Representing quantity beyond whole numbers: Some, none, and part

Representing quantity beyond whole numbers: Some, none, and part

Ellen Bialystok

Abstract Previous research has demonstrated how children develop the ability to use notational representations to indicate simple quantities. These studies have shown a developmental shift from the use of idiosyncratic, to analogical, to conventional, numerical notations. The present paper extends these findings by reporting the results from a study in which children from 3 to 7 years old were asked to write a representation to indicate a quantity presented in a game-like scenario. Three kinds of quantities were included: whole numbers, zeros, and fractions. The children’s notations were shown to them shortly after they were produced and then again two weeks later to see if children could interpret them. The results showed the familiar developmental pattern towards increased use of conventional notations for all quantities. The ability to read the notations was greatest for conventional numbers where performance was at ceiling, lower for analogue representations, and very poor for idiosyncratic global recordings. Children’s choice of a notational format was influenced almost entirely by their age and not by the quantity being represented. Children were able to solve the zero problems almost as well as they could the whole numbers, but their understanding and use of representations for fractions was very limited.

Young children are enthusiastic about counting and by about 3 or 4 years old apply this routine to many things in their environment, such as their age, the number of candies in a package, and the number of pieces of cake to be shared (Baroody, 1992; Fuson, 1988, 1991, 1992; Gallistel & Gelman, 1990, 1992; Gelman, 1993; Gelman & Gallistel, 1978). Nonetheless, their counting is not always successful and they do not necessarily understand the reference or meaning for the counting words they use (Fuson, 1988; Gelman & Gellistel, 1978; Wynn, 1992). This relationship between numbers and an abstract notion of quantity is part of the cardinal principle (Gallistel & Gelman, 1990, 1992; Gelman & Gallistel, 1978; Klein & Starkey, 1987). Therefore, the ability to count may overestimate children’s grasp of cardinality, a crucial foundation of understanding concepts of quantity.

Children also spend the preschool years developing knowledge of the written forms that correspond to the numbers in the counting sequence (Tolchinsky Landsmann & Karmiloff-Smith, 1992). By the time they begin school, they are able to identify most of the numerals by naming the quantity to which they refer. But if children are not clear about the cardinal meanings of the numbers they have learned to count, then what do they believe is indicated by written notations for quantity? Just as children who can count (that is, know the spoken number names) do not necessarily understand the significance of those forms, so, too, children who appear to recognize these written representations may not fully understand what they indicate. What do children believe that written representations of quantity mean?

Researchers have found a discrepancy between children’s ability to produce numerical notations and their understanding of what those notations represent (Bialystok & Codd, 1996; Fuson, Fraivillig, & Burghardt, 1992; Hughes, 1986; Kamii, 1981; Neuman, 1987; Sinclair, 1988, 1991; Sinclair, Siegrist, & Sinclair, 1982). The process of learning the relevant meanings for the notations is slow and continues to be refined during the time that children appear to use numerals with some facility. Fuson and Kwon (1992) propose that children learn the written patterns in the same way that they learn the spoken ones: Each numeral is associated with a particular number-word in the sequence to build up a set of meanings for the notations. With this development, the written notations become symbols for meaningful concepts of quantity.

Allardice (1977) conducted an early study on the origins of numerical symbolism for young children. She asked 3- to 6-year-olds to create written representations for some mathematical ideas. She placed small plastic mice on the table and asked the child to put something on paper to show how many there were. Almost half the 3-year-olds and 3quarters of the 4-year-olds used tallies, circles, or pictures to indicate quantity. The other children used more global representations that were not adequate for conveying quantitative information (i.e., individual letters or scribbles with writing-like characteristics which could not be interpreted). These children were questioned in a subsequent session to ascertain if perhaps they could interpret their responses. Typically, the child announced there was meaning but that they could not yet read. All the 5- and 6-yearolds were able to represent quantity, producing both analogue representations and written numerals.

Sinclair et al. (1982) pursued these developments in a similar study with children aged 4 to 6 years old. Again, children were asked to produce a representation to indicate various quantities of objects. Although most children used several notational types, there was a pattern in which the representations became increasingly complex with age. Children began by using global representations that did not correspond to quantity and then advanced to analogue representations (for example, tallies) that preserved numerical information through one-to-one correspondence. The most advanced types of representations were written numerals and alphabetically written number words, that is, numbers written as words.

Some studies have tried to provide children with more realistic situations in order to determine if context would improve performance. Sinclair and Sinclair (1984) presented children with drawings of familiar scenes or objects (for example, a birthday cake with five candles or a speed limit sign with the number 60 on it). Children were asked to identify the picture and objects in the scene. Then the number was indicated and children were asked various questions, such as “What is that?” “Can it be read?” “What does it say?” and “What does it mean?” The responses were classified as no response, numeral (the number is viewed in isolation of the scene), global (the number serves a general purpose), specific function (an understanding of the number and its function in the scene), or tag (the number is assumed to name the object on which it appears). The majority of children used global responses, for example, “It’s a birthday party,” and specific function responses, for example, “He’s 5.” Sinclair and Sinclair believe that global responses occur first in isolation and are then combined with specific responses. Through this process, the global response type should gradually disappear. Their results confirmed that the global responses decreased while specific function responses increased with age, as children came to understand that the numerical information is a significant aspect of the display.

Hughes (1986) described a comparable developmental trend in a study in which he asked children to represent the quantities 1, 2, 3 (defined by Hughes as small numbers,) 5 and 6 (large numbers) on paper. Toy bricks were placed on the table and children were asked, “Can you put something on paper to show how many bricks are on the table?” He categorized responses into four categories: idiosyncratic (scribbles), pictographic (representing the appearance of the objects as well as their numerosity), iconic (one-to-one correspondence between the number of marks and the number of bricks), and symbolic (conventional numbers). He found that the preschool children most often used iconic and idiosyncratic representations, and that there was a general move towards symbolic representations that made this the predominant choice for the 7-year-olds.

Bialystok and Codd (1996) extended this research by creating a more controlled experiment to study how children produced representations for quantity and how they subsequently interpreted their meanings. Children between 3 and 5 years old were asked to produce (production task) or select from one of three prepared alternatives (selection task) a representation to indicate the set size of a group of objects and then to use the representation later to recall the number of items in the set. The responses were categorized as global (cf. Hughes’ idiosyncratic and pictographic), analogue (cf. Hughes’ iconic), or numeric (cf. Hughes’ symbolic). As with the previous research, there was an overall trend in which children moved towards the more symbolic forms. Nearly all 5-year-olds used numerals, 4year-olds used numerals just over half of the time and were inconsistent in choosing between the other two options, and the 3-year-olds showed no preference for any of the representational forms. Children were able to correctly state the number of items in the box when the notations they had produced were numeric but had more difficulty interpreting when they had produced analogue representations (production task). However, in the selection task their performance improved if they had selected an analogue from a prepared set. In general, then, children’s production of symbols became increasingly conventional, and their ability to read the representation later was most reliable if the representation used was a conventional symbolic notation.

In all of these studies, there was a developmental trend in children’s ability to produce conventional numerals as representations of quantity. This developmental pattern is most likely attributable to both the cognitive maturity with increased age and the effects of schooling and more formal contexts for learning about these notational systems. It is unlikely that children would spontaneously begin to use the Arabic numeric system without formal instruction (teary, 1995). Another pattern that was observed, however, was that children who did not produce conventional numerals were less likely to be able to interpret them later. Even analogues that accurately conveyed quantity were not necessarily correctly “read” after they had been written. This suggests a weakness in children’s basic understanding of how representation works and casts doubt on the conclusion that the children’s notations were necessarily functioning as symbols. Furthermore, all the tasks have been based on small quantities, so children may have succeeded to some extent because the correspondences between the quantity and the notation were familiar, not because they knew the meaning of the notation. It is possible, then, that children’s ability to use and interpret written representations for quantity in general has been overestimated by these earlier studies. Therefore, it is necessary to establish a more rigorous assessment of children’s understanding of how notational representations indicate quantity by asking them to represent quantities that are less familiar or less practised. In these cases, children could be successful in representing and retrieving values of quantity only if they understood the meaning of the quantity and the meaning of the notation. Two such types of quantity are zero and fractions.

An examination of how children develop the concept of zero and the notational forms for indicating zero is interesting because these notions also have a social-historical development. The concept of zero as a distinct number with a unique representational form evolved much later than did the notational forms for indicating all other quantities. In other words, it was not spontaneously obvious in earlier cultures that zero is a quantity that could be represented.

There is some controversy about the exact origin of zero, partly because there are two uses for the notation. The first is as a placeholder in place value number systems (like ours) to indicate that a column has no value for that number (e.g., in 503, the 0 means that there are no tens); the second is as a number in its own right (e.g., a quantity indicating a null set, as in they made 0 errors). The use of zero as a placeholder was probably first developed by the Babylonians, possibly as early as 1500 BCE (Neugebauer, 1957). The use of zero as a number is more difficult to trace. Although the Alexandrian Greeks (3rd century BCE) used zero to denote the absence of quantity, it did not function as a number (Kline, 1972). The quantitative idea of zero as a number that can be entered into computations was unlikely to have existed before the 8th century CE when Hindu mathematicians created computational systems that incorporated this concept (Smith, 1963). Joseph (1991) traces the first explicit occurrence of zero in India to an inscription in 876 CE, and points out that this is more than two centuries after the first reference to the other nine numerals. Once the concept was developed and incorporated into mathematical thinking, it was spread to the Europeans through the Arabs, who had learned about it from the Hindus.

Because zero is as much a cultural creation as it is a quantitative concept, it may emerge for children on a different timetable than does the development of the concept of other numbers. The notion of absence is abstract and particularly difficult when applied to quantity – the absence of a presence. The counting procedure that children learn from an early age is based on the assumption that there is something to be counted. Zero does not fit easily into this routine. Similarly, representations for zero need to produce something (a form or notation) to indicate absence. As we saw, children’s early attempts at representing quantity tend to be analogical, producing a set of forms that correspond in number to the set of objects they represent. If this were the way in which children perceive representations of quantity, then producing a representation for zero would be very difficult.

The concept of fractions and the ability to produce a notational representation for fractions is obviously a later development than the corresponding achievements with whole numbers and even zero. Fractions are conceptually complex because they involve ratio quantities, and their notations are specialized and require specific instruction (Hunting & Sharpley, 1991; Siegal & Smith, 1997; Sophian, 1996; Sophian & Kailihiwa, 1998). Hence, fractions are introduced later in the school curriculum than are whole numbers and we do not expect children to spontaneously develop control over these concepts (Resnick, 1995; Resnick & Singer, 1993). Nonetheless, fractions are prevalent in children’s preschool lives. Children agree to eat half their vegetables, grudgingly give up half of their cookie for a younger sibling, and even announce that they are three and a half years old. Clearly they understand something about partial quantities. Therefore, their attempts to create representations for these quantities should provide some insight into how they believe representations indicate quantity.

Although very early studies showed that children have an incomplete or imperfect understanding of fractional quantities at the time they begin learning about these concepts (Gunderson & Gunderson; 1935; Polkinghorne, 1957), the concepts may nonetheless begin to develop in infancy. For example, Baillargeon and her colleagues showed that sixand-a-half month-old infants could judge the degree to which an object extended beyond a supporting object Baillargeon, Kotovsky, & Needham, 1995). By 12 months old, infants were capable of anticipating the weight of an object and adjusting for the right amount of force after having lifted the object only twice (Mounoud & Bower, 1974). By 15 months, infants who were given a series of rods of different lengths were able to anticipate the weight of a rod on the basis of their experience with the other rods in the series. In all cases, infants could respond appropriately to a task in which judgments of proportion were required.

Mix, Levine and Huttenlocher (1999) report that the development of children’s understanding of fractions appears to parallel the development of their understanding of whole numbers, though with a great deal of lag time in between. They suggest that it is the learning of whole numbers, and the awareness of what appears to be the same notation system that interferes with the learning of fractions. Others also support this notion of interference (Kamii, 1981; Sophian, Garyantes, & Chang, 1997). Siegal and Smith (1997) propose that teaching methods and strategies that disentangle written representation from the learning of fractions enhance children’s ability to acquire these skills.

Some fractional quantities appear to be more accessible to young children than others. Hunting and Sharpley (1991) showed that children had a qualitative conception of “half” but had little understanding of “quarter” or “third.” Hunting and Davis (1991) point out that “half” may be more familiar because of children’s experience with sharing between two individuals. Parrat-Dayan and VonPche (1992) note, however, that very young children consider “half” to be some portion of a whole but not necessarily half the quantity.

Gelman (1991) described a study that combined an assessment of children’s understanding of fractional quantities with their understanding of the notational forms for these quantities. She showed kindergarten to Grade 2 students written fractions such as 1/2 and 1/4 and asked them to read them and choose which was more. Children had difficulty with the written form, for example, reading 1/2 as “one and two’ or “one plus two,” or even “twelve.” They also treated the numbers as whole quantities, therefore misjudging 1/4 to be larger than 1/2. Gelman argues that the problem is difficult because children’s knowledge of counting is a source of confusion. Gallistel and Gelman (1992) elaborate on this point and suggest that fractions are the first numbers that children encounter which cannot be generated as a result of counting. Nonetheless, the children were familiar with the correspondence between the written and verbal representations and could match 1/2 with “one half.” Even in the absence of complete understanding of the quantitative meaning of fractions, children could interpret some of the representational forms.

The purpose of the present study was to examine the development of children’s representations of quantity and their conceptions of the meaning of those representations. The study extended the previous research by Bialystok and Codd (1996) by using more detailed assessments for children’s understanding of quantity and notation, and by including zero and fractions in the set. The task explored children’s understanding of the quantitative concepts for whole numbers, zero and fractions and their ideas about how to notational forms represent these quantities.

Method

PARTCIPANTS

The study included 75 children, comprised of 15 children in each of 5 age groups: 3-year-olds (mean age 3.5 years), 4-yearolds (mean age 4.5 years), 5-year-olds (mean age 5.4 years), 6year-olds (mean age 6.4 years), and 7-year-olds (mean age 7.5 years). All children lived in middle-class neighbourhoods and were attending daycare centres or primary schools.

MATERIALS AND PROCEDURES

All of the children were tested individually in a quiet place in the school. The children were visited again two weeks later for the delayed-recall component of the task.

Five dolls representing “Sesame Street” characters were lined up on a table with two boxes in front of each character. The experimenter asked the child to help give out cookies for each of the characters to have for lunch and for an afternoon snack. The first box was for the lunch cookies and the second was for the snack. The experimenter specified the amount each character was to receive for each occasion. The experimenter and the child progressed down the row of boxes one at a time in this way. For items that involved whole numbers, the experimenter would say: “Give Big Bird 2 cookies for lunch.” In the case of fractions, the experimenter would say, for example: “Give Big Bird a half of a cookie for lunch.” And for questions that could be represented with a zero, the child was asked: “Give Big Bird no cookies for lunch.” The child actually distributed real cookies into each box. The quantities tested were 2, 5, 1k, 14, and 0. With 5 characters and 2 boxes per character, there were 10 items in all. Each of the S quantities was presented twice.

After distributing all of the cookies, children were given a 12 cm x 7 cm Post-it note and a crayon. The experimenter then asked: “Can you put something on this paper to help you remember how many cookies are in the lunchbox?” The lid was then closed and the post-it note was placed on top. The boxes were removed from sight because previous research has shown that location acts as a cue to memory (Bialystok & Codd, 1996).

When this part of the task was completed, the child was engaged in an unrelated task for about 20 minutes. Then the boxes were returned to the table. Children were asked to look at the Post-it note that they had placed on the lid and see if they could use it to say how many cookies were in the box. Two weeks later, the experimenter returned and met with each child and again showed him or her the same boxes and Post-it notes and again asked the child if he or she could say how many cookies were in each box.

Scoring was carried out separately for each of the distribution, symbol production, and recall components (immediate and delayed) of the task. For distribution, the number of cookies children placed in each box was recorded, and 1 point was assigned if it was the correct amount. The cookies were large and easy to break into either roughly half or quarter portions. The divisions did not need to be exact, and there were no instances when it was not possible to decide on the fraction that the child was attempting to produce. Children’s comments corresponded to their actions and all were noted. For symbol production, children’s notations were classified as one of three types of representation. The first, symbolic, referred to representations that used conventional notations, either numerals (e.g., “5”), or words (e.g., “five”). The second, analogue, were representations in which a discrete mark or object was recorded to represent each item in the set (e.g., 5 circles to indicate 5 cookies). The third, global notations, were any other pictographic or idiosyncratic representations (e.g., drawing a picture of a cookie).

All the responses were scored separately by two raters, and any discrepancies were mediated by a third rarer. Children’s comments were used to help verify decisions in cases of ambiguity; for example, a child leaving the Post-it note blank and saying, “there are none cookies, (sic)” was scored as attempting to represent zero. There were few disputes in the ratings.

The recall portion of the task was quantified in terms of the number of items out of 10 that children correctly stated from the post-it notes in each of the immediate and delayed recall conditions.

Results

The task involved three steps: distributing the cookies, recording the quantity, and reading the notation in both an immediate and delayed condition. The results from each of these will be presented separately. The minimal level of confidence for a significant effect was chosen prior to analysis to be p

DISTRIBUTION

The question for the distribution phase of the task is whether or not children could correctly place the required quantity of cookies in each box. This was to establish their knowledge of the meaning for each quantity independently of their ability to create notations to represent it. The proportion of correct distributions that children at each age were able to complete is shown in Figure 1. Preliminary analyses showed no differences between the two whole number values (2, 5) but frequent differences between the two fractional values (1/2, 1/4). Therefore, the two whole numbers have been combined in all the subsequent analyses but the fractions are kept separate.

A two-way analysis of variance for age (3, 4, 5, 6, and 7 years) and quantity (zero, whole numbers, half, and quarter) showed main effects of age, F(4,70) = 22.65, p

REPRESENTATION

Once the cookies were distributed, children had to create a notation to indicate the quantity. These notations were classified as symbolic, analogue, or global. The overall choice of each of the three notation types is presented in Figure 2. The data cannot be analyzed across the chosen notational categories because they are distributions that add up to 100% and are therefore not independent observations. Instead, a two-way ANOVA for age and quantity (whole, zero, half, quarter) was conducted separately for each notation type (symbolic, analogue, and global). All three ANOVAS revealed a significant effect of age and no effect for quantity. For symbolic representations, there was a main effect of age, F(4,70) = 29.85,p

Children’s representations of zero and fractions were examined in more detail and compared to their representational choices for whole numbers. Consider first the representations for zero. The most common nonsymbolic response for representing zero was to leave the post-it note blank. These responses were classified as analogue, since the blank post-it note corresponds iconically to the emptiness of the box containing no cookies. Furthermore, children were questioned about their representations, and in the case of those who left blanks when the box had zero cookies frequently said, “it means none cookies, (sic)” or “because there is no cookies (sic).” Hence, it was a deliberate strategy to convey the absence of quantity.

Children’s representations of zero were compared to their strategies for representing whole numbers by means of a chi-square analysis. The purpose was to assess whether children were applying the same principles of representation to these different quantities. This would help answer the question about what children believed their representations of whole numbers meant. Since representing zero is less familiar than representing whole numbers, a more intentional approach to the representational problem is needed. Children’s responses to zero were categorized as symbolic (“0,” “zero,” or “none”), analogue (blank), or global (miscellaneous), and compared to the three notation possibilities for whole numbers, namely, symbolic, analogue, or global. The frequency distribution is shown in Table 1. The chisquare analysis revealed a significant relation between children’s choices for representing each of these types of quantities, xZ (4, N=74) = 69.76, p

The relation between children’s representations for fractions and their representations for whole numbers was examined in the same way, but an additional category was added for the fractions. Some children attempted to convey a fraction by using a whole number, such as 2 for 1/2. These were part of the symbolic classification in the general analysis but were separately classified as “numeral” for the chi-square analysis. Therefore, a chi-square analysis examined the relation between representing fractions as symbolic (“1/2” or “half”), numeral (“2”), analogue, or global, and children’s representations of whole numbers as symbolic, analogue, or global. The frequencies are shown in Table 2. The chi-square analysis again showed a significant relation between children’s choices for these quantities, (chi)^sup 2^ (6, N = 74) = 87.47, p

RECALL

The third set of analyses examined children’s ability in the use of written notations to retrieve the quantity in each box, both immediately (see Figure 4) and two weeks later (see Figure 5). Children’s ability to correctly state the number was examined by a three-way ANOVA for time (immediate, delayed), quantity (whole number, zero, half, quarter), and age. There were main effects of time, F(1, 66) = 5.76, p

There was an interaction of age by quantity, F(12, 198) = 4.31, p

The data were examined as well in terms of the type of notation used. Again, these data cannot be analyzed because they cross over notational categories, such that individuals have different numbers of entries in each of the categories. Nonetheless, the pattern is presented in Figure 6. The data are collapsed across time and indicate the percentage of correctly recalled items as a function of the notation used. It is clear from these data that the probability of correctly recalling an item was greatest if the representation was symbolic rather than any other form.

Discussion

Children were asked to distribute a given quantity, create a representation to indicate that amount, and then use the representation to recall the quantity. The two main questions were, how children between the ages of 3 and 7 years old learned to carry out these three tasks, and whether the quantity involved changed either their success or their strategy. Specifically, children’s responses to the less familiar quantities of zero and fractions could shed light on their performance with the more familiar whole numbers. If children understood the relation between the notation and quantity, that is, why the notation indicated the quantity it did, then they should be able to adapt and apply these notations to the new quantities. If they did not understand the basis of notational representations for quantity, then they would have had more difficulty in creating an appropriate representation to indicate the less familiar values.

Consider first the differences between the three parts of the task. Children were very successful at distributing the cookies into the boxes. Even the 3-year-olds were correct at least half of the time for all but the most difficult quantity, quarter. In fact, quarter remained difficult to distribute until 7 years of age. The 3-year-olds found the quantities zero and half to be more difficult than whole numbers, but by 4 years of age, children were doing very well on this part of the task for all quantities except quarter. These results signify that children had-a reasonably good understanding of what these quantities mean (except for quarter) and could carry out a procedure to produce that quantity correctly. By the time children were 4 years old, their understanding of zero as a quantity was about as good as their understanding of whole numbers. Children understood the meanings of the quantities that were tested in the study.

Children’s ability to produce an accurate representation of those quantities developed more slowly. Even though all the children were able to distribute half a cookie by the time they were 3 years old with reasonable success, the only children who could represent this amount were the 7-yearolds. Conversely, children could produce meaningful representations for whole numbers and zero almost as soon as they could accurately distribute those quantities. Although children appeared to understand all the quantities, they were not able to represent them.

Finally, children’s ability to read the notations depended on both the quantity that was indicated and the type of notation that was chosen. Children were very successful in reading their representations for whole numbers and zero but found the two fractions to be very difficult until they were 7 years old. Similarly, quantities that were written with a symbolic notation were more likely to be read correctly later, irrespective of the quantity that was indicated.

In summary, children’s ability to understand, represent, and recall quantities, indicated that they understood the significance of the quantities before they could use notational forms to represent them. Conversely, children were able to produce notational forms that they were unable to read later. Children clearly understand something about notation: Minimally, they understand that written forms can encode quantities, but without an understanding of hooted the notation works or the rules by which it encodes those quantities, they produce forms that ultimately have no meaning. Therefore, there is a disparity between children’s knowledge of quantity and their mastery of notational forms to represent quantity. One should not be taken as evidence of competence for the other.

The second issue is how the quantity influenced children’s ability to solve the problems. Throughout all phases of the task, there were large differences between children’s success in handling whole numbers and fractions but virtually no differences between their success with whole numbers and zero. Zero and fractions were interesting to examine for different reasons: zero because it is an abstract notion meaning absence, and fractions because they are technical computations derived from wholes. Zero turned out not to be a problem for children, but fractions were different. Very young children had some understanding of the concept of half as a quantity, but only the oldest children could create a display containing the quantity quarter, and none of the children could represent these quantities until they were 7 years old. These oldest children were in second grade and were the only ones receiving formal instruction in fractions. The pattern suggests that the concept of zero develops in concert with children’s growing knowledge of number and quantity, but the concept of fraction, particularly quarter, and the representation of fractional quantities, requires special instruction. It appears that children consider zero to be a position on a dimension of quantity, presumably less than 1. It is less clear that they understand that fractions can also be positioned on this scale between the values of 0 and 1. Gelman (1991) also reported evidence that children failed to understand this positional aspect of the meaning of fractions. This misunderstanding raises interesting questions about how young children interpret the contextual uses of fractions and parts, such as half the cookies, some of the cookie, and 3 1/2 years old.

Children’s decisions about which notation type to choose were driven by their age and not by the quantity they were trying to represent. There was virtually no attempt to create or adapt representations to reflect the difficult quantities; they simply applied the strategy they had always used even if it was not appropriate. Hence, there was no effect of quantity in the analyses of representational form chosen, and the chi square analyses showed that children made the same choices for zero and fractions as they did for whole numbers. Children’s strategies for choosing a notation type changed over the years studied, and age remained the primary factor in determining the notation type. Discounting global representations because they provided only questionable indications of quantity, the 3- and 4-year-olds preferred analogue notations, the 6- and 7-year-olds preferred symbolic, and almost all of the 5-year-olds used both. There were individual 5-year-olds, however, who showed a preference for only one of the two.

The choices children made for representing the quantities were important because they helped to determine whether or not children would be able to read the notation later. Both the quantity and the type of notation chosen predicted whether children would be successful in the recall portion of the task. Children’s best success in reading the notations occurred with symbolic representations. Even analogue representations that correctly depicted the quantity were less likely to be interpreted later. This replicates the finding reported in an earlier study (Bialystok & Codd, 1996) where we argued that the reason for this was that the analogue representations did not function as symbols for the children.

Instead, they were treated as pictures and therefore did not have the same referential meaning that symbols do.

The results of this study are largely descriptive. The data do not always allow for the kind of detailed analyses that are necessary to offer reliable conclusions. Nonetheless, several important patterns are clear in the data, and taken together, these patterns contribute to our conception of how children develop an understanding of various quantities, and the relationships between quantity and the representational forms used to indicate them.

First, children appeared to understand the quantitative meaning of zero from about as early as they understood whole numbers. Even though the status of the quantity zero is different from that of whole numbers in that it lacks a concrete reality, children accepted it as a number and applied their knowledge of representation effectively to this quantity. This is not the case for fractions. An understanding of the significance of these quantities developed more gradually, and children’s competence with fractions was not really evident until they were in second grade and learning about these values formally.

Second, children’s grasp of the two fractions examined in this study is different, replicating the claims of Hunting and Sharpley (1991). Even the youngest children had an informal understanding of the meaning for half and could respond to the distribution question correctly. Nonetheless, producing representations for half or reading those representations later remained difficult until children were 7 years old. The quantity quarter was more problematic in that children could not even distribute that quantity until they were 7 years old. The difference between these two fractions may be in their familiarity and the experience children have had with each of them. Children may frequently have been told they could eat half a cookie or watch half a movie, but they may less likely have been instructed to sample a quarter of the whole. In mathematical terms, there is no obvious difference between the fractions in terms of their complexity, but the exposure to one of them made it much more accessible to children. Hence, children’s mastery of a formal system like this seems to be strongly influenced by their experience.

Third, children’s ideas about representations for quantity include a few procedures for indicating quantity, but probably include very little about why the representation indicates the quantity it does. Children were not able to adapt their representations to indicate the difficult quantitative concepts. Instead, they persisted in applying the same strategies that they used for the simpler quantities. Even without knowing the conventional notation forms for fractions, children who understood the concept of representation could have created a notation that depicted the correct (or approximately correct) portion. Children who were successful created such analogue representations for fractions as a slice of a pie or a half moon or, in one case, a picture of a 25-cent coin. Most children, however, did not adapt the notation to convey the relevant information. For example, children who had used analogue representations for whole numbers by indicating a tick mark to correspond to each item in the set used the same strategy for fractions, producing some (random) number of tick marks; children who had used symbolic representations for whole numbers by writing the corresponding numeral continued to use a single numeral to indicate a fraction. These errors suggest that children’s understanding of why the representations worked in the case of whole numbers was incomplete.

Finally, replicating our earlier research with a similar problem (Bialystok & Codd, 1996), the effectiveness of a representation of quantity depended on the type of notation that was used. Although we were unable to analyze these data formally, there were large differences between the success with which children could read the notation as a function of the type of notation produced. Even when an analogue notation correctly conveyed the quantity, children were less able to decode its meaning than they were for a symbolic representation. In some fundamental sense, the analogue representation did not signify meaning.

The children in this study covered a large age span, from 3 to 7 years. Children during these developmental years acquire a great deal of knowledge about the concepts of quantity, representation, and the relations between them. The major transition period for these developments appears to be at 5 years of age. This is the age at which children’s strategies change from analogue to conventional notations, and their success in all parts of the task becomes more reliable. These children are in kindergarten and undoubtedly are receiving more formal instruction in the properties of the number system. Clearly, this educational experience is a significant factor in changing the way children approach these tasks at this age. Still, even by 7 years old, the children are struggling to apply this knowledge to the more difficult fractional quantities. Further investigations examining these developments in more detail should help to complete our understanding of how children acquire these difficult concepts and their representations.

This research was funded by Grant A2559 from the Natural Sciences and Engineering Council of Canada to the first author. We are grateful for the assistance of Amanda Tessaro who collected the data.

Sommaire

Les jeunes enfants comptent avec enthousiasme et, vers Page de trois ou quatre ans, appliquent cette routine a de nombreux objets de leur milieu, comme leur age et le nombre de bonbons duns un paquet (Baroody, 1992; Fuson, 1988, 1991, 1992; Gallistel et Gelman, 1990, 1992; Gelman, 1993; Gelman et Gallistel, 1978). Cependant, ils ne matrisent ce processus que vers l’age de quatre ou cinq ans.

Its consacrent aussi leurs annees prescolaires a acquerir une connaissance des formes ecrites correspondant aux nombres de la sequence de comptage (Tolchinsky Landsmann et Karmiloff Smith, 1992). A leur arrivee a 1’ecole, ils peuvent identifier la plupart des chiffres en nommant la quantite correspondante. Mais si les enfants comprennent mal les significations cardinales des nombres qu’ils ont appris a compter, alors que signifient, a leurs yeux, les inscriptions de quantites? Tout comme les enfants qui savent compter (c: 3-d. connaissent les noms des chiffres prononces) ne comprennent pas forcement la signification de ces formes, de meme, ceux qui semblent reconnaitre ces representations ecrites ne saisissent pas necessairement tout ce qu’elles designent. Aux yeux des enfants, que signifient les representations ecrites de quantite?

Les chercheurs ont decouvert un ecart entre la capacite des enfants de produire les elements du systeme numerique et leur comprehension de la signification de ces elements (Bialystok et Codd, 1996; Fuson, Fraivillig et Burghardt, 1992; Hughes, 1986; Kamii, 1981; Neuman, 1987; Sinclair, 1988, 1991; Sinclair, Siegrist et Sinclair, 1982). Ces etudes ont montre un decalage de developpement qui est passe successivement des elements idiosyncrasiques aux analogiques, puffs aux elements conventionnels du systeme numerique.

Le present article developpe ces conclusions en communiquant les resultats dune etude pour laquelle on a demande a des enfants de trois a sept ans d.ec.re une representation indiquant une quantite presentee a la maniere d’un jeu. II y await trois genres de quantites : des nombres entiers, le zero et des fractions. Les notations des enfants leur ont ete montrees peu apres qu’ils les aient realisees, puffs une seconde fois deux semaines plus tard, pour voir si les enfants pouvaient les interpreter. Les resultats ont montre le mode familier de developpement vers un usage accru de notations conventionnelles pour toutes les quantites. La capacite de lire les notations etait la plus prononcee pour les nombres conventionnels ou les resultats etaient les meilleurs, plus faible pour les representations analogues et tres faible pour I’enregistrement global des idiosyncrasies. Le choix du format de notation par les enfants a ete influence presque exclusivement par leur age et non par la quantite representee. Les enfants ont pu resoudre les problemes du zero presque aussi bien que ceux des nombres entiers, mais leur comprehension et leur usage des representations pour les fractions etaient tres limites.

References

Allardice, B. (1977). The development of written representations for some mathematical concepts. Journal of Children’s Mathematical Behavior, 4, 135-148.

Baillargeon, R., Kotovsky, L., & Needham, A. (1995). The acquisition of physical knowledge in infancy. In D. Sperber, D. Premark, & A. Premark (Eds.), Causal cognition: A

multi-disciplinary debate (pp. 79-116). New York: Oxford. Baroody, A. J. (1992). The development of preschoolers’ count

ing skills and principles. In J. Bideaud, C. Meljac, & J-P. Fischer (Eds.), Pathways to number: Children’s developing numerical abilities (pp. 99-126). Hillsdale, Nj: Erlbaum.

Bialystok, E. (1991). Letters, sounds, and symbols: Changes in children’s understanding of written language. Applied Psycholinguistics, 12, 75-89.

Bialystok, E., & Codd, J. (1996). Developing representations of quantity. Canadian Journal of Behavioural Science, 28, 281-291. Davis, A., Bridges, A., & Brosgall, A. (1985). Showing how

many: Young children’s written representation of number. Educational Psychology, S, 303-310.

Durkin, K. (1993). The representation of number in infancy and early childhood. In C. Pratt & A. F. Garton (Eds.), Systems of representation in children’s development and use (pp. 133-166). West Sussex, UK: Wiley.

Fuson, K. C. (1988). Children’s counting and concepts of number. New York: Springer-Verlag.

Fuson, K.C. (1991). Children’s early counting: Saying the number-word sequence, counting objects, and understanding cardinality. In K. Durkin & B. Shire (Eds.), Language in mathematical education: Research and practice (pp. 27-39). Buckingham, UK: Open University.

Fuson, K. C. (1992). Relationships between counting and cardinality from age 2 to age 8. In J. Bideaud, C. Meljac, & J-P. Fischer (Eds.), Pathways to number.: Children’s developing numerical abilities (pp. 127-149). Hillsdale, Nj: Erlbaum.

Fuson, K. C., Fraivillig, J. L., & Burghardt, B. H. (1992). Relationships children construct among English number words, multi-unit base-ten blocks, and written multi-digit addition. In J. I. D. Campbell (Ed.), The nature and origins of mathematical skills (pp. 39-112). Amsterdam: Elsevier Science.

Fuson, K. C., & Kwon, Y. (1992). Learning addition and subtraction: Effects of number words and other cultural tools. In J. Bideaud, C. Meljac, & j: P. Fischer (Eds.), Pathways to number.* Children’s developing numerical abilities (pp. 283-306). Hillsdale, NJ: Erlbaum.

Gallistel, C. R., & Gelman, R. (1990). The what and how of counting. Cognition, 34, 197-199.

Gallistel, C. R., & Gelman, R. (1992). Preverbal and verbal counting and computation. Cognition, 44, 43-74.

Geary, D. C. (1995). Reflections of evolution and culture in children’s cognition: Implications for mathematical development and instruction. American Psychologist, 50, 24-37.

Gelman, R. (1993) A rational-constructivist account of early learning about numbers and objects. In D. L. Medin (Ed.), The psychology of learning and motivation: Advances in research and theory, New York: Academic Press.

Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge, MA: Harvard University Press. Hughes, M. (1986). Children and number. Oxford: Basil Blackwell.

Hughes, M. (1991). What is difficult about learning arithmetic?

In P. Light, S. Sheldon, & M. Woodhead (Eds.), Learning to think: Child development in social context, pp. 184-204. London: Routledge.

Hunting, R. P., & Sharpley, C. F. (1991). Pre-fraction concepts of preschoolers. In R. P. Hunting & G. Davis (Eds.), Early fraction learning (pp. 9-26). New York: Springer-Verlag.

Joseph, G. (1991). The crest of the peacock. New York: LB. Tauris & Co.

Kamii, M. (1981). Children’s ideas about written number. Topics in learning and learning disabilities, 1, 47-59.

Klein, A., & Starkey, P. (1987). The origins and development of numerical cognition: A comparative analysis. In J. A. Sloboda & D. Rogers (Eds.), Cognitive processes in mathematics (pp. 1-25). Oxford: Clarendon.

Kline, M. (1972). Mathematical thought from ancient to modern times. New York: Oxford University Press.

Mix, K. S., Levine, S. C., & Huttenlocher, J (1999). Early fraction calculation ability. Developmental Psychology, 35, 164-174.

Mounoud, P., & Bower, T. G. (1974). Conservation of weight in infants. International Journal of Cognitive Psychology, 3, 29-40.

Neugebauer, O. (1957). The exact sciences in antiquity. Providence, RI: Brown University Press.

Neuman, D. (1987). The origin of arithmetic skills: A phenomenographic approach. Goteberg, Sweden: ACTH Universitatis Gothoburgensis.

Parrat-Dayan, S., & Voneche, J. (1992). Conservation and the notion of “half.” In J. Bideaud, C. Meljac, & J: P. Fischer (Eds.), Pathways to number: Children’s developing numerical abilities (pp. 67-82). Hillsdale, NJ: Erlbaum.

Resnick, L. B. (1995). Inventing arithmetic: Making children’s intuition work in school. In C. A. Nelson (Ed.), Basic and applied perspectives on learning, cognition, and development: The Minnesota Symposia on Child Psychology, idol. 28 (pp. 75-101). Mahwah, NJ: Erlbaum.

Resnick, L. B., & Singer, J. (1993). Protoquantitative origins of ratio reasoning. In T. P. Carpenter, E. Fennema, & T. A Romberg (Eds.), Rational numbers: An integration of research (pp. 107-130). Hillsdale, Nj: Erlbaum.

Siegal, M., & Smith, J. A. (1997). Toward making representation count in children’s conceptions of fractions. Contemporary Educational Psychology, 22, 1-22.

Sinclair. A. (1988). La notation numerique chez l’enfant. In H. Sinclair (Ed.), La production de notations chez le jeune enfant: Langage, nombres, rythmes, et melodies (pp. 71-97). Paris: Presses Universitaires de France.

Sinclair, A. (1991). Children’s production and comprehension of written numerical representations. In K. Durkin & B. Shire (Eds.), Language in mathematical education: Research and practice (pp. 59-68). Buckingham, UK: Open University.

Sinclair, A., Siegrist, F., & Sinclair, H. (1982). Young children’s ideas about the written number system. In D. Rogers & J. A. Sloboda (Eds.), The acquisition of symbolic skills (pp. 535-542). New York: Plenum Press.

Sinclair, A., & Sinclair, H. (1984). Preschool children’s interpretation of written numbers. Human Learning, 3, 173-184.

Smith, D. (1963). Our debt to Greece and Rome. New York: Cooper Square Publishers.

Sophian, C. (1992). Learning about numbers: Lessons for mathematics education from preschool number development. In J. Bideaud, C. Meljac, & J: P. Fischer (Eds.), Pathways to number. Children’s developing numerical abilities (pp. 19-40). Hillsdale, LIj: Erlbaum.

Sophian, C. (1996) Young children’s numerical cognition: What develops? In R. Vesta (Ed.), Annals of Child Development, Vol. 1 (pp. 49-86). London: Jessica Kingsley Publishers.

Sophian, C., Garyantes, D., & Chang, C. (1997). When three is less than two: Early developments in children’s understanding of fractional quantities. Developmental Psychology, 33, 731-744.

Sophian, C., & Kailihiwa, C. (1998). Units of counting: Developmental changes. Cognitive Development, 13, 561-585. Tolchinsky Landsmann, L., & Karmiloff Smith, A. (1992).

Children’s understanding of notations as domains of knowledge versus referential-communicative tools. Cognitive Development. 7, 287-300.

Wynn, K. (1992). Children’s acquisition of the number words and the counting system. Cognitive Psychology, 24, 220-251.

ELLEN BIALYSTOK and JUDITH CODD, York University

Please address all correspondence to the first author at the Department of Psychology, York University, 4700 Keele Street, Toronto, Ontario M3J 1P3 (E-mail: ellenb@yorku.ca).

Copyright Canadian Psychological Association Jun 2000

Provided by ProQuest Information and Learning Company. All rights Reserved