburgers were sold. How many dishes of each kind were sold?

Marie got 109 CZK. Pavla got 72 CZK.

Pizzas were 9, hamburgers were 36 and Langoses 252 pieces.

Hamburgers and Langoses

Pizzas Hamburgers Langoses

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Unidentifiable strategies It is not possible to determine the strategy used.

The role of past experience (Novotná, 1997b) Positive transfer effect: The performance on a current problem benefits from previous problem solving

Two ways how the solver can use his/her past experience of previous problems: Ignoring the previous problem, using the previous problem.

(Martin, boy, 13 years old) Problem a: Petr, David and Jirka play marbles. They have 198 marbles altogether. Petr has 6 times more marbles than David and Jirka has 2 times more marbles than David. How many marbles has each boy got? Problem b: Petr, David and Jirka play marbles. They have 198 marbles altogether. Petr has 6 times more marbles than David and 3 times more marbles than Jirka. How many marbles has each boy got?19

19 Problems were used in the given order. The Problem 2 structure is more complicated for solvers.

Altogether 198 David 1x more than David. Jirka 2x more than David. Petr 6x more than David.

Problem a

Problem balt. 198 David 1x more than David. Jirka 2x more than David.

David has 22 m. Jirka has 44 m. Petr has 132 m.

Petr 6x more than David and 3x more than

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Use of letters

Levels of dealing with letters during the period of grasping the meaning of the assignment (encoding stage), and the period of mathematization of the solving process (transformation and calculation stages)

Situation 1 – The assignment does not contain algebraic elements (Novotná, 1997a):

Four stages of the transition from an arithmetical to an algebraic way of using letters in the written record of assigned information: 1a) Solvers use one letter for labelling several values, the letter is a

symbol of a general unknown for them. 1b) Solvers use one or more letters in the encoding stage without

working with them in the transformation stage, the unknown is only used as a label for something that is to be found.

1c) Solvers consciously use letters for labelling required values and for describing assigned relationships, arithmetical models are more important and thus the arithmetical solution is used.

1d) Solvers use letters for labelling the values and algebraic operations are carried out and solved. The conditions for the successful use of algebraic methods have already been created.

(Jarda, boy, 12 years old) Problem: Petr, David and Jirka play marbles. They have 198 marbles altogether. Petr has 6 times more marbles than David and Jirka has 2 times more marbles than David. How many marbles has each boy got?

1a)

marbles . .. 198 Petr …. x 6x more than David … x Jirka … x 2x more than

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(Franta, boy, 12 years old)

Problem: Slávek has by 27 stamps more than Honza, Jakub has š times more of them than Slávek. If Slávek has 138 stamps, how many stamps have all of them altogether? (Marta, girl, 15 years old) Problem: In a packhouse a blend of coffee for 240 CZK/kg is being prepared. How can be prepares 35 kg of the blend, if two types of coffee are available, one for 200 CZK/kg and one for 280 CZK/kg?

1b)

1c)

SL. 138 J. 3 times more H. by 27 less than

X = altogether Altogether they have 663 stamps.

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Situation 2 – The assignment contains algebraic elements (Novotná – Kubínová, 2001)

Four stages of dealing with the assignment:

2a) Solvers ignore data which are not assigned as concrete numbers, their ability to work with algebraic representations is not developed.

2b) Solvers use letters only as labels for something that is to be found by calculations, the value assigned by a letter is handled as an unknown.

2c) Solvers are aware of the nature of data assigned as letters; by substituting a concrete number a letter, they change the problem into a pure arithmetical one. The symbolic algebraic description of the situation is not yet fixed in their knowledge structure.

2d) Solvers are able to work successfully with data assigned in both arithmetical and algebraic languages.

Examples - Age of solvers: 13 years

Problem: A packing case full of ceramic vases was delivered to a shop. In the case there were 8 boxes, each of the boxes contained 6 smaller boxes with 5 presentation packs in each of the smaller boxes, each presentation pack contained 4 parcels and in each parcel there were v vases. How many vases were there altogether in the packing case?

2a)

4 parcels: 4.1=4 parcels packs: 5.4=20 packs sm. boxes: 20.6=120 sm. boxes boxes: 120.8=960 packs In the case are 960 articles

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2b) 1 case

8 boxes in 6 smaller boxes in 5 packs in 4 parcels in ? v vases in

In the case are 960 vases.

2c)

8 boxes in 1 are 6 smaller boxes 1. 5 packs 1. 4 p v = 6

In the case are 3520 objects.

2d) 8 . 6 = 48 boxes 48 . 5 = 24 smaller boxes 240 . 4 = 960 parcels in 1 parcels … v vases

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7. Programme for examining pupils’ solutions

Ideas for the classroom:

This is a proposal of how to implement the analysis of students’ ways of grasping the word problem structure and of their solving strategies in a teacher’s everyday teaching.

Review and extension of ideas: Look for solving word problem process stages. Ways of recording of the assignment – reference languages, mathematical and psychological views.

Exploration and discussion: Consider the classification of various types of information given in the written records – advantages and disadvantages. The influence of the chosen way of recording information. Arithmetic and algebraic solving strategies.

Extensions/Additions: What is the role of the teacher and of the student’s previous experiences

Possible ideas for follow-up: Creation of a set of word problems, their use in your own class with

an analysis of students’ solving processes and results Finding obstacles and proposing re-educational therapies Comparison of students’ work in other domains of school

mathematics or in other subjects with the results of the analysis of the word problem solving process

Deep analysis of one (or more) discovered phenomenon. Deep analysis of types of word problems and proposed solving

strategies in textbooks

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References Brownell, W.A. (1928). The Development of Children’s Number Ideas in

the Primary Grades. The University of Chicago. Comenius, J.A. (1631). Didacta Magna. Gavora, P. (1992). Student and Text. Bratislava, SPN. (In Slovak.) Hejný, M. (1995). Grasping of word problems. Pedagogika, 4, 1995. (In

Czech.) Herscovics, N. – Bergeron, J.C. (1983). Models of Understanding.

ZDM, 2. Kubínová, M. - Novotná, J. - Bednarz, N. - Janvier, B. and Totohasina,

A. (1994). Strategies Used by Students when Solving Word Problems. Publications DMME, 9, Praha.

Novotná, J. (1997a). Phenomena Discovered in the Process of Solving Word Problems. In: Proceedings ERCME 97. Praha, Prometheus.

Novotná, J. (1997b). Using Geometrical Models and interviews as Diagnostic Tools to Determine Students‘ Misunderstandings in mathematics. In: Proceedings SEMT 97. Praha, Prometheus.

Novotná , J. (1998). Cognitive Mechanisms and Word Equations. In: Beiträge zum Mathematikunterricht 1998, Vorträge auf 32. Tagung für Didaktik der Mathematik vom 2. bis 6.3.1998 in München. Ed. M. Neubrand. Hildesheim, Berlin, Verlag Franzbecker p. 34-41.

Novotná, J. (1999). Pictorial Representation as a Means of Grasping Word Problem Structures. Psychology of Mathematical Education, 12. http://www.ex.ac.uk/~PErnst/

Novotná, J. (2000) Analysis of Word Problem Solutions. Praha, Charles University (In Czech.)

Novotná, J. – Kubínová, M. (2001). The Influence of Symbolic Algebraic Descriptions in Word Problem Assignments on Grasping Processes and on Solving Strategies. In: Proceedings of the 12th ICMI Study Conference The Future of the Teaching and learning of Algebra. The University of Melbourne, Australia, Melbourne.

Odvárko, O. et al. (1990). Methods of solving mathematical problems. Praha, SPN. (In Czech.)

Polya, G. (1962). Mathematical Discovery. John Wiley & Sons.

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Resnick, L.B. and Ford, W.W. (1981). The Psychology of Mathematics for Instruction. Lawrence Erlbaum Assoc., Publ.

Skemp, R. (1971). The Psychology of Learning Mathematics. England” Penguin Books.

Toom, A. (1999). Word Problems: Applications or Mental Manipulatives. For the learning of Mathematics 19, 1.

Verschaffel, L. - Greer, B. and De Corte, E. (2000). Making Sense of Word Problems. Sweets & Zeitlinger Publ.

Acknowledgment: The research was partly supported by Projects Grant Agency of the Czech Republic 406/02/0829: Student oriented mathem- atical education, and Research Project: MSM 13/98:114100004: Cultivation of Mathematical Thinking and Education in European Culture.