Solving Word Problems: Developing Quantitative Reasoning

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ResearchIntoPractice MATHEMATICSSolving Word Problems:Developing Quantitative ReasoningBy Dr. Randall CharlesRandall I. CharlesProfessor Emeritus,Department of Mathematics,Problem solving has been the focus of a substantial number of research studies overthe past thirty years. It is well beyond the scope of this paper to even attempt tosummarize this body of research. Those interested in significantly broader reviews ofresearch related to problem solving should see Schoenfeld (1985), Charles (1987),Charles & Silver (1988), and Lesh & Zawojewski (2007). This paper focuses on onearea of research that has been of great interest to mathematics educators: solvingmathematics “word problems.” Some relevant research and implications for teachingare discussed in this paper.San Jose State UniversitySetting the IssueThe NCTM Curriculum Focal PointsThe recently released Common Core State Standards for Mathematics consists oftwo sets of standards: the Standards for Mathematical Content and the Standardsfor Mathematical Practice. This latter set of standards, which describes the practices,processes and dispositions that teachers should look to develop in their students,highlights the continued importance that the mathematics community places on helpingstudents become proficient in solving problems and reasoning mathematically.The importance of helping students develop the abilities and skills related to solvingproblems cannot be understated. They are key foundational abilities and skills thatstudents will draw on throughout their school years, and in their professional careers.There are many types of mathematics problems that students regularly encounter inthe school mathematics curriculum. (See Charles & Lester, 1982, for a classification ofmathematics problems.) This paper focuses on a particular type of problem that manyteachers refer to as “word problems.” Some break this type of problem into “one-stepword problems” and “multiple-step word problems.” Charles and Lester (1982) callthis type of problem a “translation problem.” Word problems have been chosen asthe focus of this paper for two reasons. First, because they are the most commontype of problem-solving task found on assessments; they are likely to be prominent,if not dominant in the common assessments currently under development by thetwo consortia, the Partnership for Assessment of Readiness for College and Careers(PARCC) and Smarter Balanced Assessment Consortium (SBAC). And second,because the abilities and skills related to solving word problems are an importantfoundation for success in algebra.Dr. Randall I. Charles has dedicatedhis life to mathematics education andworks closely to train teachers at allgrade levels. He has published widelyand is a senior author with Pearson.Dr. Charles served on the writing teamfor the NCTM Curriculum Focal Points.were a key inspiration to the writers ofthe Common Core State Standardsin bringing focus, depth, and coherenceto the curriculum.

The issue with word problems is that too many students continue to be unsuccessfulat solving them! Teachers still report that developing students’ abilities to solve wordproblems is one of their most difficult and frustrating challenges. Students continue tohave anxiety about solving problems, and they know that practice alone does not helpthem improve.Here is a rather formal statement of what constitutes a word problem. This formalstatement will be helpful when discussing implications for teaching later in this paper.A mathematics word problem is a real-world situation in which mathematical quantitiesare given, values of one or more quantities are known, while the values of one ormore quantities are unknown. The relationships between or among quantities aredescribed, a question is implied or stated asking one to find the value of one or moreunknown quantities, and one or more of the four operations – addition, subtraction,multiplication, and division – are to be used to find the value of the unknown quantityor quantities and answer the question. The problem might contain extraneous data,and it might contain one or more “hidden questions”—sub-problems that need to besolved in order to provide the final answer to the problem. The answer to the questionmight be numerical (e.g., “The car costs 23,000.”) or not (e.g., “He has enough moneyto buy the car.”).Two widely taught strategies for problem solving are the key words approach andthe problem-solving steps approach. For many elementary teachers, these strategiesprovide a logical and manageable, if not formulaic, structure to teaching students howto solve word problems. However, if problem solving continues to be difficult for somany students, one can only conclude that these common teaching strategies need tobe challenged. An analysis of these two approaches can uncover their limitations.Key Words: A “key words” approach teaches students to use a particular operationwhenever they encounter a certain English word or phrase in a word problem. Forexample, students are instructed to use addition whenever the question in a wordproblem includes “in all.”A body of research is not needed to show the limited value of the key word approach.Quite often on state or national assessments, students encounter word problemswhere a key word approach is either not applicable or misleading. Problems maycontain no words that might be connected to a particular operation or they maycontain “misleading” key words (e.g., “in all” is in the question but addition is not theneeded operation). The reason problems containing no key words or misleading onesare on assessments is not to set students up for failure. Rather, the problems look tomeasure how well students are able to identify the quantities in a problem, understandthe relationship between quantities, and choose the operation(s) needed to find thesolution. They do not look to measure how well students can apply a formulaic processthat requires little thinking or reasoning. A key words approach to teaching problemsolving prepares students to solve only a very small set of problems both on stateassessments and in the real world.Problem-Solving Steps: A “steps” approach to problem solving gives students asequenced set of actions to follow to solve a problem. The thinking behind this isaligned to that of teaching a skill like long division—if one follows a set of stepscorrectly and does the sub-calculations accurately, then one will get a correct finalanswer to the problem.2

Here is an example of “steps” for solving problems found in many instructionalmaterials:Step 1: Understand the problem.Step 2: Plan a solution.Step 3: Solve the problem.Step 4: Check your work.The origin of the steps approach to problem solving goes back to 1945 when GeorgePolya, a mathematician at Stanford University, published a book on problem solvingcalled How to Solve It. One the many powerful elements in that book is Polya’s analysisof the phases of the problem-solving process, for which he identified four:Understanding the problemDevising a planCarrying out the planLooking backPolya’s use of the word phases is intentional and noteworthy for he considered eachpart or phase of problem solving to be a thoughtful, reflective experience, not a seriesof steps. Using the word steps promotes at least two misconceptions. First, “steps”suggests (like climbing stairs) that one completes one step and moves off it and ontothe next. This is not the way mental processing proceeds for problem solving. One’sunderstanding of a problem continues to expand and evolve as one works throughthe phases. Another misconception is that problem solvingFigure 1is like a computational algorithm where there is a sequenceof actions to use, which if followed correctly, will lead to thecorrect solution. Experience shows that problem solving isnot an algorithm; there is not a series of steps that guaranteesuccess. Problem solving is a process grounded in sense makingand reasoning. Certainly successful problem solvers are skilledat reading and comprehending the words and doing the neededcalculations correctly, but problem solving is not a skill.The message for teachers that should be taken from Polya’swork is that approaching problem solving in a systematic waycan be helpful in solving problems but it does not guaranteesuccess. Problem-solving guides based on Polya’s work like thatone shown in Figure 1 can be helpful in getting students to thinksystematically about solving problems, but they should not bepresented as “steps for finding the correct answer” for thereasons discussed above.One reason so many teachers have used a key words or a stepsapproach to teaching problem solving may be that there arefew alternative instructional strategies. But finally, there is nowa body of research that provides a new direction for teachingmathematics word problems that will produce success.( Pearson Scott Foresman Publishing,2011/ 2011 Pearson Education, Inc.)3

A Visual Approach to Teaching Word ProblemsAs was noted earlier, problem solving is a process grounded in sense making andreasoning, in particular, quantitative reasoning. Notably, reasoning quantitatively is oneof the Standards for Mathematical Practice from the Common Core State Standards.Reasoning quantitatively “entails habits of creating a coherent representation of theproblem at hand; considering the units involved; attending to the meaning of thequantities, not just how to compute them; and knowing and flexibly using differentproperties of operations and objects.”1 In other words, quantitative reasoning involvesidentifying the quantities in a problem and using reasoning to identify the relationship(s)between or among them.Here is a word problem whose solution requires quantitative reasoning.Carrie has 125 U.S. stamps. She has 3 times as many foreign stamps as U.S. stamps.How many stamps does she have altogether?The quantities in this word problem are: the number of U.S. stamps (a known value, 125) the number of foreign stamps (an unknown value) the total number of foreign and U.S. stamps (an unknown value)The challenge in solving word problems is not often in identifying and determining theknown and unknown quantities. Rather, it is in articulating the relationships betweenquantities, understanding those relationships, and determining the appropriateoperation or operations to show those relationships.The relationships in this problem are: There are 3 times as many foreign stamps as U.S. stamps. The total for the number of foreign stamps and the U.S. stamps.“The challenge forteaching word problemsis in helping students usequantitative reasoning.”We know from research that just because a learner can read a word problem,knows all vocabulary in the problem, and can identify the relationships statedin the problem, it does not mean that he or she can find the solution (Knifong& Holton, 1976, 1977). Rather, children who understand operation meaningsand can associate relationships between quantities given in word problemswith those operation meanings are better problem solvers (see Sowder,1988). So, the challenge for teaching word problems is in helping studentsuse quantitative reasoning—that is, helping them use reasoning to identifythe relationships among the quantities in the problem and connect thoserelationships to appropriate operations.Three research findings provide guidance for a way to develop students’ quantitativereasoning abilities.(a) Encourage students to meaningfully represent mathematical word problems. Whenstudents begin by representing word problems pictorially, rather than to directlytranslate the elements of the problems into corresponding mathematical operations,they may more successfully solve these problems and better comprehend themathematical concepts embedded within them. (Pape, S.J., 2004).14Common Core State Standards, 2010, p. 6

(b) Have students represent problems visually to improve problem-solvingperformance. Research suggests that having student represent a problem visually resultsin improved problem-solving performance. (Yancey, Thompson, and Yancey, 1989)(c) Emphasize to students the importance of the problem structure (i.e., the quantitiesinvolved and their relationships) rather than surface features (like key words).(Diezmann and English, 2001, p. 82)An approach to solving word problems derived from the research findings discussedearlier is to use bar diagrams as visual representations to show how quantities ina word problem are related. Seeing those relationships and connecting those tooperation meanings helps students to select an appropriate operation for solving theproblem. “A diagram can serve to ‘unpack’ the structure of a problem and lay thefoundation for its solution” (Diezmann and English, 2001, p. 77). Nickerson (1994)found that the ability to use diagrams is integral to mathematics thinking and learning.U.S. Stamps125Foreign Stamps1251251253 times as manyHere is a bar diagram representing the quantities and their relationships for the wordproblem given above.The relationships between the quantities in the problem can be seen in thebar diagram. There are 3 times as many foreign stamps as U.S. stamps.The set of 3 boxes that each contains 125 shows this relationship.Three equal groups are being joined. The total for the number of foreign stamps and the U.S. stamps.The combination of all four of the boxes shows this relationship. The totalfor the three equal groups is being joined with the amount for the onegroup, but since all groups have the same amount, four equal groups arebeing joined.Translating these relationships to numerical expressions requires an understanding ofthe meanings of the operations. For the first relationship, three quantities are beingjoined. When quantities are being joined, addition can be used to find the total. Whenthe quantities being joined are equal, multiplication can be used to find the total andis usually more efficient than addition. So, the numerical expression associated withthe three boxes representing the number of foreign stamps is 3 125. The numericalexpression that shows the joining of the number of foreign stamps and the numberof U.S. stamps is (3 125) 125. This expression can be simplified to 4 125; the 4groups of 125 can easily be seen in the bar diagram. The answer to the problem is thatCarrie has 500 stamps altogether.5

Figure 2 shows a collection of common one-step word problems; each can be solvedusing one of the four basic operations. For each, a bar diagram shows the relationshipbetween the quantities. Then one or more equations are given showing the operationor operations that can be used to find the answer. It is important to recognize that arelationship in some word problems can be translated into more than one appropriateequation. For example, Example B shows that how one thinks about the relationshipbetween the quantities in the problem leads to either an addition or subtractionequation; one can add on to 57 to get to 112 or one can subtract 57 from 112.Figure 2: Bar Diagrams for Addition and Subtraction SituationsExample ATotal AmountUnknownProblem TypeJoiningDiagram Showingthe RelationshipDescription ofthe RelationshipNumberSentenceProblem TypeSeparatingDiagram Showingthe Relationship6Kim has 23 antiquedolls. Her Father givesher 18 more antiquedolls. Now how manyantique dolls doesshe have?Example BAmount JoinedUnknownExample CInitial AmountUnknownDebbie has saved 57. How much moremoney does she needin order to have 112?Tom had some moneyin his savings account.He then deposited 45into the same account.Then he had 92 in all.How much did he havein his savings accountto start?2311218The two unequalamounts (23 and 18)are known and beingjoined and the totalis unknown.5792?45The initial amountis known (57). Theamount being joined tothat is unknown. Thetotal is known (112).The initial amount isunknown. The amountbeing joined to that isknown (45). The totalis known (92).23 18 ?57 ? 112112 57 ? 45 9292 45 ?Example DAmount RemainingUnknownExample EAmount SeparatedUnknownExample FInitial AmountUnknownCarrie has 45 CDs. Shegives some to Jo. NowCarrie has 27 left. Howmany did she give to Jo?Alan has some marbles.He lost 12 of them.Then he had 32 left.How many did he havebefore he lost some?45?Steven has 122 jellybeans. He eats 71 ofthem in one weekend.How many jelly beansare left?12271?271232Description ofthe RelationshipThe total amount isknown (122) and theamount separated fromthat is known (71).The amount remainingis unknown.The total amount isknown (45) and theamount separated fromthat is unknown. Theamount remaining isknown (27).The total is unknown.The amount separatedfrom the total isknown (12) and theamount remaining isknown (32).NumberSentence122 71 ?45 – ? 27? 27 45? 12 3212 32 ?

Figure 2: (cont’d) Bar Diagrams for Addition and Subtraction SituationsProblem TypePart-PartWholeDiagram Showingthe RelationshipDescription ofthe RelationshipNumberSentenceProblem TypeComparisonDiagram Showingthe RelationshipDescription ofthe RelationshipNumberSentenceExample ATotal AmountUnknownFourteen cats and 16dogs are in the kennel.How many dogs andcats are in the kennel?Example BInitial AmountUnknownExample CAmount JoinedUnknownSome adults and 12children were on a bus.There are 31 peoplein all on the bus. Howmany adults were onthe bus?Forty-nine people wenton a hike. Six wereadults and the restwere children. Howmany children went onthe hike?3149?1416Each unequal part isknown (14 and 16); thewhole is unknown.?12The first part isunknown, but thesecond part is known(12). The whole isknown (31).6?The whole is known(49) and the initial partis known (6). The otherpart is unknown.14 16 ? 12 3131 12 ?6 ? 4949 6 ?Example JAmount More (or Less)UnknownExample KSmaller AmountUnknownExample LLarger AmountUnknownFran spent 84, whichwas 26 more thanAlice spent. How muchdid Alice spend?Barney has 23 old coins.Steve has 16 more oldcoins than Barney. Howmany old coins doesSteve have?84?Alex has 47 toy cars.Keisha has 12 cars.How many more carsdoes Alex have?4712?Two known amounts(47 and 12) are beingcompared. The amountmore/less is unknown.47 12 ?262316The larger amount isknown (84), and smalleramount is unknown.The amount morethe larger is than thesmaller is known (26).One smaller amountis known (23), and thelarger amount is notknown. The amountmore the larger is thanthe smaller is known(16).84 ? 2684 26 ?23 16 ? 23 167

Figure 2: (cont’d) Bar Diagrams for Multiplication and Division SituationsExample MTotal AmountUnknownProblem TypeJoining EqualGroupsDiagram Showingthe RelationshipDescription ofthe RelationshipKim has 4 photoalbums. Each albumhas 85 pictures. Howmany photos are in her4 albums?SeparatingEqualGroupsDiagram Showingthe RelationshipDescription ofthe RelationshipNumberSentence8Example ONumber of GroupsUnknownPam had 4 bags andput the same numberof apples in each bag.She ended up with 52apples in bags. Howmany did she put ineach bag?Fred bought somebooks that each cost 16. He spent 80altogether. How manybooks did he buy?5285 85 85 85Four equal knownamounts (85) arebeing joined to find theunknown total.NumberSentenceProblem TypeExample NAmount per GroupUnknown?80?A known number (4)of unknown but equalamounts are beingjoined to give a knowntotal (52).?A known amount(16) is being joined anunknown number oftimes to itself to get aknown total (80).4 85 ?4 ? 5252 4 ? 16 8080 16 ?Example PAmount per GroupUnknownExample QNumber of GroupsUnknownExample RTotal AmountUnknownA total of 108 childrensigned up for soccer.How many 18-personteams can be made?Kim had some cards.She put them into pilesof 35 and was ableto make 4 piles. Howmany cards did shehave to start?Byron has 45 pigeons.He keeps them in 5pens with the samenumber of pigeonsin each. How manypigeons are in eachpen?45?180?The total is known (45)and being separatedinto a known numberof equal groups (5)but the amount in eachgroup is unknown.45 5 ?18?35 35 35 35The total is known(108) and beingseparated into equalgroups of a knownamount (18). Thenumber of equal groupsneeded to match thetotal is unknown.The total amountis unknown. It isseparated into a knownnumber of groups (4)with a known equalamount in each (35).108 18 ?18 ? 108? 4 354 35 ?

Problem TypeComparisonExample SLarger AmountUnknownAlex has 17 toy cars.Keisha has 3 times asmany. How many carsdoes Keisha have?Example TSmaller AmountUnknownExample UNumber of Times asMany UnknownBarney has 24 old coins.This is 3 times morecoins than Steve has.How many old coinsdoes Steve have?Ann’s teacher is 39years old. Ann is 13years old. Ann’s teacheris how many times asold as Ann?Diagram Showingthe RelationshipDescription ofthe RelationshipNumberSentence171724173 timesas many17?39?3 timesas many?13? timesas many13The smaller amountis known (17) and thelarger amount is a givennumber of times more(3). The larger quantityis not known.The larger amount isknown (24) and is agiven number of timesgreater than the smallamount (3). The smalleramount is not known.The larger amount (39)and the smaller amount(13) are known. Howmany times more thelarger amount is thanthe smaller amount isnot known.3 17 ?3 ? 2424 3 ? 13 3939 13 ?For multiple-step problems such as the one cited earlier involving Carrie’s stamps,multiple bar diagrams are used to help answer the hidden question (i.e., sub-problem)and then answer the final question. In the Carrie problem, the hidden question was tofind the total number of foreign stamps; the answer to that was then used to answerthe question stated in the problem.One of the powerful attributes of these bar diagrams is that all show relationshipsbetween parts and wholes. This coherence in visual representations helps studentssee not only the connections between the diagrams but also connections betweenand among operations. An important part of understanding operations is to know allrelationships between and among the four operations.Suggestions for TeachingHere are a few suggestions for how bar diagrams can be an integral part of teachingand learning mathematics (Diezmann, & English, 2001). Model bar diagrams on a regular basis; not just in special lessons but frequentlywhen word problems are encountered. Discuss the structure of bar diagrams and connect them to quantities in theword problem and to operation meanings. Use bar diagrams to focus on the structure of a word problem, not surfacefeatures like key words. Encourage students to use bar diagrams to help them understandand solve problems.9

The mathematics community has consistently promoted the use of visualrepresentations to foster students’ mathematical reasoning (NCTM, 2000). TheStandards for Mathematical Practice in the Common Core State Standards forMathematics are the latest reminders of the importance of helping students “see”mathematical concepts in order to understand the structure of a problem situation. Bardiagrams provide that visual structure to make mathematics accessible to a much widergroup of students. They are a powerful tool for developing visual literacy and successwith problem solving for ALL students.10

REFERENCESCharles, R.I., & Lester, F.K. (1982). Teachingproblem solving: What, why and how. Palo Alto, CA:Dale Seymour Publishing Company.Charles, R.I. (1987). “Solving word problems.”What works: Research about teaching and learning,2nd Edition. Washington, D.C.: U.S. Departmentof Education.Charles, R., and E. Silver (Eds.). (1988).The Teaching and assessing of mathematicalproblem solving: Research agenda for mathematicseducation. Research Monograph. Research AgendaProject. Reston, VA: National Council of Teachersof Mathematics.Diezmann, C., and L. English. (2001). Promotingthe use of diagrams as tools for thinking. InA.A. Cuoco and F. R. Curcio (Eds.), The role ofrepresentation in school mathematics. Reston,VA: National Council of Teachers of Mathematics,77-89.Diezmann, C. (1995). Visual literacy: Equity andsocial justice in mathematics education. Paperpresented at the Australian Association forResearch in Education Conference, November26–30, 1995, Hobart, Tasmania.Knifong, J. D. and B. Holton. (1976). “An analysisof children’s written solutions to word problems.”Journal for Research in Mathematics Education,7(March 1976), 106-12.Knifong, J. D. and B. Holton. (1977). “A search forreading diffi culties among erred word problems.”Journal for Research in Mathematics Education,8(May 1977), 227-30.National Council of Teachers of Mathematics(2000). Curriculum and evaluation standards forschool mathematics. Reston, VA: NCTM.National Governors’ Association Center for BestPractices and Council for Chief State SchoolOfficers (2010). Common Core State Standardsfor Mathematics.Pape, S.J. (2004). “Middle school children’sproblem-solving behavior: A cognitive analysisfrom a reading comprehension perspective.”Journal for Research in Mathematics Education,35:3, pp. 187–219.Polya, G. (1945). How to solve it. New York:Doubleday.Schoenfeld, A. H. (1985). Mathematical problemsolving. New York: Academic Press.Sowder, L. (1988). “Choosing operations insolving routine word problems.” In R. I. Charlesand E. A. Silver (Eds.), The teaching and assessingof mathematical problem solving, Reston, VA:Lawrence Erlbaum Associates and NationalCouncil of Teachers of Mathematics, 148–158.Yancey, A. V., C. S. Thompson, and J. S. Yancey.(1989). “Children must learn to draw diagrams.”Arithmetic Teacher, 36 (7), 15–23.11

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teachers refer to as “word problems.” Some break this type of problem into “one-step word problems” and “multiple-step word problems.” Charles and Lester (1982) call this type of problem a “translation problem.” Word problems have been chosen as the focus of this paper

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