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Correspondences and Numerical Differences between Disjoint SetsAuthor(s): Tom HudsonReviewed work(s):Source: Child Development, Vol. 54, No. 1 (Feb., 1983), pp. 84-90Published by: Blackwell Publishing on behalf of the Society for Research in Child DevelopmentStable URL: http://www.jstor.org/stable/1129864 .

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Correspondences and Numerical Differencesbetween Disjoint SetsTom HudsonUniversityof Pittsburgh

HUDSON, TOM. Correspondences nd Numerical Differencesbetween Disjoint Sets. CHILD DE-VELOPMENT,983, 54, 84-90. Young children'sunderstandingof correspondencesand nu-mericaldifferencesbetween disjointsets was studied in a series of 3 experiments.In the first2 experiments,64 children between 4 and 8 years of age were shown pairs of sets and wereaskedboth standard("How manymorebirds than wormsare there?")and nonstandard "Howmanybirdswon'tget a worm?")numericaldifferencequestions.The children'sobservedsuccessin answering he Won'tGet questionsindicatesthat many young childrenare skillful at estab-lishingcorrespondences nd determiningexact numericaldifferencesbetween disjointsets; theirpoor performanceon the standardquestions apparentlyreflects a misinterpretation r inade-quate comprehensionof comparativeconstructionsof the generalform "How many . . . [com-parative term] . . than . .. ?" The final experiment,involving 30 additionalkindergartenchildren,dealt with children'ssolutionstrategiesin answeringWon't Get questions.The mostfrequentlyobservedsolutionstrategywas a sophisticated ndirectcountingstrategyratherthana perceptuallyguided pairing strategy.Takentogether,the presentfindingsrestrictthe domainof applicabilityof the theory that young children are limited to perceptuallybased forms ofmathematicalreasoning.

A number of young primary-grade chil-dren perform poorly when responding to nu-merical difference questions of the form "Howmany more ---- than . . . are there?" (seeGibb 1956; Riley, Greeno, & Heller 1982).Instead of stating the correct numerical differ-ence, the unsuccessful children often respondby simply stating the size of the larger set.One potential explanation of the children'spoor performance is that, consistent with Pia-get (1965), the children may be unable toestablish suitable one-to-one correspondencesbetween the given sets. An alternative expla-nation is that, although the children can estab-lish correspondences and determine numericaldifferences between disjoint sets, they do not

do so because they misinterpret the "Howmany more - than ... ?" construction.To investigate these alternative explana-tions, the first of three experiments we under-

took employed two different question formatsconcerning numerical differences between dis-joint sets. The first format-"How many more-than . . . ?"-is the wording commonlyused in psychometric mathematics achievementtests and in elementary school mathematicstextbooks. The second format-"How manywon't get a . . . ?"-was specificallydevised for use in this study in an attempt tocircumvent potential linguistic difficulties asso-ciated with the "How many morethan . . . ?" construction.This paper is based on my dissertationresearchat IndianaUniversityand on subsequentresearchconducted at the Universityof Pittsburgh.I wish to extend special appreciation oR. N. Aslin, who directedmy dissertationresearch,and to L. B. Smith, who provided addi-tional guidancein the completionof that researchand the preparation f this paper.Thoughtfulcommentsby the reviewersof the manuscriptwere especially valuable.Additionalthanks aredue to the children,parents,teachers,and administrators f the Monroe County CommunitySchools.Preparationof this article and portionsof the researchreported n it were supportedby funds from the Learning Researchand DevelopmentCenter, supportedin part by fundsfrom the National Instituteof Education.The opinionsexpressedherein do not necessarilyre-flect the positionor policy of NIE, and no officialendorsementshould be inferred.The thirdstudy reported n the articlewas previouslyreportedat the biennial meeting of the Society forResearchin Child Development,Boston, April 1981. Schematicdiagramsof the stimuli maybe obtained from Tom Hudson, Departmentof MathematicalSciences, ManchesterCollege,

North Manchester, ndiana 46962.[Child Development, 1983, 54, 84-90. @ 1983 by the Society for Research in Child Development, Inc.All rights reserved. 0009-3920/83/5401-0010501.00]

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Tom Hudson 85Experiment 1Method

Subjects.-The subjects were 28 first-grade children from a rural Midwestern ele-mentary school. The mean age of the childrenwas 7-0 (range: 6-6 to 7-8).

Materials.-Two series of eight 30 X 13-cm illustrations were drawn. Each drawingshowed two sets of items whose numerical dif-ference was either one, two, or three. Tworandom orders of the eight number pairs--3:2, 4:3, 5:4, 5:4, 3:1, 5:3, 4:1, and 5:2-were used to form the two sequences of setsizes needed for the two series of drawings.The items in the two sets within each drawingwere positioned so as not to form an obviousvisual pairing of the elements in the two sets(see fig. 1); the larger of the two sets wasalways on the left. The following eight pairsof items were used in two different orders forthe two series of drawings: squirrels and nuts,kids and bikes, bugs and leaves, people andhats, birds and worms, butterflies and flowers,dogs and bones, people and cookies.

Procedure.-Each child was tested indi-vidually on both the More and Won't Get tasks,separated by a short break. Half of the chil-dren received the More task first; half receivedthe Won't Get task first.

In the More task, each child was testedusing one of the two series of eight drawingsdescribed above. As each drawing was pre-sented, the experimenter said, for example,"Here are some birds and here are someworms. How many more birds than worms arethere?" In the Won't Get task, each child wastested using whichever series of drawings heor she did not receive during the More task.

0 0

* *0 0

FIG.1.-Examples of the spatial arrangementof pairsof sets in the More and Won'tGet tasks ofexperiments1 and 2 (upper drawings); examplesof the spatial arrangementof pairs of sets in theNumericalDifferencetask of experiment3 (lowerdrawings).

Except for the questions asked, the procedurewas identical to that used in the More task.The wording of the questions in the Won't Gettask was as follows: "Here are some birds andhere are some worms. Suppose the birds allrace over, and each one tries to get a worm.Will every bird get a worm? . . How manybirds won't get a worm?" In either task if thechild responded nonverbally by pointing to oneor more individual items, the appropriate"Howmany... ?" question was repeated.Scoring.-A comparison involving fivebirds and four worms will be used here toillustrate the scoring procedure. Within eachtask a child's response to a drawing was scoredas correct if the child answered one, one more,

one won't, four will and one won't, and soforth; as absolute if the child answered five,five birds, five birds and four worms, morebirds, the birds (signified verbally or by point-ing), and so forth; and as a processing errorif the child made any other response (such asa counting error).Results and DiscussionIn each of the two tasks, a child was saidto respond correctly on a given task if six ormore of the child's eight responses were cor-rect. In the Won't Get task, 100%of the childrenresponded correctly. Thus it appears that allof the first-grade children were able to usetheir knowledge of correspondence to deter-mine exact numerical differences between dis-joint sets. In contrast, only 64%of these samechildren responded correctly to the More task,which used drawings of the same sort usedin the Won't Get task. The contrast in perfor-mance on the two tasks was significant by asign test, p < .001. The order of presentationof the two tasks did not affect the children'sperformance; the above percentages were iden-tical for the two orders. (The contrast in per-formance on the two tasks remains significant,p < .001, when perfect performance is used asthe success criterion; the percentages of chil-dren responding perfectly to the two tasks were79% and 39%, respectively.) Since the youngchildren displayed an ability to establish corre-spondences between the nonaligned sets in theWon't Get task, their failure to do so in theMore task apparently involved linguistic diffi-culties.

Consistent with this view, children's errorsin the More task were not minor deviationsfrom the exact numerical differences, as mighthave been predicted if the children's poor per-

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86 Child Developmentformance had arisen from a lack of techni-cal accuracy in establishing correspondences.Rather, all but two of the 80 responses of the10 children who failed the More task wereabsolute responses; seven of the 10 childrenalways stated the size of the larger set whengiving an absolute response, and the three re-maining children most often stated the sizes ofboth sets. Thus it appears that many of theunsuccessful children interpreted the givencomparative questions to be requests for sim-ple enumerations of the displayed sets.

Experiment 2The first experiment provided evidencethat first-grade children possess a nontrivialknowledge of correspondences and numericaldifferences between disjoint sets and that chil-dren's incorrect responses to standard "Howmany more - than . . . ?" questions mayreflect systematic misinterpretations of thosequestions. A second experiment was conductedin order to determine two matters: (1) whether

younger children display a similar pattern ofsuccess in determining numerical differencesand failure in answering the standard numer-ical difference questions; and (2) whetherchildren's difficulty with "How many morethan . . . ?" questions is a special caseof a general linguistic difficulty with "Howmany ... [comparative term] ... than ... ?"constructions.Method

Subjects.-The subjects were 12 nurseryschool children and 24 kindergarten childrenfrom a middle-class elementary school. Meanages at the two grade levels were 4-9 (range:4-3 to 5-9) and 6-3 (range: 5-9 to 6-6).Materials.-The comparative-terms task

consisted of two subtasks: (1) sets displayedand (2) sets not displayed. In the display sub-task each drawing consisted of two verticalstacks or two horizontal rows of 2.5-cm squareblocks. Within each stack or row there wasno separation between the blocks; the between-row separation was .5 cm. The bottom end ofthe stacks and the right-hand end of the rowswere aligned in order to visually highlight ap-propriate one-to-one correspondences betweenthe displayed sets. There were four drawingsof horizontal pairs of rows and four drawingsof vertical pairs; the set sizes of the four pairswere 5:4, 6:5, 5:3, and 6:4. Within each row,all blocks were the same color--either red,

green, yellow, or orange. Each block was out-lined in black.In the nondisplay subtask, no sets weredisplayed. Instead, each drawing depicted a

pair of numerals-3:2, 4:3, 5:3, or 6:4. Thelarger numeral appeared in the upper left-handcorner of the drawing and the smaller one inthe upper right-hand corner. Below each nu-meral was either a hand-drawn face, a blankspace, or a sketch of an everyday object--de-pending on the question being asked.

Procedure.-Each child was tested indi-vidually in the spring of the school year onthree tasks-the More task, the Won't Get task,and the Comparative-Termstask. The order ofthe three tasks was counterbalanced acrosssubjects. The More and Won't Get tasks wereadministered according to the same proceduresused in the first experiment.

In the display subtask of the Comparative-Terms task, each child was asked four questionsof the form "How many more red blocks thangreen blocks are there?"; two questions of theform "How many blocks taller is the red stackthan the green stack?"; and two questions ofthe form "How many blocks longer is the redrow than the green row?"The more and taller/longer questions were presented alternately.The numerical differences (either one or two)were equally distributed across the more andtaller/longer questions.

In the nondisplay subtask, each child wasasked two questions of each of the followingfour types: "Jeff is 5 years old, and Sue is 3.How many years-older is Jeff than Sue?";"Howmany more is 3 than 2?"; "The shirt costs 4cents, and the hat costs 3 cents. How muchmore does the shirt cost than the hat?"; and"I have six boats and four cars. How manymore boats than cars do I have?" The first fourdrawings presented the four question types inrandom order; the second four drawings repeat-ed that sequence of situations, but they alteredthe order of the four pairs of digits-3:2, 4:3,5:3, 6:4-so that each situation would be pre-sented using digit pairs differing by both oneand two. The names Jeff and Sue were re-placed by Alan and Betty during the secondpresentation of the age situation; the settingsof the two question types involving money andsets were also varied. In the Comparative-Termstask, questions from the two subtasks werepresented alternately.

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Tom Hudson 87Scoring.-Children's responses to individ-ual items in each task were scored as in thefirst experiment. Thus, in the Comparative-Terms task, the category of absolute responsesincluded such responses as "five blocks," "thered blocks," "Jeff is older," and so forth.

Results and DiscussionChildren's performance on the More andWon't Get tasks-As in the first experiment,a child was said to respond correctly on eithergiven task if the child gave at least six correctresponses on that task. The percentages ofchildren responding correctly to the Won't Gettask at the nursery school and kindergartenlevels were 83% and 96%,respectively. In con-trast, the percentages of children respondingcorrectly to the More task were 17%and 25%.Indeed, every child who responded correctlyto the More task also responded correctly tothe Won't Get task. Thus, since 25 (69%)of thechildren passed the Won't Get task withoutpasing the More task, whereas no childrenperformed in the reverse manner, the Moretask was significantly more difficult for thegroup of children, sign test, p < .001.

The 28 children who failed the More taskwere quite systematic in the types of incorrectresponses they gave: 68% of these childrenalways gave absolute responses in that task,and an additional 18%gave all absolute re-sponses except for a single counting error; 10of the 28 children always stated the size ofthe larger set when giving an absolute re-sponse, and eight additional children alwaysstated the sizes of both sets.

Children's performance on the Compara-tive-Terms task.-It was found that childrenperformed poorly on "How many . . . [com-parative term] . . . than . .. ?" questions bothwhen dimensional comparative terms such as"taller"were employed and when the generalcomparative term "more" was employed. Thepercentages of correct responses for the variouscomparative adjectives were 26%("more") and27% ("taller"/"longer") in the display subtask,and 28% ("more") and 29% ("older") in thenondisplay subtask. Thus, while the variety ofmeanings of the general comparative termmore" may be the source of young children'sdifficulty in certain numerical reasoning tasks(e.g., "more"can mean additional or added-see Brush [1976]; Gelman & Gallistel [1978,p. 227]), the special ambiguity of "more"rela-tive to other comparative terms does not ap-pear to be the essential source of young chil-

dren's difficulty with "How many morethan... ?"questions.From the display subtask data, it can beseen that children responded incorrectly to

"How many more than . . . ?" ques-tions even when appropriate one-to-one corre-spondences were visually highlighted. As inthe More task, a high percentage of children'stotal responses were absolute-63% ("more")and 62% ("taller"/"longer") in the displaysubtask, and 57% ("more") and 59% ("older")in the nondisplay subtask. Since the corre-spondence skills needed to solve this subtaskwere minimal, these data provide additionalsupport for the view that young children's in-correct responses to "How many morethan . . . ?" questions are not the result of alack of technical skill in establishing corre-spondences between sets.Experiment 3

The second experiment provided evidencethat even prior to first grade many young chil-dren possess some knowledge of correspon-dences and numerical differences between dis-joint sets. The depth of that knowledge cannotbe determined, however, from those data. Onthe one hand, it appears that the children'sunderstanding is well established in that thechildren were able to determine the exact nu-merical differences without any training, feed-back, or visual cues. On the other hand, itmight be argued, success in the Won't Get taskdoes not require a deep level of mathematicalunderstanding; the children could have ob-tained the exact numerical differences by mim-icking by rote the actions described by theproblem context-"Suppose the birds all raceover, and each one tries to get a worm." Inorder to determine more fully the level of chil-dren's understanding of correspondences andnumerical differences, a third experiment wascarried out that permitted a detailed analysisof children's strategies for establishing corre-spondences between disjoint sets.Method

Subjects.-The participants were 30 first-semester kindergarten children from a middle-class elementary school. The mean age of thechildren was 5-3 (range: 4-8 to 5-10).Materials and procedure.-Each child was

administered a pretest, a Conservation task, anda Numerical Difference task. Half of the chil-dren received the Conservation task first, and

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88 Child Developmenthalf received the Numerical Difference task first.The pretest involved the color names red,white, and blue and the quantitative terms"more" and "same number." The two quanti-tative-terms questions were of the form "Arethere more red chips, more white chips, or arethere the same number of red and white chips?"and involved comparisons of one red chip versusthree white chips and two white chips versustwo blue chips.

The Conservation task employed two con-servation trials-seven versus seven and sixversus six. The following procedure was em-ployed. The interviewer put down seven pairsof chips, 4 cm in diameter, to form two paral-lel rows of red chips (far row) and blue chips(near row). The between-row and within-rowseparations were 4 cm and 1 cm, respectively.The child was asked, "Arethere more red chips,more blue chips, or are there the same number ofred chips and blue chips?" The far row wasthen uniformly extended so that the far rowappeared to have an extra chip at each end.The initial conservation question was then re-peated. The second conservation trial, six ver-sus six, was like the first except that the colorsred and blue were reversed in the physicaldisplay and in the conservation questions.

The Numerical Difference task employeda series of nine drawings. Each drawing dis-played a pair of sets that were arranged so asnot to form an obvious visual pairing of theelements in the two sets. The larger of the twosets was always on the left and vertical; thesmaller of the two sets was on the right andhorizontal. In order to ensure further thatlength would not be a sufficient cue for suc-cessful performance, the interitem distancesdiffered for the two sets in each drawing. Foreach pair of sets, the two set sizes differed byeither one, two, or three. The nine pairs of setsizes (in order of presentation) were 3:2, 5:3,5:2, 6:4, 6:5, 7:4, 8:6, 9:7, and 7:6; the useof large sets in the present experiment was in-tended to encourage the children to use overtsolution strategies. The types of items in thenine pairs of sets were birds and worms, dogsand bones, butterflies and flowers, and so forth.Each drawing was accompanied by the samemode of questioning used in the Won't Get taskin the two previous experiments. Incorrect re-sponses were not corrected; correct responsesto the first one or two drawings were positivelyreinforced if the child seemed unsure of thecorrectness of his or her responses. At some

point during the interview, since the childrenwere often reluctant to touch the pictures orto count aloud, the interviewer pointed to theitem in the lower right-hand corner of one ofthe drawings and told the child, "It's okay totouch the pictures to help you figure out theanswer." This statement typically accompaniedthat drawing for which the child first seemedto hesitate in making a response. The verbalprompt was worded in a way that was in-tended not to bias the children toward the useof a counting strategy.Scoring.-In the Numerical Difference task,each of the children's nine responses wasscored both in terms of the accuracy of thatresponse (correct or incorrect) and in terms of

the observed solution strategy, if any. A re-sponse was scored as being correct if the childstated (or pointed to) the exact numerical dif-ference between the given sets. Each observedstrategy was classified as being a pairing strat-egy, a counting strategy, or a covering strategy.A pairing strategy was one in which the childdrew imaginary lines between correspondingitems in the two sets. The counting strategieswere of two types-counting out an equivalentsubset and counting whole sets. To be consid-ered a counting-out-an-equivalent-subset strat-egy, the strategy had to involve counting outa subset of the larger set that was numericallyequivalent to the smaller set. An example ofcounting out an equivalent subset was to countthe smaller set, count off the same number ofelements in the larger set, and then state thesize of the remaining difference set. For sim-plicity, the strategy just described will be de-noted by SL', where S is the smaller set, Lis the larger set, and L' is a subset of L nu-merically equivalent to S. Other examples ofcounting out an equivalent subset were LSL',L', SLSL', and LSSL'. In the counting-whole-sets strategy, the child counted all of the ele-ments in both of the given sets. Thus, examplesof counting whole sets were SL and LS. Final-ly, a covering strategy involved covering asubset of the larger set--either one that wasequivalent to the smaller set or one that wasequal in size to the exact numerical difference.Results and DiscussionOf the 270 responses to the NumericalDifference task, 218 (81%) were correct. The218 correct responses included 97 responsesfor which a solution strategy was observed.These 97 observations are the focus of thepresent analysis.

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Tom Hudson 89Only 22 of these 97 observations could beclassified as being a pairing strategy. The pre-dominant observed strategy for establishing aone-to-one correspondence between the givensets was counting out an equivalent subset,which encompassed 57 of the 97 observations.The remaining observations included 10 in-stances of counting whole sets, seven instancesof a covering strategy, and one difficult-to-clas-sify mixed strategy in which the child counted"1, 2" in the smaller set, "1, 2" in the largerset, "3, 4" in the smaller set, then "3, 4" in the

larger set. The specific frequencies of the ob-served counting strategies were SL' (37 ob-served instances), LSL' (12), L' (5), SLSL'(2), LSSL' (1), SL (7), LS (3).An analysis in terms of individual childrenlikewise indicates the dominance of countingout an equivalent subset. Of the 30 childrenin the study, 20 used an observable strategyin at least one of their correctly solved prob-lems. Of these 20 children, 80%were observedto use counting out an equivalent subset in thecorrect solution of a problem; only 25%wereobserved to use a pairing strategy in a correctlysolved problem. Moreover, posttest questioningindicated that counting strategies were beingused by several of the children who had usedno observable solution strategies during theNumerical Difference task.The children's use of sophisticated solu-tion strategies cannot be attributed to a generalmathematical precociousness on the part of thechildren since 477 of the 30 kindergarten chil-dren failed both trials of the Conservation task.

Indeed, among the 16 children who were iden-tified as correctly using counting out an equiv-alent subset, 56% failed both trials. (The Con-servation task was a conservative one forpresent purposes; it potentially overestimatedchildren's conservation level since the last re-sponse within each conservation question wasthe correct one [see Siegel & Goldstein 1969].)

This study provides evidence that manyyoung children know that a one-to-one corre-spondence between two sets necessarily existsif the two sets can be counted out to the samenumber. The design of previous studies investi-gating children's number knowledge do notappear to permit this conclusion. For example,Gelman and Gallistel (1978) show that a child,by counting out each of two sets to the num-ber "three," can classify each of the two setsas being a "winner"-that is, as having threeelements. That the young child also recognizes

that a one-to-one correspondence between theitems in the two sets necessarily exists is nota permissible inference (see Fuson 1979). Incontrast to the Won't Get questions of the pres-ent study, which specifically refer to the pair-ing relation of "getting," the questions used inthe Gelman study do not make any referenceto a pairing or correspondence of the items inthe two sets presented to the child.General Discussion

The evidence of the present series of ex-periments suggests that children's previouslyreported difficulty with "How many more--- than . .. ?" questions does not involvea lack of appropriate correspondence skills, butinstead involves a misinterpretation of com-parative constructions of the general form"How many . . . [comparative term] . . . thanS.?" irst, children's success in answering"How many - won't get a . .. . ?" ques-tions demonstrated that young children typi-cally possess the requisite correspondence skills.Second, children responded incorrectly to "Howmany more -- than ... ?" questions evenwhen the given sets were block rows placedside by side so that appropriate one-to-one cor-respondences were visually underscored. Third,

children's performance was also poor when the"How many more than . . ?" ques-tions were replaced by comparative questionsinvolving "taller," "longer," and older. Thusa linguistic factor-childrens limited compre-hension of the comparative construction "Howmany ... [comparative term] ... than ... ?"-may account for young children's apparent lackof quantitative reasoning ability when asked tofind how many more items are in one set thananother. This interpretation is consistent withrecent evidence indicating that the range ofcognitive abilities elicited by cognitive-assess-ment tasks can be significantly affected by thelanguage employed by those tasks (Donaldson1979; Gelman & Gallistel 1978; Siegel 1978).

The third experiment provides persuasiveevidence that young children's understandingof correspondences and numerical differencescannot be viewed as consisting merely of per-ceptually driven rote procedures. When estab-lishing one-to-one correspondences betweendisjoint sets, many young children do not usedirect pairing strategies; instead they use so-phisticated counting strategies that reflect theknowledge that a one-to-one correspondence

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90 Child Developmentbetween two sets necessarily exists if the twosets can be counted out to the same number.Thus, young children's knowledge of numberand correspondence can undergo a sophisti-cated elaboration on an abstract level evenprior to success on Number Conservation tasks.These findings appear to restrict the domain ofapplicability of the theory that young childrenare limited to perceptually based forms ofmathematical reasoning (Piaget 1965).ReferencesBrush,L. R. Children'smeaningsof "more."Jour-nal of Child Language,1976, 3, 287-289.Donaldson,M. Children'sminds. New York:Nor-

ton, 1979.Fuson, K. C. Review of The child's understandingof number by R. Gelman & C. R. Gallistel.Journal for Research in MathematicsEduca-tion, 1979, 10, 383-387.Gelman, R., & Gallistel, C. R. The child's under-

standing of number.Cambridge,Mass.: Har-vard UniversityPress, 1978.Gibb, E. G. Children'sthinking in the process ofsubtraction.Journal of ExperimentalEduca-tion, 1956, 25, 71-80.Piaget, J. The child's conceptionof number.NewYork:Norton,1965.Riley, M. S.; Greeno,J. G.; &Heller,J. I. The de-velopment of children'sproblemsolving abil-ity in arithmetic.In H. P. Ginsburg (Ed.),The development of mathematical thinking.New York: Academic Press, 1982.Siegel, L. S. The relationshipof language andthought in the preoperational hild: a recon-siderationof nonverbalalternatives o Piaget-ian tasks. In C. J. Brainerd & L. S. Siegel(Eds.), Alternativesto Piaget: critical essays

on the theory. New York: Academic Press,1978.Siegel, L. S., & Goldstein,A. G. Conservationofnumberin young children:recencyversus re-lational response strategies. DevelopmentalPsychology, 1969, 1, 128-130.