Word Problems - Common misconceptions of language

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Topic (Pedagogical Content Knowledge Project) by Patricia Reyes, Rica Delgado, Evelyn Lim, Pei Amy Chen (UCIrvine, August 2008)



Contents

Cognition and Learning Background

The National Research Council

The National Resarch Council (2002) considers understanding of mathematics as a significant part of education for the students. This is because children’s success in the future job market will require them the ability to apply mathematical knowledge to solve problems. Without proficiency in mathematics, students will be more likely to experience difficulty completing other more advanced branches of mathematics (e.g., algebra) and be unprepared for many occupations (Riccomini, 2005). Since schools has provided children with the opportunity to learn mathematical knowledge, skills, and confidence they need in the highly technical world, teachers will need to help students view mathematics as a tool they can use everyday. With the emphasis on conceptual understanding and higher order problem solving skills, teachers also need to emphasize the importance of computation (Riccomini, 2005). Although knowing the basic number facts for addition, subtraction, multiplication, and division is essential for students, their fluency and accuracy in methods of comprehension and computation are equally important (Riccomini, 2005).

According to National Research Council (2002), mathematical proficiency has five strands:
1. Understanding: Comprehending mathematical concepts, operations, and relations—knowing what mathematical symbols, diagrams, and procedures mean.
2. Computing: Carrying out mathematical procedures, such as adding, subtracting, multiplying, and dividing numbers flexibly, accurately, efficiently, and appropriately.
3. Applying: Being able to formulate problems mathematically and to devise strategies for solving them using concepts and procedures appropriately.
4. Reasoning: Using logic to explain and justify a solution to a problem or to extend from something known to something not yet known.
5. Engaging: Seeing mathematics as sensible, useful, and doable—if you work at it—and being willing to do the work.
(National Research Council, 2002, p.9),

The most important characteristic of mathematical proficiency is that these five strands are interrelated (National Research Council, 2002), therefore students must utilize their basic mathematical skills properly to become successful in mathematics.

The Role of Understanding in Solving Word Problems (Cummins, et al. 1988)

MathPhobic.jpg

Word problems are notoriously difficult to solve. Much of the difficulty children experience with word problems can be attributed to difficulty in comprehending abstract or ambiguous language.

Because problem difficulty patterns change with age, many researchers have adopted a Piagetian view of solution performance characteristics. According to this view, a problem proves troublesome for a child only insofar as the capacities required to process the problem are not yet possessed by the child. While this general view is fairly uncontroversial, researchers disagree as to which capacities develop over time to improve solution performance. Explanations generally fall into two camps: those that attribute improved solution performance to the development of logico-mathematical knowledge and those that attribute such improvement to the development of language comprehension skills.

The logico-mathematical development view. According to the logico-mathematical explanation of solution difficulty, children fail to solve certain problems because they do not possess the conceptual knowledge required to solve them correctly.
The linguistic development view. The linguistic development view holds that certain word problems are difficult to solve because they employ linguistic forms that do not readily map onto children’s existing conceptual knowledge structures. For example, a child may understand part-whole set relations and yet be uncertain as to how the comparative verbal form (e.g., How many more X’s than Y’s?) maps onto them. If this were the case, we would say that the child had not yet acquired an interpretation for such verbal forms.
Importantly, the linguistic development view implies that word problems that contain certain verbal forms constitute tests of verbal sophistication as well as logico-mathematical knowledge. Accordingly, solution errors on these problems may reflect deficiencies in semantic knowledge, logico-mathematical knowledge, or both.

Cognitive Obstacles and Common Misconceptions

Wordproblem comic.jpg


1. Linguistic Misconceptions:

True or False: Teaching children key words or prompts in solving mathematical word problems is helpful for teaching students to problem solve.

Answer: FALSE!!!

There is a long-held belief that mathematics is a language, with its own unique vocabulary and syntax. When learning about mathematics, many students feel as though they are learning a “foreign” language, and not just mathematical ideas, theories, and concepts. As Krussel (1998) concurs, “One often hears this sentiment in conversations among mathematicians, and many students resonate to the idea of mathematics as a "foreign" language (Krussel, L., 1998, p. 436).
When negotiating meaning in mathematics, past research shows that young children are faced with many obstacles. In relation to language, it is believed that one of the main reasons children experience difficulty in mathematics is “in understanding the nuances of the mathematical language” (Warren, E., 2006, p.169). Warren (2006) states “Children need to understand and negotiate a mathematical register, come to some understanding of its own syntax, semantics, symbols, vocabulary and grammar, and reinterpret words in a mathematical context. Therefore, words they hear and understand in its normal, real word context (such as less, fewer, more, each, every) have a different, more narrow and precise meaning in mathematics.
Consequently, a danger exists for misinterpretation and misunderstanding caused by confusion about whether words have their everyday English meaning or their more technically precise mathematical meaning. Krussel (1998) offers the following example to illustrate: I may ask a student, "If you substract zero from zero, what's the difference?" While answering with the mathematical term "zero," he or she may be thinking in plain English, "That's right! Who cares? What's the difference?" (p. 441). We must be careful to point out the differences between the English use and the precise mathematical use.


English vs. Other Languages

  • There is an importance in understanding place value and the base 10 system which it is based upon in order to develop adequate number skills (Towse, J.N. & Saxton, M., 1997, p.362)
  • Asian language such as Chinese and Japanese use an explicit compound structure for all two digit numbers, comprising and “A” ten “B” format, where “A” describes tens and “B” describes ones or units. In contrast, the connection between a value and the base 10 structure is less explicit in English (Towse, J.N. & Saxton, M., 1997, p.363).
For example, in Asian languages 37 takes the form three-ten-seven; in English the 3 and 10 occur in corrupted form. In addition, English numbers between 11 and 19 do not follow the base 10 rule in a straightforward fashion.

Addition/Subtraction Word Problems

  • Younger children do not understand less/fewer, they think that these words mean more. (Fuson, Carroll, & Landis, 1996)
Children learn larger comparative terms like taller, heavier, longer, etc. than the less frequent words such as shorter, lighter, etc. (Fuson & Abrahamson)
Example: Ali has 9 balloons. Lisa has 4 more than Ali. How many balloons does Lisa have? --- means two things: Lisa has more and she has 4 more. Young children only hear more and do not process how many more (4)
While 'more' and 'less' are essential parts mathematical language and vocabulary, their meaning cannot be divorced from home practices, especially in the early years (Warren, 2006). These conjectures are based on two studies (Walkderine, 1988, 1990) involving an analysis of discourse observed between 30 mothers and daughters from low socio-economic families in Britain in the 1980s. In these contexts 'less' was rarely used in the home environment. For these families, the opposite of ‘more’ was related to food regulation, where food was considered a rare and expensive commodity; therefore, the opposite for ‘more’ was ‘no more’ rather than ‘less.’ A conclusion reached from these studies was that children in the school environment would fine ‘more’ easier than ‘less’ because ‘less’ is not part of the home practices. (Warren, 2006).
  • Fewer/less issue
According to Fuson and Abrahamson (2008), teachers should use less instead of fewer.

Multiplication/Division Word Problems

  • Each is a difficult word for many students, it means every or all. (Fuson & Abrahamson, 2008)
Example: Every cousin will get 2 puppets or All cousins will get 2 puppets.
In some languages, like Hebrew, do not use a word like each but use all: All cousin will get 2 puppets or Al one of cousins will get 2 puppets.

Word Order Matching

  • In word order matching, the student assumes that the order of key words in the problem will map directly into the order of symbols appearing in the equation.
Example: There are six times as many students as professors at this university. Use S for the number of students and P for the number of professors. → many students wrote 6S=P instead of 6P=S

"Equal”

  • Past research has indicated that many children have limited understanding of 'equal' as quantitative sameness (Warren, 2006).
  • In mathematics, the use of the equal sign appears to fall into four main categories (Warren, 2006).
  1. The result of a sum (e.g., 3 + 4 = 7)
  2. Quantitative sameness (e.g., 1 + 3 = 2 + 2)
  3. A statement that something is true for all values of the variable (e.g., x + y = y + x)
  4. A statement that assigns a value to a new variable (e.g., x + y = z)
  • In the elementary school, the focus is often centered on the first two categories
  • In terms of quantitative sameness, equals means “both sides of an equation are the same and that the information can be from either direction in a symmetrical fashion” (Warren, 2006, p. 172). However, most students do not have this understanding of the equal sign. Instead, the idea of the equals sign is either a syntactic indicator (a symbol indicating where the answer should be written) or an operator sign (a indicator to do something). (Warren, 2006).
  • If classroom practices and discourse focus predominately on routine calculations and finding answers, then the meaning associated with the equal sign may be very narrow. For example, children's interpretation of 'equal' meaning the 'answer' reflects classroom practice where children predominately solve problems such as "2 + 3 =?" (Warren, 2006).

2. Language Comprehension Errors

  • Comprehension errors made when attempting mathematical word problems have been noted as one of the high frequency categories in error analysis (Clarkson & Campus, 1991).
  • The ability in a student’s first language will also influence their mathematical achievement (Clarkson & Galbraith, 1992).
  • Teachers associate mistakes in students’ work as carelessness will misinterpret the underlying definition of these errors (Clarkson & Campus, 1991).

3. Pre-service Elementary Teachers’ Misconceptions about Multiplication and Division

  • The belief that “multiplication always makes bigger” and “division always makes smaller,” are commonly held by pre-service elementary teachers (Tirosh & Graeber, 1989).
    • According to Tirosh & Graeber (1989), a significant number of pre-service teachers had difficulty selecting the correct operation to solve multiplication and division word problems involving positive decimal factors less than one because they held explicit misbelieves about the operations.
  • Preservice teachers’ reliance on procedural knowledge of the algorithm may support their misconception about multiplication and division.
    • According to primitive model hypothesized by Fischbein et al. (1985), multiplication is repeated addition. In the domain of whole numbers, where instruction usually begins, the idea of primitive multiplication model can be a source of the misconception that “multiplication always makes bigger” (Tirosh & Graeber, 1989).
    • In the primitive partitive model of division described by Fischbein et al. (1985), division is dividing an object or collection of objects into a given number of equal parts or sub-collections. The primitive measurement model is used to determine how many times a given quantity is enclosed in a larger quantity. This primitive model imposes two impressions on the operation of division, and they are:
  • The divisor “must” be a whole number;
  • The quotient “must” be less than the dividend.

These impressions can be the source of pre-service teachers’ misconception that “division always makes smaller” (Tirosh & Graeber, 1989).

Pedagogical Tools and Strategies

1. Linguistic Misconceptions

Krussel (1998) believes that teachers should encourage a balance of drill and practice of the language beginning at an early age. They should study its vocabulary and structure; and “practice in conversation, reading, and writing” (Krussel, L., 1998, p. 440).

English vs. Other Languages

  • Students can use base ten blocks to aid them in understanding the base-ten concepts.

Addition/Subtraction Word Problems

  • The use of linguistic and visual support helps students use the correct operation for word problems especially for word problems that use comparison. (Fuson & Abrahamson, 2008)
    • Strategies: picture graphs and bar graphs, which facilitate visual comparison. For Grades 2 to 5, comparison bars are also very successful in helping students determine the correct operant for the word problem.
  • Rephrasing the problem using equalizing language.
    • Students can either rephrase the question asked or the teacher can rephrase it.
    • Example: Ali has 9 balloons. Lisa has 4 more than Ali. How many balloons does Lisa have?
    • Rephrased: How many balloons does Ali need to get to have as many as Ali? or How many balloons does Lisa have to give away to have as many as Ali? (Fuson & Abrahamson, 2008)
  • Specifying the unknown in the problem
    • Specifying the unknown helps students and teachers understand the situation. (Fuson & Abrahamson, 2008)

Multiplication/Division Word Problems

  • Instead of using the word each, the per word or an can be used: Michael ran 2 miles an hour. It makes the one in each more implicit: Each 1 cousin, per 1 cousin, 2 miles in 1 hour.
  • Comparison Bars are helpful in showing comparison situations.

Word Order Matching

  • There are two key ideas in successful solutions to these types of problems (Clement, J., 1982, p.28-29).
  1. Remembering that variables stand for numbers rather than objects in these problems.
  2. Being able to invent a hypothetical operation on the variables that creates an equivalence.

“Equal”

  • When modeling how to solve word problems using equations, teachers should be mindful of where they place the equal sign. They should write equations in a variety of ways, some with the answer on the right side of the equation sign, and some with the answer on the left side of the equation sign.

2. Language Comprehension Errors

  • Newman (1997) suggested that in solving one step word problems, students process through five stages:
    • Reading: Reading the question aloud.
    • Comprehension: What is the question asking you to find or to do? And what do the symbols mean?
    • Transformation: What do you have to do to find the answer?
    • Process Skill: Working out the problem to get the answer.
    • Encoding: Transferring the answer to the proper area.
  • And at each stage, a student may make an error which could prevent a correct solution being found for the problem (Clarkson & Campus, 1991).

3. Using Diagrams or Graphic Representations

Mathematical problem solving is a complex cognitive activity that involves a number of processes and strategies. “Problem solving has two stages: problem representation and problem execution” (Montague, M., 2005, p. 2). The problem solver must be about to first represent the problem in order to successfully solve the problem. If the problem solver has appropriately represented and understood the problem, it will guide the student toward the solution. According to Montague (2005), if a student has difficulties when representing the math problem then the student will have difficulty solving the problem (Montague, M., 2005, p. 2).


According to Scheuermann and Van Garderen (2008), if a student struggles in mathematics, one area that should be assessed is a student’s ability to generate and use representations. Part of the assessment includes determining the relevancy as well as the quality and completeness of the representation (p.477).

Picture assessment.png

The national curriculum in New Zealand stresses the importance of teaching students how to understand diagrams and considers the use of diagrams as a communication tool (Uesaka, Y., et al., 2007, p. 324). One important technique employed by New Zealand teachers was teaching students how to construct diagrams and use diagrams for explanations of the problem (Uesaka, Y., et al., 2007, p. 333). In a study done by Uesaka, Manalo, and Ichikawa (2007), they found that there was a significantly higher percentage of New Zealand students that used diagrams and obtained the correct answer versus the Japanese students that participated in the study (p. 332). Although using diagrams may not guarantee yield the correct answer, the findings from the study done by Uesaka, Manalo, and Ichikawa show that using diagrams in word problem solving can be largely beneficial.

Advantages: (Beckmann, S., 2004, p. 43-44)

  • Without a diagram, the problem becomes much more difficult to solve.
  • Diagrams prompt children to use the appropriate operations on solid conceptual grounds.
  • Diagrams, such as strip diagrams, make it possible for children who have not studied algebra to attempt remarkably complex problems.

3. Classroom Strategies

The Learning Path Classroom Framework for Algebraic Word Problem Solving (Fuson & Abrahamson, 2008)

I. Classroom Activities Supporting Meaning-Making and Linking the Phases
Enhance meaning-making and connect A, B, C, D by Math Talk: Discuss drawings and solutions
A. Relate word problems to linguistic and real-world knowledge
B. Use math drawings to present a mathematized or numericalized situation
C. Focus on the unknown: Turn the numericalization into a solution plan
D. Carry out solution method (check it)
II. Learning Paths for a Given Problem Type: Stay in the Class Learning Zone
i. Problem language and particular real-world situation become more difficult/less familiar
ii. Easy unknown(s) to more difficult unknown(s) [immediate/almost immediate for multiplication/division and for addition/subtraction for students! Grade 2]
iii. Solve extra and missing information problems; solve multistep problems
iv. Facilitate more-advanced solution methods: do related numerical classwork on mathematically desirable and accessible solution methods
v. Numbers become larger or more difficult (multidigit whole numbers, decimals, fractions)
III. Phases of Teaching a Problem Type
Phase 1: Introduce a problem type, elicit understandings of the situation (A), elicit and discuss students invented math drawings and solution methods; do a few more within learning zone
Phase 2: Introduce mathematically-desirable and accessible numerical situation drawings; begin learning paths i and ii; work at each of A, B, C, D (in I above) across similar problems as well as working across all phases of problem solving for a given problem; students write word problems
Phase 3: Mix with other problem types; do Learning Path iii (delayed for Grade 1 and 2)
Phase 4: Across months: Facilitate remembering by occasional practice with feedback; deepen analysis across problem types by relating new problem types to old problem types
IV. Classroom Inquiry Zone to Support Meaning-Making
a. Explain thinking: Students listen to each other; errors are opportunities for everyone to learn.
b. Help is available: classmates and teacher model and help during Math Talk, informal and formal partner and teacher help during solving; everyone uses the seven responsive means of assistance
c. Differentiated instruction within whole-class activities: Students solve at own level and relate math drawings and solutions; math drawings support explaining and listener understanding of the whole learning path of solutions; Phase 2 above insures that all students move along a learning path


Children’s Math Worlds Algebraic Approach to Word Problem Solving: Learning to Express Situations in Algebraic Equations and Find the Unknown (Fuson & Abrahamson, 2008)

1. Use all unknowns in all types of word problems and use situation equations:
Build all situational meanings of the = sign (becomes, is identical to, is the same number as) and of the mathematical operations + - x ÷ by solving, discussing, and writing all three kinds of addition/subtraction situations (Change, Collection, Additive Comparison) and later all four kinds of multiplication/division situations (Equal Groups, Array/Area, Combination, Multiplicative Comparison); give experience in understanding and informal solving of many forms of an equation by varying the unknown number as any of the 3 quantities in the situation and using situation equations such as 8 + 􀀀 = 14 from Grade 1 on (and letters as unknowns from Grade 4 on); relate situation equations and solution equations but allow students to use whichever equation best shows their thinking (maybe both).
2. Teach one powerful and general strategy for solving word problems:
Read to comprehend the situation and make a math drawing if it will help. Math drawings can be invented by students, but also research-based mathematically-desirable and accessible numerical situation drawings should be shown to and used by students. Student math drawings may reflect conceptions in any phase of problem solving (a linguistic and real-world or mathematized situation or numericalized situation or solution method conception or a solution action sequence).
3. Implement the Learning Path Classroom Framework for Algebraic Word Problem Solving (See previous strategy):
Use classroom activities (in Figure 9 below) to support meaning-making and linking of each of the phases of the Conceptual Phase problem solving model (in Figure 8 below)
Stay in the Class Learning Zone by using the five learning paths
For each problem type, use the four phases of teaching
Implement the Classroom Inquiry Zone to support meaning-making
This framework supports individualized instruction within whole-class activities because of the Math Talk inquiry environment in which all levels of situation representations and solutions arise and are discussed, the conceptions for all phases of problem solving are supported and related, and equations are related to other math drawings so that various equation forms take on all of the possible situational and operational meanings including where the Total and Addends are in + and – equations and where the Product and Factors are in x and ÷ equations.
4. Continually intertwine word problems and calculation to support embedded number conceptions by finding all the “break-apart partners” hiding inside a number with equations of the form 5 = 3 + 2 (i.e., 5 can be made from 3 and 2) and 5/7 = 3/7 + 2/7 subtraction as unknown addition and division as unknown multiplication knowledge of equivalent equation forms by using equations chains (e.g., 10 = 9 + 1 = 17 – 7 = 1 + 1 + 1 + 3 + 4 = 20 – 10) and by finding 8 true equations for a given triad of numbers rather than just the usual 4 “fact families” with one number on the right (the top row below):
Picture 5.png
understanding of multidigit multiplication and distributivity by using an area model; this facilitates generalizing single-digit multiplication/division situations to multidigit numbers

Table 8.png

Table 9.png

4. Teaching word problems to students with Learning Disabilities (LD)

Based on the 1992 NAEP, it appears that mathematical word-problem solving is difficult for students of all ages and ability levels. Specifically, students with disabilities and at-risk students who have difficulties with reading, computation, or both are likely to encounter difficulties with word-problem solving (Dunlap, 1982 as cited by Jitendra, A. & Xin, Y.P., 1997, p. 413). Textbooks are usually not helpful when it comes to teaching students how to solve math problems because “they typically provide a four step formula: (a) read the problem, (b) decide what to do, (3) compute, and (4) check your answer” (Montague, M., 2005, p. 1-2). Students need to read the problem for understanding, which involves being able to understand the relationship between numbers, words, and symbols in the problem (Montague, M., 2005, p. 2). Students often acquire the skills and strategies needed to read the problem and come up with the necessary procedure to solve it at an early age. According to Montague (2005), many students with LD or other cognitive impairments do not easily acquire these skills and strategies (Montague, M., 2005, p. 2). Therefore, students with LD need explicit instruction in mathematical problem solving skills and strategies to solve problems in their math textbooks and in their daily lives.

I. Representational techniques, which may include pictorial (e.g., diagramming); verbal (e.g., analogies and metaphors); or physical (e.g., manipulatives) aids, facilitate learning (Jitendra, A. & Xin, Y.P., 1997, p. 415). “Representational techniques permit "interpretation of ideas" or information found in a problem” (Prawat, 1989, p. 3 as cited in Jitendra, A. & Xin, Y.P, 1997, p. 415).

  • Van Garderen (2007) highlights three instructional phases that each incorporate principles of explicit instruction, such as teacher modeling and demonstration, questioning, guided and independent practice, rehearsal, reinforcement, and feedback (p. 544).
    • Phase 1: Instruction for generating diagrams. This phase of instruction focuses on the understanding of what a diagram is, why to use a diagram, and how to generate a diagram (van Garderen, 2007, p. 544). “During this phase, the students are taught (a) a definition of what a diagram is, (b) reasons to use a diagram for solving word problems, (c) general rules to use when generating a diagram, (d) what symbols and graphic codes are and how to use them to represent things or people, (e) how to use a symbol such as a question mark to indicate what is unknown, and (f) two diagram types that can be generated and when to use them for different word problems” (van Garderen, 2007, p. 544). A line diagram can be useful for putting things in order, while a part/whole diagram can be useful for grouping things together (van Garderen, 2007, p. 545).
    • Phase 2: Strategy instruction for one-step word problems. This phase introduces students to the “Visualize” strategy (van Garderen, 2007, p. 545). This strategy puts emphasis on the cognitive process of visualization by first drawing the diagram and then arranging the diagram to show how the various parts of the diagram are related (van Garderen, 2007, p. 545). Phase 2 incorporates the strategies taught in Phase 1.
    • Phase 3: Instruction for two-step word problems. This phase has students apply the “Visualize” strategy to two-step word problems. In this phase students use the backward chaining procedure. The backward chaining procedure is “an approach that requires the student to first identify the overall primary goal of the problem, or the ‘final answer’” (van Garderen, 2007, p. 545). The students are then needs to identify the secondary problem, or “partial answer,” necessary to get the final answer (Goldman, 1989; Jitendra et al., 1999 as cited in van Garderen, 2007, p. 546). The students are made aware that there are two unknowns and use two symbols, such as questions marks, to represent the unknowns. Moreover, the first question mark represent the partial answer, while the second one represents the final answer. “To help delineate which is which, the students were taught to write ‘PA’ for the secondary missing element and ‘FA’ for the primary goal (van Garderen, 2007, p. 546).

II. Montague (1997) suggests more strategies that teachers can use to teach students with learning disabilities (Montague, M., 1997, p. 169).

  • Model strategy recitation and application using an overhead projector while solving problems.
  • Exchange roles with students during demonstration activities
  • Give positive corrective feedback during guided practice sessions.
  • To summarize, instructional procedures for strategy application should include independent practice, practice in pairs, teacher/student role exchange during demonstration exercises, and explanations of problem solving by students using board work or overhead projector.

Primary-figure1.gif

The difference between a pictorial and a schematic representation (Montague, M., 2005)


III. Task Variations

Task variation is the manipulation of word problem tasks (Jitendra, A. & Xin, Y.P., 1997, p.426). It is recommended that word problem solving instruction should be sequenced such that easier skills are taught before more difficult ones in order to alleviate student errors and frustration. “This may entail instruction first in relatively simple story problems that lend themselves to phrase-by-phrase translation; later, more complex problems that require advanced cognitive processing are presented” (Jitendra, A. & Xin, Y.P., 1997, p.427). Consequently, sequencing entails a gradual progression from establishing concrete understanding to an abstract level. Additionally, task variations can include “systematic manipulations of language, vocabulary, context, and size of the numbers used in word problems” (Jitendra, A. & Xin, Y.P., 1997, p.427).

IV. Schema-Based Strategy

Since students with learning disabilities can develop cognitive overload, which can shutdown their ability to verbalize processes and strategies, teachers should use schema-based strategy instruction (Montague, M., 1997, & Montague, M., 2005, p.5). Schema-based strategy instruction teaches primary school students how to solve change, group, and compare problems (Jitendra, Griffin, Haria, Leh, Adams, and Kaduvetoor (2005) as cited by Montague, M., 2005, p. 5). Below is the four-step to the schema-based strategy (Jitendra, Griffin, Haria, Leh, Adams, and Kaduvetoor (2005) as cited by Montague, M., 2005, p. 5):
  • Find the problem type.
  • Organize the information in the problem using the diagram.
  • Plan to solve the problem.
  • Solve the problem.

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Diagram and check list for a change problem using the schema-based strategy (Montague, M., 2005)

Curricula and Technological Resources

ThinkingBlocks (http://www.thinkingblocks.com/)

Thinking Blocks is an engaging, interactive math tool developed by classroom teachers to help students learn how to solve multistep word problems. Using brightly colored blocks, students model the relationships among the components of each word problem. With the help of a virtual teacher, students walk through a simple problem solving process and arrive at a solution. When building the models, students must identify information that is given as well as information that is unknown. Identifying and solving for an unknown quantity is a key concept in algebra. Thinking Blocks encourages students to look beyond the surface to discover the concepts and relationships that are at the core of every math problem.

Mathplayground (http://www.mathplayground.com/wordproblems.html)

Learn how to solve multistep word problems with the challenging activities on this page. Many activities contain videos that explain the problem solving process step by step. There are word problems for students in grades 2 to 8. Some programs on this site include:
  • ThinkingBlocks (mentioned above)
  • Math TV: Video Word Problems (http://www.mathplayground.com/mathtv.html)
    • Each math problems comes with step by step video solution (which gives students a model of how to solve a problem), follow up problems (so students can try out similar problems on their own), an online calculator, and sketch pad (accessories used to aid students with calculations and problem solving).
  • Math Hoops (http://www.mathplayground.com/mathhoops_Z1.html)
    • Math Hoops provides word problem practice for students in grades 3 to 5. All word problems use whole numbers but the problems range from single step addition to multistep equations. There are also problems that require students to interpret remainders. Students who answer 5 questions correctly get a chance to play some basketball.

Annotated References

  • Beckmann, S. (2004). Solving algebra and other story problems with simple diagrams: A methods demonstrated in grade 4 – 6 texts used in Singapore. The Mathematics Educator, 14, 42-46.
  • The article talks about the benefits of diagrams in Singapore textbook and how it benefits students learning in solving algebra and word problems.
  • Clarkson, P. C., & Campus, M. (1991). Language comprehension errors: A further investigation. Mathematics Education Research Journal, Vol. 3, No. 2, 24-33.
  • This research provides some support for the linkage of comprehension errors made when attempting mathematical word problems to measures of language competency.
  • Clarkson, P.C., & Galbraith, P. (1992). Bilingualism and mathematics learning: Another prespective. Journal for Research in Mathematics Education, Vol. 23, No. 1, pp. 34-44.
  • The research shows the correlation of the language proficiency in ELL students’ first language and their academic achievement.
  • Clement, J. (1982). Algebra word problem solutions: Thought processes underlying a common misconception. Journal for Research in Mathematics Education, 13, 16-30.
  • This article describes test data showing that a large proportion of science-oriented college students were unable to solve a very simple kind of algebra word problem. The data indicate that relatively advanced students can experience serious difficulties in symbolizing certain meaningful relationships with algebraic equations.
  • Cummins, D., Kintsch, W., Reusser, K. & Weimer, R. (1988). The role of understanding in solving word problems. Cognitive Psychology, 20, 405-438.
  • The Role of Understanding in Solving Word Problems focuses on how the important language is when solving word problems. It also sheds a light on the how language can develop misconceptions and ambiguity when it is not well understood.
  • This paper talks about several approaches and strategies developed in the 15-year-long Children's Math Worlds (CMW) that has been used in several classrooms.
  • Jitendra, A. & Xin, Y.P. (1997). Mathematical word-problem-solving instruction for students with mild disabilities and students at risk for math failure: A research synthesis. The Journal of Special Education, Vol. 30, No. 4, 1997, Pp. 412-438.
  • This article reviews published research on mathematical word-problem solving instruction that involves students with mild disabilities and students at risk of math failure.
  • Krussel, L. (1998). Teaching the language of mathematics. Mathematics Teacher, 91, 436-441.
  • This article discusses explicit versus implicit language instruction in Mathematics, and it makes assertions about the nature of literacy related to Mathematics. The authors believe that literacy is the goal of education and that more effort is needed in brining all students to a fuller understanding of, and appreciation for, the language of Mathematics.
  • Montague, M. (1997). Cognitive strategy instruction in mathematics for students with learning disabilities. Journal of Learning Disabilities; v30 n2 p164-77 Mar-Apr 1997.
  • This article uses cognitive strategy instruction to improve students’ performance in mathematics. Moreover, these strategies are assessed and taught to middle school students with learning disabilities.
  • Montague discusses effective strategies that allow students with learning disabilities to learn word-problem solving.
  • National Research Council (2002). Helping Children Learn Mathematics. Washington, DC: National Academy Press.
  • The major observations and recommendations of this book establish new goals for mathematics learning and lay out a course of action for achieving those goals.
  • Riccomini, P. J. (2005). Identification and remediation of systematic error patterns in subtraction. Learning Disability Quarterly, Vol. 28, No. 3, p. 233-242.
  • This study investigated 90 elementary teachers’ ability to identify systematic error patterns in subtraction and their prescribed instructional focus.
  • Scheuermann, A. & Van Garderen, D. (2008). Analyzing students’ use of graphic representations: Determining misconceptions and error patterns for instruction. Mathematics Teaching in the Middle School, 13, 471-477.
  • This paper focuses on the importance of looking into students work through diagrams or graphic representation in order to understand the misconceptions the student has when solving word problems.
  • Tirosh, D., & Graeber, A. O. (1989). Preservice elementary teachers’ explicit beliefs about multiplication and division. Educational Studies in Mathematics, Vol. 20, No. 1, pp. 79-96.
  • “This study, conducted in the United States, was designed to assess the extent to which the beliefs, “multiplication always makes bigger” and “division always makes smaller,” are explicitly held by preserves elementary teachers.
  • Towse, J.N. & Saxton, M. (1997). Linguistic influences on children’s number concepts: Methodological and theoretical considerations. Journal of Experimental Child Psychology, 66, 362-375.
  • From observations of how children match numerals to number tokens, previous research has suggested that cognitive representations of numbers vary with the linguistic boundary of numerals. It is argued that this prototype does not always support the idea that language affects number concepts and that children’s performance is shaped by other constraints.
  • Uesaka, Y., Manalo, E., & Ichikawa, S. (2007). What kinds of perceptions and daily learning behaviors promote students' use of diagrams in mathematics problem solving? Learning and Instruction, v17 n3 p322-335 Jun 2007.
  • This research attempts to determines the effectiveness of the use of diagrams in daily class activities, and compare Japanese and New Zealand students.
  • Van Garderen, D. (2007). Teaching students with LD to use diagrams to solve mathematical word problems. Journal of Learning Disabilities, Vol. 40, No. 6, 540-553.
  • This study examines the effectiveness of instruction on teaching students with learning disabilities (LD) to solve 1- and 2-step word problems of varying types.
  • Warren, E. (2006). Comparative mathematical language in the elementary school: A longitudinal study. Educational Studies In Mathematics, 62, 169-189.
  • This paper examines the change in young children’s understanding of “equal,” “more,” “less,” and “between.” The results suggest that many children have limited understanding of “more,” “less,” and “equal.”