Countable entities: Developmental changes

Cognition,34 (1990) 109-136

Countable entities: Developmental changes*

ELIZABETH t=.SHIPLEY BARBARA SHEPPERSCiN University of Pennsylvania Received May 4, 1987, final revision accepted June 26, 1989

Abstract Shipley, E.F., and Shepperson, B., 1990. Countable entities: Developmental 34: 109-136.

changes. Cognition,

The canonical countable entity for 3- and 4-year-old children is a discrete physical object. When children were asked to count labeled entities such as “‘forks”‘,they counted each detached part of a fork as a separate entity. When asked to count kinds (“How many kinds of animals?‘) or properties (“How many colors?“), where each kind or property was exemplified by several separate objects, they included each discrete object in their count. Their counts of classes were more accurate in the absence of objects, or in the presence of a single member of each class, than in the presence of several members of each class. Young children are evidently predisposed to process discrete physical objects. Evidence is presented that, developmentally, this bias precedes learning to count. It is proposed that this discrete physical object bias facilitatesmastery of counting.

The universality of counting, as well as its spontaneous use by young children without explicit training, has been convincingly documented by Gelman and Gallistel (1978). Apparently the human organism is preprogrammed to learn to count, much as it is innately prepared to learn to talk. *This work was supported by Grant BNS-8310009 from the National Science Foundation. We are grateful to the children, teachers, and parents of Penn Children’s Center, Chestnut House, Germantown Friends Schuol, Gateway School, Swartbmore Friends School, and Trinity Cooperative Day Nursery. Thanks are due tti Ruth Rosenberg for her thoughtful assistance in data collection and data analysis. Discussions at various stages of this work between EFS and Rachel Gelman, Patrice Hartnett. Ivy Kuhn. Barbara Landau, the MLU group, Thomas F. Shipley, and Elizabeth S. Spelke were more helfiul than they could ivtr iraiize. However. helpful discussions imply neither agreement nor responsibility for short-comings. Lila Gleitman made helpful comments on an earlier version of this paper. Requests for reprints should be sent to E.F. Shipley, Department of Psychology, University of Pennsylvania, 3815 Walnut Street, PA 19104. U.S.A.

OOlO-0277/90/$8,90 0 1990, Elsevier Science Publishers B.V.

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Gelman and Gallistel(l978) offex*edfive principles that “govern and define counting” (p. 77) to characterize the n;a:ure of counting: a one-to-one correspondence between counted entities and tags (usually numerals), a stableor&r of tags, cardinality of a set given by the final tag, the irrelevance of the order in which things are counted, and the abstraction principle that specifies that the other counting principles “Cal”.be applied to any array or cohection of entities” (p. 80). A series of compelling experiments supports the use of the counting principles as a characterization of counting. For Gelman and Gallistel (1978) counting principles do more than characterize counting. Implicit knowledge of the first three principles, the “How-toCount” principles, serves to guide, structure, and motivate counting behavior (p, 208). The abstraction principle is a “What-to-Count” principle, which serves as a boundary condition on the application of the other principles, rather than as a guide to how to count. As a boundary condition, the abstraction principle is hypothesized to allow the child to include everything in a count to which she can apply the term “thing” (p. 215). More recently, Gelman (1986) has argued that the abstraction principle (characterized in this later work as the “Item-indifference” principle), in conjumtion with other counting principles, enables the child to recognize count terms and guides the chi?d’s induction of the correct procedure for counting from observation of other people counting. As a consequence bf the work reported here, we wi!! argue that a more basic disposition than the aijstrslction principle limits what the young child can count. This disposition is a bias to process discrete physical objects. Further, we will argue that this disposition is noi limited to counting. More generally, we will suggest that dispositions more elemental than the counting principles may underlie the rapid mastery of counting, and that the individual counting principles may be decomposable into these more elemental dispositions.

Prior research has found few, if any, limitations on the kinds of discrete objects children can count. They can count familiar objects, pictures of objects, or abstract figures. They can count both animate and inanimate objects in a single count (Fuson, Pergament, & Lyons, 1985; Gelman, 1980). However, the arrays that young children have been shown to count are far more restricted than “any array or collection of entities” or even entities that can be called “things” or “objects”. Children explicitly asked to count “everything in the room” did not count such things as colors, types of objects, or parts

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of familiar labeled objects-yet surely the leg of a chair, the arm OPa person, or the doorknob on a door is an entity that can be called a “thing”. And some “things” attached to other things are certainly objects-for instance shoes or shirts or doorknobs. Counts of everything in the room included only individual discrete physical objects. (Gelman, 1980).’ The studies reported here requested young children to count a variety of entities, under a variety of conditions, and found severe limitations on what the children could count. In order to tease out separate components of children’s ability to count various entities, the same children, called the Multi-Task subjects, participated in four different experiments (1, 3, 6, and 8). In order to control for order effects most experiments also included Single-Task subjects who participated in no other experiments. All subjects were seen individually, all attended nursery schools or kindergartens, and all were from middle-class families. The first two experiments included detached parts in the arrays to be counted. Then three experiments examined children’s ability to count classes of objects distinguished by kinds, for example, kinds of animals, or by properties, for example, colors. The last three experiments explored the sources of children’s difficulties in the prior experiments.

In these two experiments we examined what children count when given an array with both intact labeled objects and two detached parts that form a labeled object if combined (Figure 1). A request to count such an array can be quite explicit: “Can you count the forks?“; or it can be more general: “Can you count the things?” The explicit request can be satisfied by counting only the intact forks, or by counting the two detached parts as a single fork, along with the other whole forks. The general request can be satisfied in a variety of ways, which depends upon the counter’s interpretation of “thing”. If each discrete physical object (DPQ), and nothing else, is considered a ‘!3y a discrete physical object we mean a bounded solid that is separately movable. Discrete physical objects that are intact exemplifiers of a kind will be called “labeled objects” or “objects” when the label is clezr from context. Discrete physical objects that are labeled objects are to be distinguished from discrete physical objects that are detached parts of an exemplar of a kind; the latter will be called a “detached part of a labeled object” or a ‘,detached part”. Thus, a bottle, the neck of a broken bottle, and a lump of glass are all discrete physical objects, although not the lid attached to a bottle. The first is also a labeled object, the second is also a part of a labeled object, btti the thhJ is only a discrete physical object.

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Figure I.

A Homogeneous-Familiar array with the detached parts separated: Experiments I and 2. “Can you count the forks?”

“Can you count the things?”

“thing”, then the subject’s count should ignore the difference between whole labeled objects and detached parts of objects. However, a fork has parts that have names-a handle and tines. Each of these can certainly be considered a “thing”. Thus, Isgically, a count of the array in Figure 1 in response to a request to count things could yield the cardinal numbers 6 (if only DPOs are counted), 25 (if all handles and tines are counted), 31 (if all handles, tines and DPOs are counted), and so forth. Experiment 1: Counting labeled objects-Developmental changes

In the first study we examined the counts of preschool children and adults for arrays typified by Figure 1. Both specific and general requests were used to determine the range of entities that can be counted by subjects of different ages, as well as the relation between what is counted and the wording of the request to count. The homogeneity of the array, the familiarity of the objects in the array, and the spatial alignment of the detached parts were varied in order to assess the influences of perceptual factors upon the subject’s treatment of detached parts as countable entities. Method Subjects. Four groups of children participated, with 6 girls and 6 boys in each group: Threes, mean age 3;3, range 2;lO to 3;7; Fours, mean age 3; 11,

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range 3;s to 4;2; Fives, mean age 5;8, range 5;6 to 510; and Sixes, mea3 age 6;5, range 6;2 to 6;9. All participants could count ten DPQs with an accuracy of plus or minus one. These 48 children were the Multi-Task subjects. This was their first task. Sixteen adults, all parents of young children, subsequently participated in an abbreviated version of this experiment. Each stimulus array included three to five intact objects and two detached parts formed by a single cut of a whole object. Most detached parts could not be labeled with a single label. Three types of arrays were USC&Heterogeneous-Familiar, in which each whole object was a different familiar object, as was the whole formed by the two detached parts; Homogeneous-Familiar, in which each whole object was the same and the two detached parts formed an identical familiar object; and Homogeneous-Unfamiliar, in which each “whole” was an identical nonsense object constructed for use in this experiment. “Intact” nonsense objects consisted of small wooden shapes glued together or held together by pipe cleaners; the two “parts” of a nonsense object, if joined, would nave formed an object identical to the “intact” objects in the array. Stimuli.

Procedure. Two different request forms were used: Specific and General. On trials with a Specific Request the experimenter arranged the objects in

homogeneous arrays, and the superordinate term “toys” for heterogeneous arrays. For unfamiliar stimuli a nonsense word was used to introduce the objects and to request a count. The General Request excluded these labels: “Can you count these things?” The objects to be counted were presented in a line, either in an unaligned arrangement, arrangement, with the two parts about an inch apart (if moved directly together they would have formed a complete object in the same as the whole objects, see Figure 2). Each child had a block of six trials with a General Request and a block of six trials with a Specific Request. The order of request forms was counterbalanced over age. Within each block each type of stimulus set was presented twice, once in an aligned arrangement, Arrangement pairs was random. If the child asked (as some did) whether the parts should be counted, he or she was answered with a noncommittal

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Figure 2.

A Homogeneous-Familiar array with the detached parts aligned: Experiment 1.

When a child asked why the objects were broken, we responded evasively: “They came that way.” The adult subjects were told explicitly that we were interested in adult performance in a task we had used with children. Each adult was asked to count six arrays of familiar objects. These were Separated Homogeneous arrays for 8 adults, and Separated Heterogeneous arrays for the other 8 adults. Both Specific and General requests were used with eacI2 subject. Results

Conveniently, all children spontaneously pointed while counting. So we could base our judgments of response type upon these readily observed points. each part and each whole Three response types occurred: (1) CountCount-Combined: the two object were included in a single count (73%); t also included each whole parts were treated as a single entity in a count object (24%); and (3) Count-Wholes: only the whole objects were counted (2%). (The corresponding cardinal numbers for Figures 1 and 2 are 6,s and 4.) Two-thirds of the children in each age group were inconsistent among these response types from trial to trial. Only one child, a 6-year-old, limited his counts to whole objects. We expected Count-All responses to be made with General Requests to count “things” but not with Specific Requests. The majority of the adults performed as expected.2 However, for the children the frequency of Count‘Eleven of the 16 adults counted all the DPOs separately when asked to count the “things” and combined the two parts with more specificrequests, as we expected. Three adults combined the parts with both requests. However, one adult counted each part separately with both requests, just as the children did, and another adult refused to include any miniature animals in her counts, even when she was asked to count “things”!

Countable entities

Subjects __-

Count-Al!: Wholes plus each part Multi-Task Threes Fours Fives Sixes Single-Task FoursC ~_____

Response types: Entities counteda -

-

-

Count-Combined: Wholes plus combined parts

Count-Wholes: Wholes only

____

115

--

Violations of one-to-one relationsb

88 80 64 60

11 19 35 31

Experiment 1 1 0 1 8

0 0 0 0

38

29

Experiment 2 24

10

“Cell entries are percentages. bSee text. Tinal responses.

All responses with the two different types of requests was similar overall (74% on General Request trials, 72% on Specific Request trials), as well as within each age group. Further, Request Form had no significant effect on the frequency of any other type of response. The main effect of Age was significant. The older the child, the fewer the Count-All responses, F(3,40) = 3.706, p < .02 (Table 1). The Sixes and Fives each counted significantly fewer detached parts separately than either the Threes or Fours (Tukey test, p < .Ol). ‘The other pair-wise age comparisons did 2ot reach significance. There was no interaction with age. T,lle main effect of Stimulus Type was also significant. Parts were most likely to be counted separately on Heterogeneous-Familiar trials (77%); they were least likely to be counted separately on Homogeneous-Familiar trials (68%), F(2,80) = 4.447, p < .02. The third main effect, Arrangement, was also significant. Parts were more likely to be counted separately when physically separated than when aligned (89% compared to 56%), F(1,40) = 48.00, p < .001. Thus, perceptual factors which emphasized that the parts were not complete objects, either by contrasting them with a homogeneous set of familiar objects or by a spatial arrangement appropriate for combining the parts into a whole object, reduced the children’s tendency to count each part separately. Adults, however, were unaffected by the nature of the array. The children were not oblivious to the difference between detached parts

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and whole named objects. On some trials they mentioned “broken” things. Even when such comments were made, each part was counted separately on a majority of trials (57%). However, separate counts of parts were more frequent on trials without comments (74%). Each age group showed this difference. A number of children who counted the two parts separately commented that the two parts were pa:t~ of a single object. Thus, counts of separate parts cannot be attributed to a belief illat each part represented a different whole named object and hence each part contributed to the total number of whole objects. Notice that the different types of responses differ in their processing requirements. The most popular response for the children, Count-All, was a count in which differences between whole objects and detached parts could be ignored. The least popular response for the children, Count-Wholes, was a count limited to intact objects such as whole forks. For such a count the ‘child must (a) decide for each DPO whether or not it is an entity to be counted, and (b) count the intact objects, while (c) ignoring the two detached parts. The processing requirements for a count of intact forks plus the fork forrlaed by :;he detached pieces, Count-Combined, are the most demanding. The child must (a) decide for each DPO whether or not it is to be counted as a single entity, (b) combine the two detached parts to form a single countle entity, and (c) count the whole objects and the entity formed by the etached parts in a single count, while (d) ignoring the discreteness of the detached parts. In general, the children’s counts were characterized by a failure to ignore any DPO, even when such an omission was dictate6 by the wording of the request to count. The adults on the whole counted as the children did when asked to count “things”; however, most adults combined the parts when the entities to be counted were named explicitly. Compared to a s, our child subjects were much less sensitive to the difference between ole labeled objects and detached parts, and to the form of the request to count. In sum, young children have a strong tendency to treat each DPO as a countable entity, even a DPO which is part of an entity explicitly named in the request to count. Although the tendency to count each DPO varied with the display and diminished with age, even 6-year-olds counted detached parts as separate entities on a majority of the trials,

Countable entities

Experiment 2: Counting labeled Objects-Detached

117

parts identified

In this experiment, a partial replication of the previous one, a child who counted two detached parts separately (i.e., made a Count-All response) was told explicitly that the two parts formed one labeled object. We wanted to ensure that the child knew that the two detached parts formed a single entity to be counted. Methcd

Twelve children, 6 girls and 6 boys, participated: mean age 3;10, range 3;5 to 4;2. Subjects.

Stimuli. Six homogeneous arrays from the previous experiment were used. Three of these contained familiar objects, and three contained “nonsense” objects. Procedure. The experimenter placed the stimuli with great precision in an unaligned arrangement. The few attempts by subjects to rearrange the stimuli were discouraged. All requests were specific. The procedure was the same as in Experiment 1 until the subject responded. If the child counted each part separately, then the experimenter said, “Let me show you something. This and this (pointing at the two separated parts) together make a [fork.]. So this and this (pointing again at the two parts) are one [fork]. Can you count the [forks] again?” The child’s initial response and the response following the experimenter’s elaboration of the instructions were scored as in Experiment 1. On trials where the subject immediately responded correctly (i.e., made either a Count-Combined response or a Count-Whole response) the experimenter did not elaborate the instructions. In these cases the subject’s single response was scored as both the initial and the final response. Results

On the first trial, before any elaboration of the task, 83% of the subjects’ initial responses were Count-All responses. This is similar to the responses of the Threes and Fours in Experiment 1 with comparable stimuli. Unlike in Experiment 1, the familiarity of the stimulus items had no effect. The children’s performance changed after the experimenter indicated that the two detached parts formed a single whole labeled project. Over the entire experiment, 67% of the initial responses were Count-All responses, but only 38% of the final responses were Count-All responses.

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Compared to Experiment 1, the experimenter’s identification of the detached parts as a single entity had several effects (Table 1). It not only reduced the frequency of Count-All responses, it also increased the frequency of Count-Whole responses (in which the parts are ignored), and led to two types of errors that violated the one-to-one relation of objects and numerals. One error was to omit one whole object. Such a response suggests that the child understood that a DPG had to be omitted from the count, but did not understand why, and hence did not know which one to omit. The other type of error was a double count of a DPO, either a whole or a deiached part. This response suggests that the child understood that the one-to-one relation between DPOs and the count numerals had to be modified, but did not understand why. In ‘brief, these errors suggest the children were focused on the DPO and count numeral relatiorr, rather than the nature of the DPOs, when they attempted to modify their performance. Such errors provide additional evidence of the saliency of DPOs tor purposes of counting. Summary: Part-whole distinctions

When preschool children were asked to count an array which contained both whole labeled objects and detached parts of comparable objects, they tended to count each part as if it were a whole object. They treated part and whole as equivalent. This bias has been observed previously in studies of measurement. For instance, when 6-year-olds were asked to measure fluids, tbey treated all containers as equivalent even when explicitly told that one type of cont.Crer held twice as much as another type (Gal’perin & Georgiev, 1969). Similarly, when 3- and 5year-olds were asked to distribute equal amounts of chocolate to several recipients, they gave each recipient the same number of pieces of chocolate, even when the pieces differed in size-they treated all pieces as equivalent. If one recipient had fewer pieces than the others, they “increased” its share of chocolate by breaking one of its pieces into smaller pieces (Miller, 1984)! sses as co Given that young children have a strong commitment to discrete physical objects as countable entities, they should consistently respond wrongly when asked to count the classes in an array when each class contains several DPOs. For instance, they should not be able to answer the questions “How many kinds of animals do I have here?” when each kind is exemplified by several objects. A request to count the classes in an array of objects requires that (1) each

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object be assigned to a class, (2) a representation of each class be stored, and (3) the representations of the separate classes be counted while (4) the SF& be ignored. Thus, children’s performance when asked to count classes might reflect the ease of assigning objects to classes, the ease of storing and counting representations of the classes, as well as the demand quality of DPOs for purposes of counting. It is known that classification is more accurate when the classes differ in form or kind rather than in the (ralue of a single property, such as color (e.g., Kagen & Lemkin, 1961). Further, classification performance is known to be better with basic level categories, such as dogs and cats, than with sub-basic level categories such as kinds of dogs (Markman & Callanan, 1983; Waxman, in press). If the difficulty of classification contributes to the difficulty of roranting classesj then comrting perfomance should vary with the nature of the classes to be counted. Hence, in the following experiments we varied the nature of the classes to be cotinted. ?- r anticipate, the nature of the classes had no effect upon children’s ability i,o count classes. In all, we examined children’s ability to count classes in five experiments with a variety of conditions. The method and results of the first experiment will be described in detail. The rationale and method of the four additional ex eriments are summarized in the text, the results are summarized in Table 3. 9 Experiment 3: Counting classes-tievelopmental changes

This task of counting classes in the presence of several exemplifiers of each class shared with the prior experiments the need to ignore the discreteness of individual objects in making a correct count of the requested entities. The Multi-Task subjects participated in both Experiment 1 and in this experiment, which permitted a within-subject comparison of performance in these two tasks with overlapping task demands. Comparison of Multi-Task subjects and Single-Task subjects (who participated only in this experiment) eiraluated the effect of prior experimental experience counting individual objects upon the ability to count classes. Method Subjects.

Multi-Task subjects, who had participated in Experiment 1 a week earlier, were run in Experiments 3 and 6 in the same session. One-half

3A more extensive descripdir:l ur’ these studies is available from the authors.

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of the subjects in each age group participated in Experiment 3 first. (No order effects were found for either Experiment 3 or 6.) Two groups of §ingle-T.i ,k subjects participated: 3-year-olds (6 girls and 2 boys), mean age 3;3, range 2;8 to 3;lO; 4-year-olds (6 girls and 6 boys), mean age 4;4, range 4;0 to 4;8. Stimuli. Six sets of stimulus objects were used. Each set consisted of either two or three homogeneous classes with two to four items in each class. The stimulus objects were small toys. Stimulus sets contrasted either at the basic level (e.g., different kinds of toys), or at the sub-basic level (e.g., different kinds of boats), or ii1 the single property of color. Procedure.

To present the stimulus objects the experimenter poured them from a bag to form a haphazard arrangement. As she presented each set, she described it and asked a question that required a count of the classes in the set. For instance, “Here are some airplanes and some cars. How many different kinds of toys do I have here?” or “Here are some red ducks, and some green ducks, and some yellow ducks. How many different colors do I have here?” The children were allowed to touch, pick up, and move the objects. The stimulus sets were presented in a different random order to each subject. Results

We first consider the Multi-Task subjects. Five types of responses were observed: (1) a correct count of the classes (31%); (2) a count of all the individual objects (37%); (3) separate counts of the objects in each class, fol example, “One, two, three red ducks” (10%); (4) naming each class, for example, “blue ducks, green ducks and red ducks” (10%); and (5) naming individual objects, for example, “This is a blue one” (8%). Subjects failed to respond on 4% of the trials. The frequency of correct responses varied with age, F(3,32) = 15.123, p c .OOl (Table 2). The Fives and Sixes made more correct counts than either the Threes or Fours (Tukey test, p < .Ol). Other pair-wise age comparisons were not significant. Thus, accuracy of performance paralleled that in Experiment 1. (Experiments 1 and 3 are compared further in Section III.) Even in their erroneous responses the older children indicated awareness of the classes-they named the classes or counted separately the objects within each class (Table 2). Qualitatively, the Fives and Sixes who responded correctly appeared to be surprised to be asked a question they found so easy; the children who answered wrongly answered prompt1.yand with confidence. There was no effect of stimulus type: it is no easier to oount basic level

Countable efitities

Table 2.

121

Counting classes: Experiment 3 Subjects

Response type” ___-

Multi-Task Threes Fours Fives Sixes Single-Task Threes Fours

Correct

Count all objects

Count objects in each class

Nme objects

Name classes

3 0 54 67

72 54 8 14

4 10 8 19

4 25 1 0

4 8 28 0

0 3

28 8

2 3

34 25

23 60

YIell entries are percentages. Refusals and unscorable responses occurred on the other trials.

classes (31% correct), than to count sub-basic level classes (29% correct), or classes based upon the single property of color (31% correct). Thus, difficulty in counting classes is evidently not to be attributed to difficulties in assigning the objects to classes. Performance was bimodal. Thirty-one of the 48 subjects never responded correctly; 13 were correct on every trial. Of the Threes and Fours, only two made a single correct response. If these children expected, as the result of participation in Experiment 1, that all our tasks required a count of individual objects, then their failure to count classes might merely reflect this expectation. The performance of the Single-Task subjects indicated that this conjecture is wrong. The number of correct counts did not differ significantly for the SingleTask subjects and the Multi-Task subjects. However, the type of error did. Multi-Task subjects made more responses to indivAra1 objects, either naming or counting them, than did Single-Task subjects, whose modal error was to name the classes (Table 2). Thus, participation in prior counting tasks strengthened children’s bias to interact overtly with DPOs. It is possible that the preschoolers’ apparent inability to count classes merely reflected their expectations abolrt the nature of counting, namely that individual objects are counted. Hence, in two subsequent experiments we provided more information about the task.

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Shipley and B. Shepperson

Experiment 4: Counting classes-Identification

of the entities to be counted

Trials began as they did in Experiment 3. If a child answered wrongly, then the experimenter placed the objects in separate groups by class, labeled each group and repeated the question. If the subject again failed to answer eorrectly, the experimenter touched and identified each group, for example: “This is one kind of animal”, and repeated the original question. We thus attempted to inform the subject of the nature of the required response, stopping just short of a demonstration of the correct response (but see Experiment 5). Identification of the entities to be counted was of some help (Table 3). However, the children’s initial responses were counts of individual objects on over 90% of the trials and only two of 24 children appeared to gain insight into the task. The children gave their wrong answers quickly and with confidence. When the experimenter tried to assist them, they were puzzled by the repetition of the request.

Table 3.

Summary of experiments requesting counts of c!asses _ ._.___ ___ ___. Experiment

Subjects Mean age

N

Correct responsesa Never correctb

Initial

FinalC

83 100 25 33

3 0 33 33

100 83 38 23

0 3 9 29 First session 19 79 Second session 38 79

Multi-task 3: Single request 6: No objects

3: Single request 4: E identified classes 5: E counted classes

7: One object/class and question repeated

3;3 3;ll 3;3 3:ll

3;3 4;4 3;lO 3;ll

3;9

12 12 12 12 Single-task 12 12 24 13

12

0 .___-_

_____

Tell entries are percentage of responses. bCell entri es are percentage of subjects. ‘Initial correct responses were also counted as final correct responses.

28

100

-___

~-

Countable entities

Experiment 5: Counting classes-Demonstrations

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of cy!rect counts

Again, the presentation of stimulus arrays and the experimenter’s original question were the same as in Experiment 3. If the child seemed puzzled the experimenter repeated the question. If the child answered wrongly by naming the classes the experimenter asked a leading question: “How ‘*manyis that?” If the child gave some other wrong response the experimenter provided infor.mation as in Experiment 4. If the child “igain answered wqongly the experimenter counted the classes (“Here’s one kind, two kinds, three kinds”) while pointing to the groups, and then repeated the question. Leading questions and demonstrations continued until the child answered correctly or gave the same wrong answer two times in a row. The procedure was repeated a week later to obtain a more sensitive index of insight into the task. Demonstrations were helpful, although not for every child (Table 3). Further, no child’s initial responses were correct on all trials. Individual subjects demonstrated three patterns of errors: early insight (23 %)-a few errors on the first trials of the first session; late insight (54%)-consistent errors in the first session; or no insight (23%)-at most one correct final response in the second session. The three types of subjects differed in the types of errors they made. The early-insight subjects miscounted classes, the late-insight subjects named classes, and the no-insight subjects made overt responses to individual objects-they consistently exhibited a DPO bias in spite of seeing 13 or more demonstrations of a correct count. Again we have evidence for a strong bias to count DPOs. Experiment 6: Counting names in the absence of objects Is it the multiplicity of visible members of each class that makes it difficult for young children to count the number of classes in an array of physical objects? In this procedure no objects were presented; to answer correctly the child had to count the words in a list: words that were either the names of kinds, proper names or nonsense words. (Performance with nonsense words was significantly worse and is not considered further.) The experimenter provided a context for a list of two or three items, and then asked how many items were on the list. For instance, “I know a farmer who has horses and dogs [and cows] on his farm. How many kinds of animals does he have?” Thus, just as in Experiments 3,4, and 5, the entities to be counted were each named b&m the “How many” tpestion was asked. However, the procedure most closely resembles that of Experiment 3 in that no additional assistance was given if a child answered wrongly. We did repeat the question if the child requested it, as frequently happened on the first trial.

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The Multi-Task subjects participated and age had its usual significant effect upon the frequency of correct responses, F(3,32)=11.563, p C .OOl. Both the Fives and the Sixes gave significantly more correct responses than either the Threes or the Fours (Tukey test, p C .Ol) and no other pair-wise age comparisons were significant. Performance was better in the absence of objects (Table 3). The Age X Task interaction was not significant. Even 3-j.?ar-olds counted more accurately in the absence of objects, F(1,32) = 5.818, p < .05 (Planned Comparison). Could the superior performance in the absence of objects have bzen due to chance? After all, if the Threes and Fours had consistently guessed “Two” or “Three” rather than counted, they would have been correct on 50% of the trials. However, the possibility that the youngest children were guessing when asked to count names is ruled out by comparison of their counts and the guesses of a control group. The control group was asked to guess the answers to questions comparable to the experimental questions. The guesses were distributed evenly among the first five digits, the counts clustered around the actual number of names. We concluded that even the youngest children had a greater ability to count the names of classes in the absence of objects than in the presence of objects. Experiment 7: Counting classes: One exemplar per class

If young children ha ve a strong bias to count DPOs, and if this bias interfe?-es with their counting classes in the presence of several members of each C&S, then their performance should be better when only one member of each class is present. To test this prediction we presented 3 or 4 objects, each of a different kind, and again asked the child “How many kinds?” The question was repeated if the child answered incorrectly. The most common incorrect response was to name the individual objects on 58% of the trials (we suspect the children were observing a conversational convention). Upon repetition of the “how many” question, all children answered correctly. Thus, young preschool children had no difficulty in counting the number of classes in an array in the presence of a single exemplar of each class. In contrast, more heroic efforts than merely repeating the question were required to achieve a much lower frequency of correct responses in Experiments 4 and 5 where several members of each class were present (Table 3). Clearly, the multiplicity of visible members of each class interferes with counting the number of classes.

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e counting classes tas

Is the bias to count each DPO the result of learning to count or does it precede learning to count? To answer this question we examined the performance of 2-year-old children who could not count, as well as older children who could count, when they were asked to name the classes exemplified in an array. We also considered a second question: is the bias to count each DPO the only source of difficulty when children are asked to count classes? ‘To answer this question we compared the performance of the Multi-Task subjects in different tasks. Experiment 8: Naming classes-Age and counting ability

There are at least two possible explanations for the prevalence of responses to DPOs in counting tasks: (1) young children incorporate part of their first learned counting procedure, a procedure which requires that each object be processed separately, into every counting task; (2) young children have a bias to interact with individual DPOs which is neither limited to nor derived from counting. To evaluate thk:se two alternatives we asked children to name the kinds or the property values Lxemplified in a stimulus array. Each specific kind, or each value of a property, was represented by several instances. This procedure shared with the Counting Classes task the need to examine each item, to assign it to a class, and to store a representation of each class. However, this procedure was simpler than counting classes -the resulting representation needed only to be named, not counted. Method Subjects.

Eighty-four children participated: (1) the 48 Multi-Task subjects who had participated in counting tasks in Experiments 1, 3, and 6 in prior sessions; (2) 12 Single-Task subjects who could count: 6 girls and 6 boys, mean age 4;4, range 4;0 to 4;s; and (3) 24 Single-Task subjects who could not count: 12 girls and 12 boys, mean age 2;9, range 2;7 to 3;0. The 2-year-olds were assigned to two groups of 12, matched on age and sex. One group participated in the Naming Classes task, the other in a control condition-the Put-Away task. Stimuli.

Three sets of stimulus objects were used with nine objects in each set: airplanes of three different colors, blocks of three different shapes, and toy animals of three different kinds.

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Procedure. The experimenter presented a set of objects in a haphazard array and asked “What colors (shapes, kinds of animals) do I have here?” The question was repeated if the child did not respond. In the Put-Away control task the child was asked to place all the objects of a stimulus set in a box through a small opening. The opening was just large enough to insert one object at a time. The experimenter recorded the order in which the child placed the objects in the box. Results

To satisfy the experimental request the child needed only to provide the names of three classes, for example, “Red, yellow, and blue”, without touching or pointing at any specific stimulus object. These responses we called Name-Class responses. overall, the Multi-Task subjects gave Name=-Class responses on a bare majority of the trials (57%). In addition to the NameClass responses, the children either: (a) named individual objects (26%), (b) counted individual objects in a single count (lo%), (c) named and counted the members of each class (4%), or (d) counted the classes (2%). See Table 4 for the response frequencies in each age group. Again age had its usual significant effect, F(3,40) = 14.562, p < .OOOl. The Fives and the Sixes gave significantly more Name-Class responses than either the Threes or the Fours (Tukey test, p c .Ol) and no other pair-wise age group differences were significant. The older children evidently felt no need to supply information about individual objects. In contrast, the majority of responses of the Threes and Fours were to individual objects, either naming or counting them (Table 4). The occurrence of unsolicited counting responses suggested that the prior Table 4.

Naming classes: Percentage of each response type, Experiment 8 .__ Subjects

Multi-Task Threes Fours Fives Sixes Single-Task Twos: Non-counters Fours: Counters ___~

Response type Name t&es

Name objects

Count

17 36 92 83

44 50 3 8

33 14 6 8

3 75 ~___

97 19

0 6 -__-

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counting tasks biased the children’s performance toward counting, and perhaps toward processing individual objects. The performance of the 4-yearold subjects who had no prior counting tasks supported this conjecture (Table 4). They gave significantly more Name-Class responses than the Multi-Task Fours, t(22) = 2.57, p < .02. The fact that prior counting tasks predisposed children toward an overt response to individual objects provides additional evidence for young children’s commitment to DPOs when counting. However, the question remains: is this bias the result of having learned to COUL: or does it precede counting? TQ answer this question we examined the performance of 2-year-olds who could not count in the Naming Classes task. They consistently touched and named individual objects (Table 4). Only one child, on only one trial, named three classes without touching any object. Did the 2-year-olds’ apparent preoccupation with individual objects reflect a misinterpretation of the task, perhaps as a task where individual objects were to be classified? We think not. Examination of the order of the children’s responses indicated they were attempting to answer the experimenter’s question, namely to provide the names of all the kinds or properties in the array. Objects from three different classes in the stimulus array were touched and named first on 35% of the trials. By chance, such a pattern is expected on less than 5% of the trials. The evidence that the children were attempting to answer the experimenter’s question is even stronger when the Naming Classes and Put-Away tasks are compared. Tn the Naming Classes task only one child ever initially touched three identical objects. In the Put-Away task 9 of 12 children initially picked up identical objects on one or more trials (p < .Ol, exact chi-square). That is, when asked to name classes children selected objects from different classes in their initial responses; when asked to put away objects children touched similar objects in sequence, as they do in other contexts (e.g., Mandler, 1986; Nelson, 1973). Thus, the &year-old subjects in th(y Naming Classes task were sensitive to the requirement that the range of properties or kinds in the stimulus array was to be reported. However, they were unable to abstract properties or kinds away from the individual objects in which they were exemplified. In sum, the bias to interact with each DPO is not a consequence of knowing how to count; it precedes counting.

What processing demands make it difficult for young children to count classes? To answer this question we compared the performance of the Multi-Task

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subjects in three different tasks that overlapped in processing de;pands. The parts must be combined or tasks are Counting Objects, in which ch representations of classes ignored (Experiment l), Counting Clas must be stored and counted while indi 0s are ignored (Experiment ations must be stored and 3), and Counting Names, in which counted (Experiment 6). ired in two tasks for a correct If a process, such as ignoring DPOs, is performance, then children who are sue ignoring DPOs should perform well in both tasks and children e difficulty ignoring DPOs should perform poorly in both tasks. However, we already know that age is closely related to performance in the experiments we have run. Hence, to evaluate the effect of the ability to ignore DPOs we must remove the effect of age upon performance. This was done by computing partial correlations between performance on two tasks with the effects of age removed statistitally .

The pair-wise correlations of correct pe ce on Experiments 1, 3, and 6, with age partialled out, are given i 5. When correct performance required the child to ignore DPOs (Experiments 1 and 3 ) performance is positively correlated. However, when correct performance had no processing components in common other than an ability to c:ouni {Experiments 1 and 6) therz is no correlation in performance. Hence the positive correlation of performance in Experiments 1 and 3 must be due to the common component of these two tasks- the need to ignore DPOs-rather than to some general ability. A second source of difficulty in children’s performance in the counting classes task is also revealed by the partial elations. Performance on the two tasks that required a count of represe ons (Experiments 3 and 6) is positively correlated (Table 5). Hence, the to store and count represenTable 5.

Partial correlations of correct counts of the Multi-Task subjects with the effect of age removed. Experiments 1 and 3 both require that discrete physical objects be ignored; Experiments 3 and 6 both require a count of representations __ ____~____ __- -_ .__ Experiment 3: Count classes ~____. Experiment 1: Count objects Experiment 3: Count classes I_. *p < -02; **p c .Ol.

+.395**

Experiment 6: Count names + .029 + .335*

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tations also contributes to the difficulty of counting classes in the presence of physical objects. In brief, correlations of the performance of the same subjects in different tasks reveal that the requirement that DPOs be ignored, and the requirement that representations of classes be counted, are two separate sources of children’s difficulties when they are asked to count classes of objects. Summary: Components of the counting classes task We have asked three questions about the sources of young children’s difficulties in counting the entities we requested they count. The first was: does the multiplicity of visible members of each class make it difficult for young children to count the number of classes in an array of physical objects? The answer is clearly “yes”. Children were better at counting the names of classes in the absence of any objects (Experiment 6), as well as in the presence of a single exemplar of each class (Experiment 7), than in the presence of several exernplars of each class (Experiment 3). The second question asked: which comes first-a bias to count each DPO or the ability to count? The performance of the 2-year-old non-counters asked to name the classes exemplified in an array (Experiment 8) provides a clear answer. The DPO bias precedes counting. Our third question asked about the processing demands that hinder performance when children are asked to count classes in the presence of several exemplars of each class. Partial correlations revealed that both a bias to attend to DPOs and difficulty in counting representations contribute to the difficulty of counting classes in the presence of objects. General discussiou Summarizing our overall findings, we have shown that preschoolers have a strong bias to count each discrete physical object as a separate entity.4 They 41n response to a referee who asked about the effect of collective terms on children’s counts of classes. we should mention the results of a pilot study run on the Multi-Task subjects after they completed the work described in this paper. They were given precise arrangements of stimulus objects and requests to count that included collective terms such as tower, team, or family. Since collective terms increase the coherence of classes of objects in the Piagetian class-inclusion task (Markman. 1973). we expected that the use of collective terms would improve the children’s ability to treat a class containing several objects as a single entity in counting. As expected, counts of classes were much better with collective terms. However. the DPO bias continued to influence the performance of the youngest children. The Threes counted individual animals (in lines) or

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did so in a variety of situations and with a variety of instructions. Three- to 6-year-olds counted detached parts separately whether asked to count “things” or to count specific entities such as “forks”. When explicitly told that two i.etached parts formed one of the entities to be counted, over a third of the time the children persisted in counting each part separately. Three- and 4-year-olds had great difficulty in counting entities such as “kinds of animals” when each kind was exemplified by several separate objects; they counted each object separately. Even with demonstrations of correct counts, some children persisted in counting individual objects. In the absence of all objects their performance was much better. Only in the presence of a single exemplar were they consistently able to count classes correctly .

Finally, we showed that the bias to process each DPO separately is not limited to counting tasks. It was exhibited in a different task by 2-year-olds who could not count. To conclude, we will propose that the bias to process discrete physical objects is a general disposition that is manifest by humans in many contexts and at different ages and that it helps children learn to count. To support this c.onjecture we will contrast humans and chimparlzees, who apparently lack this bias, with respect to their number-related competencies. Finally, we will compare general dispositions and counting principles (Gelman & Gallistel, 1978) as ways to account for the child’s ready mastery of counting. 1. The discrete physical object bias as a general disposition

We submit that young children’s bias to count DPOs is an instance of a general disposition to process separately each individual plqysical object. Such a disposition is exhibited by young infants who find discrete physical objects salient (Spelke, 1984). This disposition could be an unde+nning for a broad range of human cognitive activities. For instance, t is disposition can explain why labels are assigned to single intact objects rather than to composite entities made up of parts of different physical objects (Chomsky & Walker. 1978; Locke 1690/1956): individual a>bj.zctsare more salient than parts of miniature players (in circles) rather than “families” or “teams” on 85% of the trials; they counted the individual blocks in the towers rather than the “towers” on one-half the trials! Further work with collective terms is in progress. Note that the individual objects were parts of collections thai formed “good” perceptual wboles-lines, circles, and especially towers. In fact, individual blocks were difficult to distinguish once the tower was built. Thus, the children who comted individuai blocks must have been using their memory of the separate components of the towers, rather than the appearance of the towers, to decide what to count. In contrast, in Piagetian conservhtion tasks (e.g., Siegler, 1981) same-aged children make their judgements on the basis of appearance, rather than their memory of prior states.

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objects, either taken singly or in combination with parts of other objects. The DPO bias can also explain why nouns are learned before verbs (Gentner, 1982): an object is more salient than any property, act, or disposition of that object. And this bias can help to explain why counting is learned so early (Gehnan & Gallistel, 1978): for the novice counter, the DPO bias unambiguously nominates the participants in a one-to-one correspondence with numerals. 2. The discrete physical bias as a basis for numerosity It seems obvious on intuitive grounds that the more salient an organism finds the “oneness” of the entities it encounters, the easier it should be for the organism to learn to ?:ount those entities. The results reported here indicE:e that the oneness of DPOs is highly salient for young children.5 In contrast, consider chimpanzees. Chimpanzees resisted learning a one-tnone correspondence in which one small cube was to be placed on each individual block. Instead of complying, they put one cube on two adjacent nontouching blocks or one cube on a tower of several blocks. Further, they failed to put cubes on slightly different items, Tuch as a larger block (Premack, 1976). Although Premack’s star pupil, Sarah. eventua!!y mastered the task, she did so only after many sessions of pcssive guidance and modeling of the correct response. In contrast, most 3year-old children who could not count mastered this task with a single demonstration (Solter, 1975). Given this difference in the saliency of oneness for chimpanzees and children, chimpanzees should find it more difficult to learn number-related skills, including the ability to count. Indeed, Premack (1976) reported that he failed in h.is attempts to teach chimpanzees to count and related this failure to the animals’ poor performance in tasks he regarded as precursors to counting, such as the one-to-one matching task. Of course, it is a matter of definition to decide when a member of a non-human species can count or has the concept of number, or even what are comparab e tasks ior members of differect species (Davis & Pt,usse, 1988). Nevertheless, it seems clear that number is a more “natural’* concept for humans than for chimpanzees (Davis & Pcrusse, 1988). “We do not mean to suggest that DPOs are the only entities whose “oneness” is higAly salient for humans. Clearly some sounds are, even for infants (Starkey, Spelke & Gelman, 1983). and perhaps some movements as well. For example, one 4-year-old in Experiment 5 watched the experimenter taunt separate piles of objects by gently placing bee hand on each pile. Unable to interpret th% action as a count of DPOs. he interpreted it as a count of movements. He then proceeded to count as he slapped objects. first in piles and then separately as they were displaced by the slaps, witb apparent indifference to which objects he touched. Whether discrete movements are highly salient for younger children and whethsr they play a role in learning to count is not yet known.

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For instance, Matsuzawa (1985) trained a chimpanzee to “identify” color, type of object and the number of objects in arrays of small physical objects by pressing keys on 2 keyboard. For the first learned numerosity discrimination - that between one object and two objects- 1,821 trials were required to reach a criterion of 90% correct. Comparable color and shape discriminations had been learned previously in far fewer trials. In contrast, 92% of children vho could not count five objects (mean age 3;9) immediately responded correctly when shown two objects and asked “how many” (Sch.?.effer, Eggleston & Scott, 1974). Moreover, humans discriminate the size of heterogeneous sets of objects in the range of one to four objects at 8 months of age (Starkey, Spelke, & Gelman, 1980). Although these tasks differ, it would be hard to argue either that the performance of the humans was inferior or that it was the result of greater training. Chimpanzees are not without quantitative competence when continuous quantities are involved. For instance, Premack’s Sarah could match probjortions of different entities, such as half of an apple and a half-filled cylinder (Woodruff & Premack, 1981). She could conserve liquids and solids. However, she could not conserve number; she used the length of a line, rather than the number of objects in each line, to decide if two lines were equal (Woodruff, Premack & Kennel, 1978). In contrast, children conserve number at a younger age than they conserve solids or liquids (Siegler, 1981). Thus, when an educated chimpanzee matched the competency of children with respect to continuous quantities, her performance was deficient in tasks that required a sensitivity to the oneness of objects. In brief, we suggest that these differences between humax and chimpanzees result from differences in sensiiivity to the discreteness-the oneness-of DPQs. We hypothesize that the absence of a DPO bias makes numerosity an obscure concept for chimpanzees and handicaps them in various numerical tasks, while the possession of the DPO bias predisposes children to perform in ways that are helpful in learning to count. 3. General disposition 2nd counting principles Two issues will be considered in a comparison of general dispositions and the counting principles of Gelman and Gallistel (1978). One is the specific relationship between the DPO bias and the abstraction principle as a specification of what to count; the second is the role of general dispositions, compared to counting principles, in the child’s mastery of counting. The DPO bias, a disposition to process separately each movable bounded solid, characterizes what young children attempt to count-at least in the realm of physical objects. To repeat, our youngest subjects had difficulty

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counting entities such as kinds which were exemplified by several DPOs; they tended to count each part of a broken ect separately when the entire object was specified in-the request to co ; and they never counted separately the attached tail of an animal, the sail a boat or the wheels of a car, even when asked to count “things”. DPG bias is sufficient to describe what the beginning counter counts. There is no need to endow the child with the abstraction principle as a guide in learning to count, even though the abstraction principle may come to characterize children’s counting at an early age. More generally, could the child’s initi counting attempts be guided by general dispositions, with mastery of the unting principles growing out of the child’s initial counting experience? First, let us consider how the counting principles could guide learning to count. Gelman’s (1986) discussion of a child exposed to both labels and 0s applied to the same set of objects provides a concrete illustrr hypothesized role of counting principles. According to Gelman “the c ng principles enable a toddler to recognize when words are being used as t words” (p. 27, our italics). The prior knowledge implied by the phrase tc recognize ISpresumably knowiedge of the counting principles. The specific counting principles that mediate the recognition of count the stable-order principle (count words as count words rather than labe saliency to ordered lists of words; invariant order) which ities can be counted in any order) evance principle (a set which identifies as equivalent different uses of the count sequence with the same set of entities, no matter what the order in which the entities are counted; and the abstraction principle (any set of entities can be counted) which identifies as equivalent different uses of the count sequence with different entities. In contrast, we suggest that general biases or dispositions could facilitate, at least to some extent, learning to count and mastery of the counting principles. One way in which this could occur would be mediated by the child’s motivation to imitate. When a non-counting child observes the behavior of a person who is counting, suppose he incorporates those aspects of thz counting routine that are in accord with his own dispositions into his imitation of the counting behavior. Thus, the DPO bias would ensure that it is individual objects that the child attempts to count- even if the model counted something else! However, a DPQ bias alone would not ensure that the child imitates all

relevant aspects of the counting situation. Two other general dispositions that may be useful are a bias to form complementary pairs and a bias to exhau.stively process objects. These further biases are exhibited by children in tasks other than counting, just as the DPO .bias is exhibited in tasks other than counting.

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The disposition to form complementary pairs is seen in what Sugarman (1983) has called “between-class correspondences” in which young children pair two types of dissimilar entities in a specific way, such as pairing a doll and a ring by placing the doll in the ring. By 18 months children form sequences of such identical pairs (Sugarman, 1983). (See also Wagner & Walters, 1982, on “correspondences”.) Evidences for an exhaustive processing bias is seen in the readiness of 30-month-old children to point at each object in an array when requested to point (Potter & Levy, 1968), in the sensitivity of young children to nonexhaustive counts (Briars & Siegler, 1984), and in the tendency to use all available objects in spontaneous constructions (Sugarman, 1983). (See also Wagner & Walters, 1982, on “list exhaustion scheme”, a tendency to exhaustively process both objects and count terms.) Taken together, these dispositions could predispose the child to imitate the appropriate aspects of a counting routine and, more generally, could prepare the child to abstract the one-to-one principle from observation of another person counting objects one by one. The DPO bias would ensure that individual objects are attended to, the e xhaustive processing bias would ensure that every object is attended to, and the disposition to form complementary pairs would ensure that each object is paired with an instance of some other kind of entity-a numeral in the case of counting. Conjectures such as those given above are a long way from establishing a primacy for general dispositions over counting principles in the young child. Further, even if the requisite dispositions were exhibited in other situations before the child could count, it would not necessarily establish the primacy of general dispositions over counting principles. The dispositions could bc considered partial manifestations of the counting principles. However, to the extent that dispositions such as the DPO bias are component% of cognitibse activities other than counting, it may be more parsimonious to say that general dispositions guide the child to counting and to the counting principles, rather than that the counting principles guide the child to learn to count. In sum, we have demonstrated the existence of a bias on the part of young children which characterizes children’s initial attempts to count. We have presented evidence that this bias precedes counting and suggested that it plays an important role in many areas of cognitive development for humans. We have argued for the usefulness of the DPO bias in learning to count by comparison of humans and chimpanzees. Finally, we have raised, but not answered, the question of the relative priority of general dispositions and counting principles in guiding the child to count.

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