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Research Into Practice MATHEMATICSSolving Word ProblemsDeveloping Students’Quantitative Reasoning AbilitiesProblem solving has been the focus of a substantial number of researchstudies over the past thirty years. It is well beyond the scope of this paperto even attempt to summarize this body of research. Those interested insignificantly broader reviews of research related to problem solving should seeSchoenfeld (1985), Charles (1987), and Charles & Silver (1988). This paperfocuses on one area of research that has been of great interest to mathematicseducators, solving mathematics “word problems.” Some relevant research andimplications for teaching are discussed in this paper.Setting the IssueThere are many types of mathematics problems that should be included inthe school curriculum. (See Charles & Lester, 1982, for a classification ofmathematics problems.) This paper focuses on a particular type of problemfound in the school curriculum that many teachers refer to as “word problems.”Some break this type of problem into “1-step word problems” and “multiplestep word problems.” Charles and Lester (1982) call this type of problem a“translation problem.” Word problems have been chosen as the focus of thispaper for two reasons. First, they are the most common type of problem-solvingtask found on most state assessments. And second, the abilities and skills relatedto solving word problems are key foundational abilities and skills for solvingword problems in algebra.Randall I. CharlesProfessor Emeritus, Department ofMathematics, San Jose State UniversitySan Jose, CaliforniaDr. Randall Charles is ProfessorEmeritus in the Department ofMathematics at San Jose StateUniversity. His primary research hasfocused on problem solving withseveral publications for NCTM.Dr. Charles has served as a K–12mathematics supervisor, Vice Presentof the National Council of SupervisorsThe issue with regard to word problems is that too many students continue tobe unsuccessful at solving them! Teachers still report that developing students’abilities to solve word problems is one of their most difficult and frustratingchallenges. Students continue to have anxiety about solving problems, and theyknow that practice alone does not help them improve.Here is a rather formal statement of what constitutes a word problem. Thisformal statement will be helpful when discussing implications for teaching laterin this paper.A mathematics word problem is a real-world context in which mathematicalquantities are given, values of one or more quantities are known, values of oneor more quantities are unknown, relationships between or among quantitiesare described, a question is implied or stated asking one to find the value ofone or more unknown quantities, and one or more of the operations addition,subtraction, multiplication, and division can be used to find the value of theof Mathematics, and member of theNCTM Research Advisory Committee.He has authored or coauthored more

unknown quantity or quantities and answer the question. Data needed (knownquantities) can be given in the context of the problem, in an outside data sourcelike a graph, or created through a data collection activity. The problem mightcontain extraneous data, and a problem might contain one or more “hiddenquestions”—sub-problems that need to be solved in orderto answer the question given in the problem statement. Theanswer to the question might be numerical (e.g., “The carcosts 23,000.”) or not (e.g., “He has enough money tobuy the car.”)“The issue with regard toword problems is that toomany students continue to beunsuccessful at solving them!”Two teaching strategies for problem solving widely usedby many teachers are the key words approach and theproblem-solving steps approach. If problem-solvingcontinues to be difficult for so many teachers and too manystudents are not becoming successful problem solvers, one can only concludethat these common teaching strategies need to be challenged.Key Words: A “key words” approach teaches students to always use a particularoperation whenever a word problem contains a certain English word or phrase.For example, a typical key word approach to teaching problem solving tellsstudents to use addition whenever the question in a word problem includes“in all.”A body of research is not needed to show that a key word approach to problemsolving has limited value. Most if not every state and national assessmentcontains word problems where a key word approach does not produce correctsolutions for many problems. Sometimes “misleading” key words are used(e.g., “in all” is in the question but addition is not the needed operation), andother times problems contain no words that might be connected to a particularoperation. The reason problems containing misleading or no key words are onassessments is not to set students up for failure. Rather, the fact is that mostREAL problems in our world do not come neatly presented with key wordstelling one how to solve them. If they did, teaching problem solving wouldsimply not be a problem! A key words approach to teaching problem solvingprepares 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 teaching problem solving givesstudents a sequenced set of actions to follow to solve a problem. The thinkingbehind this is aligned to that of teaching a skill like long division—if one followsa set of steps correctly and does the sub-calculations accurately, then one will geta correct final answer to the problem. Here is an example of “steps” for solvingproblems found in many instructional materials:2Research Into Practice Pearson

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 whenGeorge Polya, a mathematician at Stanford University, published a book onproblem solving called How to Solve It. One the many powerful elements inthat book is Polya’s analysis of the phases of the problem-solving process. Polyaidentified four phases of solving problems. Understand Plan Solve Look BackPolya’s use of the word phases is crucial and gets at the heart of the issue; hedid not use the word steps in his description of the problem-solving processbecause it promotes at least two misconceptions. First, “steps” suggests (likeclimbing stairs) that one moves off of one step and moves onto the next. Thisis not the way mental processing proceeds for problem solving. For example,one’s understanding of a problem continues to evolve as planning and solvingare underway. Another misconception that “steps” promotes, as mentionedabove, is that problem solving is like a computational algorithm where there isa sequence of actions to use, which if followed correctly, will lead to the correctsolution. Experience shows that problem solving is not an algorithm; there is noseries of steps that guarantee success. Problem solving is a process grounded onreasoning. Certainly to be a successful problem solver there are skills needed,like reading and comprehending the words and doing the needed calculationscorrectly, but problem solving is not a skill.The message for teachers that should be taken from Polya’s work is thatapproaching problem solving in a systematic way can help students solveproblems but not guarantee success. Problem-solving guides based on Polya’swork like that one shown in Figure 1 can be helpful in getting students to thinksystematically about solving problems, but they should not be presented as“steps for finding the correct answer” for the reasons discussed above.Research Into Practice Pearson3

Figure 1: Problem-Solving Recording Sheet( Pearson Scott Foresman Publishing, 2009/ 2009 Pearson Education, Inc.)The reason so many teachers have used a key words or a steps approach toteaching problem solving is that they have not had any alternative instructionalstrategies. But finally, there is now a body of research that provides a newdirection for teaching mathematics word problems that will produce success.A Visual Approach to Teaching Word ProblemsIt was mentioned that problem solving is a process grounded on reasoning, inparticular, quantitative reasoning. Quantitative reasoning involves identifyingthe quantities in a problem and using reasoning to identify the relationshipbetween them. A mathematical quantity is anything that can be measuredor counted. Here is a word problem whose solution requires quantitativereasoning.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)4Research Into Practice Pearson

The challenge in solving word problems is often not to identify the knownand unknown quantities. Rather, the challenge is to identify statements in theproblem that express relationships between quantities, to understand thoserelationships, and to choose an appropriate operation or operations to showthose 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.We know from research that just because a child can read a word problem,knows all vocabulary in the problem, and can identify the relationshipsstated in the problem it does not mean that he or shecan solve it (Knifong & Holton, 1976, 1977). Rather,children who understand operation meanings and canassociate relationships between quantities given in wordproblems with those operation meanings are better problemsolvers (see Sowder, 1988). So, the challenge for teachingword problems is how to help students use quantitativereasoning—that is, use reasoning to identify the relationshipsbetween the quantities in the problem and connect thoserelationships to appropriate operations.“A new approach to solvingword problems . . . is touse bar diagrams as visualrepresentations . . .”Three research findings provide guidance for a new way todevelop students’ quantitative reasoning abilities.(a) If students are encouraged to understand and meaningfully representmathematical word problems rather than directly translate the elements ofthe problems into corresponding mathematical operations, they may moresuccessfully solve these problems and better comprehend the mathematicalconcepts embedded within them. (Pape, S.J., 2004)(b) Training children in the process of using diagrams to [meaningfully representand] solve [mathematical word] problems results in more improvedproblem-solving performance than training students in any other strategy.(Yancey, Thompson, and Yancey, 1989)(c) “. . . teachers need to emphasize the representation of the problem structureand de-emphasize the representation of surface features.” (Diezmann andEnglish, 2001, p. 82)A new approach to solving word problems derived from the research findingsabove is to use bar diagrams as visual representations that show how quantitiesin a word problem are related. Seeing those relationships and connectingthose to operation meanings enables one to select an appropriate operationfor solving the problem. “A diagram can serve to ‘unpack’ the structure of aproblem and lay the foundation for its solution” (Diezmann and English, 2001,p. 77). Nickerson (1994) found that the ability to use diagrams is integral tomathematics thinking and learning.Research Into Practice Pearson5

Here is a bar diagram representing the quantities and their relationships for theword problem given above.U.S. stamps125Foreign stamps1251251253 times as manyThe relationships between the quantities in the problem can be seen in the bardiagram. 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 total forthe three equal groups is being joined with the amount for the one group, butsince all groups have the same amount, four groups are being joined.Translating these relationships to numerical expressions requires one tounderstand operation meanings. For the first relationship, three quantities arebeing joined. When quantities are being joined, addition can be used to find thetotal. But, when the quantities being joined are equal, multiplication can be usedto find the total and is usually more efficient than addition. So, the numericalexpression associated with the three boxes representing the number of foreignstamps is 3 125. The numerical expression that shows the joining of thenumber of foreign stamps and the number of U.S. stamps is (3 125) 125.This expression can be simplified to 4 125; the 4 groupsof 125 can easily be seen in the bar diagram. The answer tothe problem is that Carrie has 500 stamps altogether.“. . . a relationship in someword problems can betranslated into more than oneappropriate number sentence.”Figure 2 shows a collection of common 1-step wordproblems; each can be solved using one of the four basicoperations of addition, subtraction, multiplication, ordivision. A bar diagram is given for each showing therelationship between the quantities. Then one or morenumber sentences are given showing the operation oroperations that can be used to find the answer. It is important to recognizethat a relationship in some word problems can be translated into more thanone appropriate number sentence. For example, Example B shows that how onethinks about the relationship between the quantities in the problem leads toeither an addition or subtraction number sentence; one can add on to 57 to getto 112 or one can subtract 57 from 112.6Research Into Practice Pearson

Figure 2: Bar Diagrams for Addition and Subtraction SituationsExample AProblem TypeJoiningDiagram Showingthe RelationshipDescription ofthe RelationshipNumber SentenceExample BKim has 23 antiquedolls. Her father givesher 18 more antiquedolls. Now how manyantique dolls does shehave?Debbie has saved 57.How much more moneydoes she need in orderto have 112?2311218The two unequal partsare known and beingjoined and the amountin all is unknown.23 18 ?57?The initial amount isknown. The amountbeing joined to that isunknown. The totalis known.57 ? 112112 57 ?Example CProblem TypeSeparatingDiagram Showingthe RelationshipExample DSteven has 122 jellybeans. He eats 71 ofthem in one weekend.How many jelly beansare left?Carrie has 45 CDs. Shegives some to Jo. NowCarrie has 27 left. Howmany did she give to Jo?1227145?27Description ofthe RelationshipThe total is known andthe amount separatedfrom that is known.The amount remainingis unknown.The total is known andthe amount separatedfrom that is unknown.The amount remainingis known.Number Sentence122 71 ?45 ? 2727 ? 45Research Into Practice Pearson7

Figure 2 (continued): Bar Diagrams for Addition and Subtraction SituationsExample EProblem TypePart-Part-WholeDiagram Showingthe RelationshipExample FFourteen cats and 16dogs are in the kennel.How many dogs andcats are in the kennel?Some adults and 12children were on a bus.There were 31 peoplein all on the bus. Howmany adults were onthe bus?143116Description ofthe RelationshipEach part is known; thewhole is unknown.Number Sentence14 16 ?12The first part isunknown, but thesecond part is known.The whole is known.? 12 3131 12 ?Example GProblem TypeComparisonAlex has 47 toy cars.Keisha has 12 cars.How many more carsdoes Alex have?Diagram Showingthe RelationshipNumber Sentence8Barney has 23 old coins.Steve has 16 more oldcoins than Barney. Howmany old coins doesSteve have?4712Description ofthe RelationshipExample H?Two known quantitiesare being compared.The amount more/lessis unknown.47 12 ?2316One quantity is known.The larger quantityis not known. Therelationship of thelarger quantity to thesmaller quantity isgiven.23 16 ?Research Into Practice Pearson

Figure 2 (continued): Bar Diagrams for Multiplication and Division SituationsExample IExample JProblem TypeJoining Equal GroupsKim has 4 photoalbums. Each albumhas 85 pictures. Howmany photos are in her4 albums?Pam had 4 bags andput the same number ofapples in each bag. Sheended up with 52 applesin bags. How many didshe put in each bag?Diagram Showingthe Relationship?5285858585Description ofthe RelationshipFour equal knownamounts are beingjoined to find theunknown total.Number Sentence4 85 ?A known number ofunknown but equalamounts are beingjoined to give aknown total.4 ? 5252 4 ?Example KExample LProblem TypeSeparating EqualGroupsByron has 45 pigeons.He keeps them in 5 penswith the same numberof pigeons in each. Howmany pigeons are ineach pen?A total of 216 childrensigned up for soccer.How many 18-personteams can be made?Diagram Showingthe Relationship45216?18Description ofthe RelationshipThe total is known andbeing separated intoa known number ofequal groups, but theamount in each group isunknown.The total is knownand being separatedinto equal groups ofa known amount.The number of equalgroups needed to matchthe known total isunknown.Number Sentence45 5 ?216 18 ?18 ? 216Research Into Practice Pearson9

Figure 2 (continued): Bar Diagrams for Multiplication and Division SituationsExample MProblem TypeComparisonDiagram Showingthe RelationshipExample NAlex has 17 toy cars.Keisha has 3 times asmany. How many carsdoes Keisha have?Barney has 24 old coins.This is 3 times morecoins than Steve has.How many old coinsdoes Steve have?521717241717?Description ofthe RelationshipTwo quantities arebeing compared. Oneis known and the otheris a given number oftimes more. The otherquantity is not known.Two quantities arebeing compared. Oneis known and is a givennumber of times greaterthan the other. Theother quantity isnot known.Number Sentence3 17 ?3 ? 2424 3 ?10Research Into Practice Pearson

For multiple-step problems such as the one above involving Carrie, multiple bardiagrams are used to help answer the hidden question (i.e., sub-problem) andthen answer the final question. In the Carrie problem, the hidden question wasto find the total number of foreign stamps; the answer to that was then used toanswer the question stated in the problem.One of the powerful attributes of this set of bar diagrams is that they are allconnected to parts and wholes. This consistency in visual representations helpsstudents see not only the connections between the diagrams but also connectionsbetween and among operations. An important part of understanding operationsis to know all relationships between and among the four operations.Suggestions for TeachingHere are a few suggestions for how bar diagrams can be an integral part ofteaching and learning mathematics (Diezmann, & English, 2001). Model bar diagrams on a regular basis; not just in special lessons butfrequently when word problems are encountered. Discuss the structure of bar diagrams and connect them to quantities inthe word 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 understand andsolve problems.Using diagrams to promote mathematical reasoning is an essential element ina student’s mathematics education (NCTM, 2000). Bar diagrams can makepowerful mathematics accessible to a much wider group of students; “. . . toimplement the values of social justice in mathematics education, opportunitiesneed to be provided which facilitate the development of visual literacy”(Diezmann, 1995). The bar diagrams presented here are a powerful tool fordeveloping visual literacy and success with problem solving for ALL students.Research Into Practice Pearson11

Charles, R.I., & Lester, F.K. (1982). Teachingproblem solving: What, why and how. PaloAlto, CA: Dale Seymour Publishing Company.Diezmann, C. (1995). Visual literacy: Equityand social justice in mathematics education.Paper presented at the Australian Associationfor Research in Education Conference,November 26–30, 1995, Hobart, Tasmania.Polya, G. (1945). How to solve it. New York:Doubleday.Charles, R.I. (1987). “Solving wordproblems.” What works: Researchabout teaching and learning, 2nd Edition.Washington, D.C.: U.S. Departmentof Education.Knifong, J. D. and B. Holton. (1976). “Ananalysis of children’s written solutions toword problems.” Journal for Research inMathematics Education, 7(March 1976),106-12.Charles, R., and E. Silver (Eds.). (1988).The Teaching and assessing of mathematicalproblem solving: Research agendafor mathematics education. ResearchMonograph. Research Agenda Project.Reston, VA: National Council of Teachersof Mathematics.Knifong, J. D. and B. Holton. (1977). “Asearch for reading difficulties among erredword problems.” Journal for Research inMathematics Education, 8(May 1977),227-30.Sowder, L. (1988). “Choosing operationsin solving routine word problems.” In R. I.Charles and E. A. Silver (Eds.), The teachingand assessing of mathematical problemsolving, Reston, VA: Lawrence ErlbaumAssociates and National Council of Teachersof Mathematics, 148–158.REFERENCESDiezmann, C., and L. English. (2001).Promoting the use of diagrams as tools forthinking. In A.A. Cuoco and F. R. Curcio(Eds.), The role of representation in schoolmathematics. Reston, VA: National Councilof Teachers of Mathematics, 77-89.Schoenfeld, A. H. (1985). Mathematicalproblem solving. New York: Academic Press.Yancey, A. V., C. S. Thompson, and J. S.Yancey. (1989). “Children must learn to drawdiagrams.” Arithmetic Teacher, 36 (7), 15–23.National Council of Teachers of Mathematics(2000). Curriculum and evaluation standardsfor school mathematics. Reston, VA: NCTM.Pape, S.J. (2004). “Middle school children’sproblem-solving behavior: A cognitiveanalysis from a reading comprehensionperspective.” Journal for Research inMathematics Education, 35:3, pp. ) 552-2259ADV 978-1-4182-4094-3 1-4182-4094-XCopyright Pearson Education, Inc. Mat07289

operation whenever a word problem contains a certain English word or phrase. For example, a typical key word approach to teaching problem solving tells students to use addition whenever the question in a word problem includes "in all." A body of research is not needed to show that a key word approach to problem solving has limited value.

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