Part-whole number knowledge in preschool children

Journal of Mathematical Behavior22 (2003) 217–235

Part-whole number knowledge in preschool children�

Robert P. Hunting

Department of Mathematics and Science Education, 323C Austin, East Carolina University,East Fifth Street, Greenville, NC 27858-4353, USA

Abstract

Ability to reflect on a number as an object of thought, and to isolate its constituent parts, is basic to a deepknowledge of arithmetic, as well as much practical and applied mathematical problem solving. Part-whole reasoningand counting are closely related in children’s numerical development. The mathematical behavior of young childrenin part-whole problem settings was examined by using a dynamic problem situation, in which a small set of items waspartitioned such that one of the subsets remained perceptually inaccessible. Issues addressed include the problemsolving strategies successful children used, adaptations children make in response to successive administrations ofthe task over time, and characterizations of children’s mathematical thinking based on their responses to the task.© 2003 Elsevier Inc. All rights reserved.

Keywords: Part-whole number knowledge; Counting

1. Part-whole reasoning as a conceptual structure

Ability to reflect on a number as an object of thought, and to isolate its constituent parts, is basic toa deep knowledge of arithmetic, as well as much practical and applied mathematical problem solving.Resnick (1983)has argued that in order for a child to deal with many numerical situations, includingsimple addition and subtraction problems involving spontaneous counting, it is necessary to account for aninterpretation of numbers in terms of part and whole relationships.Resnick (1983)noted that a primitiveform of part-whole reasoning is present in early counting behavior, evidenced in the capacity of the child tomaintain a partition on a collection of items: those items already counted and those items yet to be counted.Resnick (1983)proposed that the child’s developing counting competence allowed the basic part-wholeschema to become elaborated into aquantitative part-whole schema. In characterizing a part-wholeschema in terms of conceiving numbers as compositions of other numbers, Resnick recognizedPiaget’s(1941/1965)definition of number in terms of operational structures involving additive compositions.

� This article is based on a paper presented at the 2002 Annual Meeting of the American Educational Research Association,New Orleans, April 1–5.E-mail address: [email protected] (R.P. Hunting).

0732-3123/$ – see front matter © 2003 Elsevier Inc. All rights reserved.doi:10.1016/S0732-3123(03)00021-X

218 R.P. Hunting / Journal of Mathematical Behavior 22 (2003) 217–235

Piaget (1941/1965)argued that underpinning the child’s developing understandings of both logicalclassification and number was an additive operation, in which parts are combined to make wholes, andwholes are divided into parts. The logic of class inclusion, and knowledge of number as a seriated class(for example, 3 contains 2, but 2 does not contain 3), involves part-whole operations. Ability to considerthe parts and the whole simultaneously, according to Piaget’s research, does not appear typically until theage of 6 or 7 years, with one exception. He concluded that in the case of small sets such as 1–5 — whatone translator called the intuitive numbers — capacity for simultaneous perception of the whole and itsparts was already present in the children he studied (who were for the most part 5 years of age or more).Subsequent class inclusion research (for example,Markman & Siebert, 1976; McGarrigle, Grieve, &Hughes, 1978; Trabasso et al., 1978; Wohlwill, 1968) suggests young children may be able to coordinateparts and wholes earlier than Piaget claimed, when varying aspects of task and task presentation areaccounted for.

Elaboration of the part-whole schema is likely to underpin meaningful understanding of the place valuesystem of decimal numeration. Being able to mentally move flexibly within and between different unitsystems — such as hundreds, tens and ones of arithmetic — is made possible by part-whole knowledge. Inparticular, the commonly taught school algorithm for subtraction — for example, 42 minus 27 — requireschildren to restructure four 10s and two 1s into three 10s and twelve 1s. Ability to coordinate systems ofunits, including dexterity within a unit system (operational use of the distributive property being a primeexample), is an important component in an individual’s mathematical toolbox at the elementary level.

Part-whole reasoning and counting are closely related in the child’s numerical development. Countingas a complex cognitive skill has been studied extensively (Baroody, 1986, 1992; Baroody & Price, 1983;Fuson, 1988, 1992; Fuson & Hall, 1983; Gelman & Gallistel, 1978; Gelman & Meck, 1983, 1986; Saxe,1977; Sophian, 1987, 2000; Sophian & Kailihiwa, 1998; Steffe, 1994; Steffe & Cobb, 1988; Steffe,von Glasersfeld, Richards, & Cobb, 1983). It seems reasonable to propose, based on the evidence, thatyoung children’s part-whole reasoning is developing along with their increasing counting abilities. Howelaboration of counting skill assists, and is assisted by, maturing part-whole operations, is not clear, atleast in the early stages.

Fuson (1988)distinguished different levels of meaning in children’s correct last-word responses tosimple counting tasks. Children seem to initially learn to say the last count word when asked, “Howmany Xs are there?” Such a response may not, for the child, involve any reference to a set of entitiesas a whole or cardinality of the set. A higher level of understanding is whatFuson and Hall (1983)called the “count-to-cardinal” transition, in which the child means the answer to the question to referto the whole set.Fuson (1988)posits in the child a mental reconstruction of the situation in which acardinal integration of perceptual unit items occurs. Integration is a shift in focus away from a countingsituation where each number word refers to a different entity, to a cardinal situation where a number wordrefers toall of the entities. The termintegration was adapted fromSteffe et al.’s (1983)theory, whichexplained integration as “the application of the attentional unit pattern to any plurality of elements. It isthe act of uniting what one may also consider distinct unitary items” (p. 67). The operation of integrationis a component of a more powerful cognitive scheme called theprogressive integration operation. Aprogressive integration operation is an integration operation applied to material that includes the resultsof a previous application of the integration operation. The notion of progressive integration operation maybe an important theoretical construct in our work, because it may delineate the ability of children to succeedin certain problem situations that have a part-whole structure. ForSteffe and Cobb (1988), achievementof progressive integration operations was necessary for the attainment of reversible part-whole operations

R.P. Hunting / Journal of Mathematical Behavior 22 (2003) 217–235 219

in the first graders they studied. In relation toFuson’s (1988)count-to-cardinal transition, if the last worduttered has cardinal meaning, made possible by application of an integration operation, the next-to-lastword uttered may well have cardinal meaning, and so on down. That is, number words used in countingcome to represent distinct pluralities — whatSteffe and Cobb (1988)called abstract composite units.1

In problem situations involving small discrete quantities, is there evidence that preschool children canconceptualize small sets of items as composite units through application of an integration operation?Can they take the results of that integration operation, and perform further integration operations, sothat, for example, a composite unit of 4 items may be conceived of as 2 composite units; each of 2items? Part-whole reasoning in this range would seem to depend on progressive integration operations,just as reversible part-whole operations in more complex numerical contexts is supposed to depend onprogressive integration operations.

Children younger than 5 years are able to deal with problem situations involving quantities in thein-tuitive range. They are able to discriminate between sets of different numerosities in this range (Starkey,1992; Wynn, 1998), have been found able to do so from the first months of life (Starkey & Cooper, 1980),are able to match small numerical sets across modalities (Mix, Huttenlocher, & Levine, 1996), and aregenerally able to apprehend the numerosity of sets of items up to 5 without needing to count — a processknown as subitizing (Clements, 1999; Kaufman, Lord, Reese, & Volkmann, 1949). Gelman and Gallistel(1978), Hughes (1986), andStarkey (1992)have found preschoolers have intuitive ideas about additionand subtraction.Gelman and Gallistel (1978)came to this conclusion based on their magic experiments.In general, when confronted with the discrepancy between an actual numerosity,A, and an expectednumerosity,E (where the numerosities ofA andE were less than or equal to 4), the children Gelmanstudied knew thatE could be converted toA by either addition or subtraction, and reliably appliedthe correct operation. When the difference betweenA and E was one, they were able to specify thenumber to be added or subtracted. As the difference betweenA and E became greater than one, thechildren reliably indicated that the number to be added or subtracted was greater than one, but wereless accurate. It is reasonable to posit some mechanism that allowed the successful children to recall orre-present a pattern corresponding to the prior perceptually accessible array of items, now not visible.Further, a comparison between the re-presented array and the perceptually accessible array would seemnecessary in order to reliably conclude that an operation of adding or subtracting would be needed toreconcile the discrepancy between the actual and expected numerosities, regardless of what role count-ing may have played. The mechanism making possible children’s accurate judgments in these contextsmay be an important key in understanding the dimensions of part-whole reasoning in other numericalsituations.

Starkey’s (1992)searchbox experiments provided similar results with regard to children’s intuitiveknowledge of addition and subtraction. He concluded that children have a numerical abstraction abilitythat precedes verbal counting by about 2 years. In the first experiment, children aged 1–4 years wereinvited to transfer small numbers of table tennis balls, one at a time, into an opaque box. One or twoseconds later the child was asked to remove all the balls. Of interest was accuracy in reconstructing theoriginal set. Relatively few errors were found for sets of 1–3, but many more errors on sets of 4 or 5.Starkey concluded from the data that children did not simply attempt to guess the numerosity of a small

1 An abstract composite unit is “the result of applying the integration operation to a numerical composite or to a symbolizednumerical composite. The child focuses on the unit structure of a numerical composite — e.g., one ten — rather than on the unititems — e.g., ten ones” (Steffe & Cobb, 1988, p. 334).


set while it was screened from view. They formed a representation of the set’s numerosity, which wasretained for a few seconds at least, and used that representation in reconstructing the screened set. In asecond experiment, after the child had placed all the available balls in the searchbox, the experimentereither took one or more balls from an opaque bag and placed them in the box, or removed one or moreballs and placed them back in the bag. Again the child was instructed to remove all the balls from the box.Children’s search behaviors indicated that they knew that the numerosities of sets had been increased by theaddition transformations, or decreased by the subtraction transformations. Starkey suggested that a kineticimagery hypothesis may account for the data, and referred to work bySiegler and Robinson (1982)where3–5-year-old children were presented with a task in which single sets of imaginary objects underwentan addition transformation, the information provided in a word problem. Three-year-old children couldsolve small set problems typically without verbal counting. Older children used overt strategies, includingcounting, on 38% of the problems, which were mainly large set problems. No visible strategy was usedon the remaining 62%, including many small set problems. Children who did not use verbal countingmay have been generating a numerically accurate representation of the imaginary set, which was thentransformed in accordance with the transformation described in the word problem. The data reportedby Starkey (1992), andSiegler and Robinson (1982), are not inconsistent with the notion of progressiveintegration operations.

To further examine the mathematical behavior of young children in part-whole problem settings, adynamic problem situation was devised whereby a small set of items was partitioned such that one of thesubsets remained perceptually inaccessible. The problem situation was presented as a story calledBugsin the Rain. Questions of interest included: What problem solving strategies did successful children use?What cognitive obstacles seemed to prevent success? What adaptations did children make in response tosuccessive administrations of the task over time. How might one characterize children’s mathematicalthinking based on their responses to this task?

2. Methodology

Fourteen 3- and 4-year-old children (median age 3 years 10 months) attending a University preschoolcenter were individually interviewed by the author, four times on average, over a 6-week period. In theprior semester, the author had spent a number of occasions visiting the preschool classroom, becomingacquainted with the children, and interacted with them as they played, sometimes introducing them toappropriate math activities in small group settings. Children selected for interview by the preschoolcenter director on a given morning were usually in transition from one activity to another in the preschoolclassroom. Interviews were conducted in a nearby separate room. A teaching assistant would accompanya child if that child felt uncomfortable working with the interviewer alone. In that case, the assistantwould observe unobtrusively. A set of part-whole tasks adapted from those used byPiaget (1941/1965)andHughes’ (1986)was used. Results from one task, calledBugs in the Rain, are presented here. Eachinterview lasted approximately 15 min, with theBugs portion of it averaging 5 min. After each interview,the video record was reviewed, and adaptations made to task demands in preparation for subsequentinterviews, to best locate each child’s zone of proximal development (Vygotsky, 1978). In this way, itwas possible to optimize task difficulty based on performance.

In Bugs in the Rain, a set of 5 “bugs” and a large opaque plastic cup are placed on the table in frontof the child. The number of bugs in the set is first established. The child is then told a story about bugs


who were out walking when it began to rain. The bugs, one after another, hide under a rock (the upturnedcup). After each bug hides, the child is asked how many bugs they can see and how many bugs are hiddenfrom view under the rock. The child is allowed to lift the cup for a “quick peek,” or, if necessary, to countthe covered bugs. In the second part of the story, the rain stops. So one bug decides to go home to Mama.The child is asked how many bugs are left under the rock, and how many bugs are now visible. Thesequestions are repeated after each bug goes home to Mama (see Appendix for task protocol). This task waspresented to 11 children three times, presented to 2 children twice, and to 1 child once. All interviewswere videotaped.

3. Results

In Bugs in the Rain, a series of partitions was created on (initially) a set of 5 items, where one subset wasperceptually accessible; the other was not. A good number of the children interviewed were successful;at least up to the step where all the bugs were hidden (seeTable 1). Six children were successful withall 5 steps, 8 children were successful with at least 4 steps, and 12 were successful with at least 3steps. How can we account for this achievement? What mental operations could be at work here? Isthere evidence for an integration operation? It is unlikely that children who used the last-count-wordstrategy in response to the “how many?” question (Fuson, 1988) would be successful in this setting,since in that case perceptually accessible items are needed. Simple recall of the last number word uttered,then saying the next word in the memorized number word sequence, is possible, though unlikely. It isunlikely because the task protocol focused the child’s attention on the collection of visible items each timebefore asking about those items that were hidden from view, thus weakening any stimulus-response link.Evidence for a cardinal understanding of these small numbers, and hence the existence of an integrationoperation, would seem to lie in these children’s ability to recall or replay actions that led to the previousdetermination of hidden items, extract and hold the numerical aspect of an action, possibly through visualmeans, unite the physically hidden but mentally active item to the representation of items hidden in theprevious step, and produce the number word and/or finger set to signify the new result. Being able toextract, hold, and store the numerosity of a set of items, while switching to determine the size of a setof perceptually available items, then switching back to mentally increase the size of that set by one,seems rather compelling evidence for an ability to conceive of these results as numbers with cardinalproperties.

Bugs in the Rain was a complex task because success in responding to a question about the numberof hidden bugs seemed to depend on success responding to a prior question. If a question was answered

Table 1Result summary for first presentation ofBugs in the Rain task (N = 14)

Numbers and mean ages (months) of children successfully completing each task step

Initial countcorrect

One bughidden

Two bugshidden

Three bugshidden

Four bugshidden

Five bugshidden

Two bugsleave

12 (86%) 14 (100%) 12 (86%) 9 (64%) 6 (43%) 9 (64%) 4a (33%)48.7 48.2 47.6 48.8 51.8 49.6 47.3

a Only 12 children were asked this question.


Table 2Frequency of enabling behaviors for first presentation ofBugs in the Rain task

Enabling behaviors Frequency of enabling behaviors for identifyingnumbers of bugs hidden (N = 14)

One bughidden

Two bugshidden

Three bugshidden

Four bugshidden

Five bugshidden

Two bugsleave

Need to peek 0 2 4 3 3 8Use of fingers with correct answer 2 5 1 1 2 1Use of fingers with incorrect answer 0 1 3 1 1 1

incorrectly, the child could take a “quick peek,” whereupon the cup was lifted just long enough for theactual number of bugs to be ascertained, then replaced — usually about a second. It could be argued thatthe numbers of children successfully completing each task step, as hidden bugs were added one by one,could be due in part to opportunities to peek the step before (seeTable 2). This was true for children whopeeked after 2 bugs were hidden, not true for any of the 4 children who peeked after 3 bugs were hidden(these children gave incorrect responses at the stage when 4 bugs were hidden), and true for only 2 of the3 children who peeked after 4 bugs were hidden.

Two enabling behaviors were observed as children responded to interviewer questions. If a child wasnot sure how many bugs were hidden from view, a “quick peek” was encouraged. The interviewer (orsometimes the child) would lift the cup long enough for the number of bugs to be determined, and thenset back in place.Table 2shows that this strategy was used by a majority of the children after 2 bugs wereremoved. Success for this step dropped significantly, compared with success on the previous step.

A spontaneous enabling behavior was use of fingers to represent number of bugs hidden. Finger setswere sometimes incorrect. Eight of the children were observed to use fingers at least once; six childrendid not use fingers at all.

4. Conceptual analysis of preschoolers’ part-whole reasoning in the context of small quantities

Based on research reported in the literature and preliminary analyses of data from this study, wesummarize a proto-model of part-whole reasoning involving four levels of cognitive competence.

4.1. Qualitative Comparison Scheme

A fundamental and innate ability to make gross comparative judgments of quantities is present in chil-dren from the earliest months (Starkey & Cooper, 1980; Starkey, 1992; Wynn, 1998). Perceptual strategiesincluding length and density can be used in making judgments (Fuson, 1992), although conservation ofdiscontinuous quantities has not yet been achieved (Piaget 1941/1965). Gelman and Gallistel (1978)andStarkey (1992)have identified an ability to make consistently accurate judgments about the effect ofincreasing or decreasing the number of items in small quantities, on the sizes of such quantities, in termsof more and less when compared with the base amount. Thus, children know that observing items beingadded to a collection will have the effect of increasing the size of the collection. A cognitive structure formaking these kinds of judgments possible has been called the protoquantitative increase/decrease schema


(Resnick, 1983). Children in division situations involving the sharing of discrete items will partition a setwith little apparent attention to equality of subsets (Hunting & Davis, 1991).

4.2. Quantitative Comparison Scheme — up to 4

Provided the number of items is perceptually accessible, and the number is in the subitizable range,young children are able to focus attention on quantitative relationships arising from physical transforma-tions of a defined collection, and will use counting to enumerate small sets if needed. When countingbehavior is observed, children typically count each item deliberately; usually in response to the adult’squestion “how many?” However, the last-word response (Fuson, 1988) may not have cardinal meaning.Children in this stage are able to perform integration operations (Steffe et al., 1983) on perceptuallyavailable material only, in the range 1–3.

4.3. Quantitative Comparison Scheme — representational stage

Small sets can be represented internally using visual imagery when such sets are not perceptuallyaccessible. Children in this stage are able to perform integration operations (Steffe et al., 1983) onrepresentations of sets of 1–3 (visualized or figural), so that they are able to perform simple calculationsmentally (Huttenlocher, Jordan, & Levine, 1994). For quantities beyond 3, perceptual material needsto be available. They also begin to communicate their ideas of quantity through use of finger patterns(Brissiaud, 1992). Thus, when asked how many toy bugs are now hiding from view under a plastic cup, achild may respond by showing the number of fingers corresponding to the number thought to be hidden.Ability to anticipate the outcomes of transformations yet to be observed is limited.

4.4. Numerical comparisons

Children at this level are fluent in solving problems involving quantitative relationships within smallcollections. Prior experiences can be drawn upon to anticipate outcomes of transformations within smallcollections. These children can be said to have coordinated the logic of class inclusion with cardinalknowledge for the “intuitive” numbers (Piaget, 1941/1965). Thus, they are aware that 1 is contained in 2,but 2 is not contained in 1, on up, to 4 or 5. InSteffe and Cobb’s (1988)terms, children have constructedprogressive integration operations in this range. Development toward reversible part-whole operations iswell advanced for problems involving small numbers. Emergence of symbolic activity does not depend onacquisition of conventional symbols (Huttenlocher et al., 1994), but such acquisition is well establishedand facilitative. In sharing situations, children use systematic many-to-one allocations, and check theirresulting shares using counting methods (Hunting & Davis, 1991).

5. Cases

Four cases follow — one girl and three boys, whose ages ranged from 3 years 6 months to 3 years10 months. These children are representative of the third stage of cognitive competence in the modeloutlined above: Quantitative Comparison Scheme — representational stage. Their behavior will be ex-amined across three interview sessions. For each case, performance is described followed by interpretivecomments.


5.1. Henry (3 years 10 months)

First interview. Henry was able to correctly identify the numbers of both visible and hiddenspiders,as he called them, at each step in the story, up to 4 hidden spiders, even though, at the beginning, hemiscounted the 5 visible items as 6.

I = Interviewer: And the last little spider wants to come under here.H = Henry: [Moves bug]

I: Get out of the rain.H: [Pushes bug under cup]I: There he goes. [Taps cup] How many spiders under the rock?

H: [Holds up right hand, five fingers extended] Four!I: [Pauses] Ok. Now. It stops raining. . . [Tilts cup reaching under]. . . and two spiders

say — I wanna go home [Retrieves bugs]H: [Moves bug to his right] Whee!I: How many spiders under the rock now?

H: [Holds out two fingers on right hand] Two. [Pauses, holds up three fingers] Three.I: Can you make. . . . Are you sure?

H: [Nods, glancing to exposed bugs]I: Alright, let’s see if you’re right [Lifts cup]

H: [Looks at bugs] Three.

Henry corrected his answer because he knew if there were 2 bugs at Mama’s, then 2 bugs under the rockwas insufficient to re-create the whole. Finger representations were observed twice in the first interview:at the step of placing the second bug under the cup, and when the final bug was placed under the cup. Theonly peek occurred at the end when the interviewer allowed Henry to see that there were indeed 3 itemshidden from view, as Henry had indicated.

Second interview (14 days after first interview). Based on his strong first performance, 6 bugs wereused. Henry was remarkably successful quantifying the hidden bugs even though he miscounted severaltimes. For 6 visible bugs, he double counted the fifth bug, concluding 7; for 5 bugs, he double countedthe second, concluding 6. For 3 bugs, he double counted, stopping at 4, paused, moved back 1 bug andcontinued “5, 6, 7.” However, when challenged by the interviewer he looked at the set, extended 3 fingers,uttering the word three. In this interview, at every step except one, Henry used fingers appropriately torepresent numbers of hidden bugs. Inappropriate use of fingers was observed after the sixth bug was placedunder the cup. On that occasion, he held up both hands with all fingers extended. When the notion of bugsgoing home to Mama was introduced, Henry wanted to transfer all the bugs at once but was restrained.Three bugs were transferred to begin with. Henry offered 3 fingers to represent remaining hidden bugs.At that stage the interviewer invited him to peek, to confirm the number of remaining bugs, which he did.After 1 more bug was transferred, Henry extended 2 fingers to indicate the number of hidden bugs.

Third interview (8 days later). Again 6 bugs were used, but in the second part of the interview, after itstopped raining and when bugs came out from under the rock to go home to Mama, a blue plastic cupwas introduced as Mama’s house. So except for transfers, bugs under the rock and bugs at Mama’s housewere hidden from view. The cognitive demands of having both subsets hidden proved quite difficult forHenry. The first part of the interview proceeded similarly to the previous interview. He did not miscount,although he came close when counting the initial set of 6 bugs. His sequence of utterances almost overtook


his acts of pointing. In the second part of the interview, to begin with 2 bugs were transferred from thered cup (rock) to the blue cup (Mama’s house). Henry called for a peek under the rock. He extended 2fingers to represent the bugs at Mama’s. One more bug was transferred. Again, Henry needed to peek tosee the bugs remaining under the rock. He extended 3 fingers to show the number of bugs at Mama’s,but also needed to peek. Two more bugs were transferred. He was not sure if there were 4 or 5 bugs atMama’s, based on his finger extensions, neither was he sure how many bugs remained under the rock,because he needed perceptual information.

Comment. Henry was not a robust counter, yet he was a strong performer when he needed to determinesuccessive numbers of items that were perceptually unavailable, provided one subset was visible. As thewhole was partitioned into 2 co-varying subsets — one visible, the other hidden — Henry was able tokeep track of the bugs that were placed under the cup. There are two possible explanations. First, hemay have been able to infer the hidden number by dis-embedding the accessible subset from the whole,which was never perceptually available except at the beginning. Second, his use of finger sets to representhidden bugs suggests that he was able to visualize the situation dynamically. For example, to recall that2 bugs were hidden, he needed to be able to re-present, or replay in his mind, the previous action, as wellas recall the number of bugs previously hidden. The latter explanation seems more plausible.

5.2. Roy (3 years 10 months)

Roy’s performance in each interview was characterized by absence of finger sets. For the first interview,Roy was able to respond appropriately for each transfer until all 5 bugs were hidden. He was unsure howmany bugs remained hidden after 2 bugs were removed, so he was given a peek.

Second interview (5 days after first interview). On this occasion, 6 bugs were introduced, and Mama’shouse was a different color cup. The context of the story was snow, rather than rain. Roy was nowsomewhat familiar with the story. He took the initiative moving the bugs, one by one, assisted by theinterviewer who tipped the cup just enough to place the bug under, yet conceal the previously placedbugs. Roy correctly stated the number of hidden bugs all the way through 5. The interviewer did not askhow many bugs were hidden after all 6 bugs were covered. But Roy said 5 bugs were hidden after thefirst bug went home to Mama’s house. After the second bug was transferred he incorrectly stated 3 bugswere in Mama’s house and 2 bugs were left hiding.

I: Want to have a quick peek? [Raises the cup representing the rock, and replaces it as Roy starts to pointto the bugs.]

R: Four.I: Uh-huh! Another little bug says, “I wanna go home to Mama!” [Roy takes a bug from the red cup to

the blue (Mama’s) cup.] Yep. Now, how many bugs are under here now? [Indicates the red cup.]R: Three.I: How many bugs under here? [Indicates the blue cup.]

R: Let’s take a. . . [Lifts the blue cup.]I: Oh!

R: Three. I just wanted to take a quick peek.I: And so, what did you see?

R: That three is in there [taps on the blue cup] and [taps on the red cup] four in there — three in there[corrects himself with a quick negative shake of the head].


I: This little bug goes home to Mama. [Together they take a bug from the red cup to the blue cup.]Whoops! Leave him under there! And how many bugs are hiding under the rock now?

R: Uh, three.I: And how many bugs are hiding under here? [Indicates the blue cup.]

R: Want to. . . [Lifts the blue cup.] Four.I: [Taps on the red cup again.] And how many under here?

R: [Hesitates.] Uh, two.I: How do you know that?

R: Because, I just know!I: You said three before.

R: Two.

Roy was not successful enumerating states after subsequent transformations. Again, no finger sets wereobserved.

Third interview. Two weeks later, Roy played theBugs game again. Six bugs were used; a red cup wasused for the rock and a blue cup used for Mama’s house. This time, the sequence of transformations inthe first phase, went 1, 2, 1, and 2 bugs, for each transfer. Roy needed to peek after the second step when2 bugs were removed from view. However, after that he was successful without looking. In the secondphase, bugs were moved to Mama’s house one at a time. Roy was able to keep count of the number ofbugs remaining hidden, without peeking. He needed to peek once, at the second step, to check the numberof bugs at Mama’s. His task may have been easier at Steps 5 and 6, because the interviewer asked firstabout the state of Mama’s house just after a bug was placed under that cup.

Comment. Roy’s performance was not perfect but did demonstrate his ability to represent mentallythe number of items hidden, apart from several occasions when he needed to peek. Those occasions,except one, occurred when both subsets were hidden from view. The sets he was able to represent inhis head were accumulated sets; that is to say, he could consistently perform an operation of addingone more to a mentally constructed set of items, and, as his performance on the third occasion showed,simple subtraction also. Here is strong evidence for possession of a cardinal understanding of Numbers1–5 made possible by progressive integration operations. Although in the second phase of the secondinterview he needed to peek to coordinate the compensating states, when both subsets were unavailable,by the third presentation, 2 weeks later, he had made progress. In the second phase of the second interview,where items in each set were perceptually unavailable except during transfer, Roy peeked to see 3 bugs atMama’s house, then concluded without looking that there were 3 remaining under the rock. After the nexttransfer, he was first asked how many bugs remained under the rock. He said 3, but after peeking under thecontainer representing Mama’s house, and seeing 4, he deduced there were just 2 bugs remaining underthe rock. How did he know? Roy may have been able to relate the larger part — 4, to the whole — 6, inorder to conclude the size of the other part was 2. Or, did he just recall that previously the interviewerseemed satisfied there were 3 bugs under the rock so the next transfer meant there was now one less?Roy would be close to moving to Level 4 in the proposed model. Absence of evidence of finger set useprovides support for this assessment.

5.3. Maz (3 years 6 months)

Maz demonstrated behavior that suggested she was capable of visualizing outcomes after adding orsubtracting 1 and 2 items to or from small sets of hidden items. Her performance in the 3 interviews


was not flawless, and she was mature enough on one occasion, when 5 bugs were hidden from view, tostate categorically that she could not remember how many there were. She used finger sets to represent2 hidden bugs in Interviews 1 and 2, but more frequently during both phases of Interview 3.

First interview. In the first phase, Maz was able to state the total number of bugs hidden after eachtransfer, except the fifth. Fingers were used to represent 2 hidden bugs only. After the third bug washidden, Maz responded to the interviewer’s question: “And how many bugs are hiding under the rock?”by pausing momentarily, looking up and to the right of the interviewer, saying 3. When the fifth bug wasremoved from view, and Maz was asked how many were hiding, she frankly admitted she didn’t knowbecause she could not remember.

After counting the bugs, the cup was replaced, and 2 bugs were removed. Maz stated that there were 3bugs remaining.

Second interview (10 days later). Again, 5 bugs were used, except another cup was introduced torepresent Mama’s house. Maz participated in moving the bugs as the story developed. She used fingers torepresent 2 hidden bugs, but could not tell the number of hidden bugs after the next transfer. So Maz wasgiven a peek. She was able to answer correctly after the fourth and fifth bugs were placed under the cup, andcould tell how many bugs remained, when 1, 1, 2 bugs, and 1 bug were removed and covered under the cupthat represented Mama’s house, even when there were distractions, as the following transcript indicates:

I: Now it stopped raining. It stopped raining. [Produces the blue cup from under the red one and sets iton Maz’s right.] This is Mama’s house. Okay?

M: [Maz points at the rim of the blue cup (at the table surface) and says something inaudible.]I: Under that blue cup. . . We’ll pretend that’s Mama’s house. [Extracts one bug from the red cup.] So

one little bug said, “I want to go home to Mama while it isn’t raining so I won’t get wet.” What doeshe do? Chu-chu-chu-chu. . . [Maz moves the bug to the blue cup and puts it under.] Okay. Now, howmany little bugs are [taps on the red cup] left hiding under the rock?

M: [Immediately] Three. Four. Four.I: Five? Four? Three? How many?

M: Four.I: Just a lucky guess?

M: Four.I: Four. And how many bugs are at Mama’s house?

M: [Immediately] One!I: And another little bug said, “I wanna go home to Mama.” [Moves a bug to the blue cup and puts it

in as Maz tilts the cup.] Now [tapping the red cup], how many little bugs are hiding under the rock?M: [Immediately] Three!

I: And how many bugs are [tapping the blue cup] hiding under here?M: [Looks at her watch and adjusts her watchband.] Two!

I: You have a pretty watch. [Extracts two bugs from the red cup.] Then,two little bugs said, “I wanna— We wanna go home to Mama.” So two little bugs go back home to Mama. There they go. . . Scootscoot scoot. Can you put them under there? [Moves them to the blue cup and puts them under as Maztilts the cup.]

I: Alright. Now. [Waves to someone behind Maz.] How many little bugs. . .

M: [Turns around and waves.] Hi! [Maz looks at interviewer and says something inaudible, to whichinterviewer responds “Okay,” and Maz turns again and remains with her back to Interviewer.]


I: How many bugs are hiding under the rock?. . . Maz?M: [Turns to Interviewer.] One.

I: And how many bugs are home with Mama?M: [Lifts the cup and slowly touches each bug once, at the same time uttering the counting sequence,

but with no correspondence to the motor acts.] One, two, three, four, five, six.I: Ah, you’re not supposed to look! [Maz covers the bugs again.] How many bugs are under there?

M: Four.

Third interview (11 days after second interview). Again 5 bugs were used, Both the rock and Mama’shouse were represented by plastic drink containers. On this occasion, the first phase of the interviewwas varied. After the first bug had been hidden, and successfully identified, 2 bugs were removed. Mazextended 4 fingers as her response to how many bugs were now hidden. Before allowing Maz a peek, theinterviewer reviewed with her the previous actions:

I: How many bugs went under here to begin with? [Taps on the blue cup.]M: [Holds up index finger.] One!

I: And then how many more bugs came under?M: [Holds up two fingers.] Two.

I: So how many is that altogether?M: [Holds up three fingers.] Three!

I: Is it three or is it four?M: I think it’s four.

I: Do you think it’s four?M: Yeah, and Ineed to count them!

I: Let’s have a quick peek! [Lifts the cup and sets it back down immediately.]M: Three!

I: Do you need to count them?M: I saw three bugs!

Maz continued fairly successfully in Phase 2 of the interview. Bugs were returned to Mama’s houseone at a time. She needed to peek under the rock container after the first bug was removed, but beforedoing so seemed to use her fingers to replay the sequence of bugs remaining under the rock:

I: What happens next?M: Then one little bug — “I wanna go in with the Mom.”

I: Because it stopped raining. Alright. So one little bug goes into Mama’s house. [Tilts the blue cupso that Maz can remove a bug and put it under the red cup.] Alright. So another little bug goes intoMama’s house [Transfers another bug]. Now, how many little bugs are hiding under the rock?

M: [Maz extends three fingers.]I: Three?

M: [Maz slightly nods yes.]I: Want a quick peek?

M: [Maz slightly nods yes.]I: [Taps on the red cup (Mama’s house).] How many bugs are hiding under here?

M: One.I: [Indicates the blue cup.] How many bugs are hiding under here?


M: [Holds up her hand to show five, then folds the thumb down to make four, and then folds down herlittle finger to make three. She is looking at this hand as she does so.] Three.

I: Three. Want a quick peek? [Gives Maz a glance under the cup.]M: Three.

Comment. We claim Maz was able to produce arithmetic units, and operate on them dynamically, byadding or subtracting one. In this way she was able to answer quantitative questions about the resultsof actions even though perceptual information was absent. One can interpret Maz’s action in the firstinterview, of looking up and away from the interviewer just before answering, as engaging in a processof visualizing or replaying in her mind’s eye, the essence of her prior experience. Her most impressiveperformance came during the second phase of the second interview. In this situation, both subsets of bugswere unavailable except for the period while 1 or 2 bugs were being transferred from one cup to the other.Interruptions during the sequence of transfers made Maz’s performance even more remarkable. She oftenrelied on finger sets as perceptual representations of not only static collections, but also to tally a sequenceof actions, as seen in the third interview, when she was observed to fold down 2 fingers of her extendedhand as she appeared to re-play the transfer of 2 bugs from the rock to Mama’s house. Engagement offingers indicates that development of progressive integration operations for addition or subtraction of 2is more challenging than for addition and subtraction of 1.

5.4. Nat (3 years 9 months)

Nat had yet to develop operations that would allow him to successfully answer questions about smalldiscrete quantities when a part of the whole was perceptually unavailable.

First interview. Nat used a non-standard number name sequence to count remaining bugs as 1 bug at atime was hidden from view. His initial count of 5 bugs: “7, 0, 6, 3, 2, 1”; 4 bugs: “0, 6, 2, 1”; 3 bugs: “0, 7,1”; and 2 bugs: “0, 3.” He correctly identified the number of hidden bugs, using his fingers to represent 1and 2, but no fingers were used for 3 bugs. When Nat identified the last visible bug as “1,” the interviewerchallenged him:

I: I thought that was zero.N: [Looking at the bug] Zero. [Looks back to interviewer]I: You said one

N: OneI: Is it one, or is it zero? [Shakes head back and forth.]

N: It’s one.

Nat’s counting sequence utterances were unusual, because he did seem to know that, for 1–3, there wasa word that signified the cardinal set, as evidenced by his ability to co-represent hidden sets of 1 and 2with fingers and appropriate number names, and a hidden set of three with the correct number name. Theconversation above indicates that whatever the utterances used to count the visible bugs stood for, theyhad no cardinal meaning. Nat did not know there were 4 hidden bugs at the next step, although he knewhe could represent items using fingers:

I: Alright, and how many bugs are under here [touches cup]?N: [Leans back, pauses, looks at right hand, and begins to adjust fingers, holding out three, then, four,

then five fingers. Looks to interviewer]


I: Show me with your fingers.N: [Holds out all fingers on both hands] Seven.

Nat was offered a peek, which he accepted, but he was more interested in placing the last bug underthe cup than focusing on the number of hidden bugs. He again held out both hands, saying “seven” whenthe interviewer asked him to enumerate the set of 5 hidden bugs.

Second interview (21 days later). As in the previous interview, 5 bugs were used. Nat’s count of theinitially visible set was “3, 7, 8, [pause] 2, 1.” After the first bug was placed under the cup, Nat tookover the story and quickly placed the remaining bugs, successively, under the cup. Another cup wasintroduced to represent Mama’s house. Nat needed to peek to check his answers at each stage, but thevisual assistance did not help him to correctly identify numbers of bugs remaining under the rock, or bugsat Mama’s house. He was observed to count sets of 4 in a conventional way on two occasions however.

Third interview (14 days after second interview). On this occasion 5 bugs were used, and 2 cups. Nat wasobserved to count the visible bugs at each step, including at the outset, using the standard number wordsequence. He co-produced finger sets and utterances for 1, 2, and three hidden bugs. The fourth and fifthbugs were hidden simultaneously, and Nat co-produced 5 fingers and said “five” when asked how manybugs were now hidden. Nat had some success in the phase of the task where bugs went home to Mama. Hepeeked and point-counted 4 bugs remaining after the first bug was transferred. He first said 5, even after theinterviewer reminded him that he had said there were 5 bugs before removing the first. Nat initiated trans-ferring 2 more bugs. He was content to place them near Mama’s house, but then placed them under the cup.

I: Now how many bugs are at Mama’s house?N: Leans back, pauses; then holds up three fingers on right hand.I: Alright. And how many little bugs. . .

N: Three.I: [Taps blue cup]. . . are hiding under the rock still?

N: [Turns to blue cup, pauses; wiggles fingers on left hand and holds up two slightly.] Two.I: Alright. What happens next?

N: I want a peek.I: [Lifts blue cup.] You were right.

Nat’s request for a peek indicates uncertainty about his answer. He then transferred the remaining 2bugs to Mama’s house. When the interviewer asked Nat how many bugs were now at Mama’s house heresponded: “I put all of them.” When then asked how many were now under the rock, he badly wantedto look, but was prevented. Nat eventually responded with 3 fingers. When challenged; “Are you sure?”he said “two.” Nat was offered a peek at the bugs under Mama’s cup, and he said “five” and extended 5fingers of his left hand. The interviewer returned to ask about the number of bugs remaining under therock. Nat’s response was 2.

Comment. We make two assertions about Nat. First, his counting scheme was just emerging as a tool toenumerate small sets. In the space of 5 weeks, we observed him first display unconventional number wordsequences, then conventional counting behavior, in the range 1–5. He did seem able to coordinate motormovement and verbal utterance. Second, he was limited in ability to either recall previous perceptualevents involving small sets, or limited to mentally representing small sets up to and including 3. Creationof perceptual sets using fingers was an enabling strategy he was observed to use frequently, although onseveral occasions his finger productions failed to correspond with any recent numerical experience.


6. Discussion

Part-whole reasoning supports children’s mathematical learning through the elementary school andbeyond. In particular, it serves the acquisition of concepts such as numeration and number sense, basicnumber facts and operations, and makes possible meaningful understanding of arithmetic algorithms. TheBugs in the Rain tasks were presented in story form, in order to investigate young children’s part-wholereasoning involving small numbers of items. They allowed a detailed examination of interactions betweenpreschoolers’ emerging counting knowledge, cardinal conceptions of small discrete quantities based onspatial patterns and subitizing, and informal addition and subtraction operations provoked by transfor-mations of subsets when perceptual information was unavailable for one or both subsets. We have notedthat in counting, utterances of number words can be a rote activity, but at some point the significanceof each utterance must become related to a physical pattern; initially a perceptual subitized whole, latersome internally represented material. This “peculiar coincidence” (von Glasersfeld, 1982, p. 212) of thenumber word as the terminal point of an interactive procedure, and as the name for a figural pattern,“provides an experiential foundation for the conception of number as aunit composed of units (vonGlasersfeld, 1982, p. 207).Von Glasersfeld (1982)cautioned that relations between quantities may beexperientially acquired. For example, discovering that a “three” can be produced by joining a “one” to a“two.” Such relations are analogous to those that characterize the conceptual system of whole numbers,but initially exist through manipulation of perceptual patterns. In the cases presented here, there is con-vincing evidence that these children were capable of much more. These children were capable of dynamicmental processes involving integration operations on self-generated material consistent with a capacityfor reflective abstraction (Piaget, 1970) — a mechanism that becomes more general and necessary in latermathematical problem solving.

A major cognitive tool at work in many of these children seemed to be an ability to visualize, notjust static configurations, but sequences of actions, when outcomes of such actions were hidden fromview. Success enumerating hidden bugs depended either on an ability to keep track of successive lev-els of outcome, unless perceptual feedback was needed by virtue of a peek, or on an ability to makesophisticated deductions based on coordination of visible parts and the original whole. For example,after a third bug was hidden, a successful child would need to know that 2 bugs were previously hid-ing under the rock, which in turn was a consequence of a second bug joining the first bug that hid.A follow-up study that incorporates questions asking children to anticipate outcomes prior to an eventmay throw further light on their capacity to visualize outcomes of operations. In children of this age,such visualizations are assumed to be predominantly spatial in nature. Can preschoolers make sets of1–5 in their mind in the absence of physical material? Evidence from this study would suggest thatthis is so. How spatial representations of quantitative transformations evolve into numerical entities (ifin fact this is universally the case) is yet to be explained.Von Glasersfeld (1981)proposed an atten-tional model for the construction of number entailing a hierarchy of conceptual sophistication deal-ing with experiential objects, ranging from awareness of boundedness, a succession of bounded ex-periences having something in common to form pluralities, framed pluralities called collections, lotsas collections without interest in specific sensory-motor content, up to arithmetic units which cor-respond toPiaget’s (1970)elements stripped of qualities — a product of reflective abstraction.VonGlasersfeld’s (1981)model is consistent with recent evidence for two non-verbal representations ofquantity (Mix, Huttenlocher, & Levine, 2002). One representation emerges in infancy and is based onrelative amount and a second that emerges in early childhood and is based on discrete number. Questions


requiring preschool children to “think forward” to anticipate the outcomes of simple transformationsmay help clarify whether imagery is due to retrieval from memory or based on capacity to generateitems.

Counting is a culturally acquired tool, and number discrimination appears to be an innate ability. Useof finger symbol sets can serve as referents for cardinal numbers.Brissiaud’s (1992)work indicatesthat some children are able to represent various numerosities of sets but not know how to name thedifferent quantities. He documented the case of his son Julien, whose pathway to number was via fingersymbol sets. Counting played a minor role initially. In contrast, the prevalent (at least in the UnitedStates) pathway is where children first learn to tag items using counting words, followed by a transition tocardinalized counting, and where finger representations are generally not encouraged.Brissiaud (1992)hypothesized that acquisition of a gesticular system is almost never a byproduct of the verbal system ofnumber names. Does this behavior indicate that if such children were to learn number names associatedwith finger sets, those number names would have cardinal meaning? We do not know, but it is quiteplausible.

With these children, we observed finger sets used as physical presentations for unavailable sets. Pro-duction of finger sets provides the benefit of both visual and kinesthetic feedback. Co-production offinger sets with oral utterances, favored by most children in this study, increases modes of sensoryfeedback one more. Are finger sets used to represent visualized material, or simply used as a standardsymbol set because visualization alone was too great a cognitive task? This we do not know. More re-search is needed. In the case of Maz, fingers were used as a perceptual aid to re-present events pastwhereby 2 bugs were transferred from a hidden collection. In the case of Henry, his finger representationswere an outcome of quite impressive mental abilities, where the critical operations had already beenenacted.

Children in this study were less successful enumerating more than 3 hidden bugs, supportingStarkey’s(1992)observation of a discontinuity between 3 and 4 in children’s numerical development. However,this discontinuity may just be relative to the ages of the children studied. If the children in this study hadbeen aged 2–3 years, there may well have been a discontinuity between 2 and 3. AsBrissiaud (1992)observed of his son: “Julien’s 3rd birthday had a catalyzing role in his learningthree” (p. 48). In general,the social importance of birthdays, along with their associated adult–child verbal interactions involvingfinger patterns and observation of sets of candles on cakes and birthday cards, is likely to influenceacquisition of cardinal meanings.

A continuing mystery requiring further study is how numerical information is processed. A capacityto switch focus from individual items to a grouping or uniting of items, depending on the person’s goals,seems fundamental. An integration operation has been proposed to explain how cardinal understandingsare made possible. We are aware of the dynamics of part-whole reasoning where a subset is cut out fromthe whole while the whole set is kept in focus. We suppose that the logical operations of class inclusionare important here. The reverse situation, where a whole is rendered a part, by the conjoining of otheritems to make a new enlarged whole, prefigures the symbolic statements we know as addition of wholenumbers.

Problem situations where 1 or 2 items are added to or removed from a small set appear to be po-tent experiences for the development of numerical part-whole reasoning, just as part-whole reasoningand integration operations serve as foundations for solving informal addition and subtraction problems.Part-whole reasoning and ability to perform simple addition and subtraction operations in whole numbercontexts are reflexive cognitive skills. That is, facility with part-whole reasoning enables children to con-


ceptualize, and be successful solvers, of addition and subtraction problems. Experience solving informaladdition and subtraction problems, like those inBugs in the Rain, enables development of part-wholereasoning, along with its critical engine: progressive integration operations.

Acknowledgments

The author acknowledges the assistance and cooperation of Ms. Nancy Lee, Director, and staff, of theChild Development Laboratory, Department of Child Development and Family Relations, East CarolinaUniversity. Work on this project was partially supported by Grant No. 200200218 from the SpencerFoundation. Any conclusions or recommendations stated here are those of the author and do not necessarilyreflect an official position of the Foundation.

Appendix A. Bugs in the Rain

Set-up:

A collection of 5 make believe bugs is presented, arranged in a line. A plastic cup or small box is neededto hide bugs as the story proceeds.

Story:One day some bugs were out walking. How many bugs do you see (allow child to count)?It starts to rain, so one bug decides to hide under a rock — watch (move the first bug under the container).How many bugs can you see now? How many bugs are hiding?Another bug decided to hide under the rock — watch (let child observe next bug be placed under thecontainer, but do not let child observe the screened bugs).How many bugs are under the rock now? How many bugs can you see? (Let child peek if necessary).

Repeat until all bugs are hidden.If child has successfully responded to these questions, say:

Some bugs decided it was time to go home, so they came out (remove 2 bugs and leave them visible).How many bugs are hiding now? (If necessary allow child to peek). How many bugs can you see?If child was hesitant or had difficulty with initial questions, remove a single bug at a time using the samestory and the same questions.Task can be varied according to the competence of each child. Two or more bugs can be added orsubtracted at a time.

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Part-whole number knowledge in preschool children