Units of counting: Developmental changes

Cognitive

Development,

ISSN 0885-20

13, 561-585

(1998)

0 1998 Ablex Publishing All rights of reproduction

14

reserved.

UNITS OF COUNTING: DEVELOPMENTAL CHANGES Catherine Sophian Christina Kailihiwa University

of Hawaii

Insofar as counting is directed toward a definite quantification goal, only items that qualify as valid instances of what is being quantified should be included in the count. Thus, the choice of what to treat as a unit to be counted depends upon one’s quantification goals. The present research examined developmental changes in the way children define units for counting. In the first experiment, children were shown arrays of toy animals and asked to count either the number of families or the number of individuals within a family. In the second and third experiments, the stimuli were objects that came apart into two pieces. Children were shown arrays composed of some intact objects and some objects that were separated into their parts, and they were asked either to count the number of wholes or the number of pieces in the entire array. Virtually all the counts children generated were based on some type of common unit, even if it was only defined by physical discreteness. However, marked age differences emerged in children’s adaptation of their counting units to what they were asked to count.

Although much progress has been made in elucidating children’s early knowledge about how to count (e.g., Fuson, 1988; Gelman & Gallistel, 1978), the extent to which very young children understand the quantitative significance of counting and the mathematical relations that give it that significance remains unclear. Much of the debate surrounding this question has turned on whether or not young children understand that the last number in a count is the numerical value of the whole set. Even 3-year-olds give evidence of treating the last number in a count as some-

The authors express appreciation to Danielle Garyantes for collecting the data for Experiment 2. Thanks are also extended to the children and staff of the following schools for their assistance with the research: Alawai Elementary School, The Carey School, Hongwanji Mission School, Kailua Baptist Preschool, Mililani Uka Elementary School, St. Anthony School, St. John’s Mililani Montessori Center, University Ave. Baptist School, and the University of Hawaii Lab school. Direct

all correspondence

Road, University of Hawaii,

to: Catherine Honolulu,

Manuscript received April 22,1997;

Sophian,

HI 96822

Department

of Psychology,

2430

Campus

.

revision accepted January 1, 1998

561

562

Sophian and Kailihiwa

how special-repeating it, or saying it with extra emphasis (Gelman & Gallistel, 1978). Yet there are marked age differences in performance over the preschool years when children are explicitly asked how many objects are in a set they have just counted (Fuson, Pergament, Lyons, & Hall, 1985; Schaeffer, Eggleston & Scott, 1974) and there is evidence that even when children respond correctly to such questions they may not fully understand how the last number in a count functions as a representation of the cardinal value of the set. Fuson (1988) asked children, when they gave a final-number response, to indicate which objects(s) it referred to. Instead of pointing to the whole set, children often indicated only the last-counted object, suggesting that they had learned to answer “how many” questions by repeating the last number without actually understanding its cardinal significance. Consistent with this conclusion, Bermejo (1996; Bermejo & Lago, 1990) showed that young preschoolers often give the last number in a count as the cardinal value of a set even under circumstances where that is incorrect (e.g., when they have been instructed to begin counting with a number other than one). Neverthless, Bermejo, like Fuson, found that performance improved markedly over the preschool period. The present research addresses children’s understanding of counting from a related perspective, viewing counting as a form of measurement (Miller, 1989) in which, as in other kinds of measurement, the choice of a unit of measure is critical. If we want to characterize the weight of something, we must use a scale to quantify it in units such as ounces or grams; whereas, if we want to know its length we will instead measure it in inches or meters, using a ruler or tape measure. In the same way. depending on how we want to quantify an array we are counting we may need to choose different units of measure. If we are interested in community participation in some event, for instance, we may wish to characterize the number of families in attendance or the number of individuals. Depending on which we choose, we will get different results, and so we need to choose a unit of measure that is appropriate to what we are trying to find out. To fully understand the quantitative significance of counting, then, children need to understand it as a form of measurement that requires the selection of an appropriate unit of measure in relation to the quantitative goal we have. In one of the few lines of research to investigate young children’s selection of units for counting, Shipley and Shepperson (1990) observed marked age differences over the preschool period. In several experiments, they asked 4- and 5-yearold children to count arrays composed of some intact objects and some that had been broken in half. Particularly at 4 years of age, children were reluctant to count the two halves of an object such as a fork as a single item, and they were no more likely to do so when explicitly asked to count forks than when asked simply to count things. In other experiments, Shipley and Shepperson asked children how many colors or how many kinds of things were present in arrays containing several exemplars of each color or kind. Again the 4-year-olds tended to count each

563

Units of Counting

individual item separately, rather than treating a group of similar objects as a single unit. Two possible interpretations of children’s difficulties in Shipley and Shepperson’s (1990) research have very different implications for their understanding of counting units. On the one hand, it could be that they do not shift from counting discrete items to counting other kinds of units because they do not understand the significance of the counting unit (Gal’perin & Georgiev, 1969). In effect, on this view they focus on individual items just because that is the most familiar way to count, and they have no understanding of the need to adopt other counting units to meet certain quantitative goals. On the other hand, it is possible that children do have some notion of the significance of the counting unit, even at 4 years of age, but discrete objects are so much more salient for young children than alternative units composed of several such objects that they fail to adapt their counting appropriately. Three experiments were carried out to evaluate these alternative views of young children’s limitations in counting units other than discrete objects. The first experiment was an extension of Shipley and Shepperson’s (1990) research on children’s counting of classes versus individual members of a class, using modified procedures designed to reduce the linguistic and perceptual difficulties of the countingclasses task. Experiments 2 and 3 used arrays composed of objects that could be separated into two component pieces to further characterize the nature of developmental limitations in children’s adoption of alternative counting units,

EXPERIMENT 1 Shipley and Shepperson’s count-classes task is of particular interest in relation to children’s understanding of counting units, because it poses a choice between two alternative units-the discrete items (which were the basis for most of the 4-yearolds’ counts in Shipley and Shepperson’s research), and the classes to which they belong (the unit dictated by the count-classes instructions). A variant of the countclasses task was therefore used to assess young children’s appreciation of the possibility of using more than one kind of unit to enumerate an array. Across problems, children were sometimes asked to count individual items and sometimes to count classes in order to evaluate their ability to adopt alternative counting units. In order to adequately address this issue, however, it is important to be sure that children can recognize the aggregate units-the classes-and understand the verbal references to those units. In Shipley and Shepperson’s (1990) work, classes were referred to as “kinds”, but a potential concern about their results is that the younger children may not have understood what a “kind” is. This concern is particularly serious because if children simply skipped over the unfamiliar word, “kind”, when asked how many kinds of things were in an array, they could easily have confused that question with a request to count the “things”, that is, the individual items. In the present research, the arrays were composed of animals of sev-

564

Sophian and Kailihiwa

era1 types and the word, “families”, was used to refer to groups of like animals. Besides the familiarity of the term “family”, an advantage of this wording was that the “families” questions did not include any explicit reference to individual items; children could be asked, “How many families” were in the array without mentioning “animals” or any other term that could be applied to the individual items. A related concern was that even if children understood the count-classes questions, they might have difficulty identifying the pertinent units in the course of conducting a count. In Shipley and Shepperson’s work, the different kinds of items were all intermixed in the array, so that to identify the distinct kinds while keeping track of the ones that had already been counted may have posed a substantial cognitive load. Children might have persisted in counting individual items, then, because they were not clear what the classes were within the arrays or because it was too difficult to identify and keep track of them as they counted. In the present research, spatial groupings were used to facilitate the identification of the aggregate units, the “families”. Moreover, problems in which children were asked to count the members of one particular family, identified by the type of animal (e.g., “How many rabbits are in the rabbit family?‘), provided independent information about children’s ability to discriminate the family units from each other. Method Participants. Twenty 4-year-old children (4 years, 4 months to 5 years, 0 months; M = 4 years, 8 months) and sixteen 5-year-olds (5 years, 1 month to 6 years, 0 months; M = 5 years, 8 months) participated in the experiment. There were approximately equal numbers of boys and girls at each age. The children were recruited from day care centers, preschools, and elementary schools they attended. They were predominantly middle-class and represented the diversity of ethnic backgrounds typical of the region. All of the children in the older group, and none of those in the younger group, attended kindergarten. Materials. Small toys representing birds, rabbits, horses, bears, and trolls were used as stimuli. There were five toys of each kind, one or two of which were slightly larger than the others but matched them in coloring and other features. The toys ranged from 1.6 cm (the smaller birds) to 9 cm (the large troll) in their greatest dimension. A yellow toy truck, with a rear platform that measured 25 cm long, was also used. Procedure. Children were tested individually by a female experimenter in a single session that lasted about 20 minutes. During this time, 16 counting trials were presented, which varied in whether the target units were the families or the individuals comprising one family and in whether the correct enumeration was 2, 3,4, or 5. For comparability to the family-count trials, individual-count problems also varied in whether 2,3,4, or 5 families were presented altogether. Each family

Units of Counting

565

within an array was represented by at least two and no more than five individuals, and at least one of the individuals within a family always differed in size from the other(s) so that none of the families was homogeneous. The numerosities of the families varied within as well as across arrays, and the particular animals used within each array, as well as the number of members comprising that family on a given trial, was determined randomly. Within each problem, which of the families that were present would serve as the focal family for that problem was also determined randomly. On individual-count trials, the focal family was the one the child was asked to count. On family-count trials, the focal family was the one the child was asked to place on the truck as an exemplar of the units to be counted. On individual-count trials, the experimenter began by identifying the family to be counted, and asking the child to put one exemplar on the truck. No child needed any assistance in carrying out this instruction. The experimenter then said (choosing the correct term for the focal family), “Good, here is one bird/rabbit/ horse/bear/troll. Count all the birds/rabbits/horses/bears/trolls. How many birds/ rabbits/horses/bears/trolls are here?’ On family-count trials, the experimenter told the child that this time they were going to count families, and asked the child to put a particular family (e.g., the family of birds) on the truck. Occasionally children initially put only a single bird on the truck, and then the experimenter prompted them, saying, “Put the whole family on the truck”. Once this instruction was carried out, the experimenter said, “Good, here is one family. Now count all the families. How many families are here?’ For both kinds of problems, the items were presented in a clustered fashion, with members of a family near each other but not systematically lined up. The experimenter recorded the child’s counting verbalizations and noted what units they were applied to. On the rare occasions when the experimenter could not tell what the child was counting, she asked the child to show her. Children were not given any feedback about the correctness of their responses. Before the experimental problems, a warm-up procedure was carried out that was designed both to introduce children to the families of animals that would be used and to familiarize them with the counting tasks. In the first phase of the warm-up, the experimenter introduced two families. She put out several exemplars of the first family (e.g., birds), asked the child what they were, provided a label if the child could not produce one, and then told the child that all the birds (bears, horses, rabbits, or trolls) were in the same family. After introducing the second family in the same way, she asked the child to count one type of animal (e.g., the birds) and then to count the families. She prompted the child, and modelled the procedure as necessary, to elicit correct counts of both kinds. In the second phase of the warm-up, she introduced the remaining three families in the same way and again elicited both an individual-item count and a family count. This time, however, she also introduced the truck and asked children to put exem-

566

Sophian and Kailihiwa

Table 1. Mean Proportions and Standard Deviations of Family-count Problems on Which Children Counted Individuals versus Families (Experiment 1) Age Group 4-year-olds

Count type

S-year-olds

-___ M

SD

M

SD

.34 .23 .04 .39

.40 .36 .I2 .46

.I2 .II .OO .71

.26 .26 .OO .41

Counted individuals Within one family Combining families Restarting at one for successive families Counted families

plars on it just as they would in the experimental

trials, so as to demonstrate

-

how

it would be used.

RESULTS

The counts children produced on each problem were classified according to whether they counted individual items or families. Individual-item counts were further subdivided into three categories: those in which children counted the individuals within just one family (the correct response on individual-count trials), those in which children combined individuals from more than one family in their counts, and those in which children counted individuals within more than one family but restarted at one for each successive family. It was possible to classify every one of children’s counts as either a family count or one of these three types of individual count. On the individual-count trials, children always counted individual items and, with only one exception (a 4-year-old, on one trial), they restricted their counts to individuals comprising just one family. Moreover, that family was invariably the correct, focal, family. On the family-count trials, a wider variety of count types were produced; the proportions of each problem type on which each type of count was produced are given in Table 1. Counts of families were produced more often than any other type of count on these problems, although among the 4-year-olds they still accounted for less than half of children’s responses. Because the distribution of family-count responses on the family-count problems was markedly bimodal, non-parametric contrasts were used to compare the age groups. The numbers of children within each age group who produced different numbers of family-count responses can be seen in Table 2. Only six of the twenty 4-year-olds (30%) produced family counts on all eight of the family-count trials, whereas twelve of the sixteen Syear-olds (75%) did so; this contrast is statistically significant, x2(1, N = 36) = 5.51, p < .Ol. Likewise, eleven 4-year-olds did not produce a single count of families (55%), whereas only two 5-year-olds failed to produce a single count of families (13%); x2( 1, N = 36) = 5.24, p < .Ol.

567

Units of Counting

Table 2. Numbers of Children at each Age who Produced Various Numbers of Family-count Responses across the Eight Family-Count Problems

Age group Number of family counts 8 7 6 5 4 3 2 1

4-year-olds 6 0 1 1 I 0 0

5year-olds 12 0 0 0 0 0

0

1

1

DISCUSSION As in Shipley and Shepperson’s previous research, the 4-year-olds in this study showed a strong preference for counting individual items rather than aggregate units. While performance on problems that required counting classes was somewhat better here than in Shipley and Shepperson’s study, a majority of 4-year-olds still produced only individual-item counts across both the individual-count and the family-count problems. Children’s difficulties in counting classes in this study were clearly not due to difficulties identifying the family units. This process was facilitated for children by grouping the members of each family together spatially, and also by using the familiar collection term, “family”, to label the aggregate units. Moreover, children’s performance on the individual-count problems makes it clear that they had no difficulty telling the families apart. On those problems, children had to differentiate between the families in order to identify the correct subset of individuals to count, and children at both ages did so virtually without error: Only one child ever included items from a non-focal family in responding to an individual-count problem, and she did so only on one problem. Shipley and Shepperson’s focus, in testing their hypothesis of a “discrete physical object bias”, was on children’s ability to count aggregate units consisting of more than one discrete item. This focus is also reflected in the problems in Experiment 1. However, another way of examining the discrete-object bias is to look at children’s ability to count parts of a single object. Must items be discrete in order for children to treat them as countable? Everyday observations of children counting body parts such as fingers and toes suggest that counting of units other than discrete objects may emerge in relation to parts before it emerges in relation to aggregates.

568

Sophian and Kailihiwa EXPER~A~ENT

2

In Experiment 2, children’s counting of objects and pieces of objects was examined by asking children to count arrays composed of toys that came apart into two pieces, like the two halves of a plastic Easter egg. In each array, some of the toys were presented with the two parts joined to form a cohesive object and some were presented as two separate pieces. As in Experiment I, different units were specified in the questions posed to children, in order to ascertain their ability to adapt their choice of counting unit. Thus, sometimes children were asked to count “pieces” of things, and sometimes they were asked to count “whole” things. An important feature of this task is that because the arrays were composed of some intact and some separated toys, a count of discrete items was not a correct response to either of the questions. To count whoies correctly, children needed to treat as a single unit some toys that were presented in two separate pieces. Complementarily, to count pieces correctly, children needed to treat as separate units some pieces that were presented as integral parts of a single object. The initial plan was to focus on 4- and Syear-olds in this study, as in Experiment 1. However, because preliminary results indicated that Syear-olds still had considerable difficulty with the task, it was decided to add a group of 7-year-olds to the experiment as well. Method Participants. Twenty 4-year-olds (4 years, 3 months to 5 years, 1 months; M = 4 years, 8.3 months), twenty Syear-olds (5 years, 4 months to 6 years, 0 months; M = 5 years, 8.5 months), and twenty 7-year-olds (7 years, 5 months to 8 years, 0 months; M = 7 years, 8.2 months) participated. None of these children had participated in Experiment I. There were approximately equal numbers of boys and girls in each group. The chiIdren were recruited from preschools and elementary schools they attended. They were predominantly middle-class and represented the diversity of ethnic backgrounds typical of the region. All of the Syearolds, and none of the 4-year-olds, attended kindergarten. All of the 7-year-olds were in second grade. Four sets of toys that could be separated into two parts were used Materials. as stimuli: blocks, whistles, fish, and eggs. There were ten ofeach: four of each of two colors, and two more that were each a different color from any of the others. For example, there were four red blocks, four yellow ones, and one blue and one green one. Two towels were used as mats on which to present arrays of toys. In addition, a small plate on a stand was used to make a platform. Procedure. Children were tested individually by a female experimenter in a single session that lasted about 20 minutes. Testing began with a warm-up procedure in which children were familiarized with the four types of toys that would be

Units of Counting

569

used and with the fact that each consisted of two pieces. Children then received 12 experimental trials. During the warm-up process, the experimenter presented one exemplar of each of the four kinds of objects to be used in the experiment. The objects were presented one at a time, beginning with the two pieces separated. “Look,” the experimenter said, holding up the two pieces, “these pieces go together to make a ...“. She put the two pieces together, and paused to give the child an opportunity to supply a label for the intact object, If the child did so, the label the child gave was used throughout the session. If the child did not offer a label. the experimenter named the object and asked the child to repeat its name, for example, “It’s a whistle, can you say ‘whistle’?’ She then asked the child how many pieces she would get if she took the object apart again. All children gave the correct answer, two, to this query for each of the four warm-up objects. On each of the experimental trials, children saw an array composed of a number of toys of one kind arranged in a row, with color alternating from one toy to the next. An example array can be seen in Figure 1. Some of the toys in each array (those toward the child’s right) were presented intact (i.e., with the two pieces comprising them joined together) while others (those toward the child’s left) were separated into their two component pieces but with the two pieces placed in close proximity to each other. The experimenter began the trial by identifying the unit to be counted and presenting two exemplars, which differed in color from the items comprising the array and were placed on a platform just beyond the array, where they remained in view throughout the trial. For count-wholes trials, the experimenter said, “This time we are going to count whole blocks (or whistles or fish or eggs), like these”, and placed two whole objects on the platform; similarly, she began count-pieces trials by saying, “This time we are going to count pieces of blocks (or whistles or fish or eggs), like these”, and placed two pieces of a single object on the platform. The child’s attention was then drawn to both the intact and the separated objects in the array (e.g., “Some of these blocks are already together and some are still separated”), and one of two questions was posed. For the count-wholes trials, the child was asked, “How many whole blocks (or whistles or fish or eggs) can we get from all these things?’ For the count-pieces trials, the child was asked, “How many pieces can we get from all these things?” As in

Figure 1.

Example of arrays used in Experiment 2

Count pieces, all: Child increments count by one for each piece of each object in the array

Count wholes, separated: Child increments count by one for each pair of separated pieces (intact objects ignored)

5 I Oo Oo 6 8 problems: problems:

1 3 0 0 2 4 on count-wholes on count-pieces

problems:

‘0 2

on count-pieces

‘0 1

problems:

0

on count-wholes

0

problems:

‘0

on count-pieces

“o

problems:

0 2

on count-wholes

0 1

problems:

on count-pieces

Count wholes, intact: Child increments count by one for each intact object in the array (separated objects ignored)

‘o ‘o 3 4 problems:

0 1 2 on count-wholes

Count wholes, all: Child increments count by one for each intact object or pair of pieces in the array 0

Example

Count Qpe

11 OO 12

OO

10

4

3

9

OO

OO

“0

6

5

OO

OO

OO

LOO) (.28) (.08)

(.36) .68 (.44)

.13 (.31) .21 (.34)

.O8 (.19) .11 (24)

.18

.OO (.OO) .oO

.O5 (.15) .12

.03

(.04)

(22) .Ol

.05

(.41)

(42) .21

.72

‘Yr

(.12) .03

.40 (.45) .OY (.26)

.08

(.26)

(.40)

(.26) .07 (.19)

.I8 (.38) .09

.28

5yr

Means

(.39) .37

Table 3. Alternative Count Patterns and Mean Proportions (and Standard Deviations) of Problems on Which They Occurred (Experiment 2)

Notes.

on count-pieces problems:

2 0

3 5 ‘0 “o 4 6 on ~ount-~~oIes problems:

1 0

on count-pieces problems:

1 3 0 0 % Oo 2 4 on count-wholes problems:

oa count-pieces problems:

0

1 3 “o “o 2 4 on cuunt-wholes problems: 0 %

6

7 OO

5 *0

.10

.I3 (27)

(24)

’ In the examples, each “0” represents an intact object (two joined pieces) and each ‘0” represents a piece of a separated object. The numerals indicate a counting sequence consistent with the count pattern being des~Tibed.

Count discrete items: Child increments count by one for each intact object and each piece of each separated object

Count pieces, intact: Child increments count by one for each piece of each mtact object (separated objects ignored)

Count pieces, separated: Child increments count by one for each piece of each separated object (intact objects ignored)

572

Sophian and Kailihiwa

Experiment 1, if the experimenter could not tell what the child was counting, she asked the child to show her. The experimenter recorded children’s counts by writing numerals for each number the child said on a diagrammatic representation of the array, in the same way that alternative count patterns are represented in Table 3. Children were not given any feedback about the correctness of their responses. The twelve problems each child received varied factorially in whether children were asked the count-wholes or the count-pieces question, in the size of the array (a total of 4,6, or 8 toys), and in whether there were more intact or more separated toys in the array. Intact toys accounted for either 1 or 3 of the toys in arrays of 4, for either 2 or 4 of the toys in arrays of 6, and for either 3 or 5 of the toys in arrays of 8.

RESULTS Counting Patterns Seven distinct counting patterns were defined a priori and used to classify the counts children produced. These counting patterns differed in whether children counted the whole objects, the pieces, or the discrete items (pieces when objects were separated and whole objects when they were presented intact) in the array, and in whether they counted the entire array or included either only the intact objects or only the separated ones in the count. Discrete-item counts, however, had to include the entire array, because the classification of a count as based on discrete items entailed considering how they counted both the separated and the intact toys. Table 3 describes the seven counting patterns and indicates how frequently they occurred, in relation to whether children were asked about pieces or whole objects and in relation to age. While other counting patterns were logically possible, including mixtures of those in the Table, in fact every one of the counts children produced unambiguously fit into one and only one of the seven categories listed. Despite substantial variation across children, individual children were remarkably consistent in the kinds of counts they produced. Thirty-seven children (nine 4-year-olds, thirteen Syear-olds, and fifteen 7-year-olds) produced the same type of count on every one of the six count-wholes problems, and 3 1 children (six 4year-olds, nine Syear-olds, and sixteen 7-year-olds) did so on every one of the six count-pieces problems. All but two children (both 4-year-olds) used a single count type on at least four of the six count-wholes problems, and all but two children (both 4-year-olds) used a single count type on at least four of the six count-pieces problems. While counting of discrete items was not the most prevalent pattern at any age, it was more common at 4 years than among the older children. Thirteen of the 20 4-year-olds (65%) produced at least one count of this type, whereas only three 5year-olds (15%) and one 7-year-old (5%) did so. Thus, discrete-item counting was

Units of Counting

573

significantly more prevalent at 4 years than at either 5 years, x2( 1, N = 40) = 8.44 (corrected for continuity),p < .Ol, or 7 years, x2(1, N = 20) = 21.61 (corrected for continuity), p < .OOl. Since the principal goal of the study was to examine children’s ability to alter the counting units they used in accordance with instructions, the primary focus of the data analyses was on the extent to which children used counting patterns based on different units on the count-wholes versus the count-pieces problems. Because of the consistency at the individual level, data analyses were based on the counting patttems individual children used predominantly (i.e., on at least four of the six problems; binomial p = .03, assuming a chance probability of .14 for each of the seven counting patterns) when they were asked to count whole objects and when they were asked to count pieces of objects. Children’s counting patterns on the two sets of problems were cross-classified, and patterns of performance across the two sets of problems were examined.’ Thirty-two children (twelve 4-yearolds, twelve 5-year-olds, and eight 7-year-olds) used the same predominant counting pattern for both the count-wholes and the count-pieces problems. Twenty others (eight 5-year-olds and twelve 7-year-olds), however, adapted their counting appropriately in that they predominantly counted whole objects on the countwholes problems and they predominantly counted pieces of objects on the countpieces problems. There were also two 4-year-olds who counted differently on the count-wholes and the count-pieces problems, but not in a way that effectively mirrored what they were asked to count. Both of these children predominantly counted discrete physical objects on the the count-wholes problems; on the countpieces problems, one counted wholes and the other counted pieces. Omitting these two ambiguous cases from the comparison, differentiated counting patterns were significantly more prevalent, compared to undifferentiated ones, among the two older groups than they were among the 4-year-olds; 4-year-olds vs. %yearolds, ~“(1, N = 32) = 4.44 (corrected for continuity), p < .05; 4-year-olds vs. 7year-olds, x2( 1, N = 32) = 9.10 (corrected for continuity), p < .Ol . The 5-year-olds and the 7-year-olds did not differ significantly, x2 < 1. Further examination of the counting patterns of the twenty 5- and 7-year-olds who adapted their counting appropriately across the count-wholes and countpieces problems reveals two contrasting forms of adaptation, which differed in prevalence across the two age groups. Five 5-year-olds and one 7-year-old focused on different parts of the arrays on count-wholes and count-pieces problems. Thus, they counted only the intact objects when they were asked to count wholes and only the separated pieces of objects when they were asked to count pieces, effectively restricting their counts to discrete physical objects that corresponded to the unit they had been instructed to count. Three 5-year-olds and eleven 7-year-olds adapted their counting without modifying which subsets of ‘The six 4-year-olds who did not generate a single predominant pattern of counting other of the two sets of problems were not included in this analysis.

on one or the

574

Sophian and Kailihiwa

objects (the intact ones or the separated ones) they counted across the two problem types. Most of these children (all but one Syear-old) included both intact and separated objects in their counts on both count-wholes and count-pieces problems; the exception was a child who included only the separated objects in the majority of his/her counts both when counting wholes and when counting pieces. Thus, within the children who adapted their counting, the 7-year-olds were significantly more likely than the Syear-olds to alter the unit they used in counting (whole objects vs. pieces of objects) without simultaneously altering what they included in their counts, x2( 1, N = 20) = 4.38 (corrected for continuity), p < .05. Outcomes

of Children’s

Counts

A complementary way of looking at the data is to consider the final values children obtained in their counts. Although this method of analysing the data does not provide as direct a picture of how children counted, it has the advantage that it does not require the classification of children according to their predominant count types but rather uses the data from all trials in a simple parametric analysis. If children adapted their counting appropriately according to whether they were asked about pieces or whole objects, then they should have reached a higher number in their counting in response to questions about the number of pieces than they did in response to questions about the number of wholes, and the difference between the two should have increased as the array size increased. These predictions were tested via a 3(age) x 2(gender) x 2(task: count wholes or count pieces) x 3 (array size: 4, 6, or 8 objects) x ‘L(composition of array: majority of objects

14

12

count pieces

~-a

-I

)

count

hholes

1

7 YEARS/

5 YEARS

4 YEARS

1

Highest Count

Term Produced

I

4

6 Total

8 Number

4 of

Items

6

In Array

1

I

6

4 (in

whole

6

8

objects)

Figure 2. Mean value of highest count terms children generated in counting arrays (Experiment 2).

575

Units of Counting

intact vs. majority of objects separated into pieces) analysis of variance, with the last three factors as repeated measures. Means corresponding to this analysis are graphed in Figure 2. There were significant effects of age, F(2, 54) = 6.007, MSE = 66.68, p < .Ol, task, F(1,54) = 32.704, MSE = 12.55,~ < .OOl, and array size, F(2, 108) = 282.32, MSE = 4.59, p < .OOl, and of all the interactions between them were also significant: age x task, F(2, 54) = 15.61, MSE = 12.55, p < .OOl; age x array size, F(4, 108) = 4.468, MSE = 4.59, p < .Ol; task x array size, F(2, 108) = 13.20, MSE = 1.83, p .c .OOl; age x task x array size, F(4, 108) = 8.647, MSE = 1.83, p < .OOl. Although the outcomes of children’s counts increased with increasing array size at all three ages (simple effect tests: at 4 years, F[2, 1081 = 22.68; at 5 years, F[2, 1081 = 16.65; at 7 years, F[2, 1081 = 38.60; all ps < .OOl); only the 7-year-olds obtained significantly larger outcomes when asked about pieces than when asked about wholes, F(l, 54) = 10.3 1, p c .002; for the younger ages, Fs < 1.09, ps > .lO. Likewise, although effects of array size should be greater when children are instructed to count pieces than when they are instructed to count wholes (because the number of pieces increases from 8 to 16 whereas the number of wholes only increases from 4 to S), this pattern was significant only for the 7-year-olds, F(2, 36) = 34.29, MSE = 1.30, p < ,001; versus for 4-year-olds, F < 1, MSE = 1.70, and for 5-year-olds, F(2, 36) = 2.35, MSE = 2.49, p > .lO. There were also significant effects of array composition, F( 1,54) = 22.38, MSE = 4.66, p < ,001, which interacted with both age and task (array composition x age: F[2,54] = 4.84, MSe = 4.66, p = .Ol; array composition x task: F[l, 541 = 5.88, MSE = 2.55, p < .05; array composition x age x task: F[2, 541 = 6.64, MSE = 2.55, p < .Ol). Particularly at the younger ages and on the count-pieces task, count outcomes were greater when the majority of the objects in the array were in pieces rather than intact (MS = 7.7 vs. 6.5 for the count-pieces task and 7.8 vs. 6.6 for the count-wholes task at 4 years; 7.8 vs. 5.8 for the count-pieces task and 5.7 vs. 5.6 for the count-wholes task at 5 years; and 10.7 vs. 10.6 for the count-pieces task and 7. I vs. 7.1 for the count-wholes task at 7 years).

DISCUSSION A tendency to focus on discrete objects was reflected in children’s counting patterns in this study in two ways. Some children combined wholes and parts in their counts by counting whichever appeared as discrete items in the array; that is, they counted the objects as wholes when they were intact and as pieces when they were separated. Others included only wholes or only pieces in a single count but omitted the parts of the array in which they were not discrete objects: that is, they omitted the separated items in counting wholes or the intact items in counting pieces. At the same time, quite a few children-even among the 4-year-olds-consistently adopted either the wholes or the parts as the focus of their counting, and included even wholes that appeared in separated form or parts that were joined to

576

Sophian and Kailihiwa

form an intact object in their counting. These counts make it clear that even the 4year-old children were not fundamentally incapable of counting things other than discrete items, whether those things were aggregates (wholes comprised of two discrete parts) or parts (pieces within a single intact object). On the other hand, where children clearly did have difficulty, particularly at the younger ages, was in switching from one unit of counting to another in accordance with the problem that was posed. No evidence was obtained for the idea that the first advance over discrete-item counting may be the counting of parts of a single object rather than the counting of aggregates. However, aggregate counting was somewhat more in evidence among the 4-year-olds in this study than among those in Shipley and Shepperson’s (1990) experiments with detached pieces of objects. The kinds of objects we used may have contributed to this increase: it is unlikely that children have ever seen two pieces of a fork assembled to form a whole fork, but they have quite likely had experience with toys that come apart into pieces and can be put back together. The demonstrations that children were given of the objects actually being assembled from the pieces also may have helped to make the wholes salient for them even when the pieces were not yet put together. But by the same token, the kinds of materials used in this study may have contributed to the relatively low frequency with which children counted pieces within intact objects. The pieces fit together tightly and, once joined to form a whole object, did not retain a visual separateness (as the fingers of a hand do). Counting of pieces might occur more often if materials were used in which the pieces retained greater distinctness within the wholes, just as the counting of wholes appears to be depend on the kinds of objects used and how they are presented. A surprising aspect of the results of Experiment 2 is that, although performance improved with age, even among the 7-year-olds a substantial portion of the sample still failed to adapt their choice of counting units in accordance with the problems that were posed. Shipley and Shepperson (1990) found that while 4-yearolds had difficulty counting units other than discrete objects, 5-year-olds were generally successful, and a comparable result was obtained in Experiment 1 of this paper. The persisting difficulties that 5-year-olds and even some 7-year-olds had with the task used in Experiment 2 therefore need to be carefully evaluated. One possibility is that these difficulties primarily reflect limitations on children’s processing of the verbal requests for counts of “pieces” versus “whole” objects. Although the presentation of exemplars of what was to be counted provided visual support for these verbal requests, it is possible that children interpreted all the exemplars as simply things, consistent with an undifferentiated interpretation of the verbal count requests. This interpretation seems improbable, however, since none of the children had any difficulty with the questions posed using the same terms during the warn-up phase. A more plausible idea is that they did not attend to the specific terms used in the counting requests, although they might have been able to differentiate appropri-

577

Units of Counting

ately between the two questions had they done so. Because children are rarely asked to count using different kinds of units, they may simply focus on whatever type of unit is most salient when they are asked any “how many” question. To address this possibility, Experiment 3 undertook to make the contrast between the two kinds of questions more salient.

EXPERIMENT 3

Experiment 3 was similar to Experiment 2, except that in an effort to make it clear to children that different kinds of counts were being requested children were asked to count both the number of pieces and the number of whole objects in each array that was presented (in counterbalanced order). Requesting both types of counts for each array was expected to help children notice the difference between the two requests. In addition, in Experiment 3 children were actively encouraged to include the entire array in their counts. If a child omitted a subset of the objects from a count, the experimenter reminded him or her to count the entire array. By discouraging the production of counts that encompassed only one type of object (intact or separated), this procedure facilitated the discrimination of discrete item counts from counts of wholes or of pieces. Correspondingly, it provided a stronger test of 5-

Table 4. Numbers of Children who Showed Different Combinations of Count Patterns Across the Count-Wholes and Count-Pieces Problems (Experiment 2) Pattern Across Problem Types

Count Type on Count-Whole Problems

Count Type on Count-Pieces Problems

Age Group 4 years

5 years

7 years

wholes, all wholes, intact wholes, separated

5 1 0

2 2 2

4 0 0

pieces, all pieces, separated

2

2

3

0

2

0

discrete items

4

2

1

pieces, separated

0

5

1

pieces, all pieces, separated

0

0

2 1

11 0

wholes, all pieces, all

I 1

0

0

0

0

1.

Undifferentiated Patterns Undifferentiated counting of whole objects Count wholes, all Count Count wholes, intact Count Count wholes, intact Count Undifferentiated counting of pieces Count pieces, all Count Count pieces, separated Count Undifferentiated counting of discrete items Count discrete items Count II. Differentiated Patterns Differentiation, different parts of arrays counted Count wholes, intact Count Differentiation, same items included in counts Count wholes, all Count Count wholes, separated Count III Other Count discrete items Count Count discrete items Count

578

Sophian and Kailihiwa

year-olds’ ability to adapt their choice of counting units in accordance with the question that was posed, since it discouraged the use of counting patterns that adapted to different questions by focusing on different parts of the array. Method Participants. Eighteen 4-year-olds (4 years to 4 years, 8 months; M = 43) and eighteen Syear-olds (5 years, 2 months to 5 years, 11 months; M = 5;lO) participated. None of these children had participated in either Experiment 1 or Experiment 2. The children attended preschools and kindergarten classes in the university area and were tested at their schools. All of the 5-year-olds, and none of the 4-year-olds, were in kindergarten. Each age group included approximately equal numbers of boys and girls. The children were predominantly middle-class and represented the diversity of ethnic backgrounds typical of the region. Materials. The same toys used in Experiment 2 were used in this experiment. The towels used in Experiment 2 were also used again. Procedure. Children were tested individually by a female experimenter in single sessions that lasted about twenty-five minutes. Testing began with a warmup procedure of two parts. The first part, like the warm-up procedure in Experiment 2, familiarized children with the four types of toys that would be used and with the fact that each consisted of two pieces. The second part of the warm-up familiarized children with the fact that regardless of whether an object was presented intact or with the pieces separated, it could be counted either as a single whole or as two pieces. Children were presented with four practice arrays, each consisting of two or three toys, all of the same type. All the items within an array were either intact (with the pieces joined to form whole objects; this type of array was used for the first two practice trials) or in parts (with the two pieces in close proximity to each other but not joined; this type of array was used for the third and fourth practice trials). On each practice trial, children were asked both, “How many whole eggs [or whatever objects comprised the array] can we get from all of these things?,” and, “How many pieces can we get from all of these things?‘. For the first two practice trials, the question about wholes was asked first and then the question about pieces; for the latter two, the order of questions was the reverse. Children were corrected and helped to count the appropriate units if they made any mistakes. By the final practice trial, all of the children counted correctly in response to both questions without any intervention from the experimenter. Children then received 12 experimental trials, on each of which they were asked about both the number of wholes and the number of pieces in the array (in the same manner as on the practice trials). Children were not given any feedback about the correctness of their responses on these trials, but they were prompted if they omitted part of the array from their counts (e.g., if they counted only the whole objects and ignored the separated ones). In that event, the experimenter

579

Units of Counting

waved her hand across the entire array and said, “Remember, we are counting all of these things.” The experimental trials were divided into two blocks, differing only in whether children were asked first about wholes and then about pieces or vice versa (with the order of the two blocks counterbalanced across children). The problem types within each block paralleled those in Experiment 2, varying in whether the array contained a total of 4, 6, or 8 toys, and in whether there were more separated toys or more intact toys in the array (i.e., 1 or 3 intact toys in arrays of 4; 2 or 4 intact toys in arrays of 6; and 3 or 5 intact toys in arrays of 8). The order of these problems was randomized independently for each child.

RESULTS Counting

Patterns

Because children were discouraged from omitting any of the objects in the arrays from their counts, we focused on just three of the seven count patterns from Experiment 2 in examining their count patterns in this study: count wholes-all; count pieces-all; and count discrete items. Table 5 indicates how often each of these three types of counting occurred when children were asked about wholes and when they were asked about pieces. As in Experiment 2, counting of discrete items was most prevalent among the 4-year-olds: All but one of the 4-year-olds (94%), versus just half of the 5-year-olds (N = 9) produced a discrete-item count at least once in the problem sequence, x2( 1, N = 36 ) = 6.78 (corrected for continuity), p < .Ol. As in Experiment 2, it was not uncommon for children to use the same counting pattern every time they were asked a given type of question. Twelve children (two 4-year-olds and ten 5-year-olds) produced the same type of count in response to all twelve of the questions about wholes; and ten (two 4-year-olds and eight 5year-olds) did so in response to all twelve of the questions about pieces. Six 4year-olds and thirteen 5-year-olds could be classified as having used a single

Table 5. Mean Proportions (and Standard Deviations) of Problems on Which Alternative Count Patterns Occurred (Experiment 3) Count Type Count wholes (all) When asked about When asked about Count pieces (all) When asked about When asked about Count discrete items When asked about When asked about

4 years

5 years

wholes pieces

.34 (.37) .16 (.21)

.x0 C.40) .l 1 (.25)

wholes pieces

.23 (.29) .48 (.34)

.07 (.25) .76 (.36)

wholes pieces

.36 (.25) .28 (2 1)

.12 (.23) .lI (.ll)

580

‘Sophian and Kailihiwa

count pattern for at least 8 of the 12 of the questions about wholes (binomial p = -06, assuming a chance probability of .33 for each of the three count patterns), and also as having used a single count pattern for a majority of the questions about pieces. Among these children, a large majority of the Syear-olds (11 out of 13; 85%) but only two 4-year-olds (33%) predominantly counted wholes in response to questions about wholes and predominantly counted pieces in relation to questions about pieces; the remaining children-four 4-year-olds (67%) and two 5 year-olds (15%)-showed the same predominant count pattern in response to the two question types. Thus the age trend was consistent with that found in Experiment 2 in showing an increase with age in the prevalence of differentiated count patterns. However, no statistical contrast could be performed because of the small numbers of children who could be cross-classified as having shown more-or-less consistent response patterns to both of the question types.” Outcomes

of Children’s

Counts

Figure 3 presents the mean values children reached as the highest number in their counts, when they were asked about the numbers of pieces and when they were asked about the numbers of whole objects in the arrays. A 2(age) x 2(gender) x 2(block: problems on which children were asked first about wholes and then about pieces vs. those on which children were asked first about pieces and then about wholes) x ‘L(question type: focusing on wholes vs. pieces) x 3 (array size: 4, 6, or 8 objects) x 2(composition of array: majority of objects intact vs. majority of objects separated into pieces) analysis of variance, with the last four factors as repeated measures, was carried out on these data. There were significant main effects of question type, F( 1, 32) = 78.83, MSE = 22.7 1, y < .OOl, and array size, F(2,64) = 333.86, MSE = 6.67, p c: .OOl; and significant interactions of question type x array size, F(2,64) = 20.53, MSE = 2.76, p < .OO1, question type x age, F( 1, 32) = 21.07, MSE = 22.71, p < .OOl,and question type x array size x age, F(2,64) = 9.83, MSE = 2.76, p < .OO1. As in Experiment 2, both 4- and S-year-olds counted higher in response to larger arrays than in response to smaller ones (simple effects of array size: at 4 years, F[2, 641 = 19.63; at 5 years, F[2, 641 = 23.54; both ps s; .OOl). In addition, in this experiment, the Syear-olds counted significantly higher when asked about pieces than when asked about wholes, simple effect of question type, F( 1, 32) = 7.80, p < .Ol; this difference was significantly smaller for the 4year-olds, however (as indicated by the Question Type x Age interaction), and not ’ Using a more lenient criterion it was possible to obtain statistical support for the age difference. Ten 4.year-olds and fifteen 5-year-olds could be classified as having used a single count pattern for at least 7 of the 12 questions about pieces and also as having used a single count pattern for at least 7 of the 12 questions about wholes. Among these children, twelve 5-year-olds (80%) but only three 4-yearolds (30%) predominantly counted wholes in response to questions about wholes and pieces in response to questions about pieces, x2( 1, N = 25) = 4.34 (corrected for continuity), p < .05. Six 4-yearolds (60%) but only two Z-year-olds (13%) showed the same predominant counting pattern in response to both questions, x2( I, N = 25) = 4.05 (corrected for continuity), I? < .OS.

Units

of

Counting

581

16

12

Highest Count Term lo Produced 8 6

4

’

I 4 (8

wholes pieces)

6 (12

I wholes pieces)

I 6 (16

wholes pieces)

Total Number of Items in Array Figure 3. Mean value of highest count terms children generated in counting arrays (Experiment 3).

significant at that age, simple effect F x 1. Likewise, among the 5-year-olds, effects of array size were significantly greater when children were asked about pieces than when they were asked about wholes, F(2, 32) = 31.48, p < .OOl; but this effect was not significant for the 4-year-olds, F(2, 32) = 1.67, p > . 10. Array composition did not have any significant effects on children’s count outcomes in this experiment. DISCUSSION

The 5-year-olds performed much better in this experiment than in Experiment 2. They gave strong evidence of understanding the need to adopt different kinds of counting patterns in accordance with different quantification goals, in that they discriminated clearly between the two question types and often were quite systematic in adopting the appropriate counting unit for each question. Nevertheless, strong age differences between 4- and 5-year-olds remained. These age differences are consistent with both Shipley and Shepperson’s (1990) research and our

582

Sophian and Kailihiwa

own findings from Experiment 1 in pointing to an important developmental change between 4 and 5 years in the way children choose counting units. Moreover, the present findings suggest that the difficulties the 4-year-olds have are not solely a function of the perceptual salience of discrete objects; rather, they appear to reflect a lack of understanding of the relation between what one is trying to find out and the counting unit one should adopt. In the present research, 4-year-olds’ counting was by no means restricted to discrete objects; they treated both the individual pieces of intact objects and the pairs of separated pieces that would together form a whole object as units of counting quite readily. However, their choices of which unit to use were not well coordinated with the questions that were posed. Inflexibilities similar to those the 4-year-olds showed in choosing counting units in this research have been observed in their rule use in many problem-solving tasks (Zelazo & Jacques, 1997). These inflexibilities in rule use have been attributed to developmental limitations on executive function, an explanation that contrasts with the present theoretical perspective in emphasizing domain-general processes rather than the knowledge children have about a particular conceptual domain. These are probably not incompatible perspectives: A possible basis for limited conceptual understanding at early ages might be the lack of sufficient processing capabilities to construct and operate with complex concepts, and a possible developmental basis for overcoming early processing limitations might be the acquisition of conceptual knowledge that can guide the selection and execution of task-appropriate cognitive operations. An issue that needs to be addressed to establish conceptual limitations, however, is whether conceptual knowledge that children do have might have been masked in the present studies by limitations on their ability to handle the executive demands of the tasks presented to them. The strongest evidence against this interpretation comes from Experiment 1, where even the youngest children clearly did adapt what they counted in one important respect: They did not count rabbits when asked about the number of birds, and so on. Yet they still did not shift from counting individuals to counting groups when asked about families. This result suggests that children had executive control over the selection of which units to include in a count but not over the way they parsed the arrays into units. In Experiments 2 and 3, children were able to parse the arrays into different kinds of counting unitsaggregates of discrete items, and pieces of items that formed a coherent whole, as well as discrete objects-but the 4-year-olds still were unsuccessful in determining when to use which type of unit. In combination, then, the results suggest that the inflexibility of children’s early counting is quite specific: although they can adapt the items they include in a count to match a stated goal, and they can treat entities other than discrete items as units to be counted, they cannot shift between types of counting units (wholes vs. pieces) in a goal-directed manner. Our belief is that it is the conceptual limitations on children’s understanding of counting units that constrain the adaptability of this aspect of their counting. If there were not a conceptual component to the

Units of Counting

583

difficulty, it is not clear why a domain-general inflexibility in cognitive processing reflecting constraints on the executive function would result in such a localized lack of adaptability in children’s counting. Gallistel and Gelman (1990) responded to Shipley and Shepperson’s (1990) findings of a discrete-object bias by acknowledging that their abstraction principle (Gelman & Gallistel, 1978), which asserts that all kinds of things can be counted, need not rule out the operation of biases in children’s selection of what to count. The present findings, however, suggest that the significance of the counting patterns noted by Shipley and Shepperson is not so much for children’s understanding of what can be counted as for their understanding of how to segment an array into counting units. Gelman and Gallistel’s (1978) one-to-one principle takes for granted the segmentation of the to-be counted array into countable items, and their cardinality principle asserts that children understand that the enumeration of those items will generate the cardinal value of the set. The present research, however, suggests that there are important developmental changes, extending throughout the preschool period, in children’s understanding of the segmentation process and its relation to cardinality. Gallistel and Gelman’s (1990) assertion that there are no principled restrictions on what can be counted, that the counting principles do not “restrict what it is that may be counted (p. 197)“, is only partly valid. While it is possible to count heterogeneous as well as homogeneous arrays, temporal events as well as physical objects, etc., the goal of obtaining a cardinal value does entail that the items we count must all be construable as equivalent units; they must have a commonality at some level of abstraction that allows us to think of each of them as an instantiation of whatever it is we are quantifying. We cannot include fork-handles as units equivalent to whole forks if we want to know how many forks are in an array. Moreover, we cannot switch units in mid-count. It would not be valid to count some of the separated objects used in Experiments 2 and 3 as two separate units and others as a single unit, because such a count would not correspond to any coherent quantification goal; having done it, we would not be in a position to say how many of anything are in the array. The fact that counting is a process of quantification or measurement, then, restricts what we can count in that it precludes combining incommensurate units in a single count. Further, insofar as any specific count is directed toward a specific quantification goal, what can be counted is constrained in that only items that qualify as valid instances of what is being quantified can apprpriately be included in the count. It is noteworthy that even the youngest children in the present research adhered to the more global of these constraints. Children virtually never counted in such a way that, within a single count, the pieces of a separated object were treated sometimes as a single item and sometimes as two items. Likewise, within any one count, they did not treat one intact object as a single item and another as two. Thus, it was possible to identify some common unit-whether whole objects, pieces, or discrete things-in virtually all the counts the children produced.

584

Sophian and Kailihiwa

Where age differences emerged was in children’s adaptation of the counting unit they used to the quantification goal, as specified by the experimenter in presenting the problem. The 4-year-olds, even when they did not restrict their counting to discrete objects, gave no evidence of adapting their choice of counting unit approp~ately. Some of them decided on a single way of parsing the arrays, and held to that regardless of the question that was posed; others vacillated among different counting units, but still without adapting the unit they used appropriately to the question they were trying to answer. In contrast, the 5-year-olds showed some awareness of the importance of considering the quantification goal even in Experiment 2; and in Experiment 3, where the contrast between the two questions was highlighted by asking both questions about each array, they adapted the counting units they adopted quite systematically to match the differing quantification goals. This developl~entai change is not simply a matter of acquiring the ability to segment arrays in different ways, because even at 4 years most children did count whole objects on some occasions and pieces on others. Rather, what develops is children’s understanding of the role of one’s quantification goal in determining how to segment an array into units to be counted. This understanding is inseparable from an understanding of cardinality, since the adoption of different counting units can result in the assignment of different cardinal values to a given array. Untif chiIdren understand that what one counts depends on what one wants to quantify, they cannot futly understand what it means to arrive at a cardinal representation of a set.

REFERENCES Bermejo. V. (1996). Cardinality development and counting. Developmentul Psychology 32, 263-268. Bermejo, V., & Lago, M. 0. (1990). Developmental processes and stages in the acquisition of cardinality. lnter~fftiona~ Journal o~~ehav~or~l ~e~elor)rnerlt, 13, 23 I-250. Fuson, K. C. (1988). Children’s counting uMd concepts of number New York: SpringerVerlag. Fuson, K. C., Pergament, G. G., Lyons. B. G., & Hall, J. W. (1985). Children’s conformity to the cardinality rule as a function of set size and counting accuracy. Child Development, 56, 1429-1436. GalIistel, C. R., & Getman, R. (1990). The what and how ofcounting. Cognirion. 24, 197- 199. Gal’perin, P., & Georgiev, L. S. (1969). The formation of elementary mathematical notions. In J. Kilpatrick & 1. Wirszup (Ed%), Soviet studies in the p.~~choZ[~~~u~~ear~jn~ and teaching mathematics: Vol. 1. The /earning of mathe~ti~a~ concepts (pp. 189-216). Chicago: University of Chicago Press. Gelman, R., & Gallistel, C. R. (1978). 7%zechild% understanding of numhel: Cambridge, MA: Harvard University Press. Miller, K. F. (1989). Measurement as a tool for thought: The role of measuring procedures in children’s understanding of quantitative invariance. Developmental Psychology, 25, 589-600.

Units of Counting

585

Shipley, E. F., & Shepperson, B. (1990). Countable entities: Developmental changes. Cognition, 34, 109-136. Zelazo, I? D., & Jacques, S. (1997). Children’s rule use: Representation, reflection, and cognitive control. In R. Vasta (Ed.), Annals of child development, vol. 12 (pp. 119176). London: Jessica Kingsley Press.