journal of educational psychology 1988, vol. 80,

8/14/2019 Journal of Educational Psychology 1988, Vol. 80,

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Journal of Educational Psychology1988, Vol. 80, No. 2 , 192-201

Copyright 1988 by the American Psychological Association, Inc.0022-066 3/88/SOO.75

Teaching Children to Use Schematic Drawings to Solve Addition andSubtraction Word Problems

Gordon B. Willis and Kare n C. FusonSchool of Education and Social PolicyNorthwestern University

Two classes of second graders of average and above-average m athematics ability w ere taught touse differing schematic drawings to represent differing categories of addition and subtractionword problems. C hildren entered the three-digit num bers used in the problems into the schematicdrawings and then were to use the drawings to facilitate the choice of the solution procedure.The children were able to make the correct drawing for a given category, usually inserted thenumbers from the problem into a schematic drawing correctly, and usually selected the correctsolution strategy for the problem. There was little support for the hypotheses that children use asingle part-part-whole schema to solve either all categories of problems or the more difficult"Chan ge" problems. T he m ost difficult problems were those in which the underlying semanticsubtractive problem category ("Change-Get-Less" and "Compare") conflicted w ith the addition

solution strategy required to solve the problem. The good-to-excellent posttest performance onmost of the possible kinds of addition and subtraction word problems indicates that most ofthese problems are within the zone of proximal development of second graders of average andabove-average mathematics ability. Thus American textbooks can include many of the moredifficult w ord problem s, as do textbooks in the Soviet U nion.

Solving addition and subtraction word problems involvesat least three aspects: represen ting the word problem situation,selecting a solution strategy, and using the solution strategyto find the answer. Initially, children solve word problems byrepresenting the problem with concrete objects and then usingthese objects to carry out the solution strategy (Briars &

Larkin, 1984; Carpenter & M oser, 1984; Fuson, 1988; Riley,Greeno, & Heller, 1983). Later, children solve problems byusing more sophisticated counting strategies that also aredirectly derived from the representation of the problem situ-ation (Carpenter & Moser, 1984; Fuson, 1988). Finally, chil-dren solve problems by choosing an arithmetic operation(addition or subtraction) and then using some particularmethod of adding or subtracting such as thinking strategies,known facts, or the multidigit addition or subtraction algo-rithms (Carpenter & Moser, 1984; Fuson, 1988). Thus thefirst two aspects of problem solving may be merged for smallnumbers or simple types of problems, but they are separatefor large numbers because these require the choice of analgorithm.

The most common method of teaching addition and sub-traction word problems ignores children's need to representthe problem situation and instead focuses only on the solutionstrategy: Children are taught to write a solution addition orsubtraction sentence (e.g., 8 + 5 = ? or 8 - 5 = ?) for aproblem and then are to write the answer for the sentence.The disadvantage of this approach is particularly strong forthe more complex kinds of word problems, for these requirenot only that children represent a problem but also that they

This research was supported by a grant from the Amoco Founda-tion to the University of Chicago School Mathematics Project.

Correspondence concerning this article should be addressed toKaren C. Fuson, School of Education and Social Policy, N orthwesternUniversity, Evanston, Illinois 60208.

reflect on that problem representation and modify it in someway in order to select a solution strategy (Briars & Larkin,1984; Riley et al , 1983). A teaching method that helpschildren to represent the problem situation would be morehelpful than the prevalent solution sentence method.

In developing and testing such a method, we decided to

focus initially on the upper end of problem solvingthatinvolving more difficult types of problems and multidigitnumbersbecause the need for representational supportseems to be most crucial here: The more difficult problemsrequire reflection on the representation, and the large num-bers require a separate selection of the solution strategy. Theteaching approach that we chose was to teach children torepresent word problems by making a schematic drawing thatmodels the semantic features of the problem situation; thenumbers in the problem can then be written in this schematicdrawing and the drawing used to decide whether to add orsubtract to find the missing problem element.

Understanding the schematic drawings used requires un-

derstanding the different types of addition and subtractionword problem situations. These are commonly divided intofour m ajor categories (see Tab le 1). Two of these are basicallyadditive. In the Change-Get-More category, some initialquantity gets some more added to it; in the Put-Togethercategory, two separate quantities are put together to form onecombined quantity. The other two categories are basicallysubtractive. In the Change-Get-Less category, there is a quan-tity from which some quantity is taken; in the Comparecategory, two quantities are compared in order to find outhow muc h greater one quantity is than another. A fifthcategory, Equalize, is identified by some researchers; theseproblems are Compare problems that explicitly mention achange needed to make the two original quantities equivalent.Very little research has been done on this category, so we didnot use it in this study. The names of the problem types in

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Table 1Classification of Word Problems

Addition situationsChange-Get-More

Missing end

Joe had 3 marbles.Then Tom gave him 5 more marbles.How many marbles does Joe have now?

Missing change0

Joe had 3 marbles.Then Tom gave him some more marbles.Now Joe has 8 marbles.How many marbles did Tom give him?

Missing start3Joe had some marbles.Then Tom gave him 5 more marbles.How m any marbles did Joe have in the beginning?

Put-TogetherMissing all

Joe has 3 marbles.Tom has 5 marbles.How m any marbles do they have altogether?

Missing first parta

Joe and Tom have 8 marbles altogether.Tom has 3 marbles.How many marbles does Joe have?

Missing second parta

Joe and Tom have 8 marbles altogether.Joe has 3 marbles.How many marbles does Tom have?

Subtraction situationsChange-Get-Less

Missing end

Joe had 8 marbles.Then he gave 5 marbles to To m.How many marbles does Joe have now?

Missing changeJoe had 8 marbles.Then he gave some marbles to Tom .Now Joe has 3 marbles.How many marbles did he give to Tom?

Missing start*Joe had some m arbles.Then he gave 5 marbles to To m.Now Joe has 3 marbles.How m any m arbles did Joe have in the beginning?

CompareMissing difference

Joe has 8 marbles.

Tom has 5 marbles.How many more marbles does Joe have than Tom?Missing big3

Joe has 3 marbles.Tom has 5 more marbles than Joe.How many marbles does Tom have?

Missing smallJoe has 8 m arbles.He has 5 more marbles than Tom.How many marbles does Tom have?

Note. These examples are adopted from Riley, Greeno, and Heller's (1983) example problems. Compare problems can also be asked w ith thewords "less" and "fewer" rather than "mo re," and w ith Equalize questions containing either "less"/"fewer" or "mo re" (e.g., "How m any m oremarbles does Tom have to get to have as many as Joe?").a In these problems, there exists a conflict between the overall category situation as additive or subtractive and the operation required to solvea particular problem subtype.

Table 1 were chosen to be comprehensible to the second-grade children participating in this study.

Because all problems involve three quantities and any ofthese quantities can be unknown, there are three possibleproblem subtypes within each main problem type (see Table1). Two of these require subtraction of the two given numbersin the problem and one requires addition of the two givens.Some of the problems created by particular unknowns maybe especially difficult because ofthe internal conflict betweenthe basic semantic structure of the underlying situation asadditive or subtractive and the solution strategy (addition orsubtraction) required to solve a given problem with a partic-

ular kind of unkn own (DeCorte & Verschaffel, 1985a; Greer,in press; Vergnaud, 1982; Willis& Fuson, 1985). For example,in a Change-Get-More situation, a missing Start problem hasthe additive Change-Get-More underlying structure, but onecan solve it by subtracting the change number from the endnumber. The conflict problems are footnoted in Table 1.Conflict problem s have been found to be particularly difficultfor children (see Carpenter & Moser, 1983, and Riley et al.,1983, for reviews). Therefore, of particular interest waswhether the schematic drawings would improve performanceon conflict problems.

Three different kinds of schematic drawings were used inthe word problem instruction (see Figure 1). The Change-Get-More and Change-Get-Less problem structures are simi-lar; both problem types involve an initial state, a change, and

a final state. Therefore the same basic drawing was used forthese two types, and the different nature of the change wasrepresented by a plus or minus symbol inserted by the child(see Figure 1). The Put-Together drawing was the part-part-whole drawing used in theDeveloping Mathematical Proc-esses series (Romberg, Harvey, Moser,& Montgomery, 1974).However, the qu antities were labeledpart, part, and all (ratherthan whole) because often there is no actual "whole" thatcomprises the two given sets in a Put-Together problem. TheCompare drawing contained big and small quantities placedadjacent to one another to facilitate their comparison; thedifference was enclosed by a broken line because tha t differ-

ence is not actually a physically sep arate entity in a Com parestructure. The three quantities in each problem were givenmnemonic labels (see Figure 1). Children were to write theletter for each label on the known or unknown quantity inthe problem and then enter the known quantities into theircorrect place in the drawing. The drawing could then be usedto determine the correct solution strategy. The choice ofsolution procedure was aided by the presence of the drawingeither through the relations among the physical sizes of thedifferent parts or by the temporal ordering implied by thedrawing.

The ultimate question, of course, is whether teaching sche-matic drawings improves children's ability to solve wordproblems. The prior question is whether children can evenlearn this method. Because of the complexity ofthis method,


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TEACHING SCHEMATIC DRAWINGS 19 5

algorithms for solving multidigit addition and subtractionproblems (Fuson, 1986a). The children used finger-patterncounting procedures for performing single-digit computationswhen an addition or subtraction fact was unknown (Fuson,1986b; Fuson & Secada, 1986; Fuson & Willis, in press); they

were taught to count up rather than down for subtractionproblems (e.g., 1 4 - 8 was solved by counting up from 8 to14). These children were in general quite competent at single-digit and m ultidigit comp utation.

Method

Subjects

Subjects were second-grade children from two public schools in asmall city near Chicago. The schools in this city enroll children ofwidely varying socioeconomic status. One of the classes contained 24students and was categorized by the teachers as containing children

with high math ability (Class HA). The other class contained 19children categorized by teachers as children of average math ability(Class AA).

Procedure

Use of theschematic drawings. The categories of word problem sused in the study appear in Table1 and the schematic drawings usedfor each category of word problems appear in Figure 1; both werediscussed in the introduction. Children were first introduced to ageneral category of word problems and shown the drawing for thatcategory. For any given problem, they were taught first to write onthe problem the letter for the verbal labels naming the three elementsin the story (the two givens and the unknown). The particular labels

in Figure 1 were substituted for Riley et al.'s (1983) labels becausethey were more ap propriate for second graders.After applying the labels, the children were taught to make the

schematic drawing that is appropriate for that general story category.The drawing contained three parts, one corresponding to each of thelabeled story elements. The two given num bers were then written inthe appropriate parts of the drawing; these locations were determinedby the labels attached to the numbers. Last, the children chose thecorrect arithmetic solution procedure by determining how to obtainthe unknown (either by adding or subtracting), given the two filledparts. The drawing facilitates this choice either by the relative sizes ofthe sections of the drawing (for Put-Together an d Com pare problems,"small + small = big" and "big small = sma ll") or by the temporalordering of events (for C hange-Get-More an d C hange-Get-Less prob-lems, "some + some more = a larger result" and "some some of

them = a smaller result").An ex amp leof how this series ofsteps is applied to a Put-Together

problem is provided in the first part ofFigure 1. The child identifiesthe problem a s a type from the Put-Together category and applies theappropriate verbal labels(part, part, an d all) to the word problem,writing P, P, and A by the appropriate words. The first part(P ) isrepresented by Jon, the second and unknown(P ) by Bill, and the all(A ) by the given altogether. The next step is to produce the drawing,and the following step is to fill in the given elements. The child thenanalyzes the relations between th etwo known num bers in the drawingand arrives at the subtraction solution to obtain the unknown (thenumbe r tha t is to fill the empty box). Thus the physically represented"part + part = all" structure is potentially capable of inducing theproper solution strategy as long as the child has filled in the sectionsof the drawing correctly. Examples of the use of the labels and

drawings for typical Change and Compare stories are also providedin Figure 1.

Worksheets and tests, Practice worksheets were constructed bothfor student practice and to provide data on problem difficulty. Pureworksheets were those containing problems from a single semanticcategory. The Put-Together, Change-Get-More, and Change-Get-Lessworksheets each contained 18 problems equally distributed acrosseach of the three missing positions. The Compare worksheets con-tained 27 problems distributed equally across the three missing posi-tions and across problems in which the wordsmore and fewerappeared (e.g., "How many fewer problems does Joe have thanTom?"). The mixed worksheets contained 24 intermixed problemsequally distributed across the four semantic categories and across thethree missing positions for each type. The equal distribution acrossproblem variables permitted analyses in which we compared therela tive difficulty of different p roblem types and different missingpositions. However, it was not an instructionally maximal treatmen t,which would have concentrated on problems with which childrenwere having the most difficulty.

Th e 10 most difficult problem subtypes were selected for inclusion

in 10-problem pretests and posttests given to children before and afterthe teaching. These tests omitted the simple "Put-Together: missingall" and the "Change-Get-More: missing end" problems because wewere concerned about fatigue and time limitations that would haveoccurred with a longer test. The easiest problems were omittedbecause we were concerned about ceiling effects on these problems.In all of the Compare problems, the wordmore, rather than fewer,was used. Problems were ordered so that no two problems from theidentical semantic category (Put-Together, Change, Compare) wereplaced sequentially. To control for problem ordering effects, weconstructed two forms in which problems were listed in the reverseorder. All numbers in problems were of three digits, and one trade(carrying or borrowing) was required to obtain the sum or difference.Children were given the test initially as a pretest and allowed to solveproblems in any way. When the test was given as a posttest, children

were told to make a schematic drawing for each problem. Eachproblem was then evaluated with respect to whether the drawingmade was of the correct problem category, the numbers were filledinto the drawing correctly, the correct solution strategy was chosen,and the solution strategy was carried out correctly.

Organization of the instruction. Teaching was divided into units;one major problem category was taught within a unit. The order ofteaching was Put-Together, Change-Get-More, Change-Get-Less,Compare, and mixed problem categories. The first unit taught toClass HA focused first on small-number problems (sums and differ-ences to 18), and small-number worksheets were completed beforelarge-number problems were introduced. This w as done because wewere afraid that beginning with the large numbers would be toodifficult. For the first unit, children were also instructed to write anumber sentence that reflected the seman tic structure ofthe problem(e.g., 6 + = 14 or + 342 = 629) before solution. Itbecame evident, however, that the writing of the number sentenceproblem equation was a burdensome and much-resisted extra step atboth number sizes and that children tended to write the equationafter solution or else omitted it entirely. Therefore, this step wasbypassed in further units.

All teaching was done by one of the investigators. In one mathperiod, story problems of a particular category were presented, theclassification and labeling were described, and the complete solutionfor different variants ofthe problem was illustrated by the instructor.Then the children spent between 2 and4 days completing the practiceworksheets for that category. Children received little systematic indi-vidualized feedback on the worksheets because these were kept forpurposes of analysis. As much as possible, particular difficulties thatchildren had were addressed initially as the instructor aided them


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Put-Together drawings made for Compare problems and viceversa were scored as correct, selection of a correct drawingrose above 90% for these two categories. The substitutions ofthese two types for each other were symmetric for this class,suggesting a more general confusion between these similar

drawings. Correct drawings were made for Change-Get-Lessproblem s markedly less often th an for the other three problemcategories; Put-Together drawings were often made for theseproblems.

Filling in the D rawing

Drawings were scored as correctly filled in if the correctsemantic type was drawn and the numbers were placed cor-rectly within the drawings, with the exception that substitu-tions of Compare and Put-Together drawings were allowedbecause of their considerable similarity. Percentages of correct

drawing fill-in are given in T able 2. Fo r Class HA, the overallaverage correct fill-in was66 % on the worksheets. This roseto 82% correct on the posttest, indicating considerable learn-ing from the worksheet practice and the review. On bothworksheets and the posttest, performance on the Compareproblems was somewhat lower than for the other three prob-lem categories. The results for Class AA posttests generallyparalleled those of Class HA, but were overall about 15%lower. It is not clear (a) how much of the decrement in thisclass in com parison with the firstis due to difference in abilityand (b) how much was produced by the less-than-adequatelearning and review tim e.

Overall Adequate Problem Representation

A second analysis of problem representation consideredonly whether the drawing produced by the child adequatelyrepresented the relationship among the problem elements,independently of the drawing initially made by the child.Thus if one would select the correct solution strategy from afilled-in drawing, the drawing was creditedas a correct overallrepresentation (see Table 2). This analysis was done becauseone can assimilate a given problem into a different problemstructure while still remaining faithful to the relationships inthe story; for example, one can think of a Change-Get-Moreproblem as a Put-Together problem in which the startingquantity and the change are the "parts" and the end quantityis the "all." We wished to ensure that children could have thisfreedom of individual interp retation; th e use of drawings wasintended to be facilitative, not prescriptive. On the posttest,only one problem category had more than three instances ofa correct overall representation tha t differed from the p roblemcategory.2 For Class AA there were six instances of a Put-Together drawing filled in correctly for a Change-Get-Lessproblem; these instances were distributed across all threemissing positions. Though the nu mb er of correctly filled-innoncanonical drawings was fairly small, such drawings didraise the percentages of correct overall representation to 79,87 , and 78 for the HA mixed worksheets, the HA posttest,

and the AA posttest, respectively.

Analysis of the Class HA worksheets by missing positionindicated that four problem subtypes were particularly diffi-cult for children to represent in this initial learning period.The "Change-Get-More: missing end," "Change-Get-Less:missing start," "Comp are: missing big," and "Com pare: miss-

ing small" problems had more than twice as many incorrectoverall representations as did the other subtypes within thesemain types. On the HA posttest, "Com pare: missing big" and"Com pare: missing small"were the m ost difficult to represent;"Compare: missing difference," "Change-Get-More: missingstart," and "Change-Get-Less: missing start" were somewhatmore difficult than the remainder of the problem subtypes.On th e AA posttest, differences in represen tation were spreadfairly equally across missing positions and main categories,though "Compare: missing big" and "Compare: missing dif-ference" were the problem subtypes represented least well.Thus there was some difficulty in representing all of theCompare missing positions, but the "missing big" problem

was particularly difficult to represent.

Use of Correct Solution Strategy

A solution strategywas scored as addition if the answer wasgreater than the two numbers given in the problem; it wasscored as subtraction if the answer was less than the largernumber. The percentages of problems on which childrenchose the correct solution strategy are given in Table 2. Onthe posttests both classes used co rrect solution strategies m orefrequently on the additive Put-Together and Change-Get-More situations than on the subtractive Change-Get-Less andCompare situations, even though the easiest problems in eachof the ad ditive categories were om itted from the po sttest. Thisdifference b etween overall additive an d subtractive categorieswas greater in Class AA than in Class HA. Overall, Class HAhad an impressively high correct choice of solution strategy(89%), and Class AA had a respectable 77%.

The pure worksheets completed by Class HA permitted ananalysis by missing position within each main category w ith-out the distractions of other problem types. Put-Together andChange-Get-More problems showed no effect of missing po-sition on correct strategy. Analysis of two pure Change-Get-Less worksheets revealed significant effects of missing positionon performance for both, Fs(2, 46) = 15.17(M SE = 0.15) and4.50 {M SE - 0.24), ps < .02, respectively; strategy scores weremuch lower when the missing element was the first one (the

start quantity). Averaged across the two worksheets, the meanperformance levels were63%, 94%, and 90% correct for thestart, change, and end qu antities, respectively. Com pare work-sheets also showed a significant effect of missing position,Fs(2, 42) = 8.30(M SB = 0.21) and 5.77 (M SB = 0.19), ps

journal of educational psychology 1988, vol. 80,

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