Elsevier

Cognition

Volume 83, Issue 3, April 2002, Pages 223-240
Cognition

The development of ordinal numerical knowledge in infancy

https://doi.org/10.1016/S0010-0277(02)00005-7Get rights and content

Abstract

A critical question in cognitive science concerns how numerical knowledge develops. One essential component of an adult concept of number is ordinality: the greater than and less than relationships between numbers. Here it is shown in two experiments that 11-month-old infants successfully discriminated, whereas 9-month-old infants failed to discriminate, sequences of numerosities that descended in numerical value from sequences that increased in numerical value. These results suggest that by 11 months of age infants possess the ability to appreciate the greater than and less than relations between numerical values but that this ability develops between 9 and 11 months of age. In an additional experiment 9-month-old infants succeeded at discriminating the ordinal direction of sequences that varied in the size of a single square rather than in number, suggesting that a capacity for non-numerical ordinal judgments may develop before a capacity for ordinal numerical judgments. These data raise many questions about how infants represent number and what happens between 9 and 11 months to support ordinal numerical judgments.

Introduction

A growing body of data suggests that non-human animals and humans share a primitive non-verbal numerical system (see Dehaene et al., 1998, Gallistel and Gelman, 2000). For example, when rhesus monkeys, young children and human adults compare the relative numerosity of visual arrays, both distance and size effects are found (e.g. Brannon and Terrace, 1998, Brannon and Terrace, 2000, Brannon and Terrace, in press, Moyer and Landaeur, 1967, Temple and Posner, 1998). The numerical distance effect is defined by faster and more accurate responding when subjects compare the relative magnitude of values that are farther apart numerically, whereas the numerical size effect is characterized by higher accuracy and faster responding for smaller numerical magnitudes when numerical disparity is held constant. The distance and size effects suggest that this shared system relies on numerical representations in the form of continuous magnitudes (Dehaene et al., 1998, Gallistel and Gelman, 2000). Although adult humans clearly have alternative computational routes available (e.g. verbal counting), it appears that under some circumstances humans bypass these alternative symbolic routes and rely instead on a system that is evolutionarily primitive (e.g. Cordes et al., in press, Whalen et al., 1999).

Although the data reviewed above suggest evolutionary continuity in the numerical representations of adult humans and animals, a more controversial question is whether there is ontogenetic continuity in numerical cognition (Carey, 2001, Gelman and Cordes, 2001). Research from the habituation dishabituation of looking time, cross-modal transfer, and violation of expectation paradigms all suggest that in some sense human infants represent number (e.g. Antell and Keating, 1983, Bijeljac-Babic et al., 1991, Starkey and Cooper, 1980, Starkey et al., 1990, Strauss and Curtis, 1981, Treiber and Wilcox, 1984, Uller et al., 1999, van Loosbroek and Smitsman, 1990, Wynn, 1992, Xu and Spelke, 2000, Koechlin et al., 1998, Simon et al., 1995, but see Clearfield & Mix, 1999). It remains unclear however, whether the numerical abilities of infants are the developmental precursors of the non-verbal numerical system displayed by adult humans and animals. One avenue towards addressing this question is to look for common features of numerical representations in non-human animals, adult humans and human infants. For example, Xu and Spelke (2000) recently showed that 6-month-old infants discriminate 8 from 16 elements but fail to discriminate 8 from 12 elements. This pattern of data might mean that the ratio of the numerosities being compared controls discrimination, however, such an account does not explain why the same aged infants can discriminate 2 vs. 3. Another prediction of the continuity hypothesis is that infants, like adults and non-human animals, should appreciate the ordinal relationships between numerical magnitudes. The experiments described here address this second question.

Very little is known about infants' knowledge of ordinal numerical relationships. To illustrate the distinction between cardinal and ordinal numerical knowledge, imagine being able to differentiate two objects from three objects but not knowing which set is numerically greater. Some authors have argued that infants first comprehend only the cardinal properties of number and then later come to appreciate ordinal relationships between numbers through observing numerical transformations in their environment (see Cooper, 1984, Dehaene and Changeux, 1993, Kitcher, 1984, Strauss and Curtis, 1984). An alternative view is that infants represent numerical ordinality from the start (e.g. Wynn, 1995). The question boils down to whether for a young infant twoness is to threeness much like a blender is to a chair, or alternatively whether even for the very young infant twoness and threeness are perceived as different values along one numerical continuum.

Only a handful of studies since the landmark work of Piaget (1952) have directly addressed the development of ordinal numerical knowledge in young children. For example, Brannon and Van de Walle (2001) recently showed that children as young as 2 years of age represent the ordinal relations between numerical values as large as 4 or 5 even when surface area is controlled (see also Bullock and Gelman, 1977, Huntley-Fenner and Cannon, 2000, Siegel, 1974, Sophian and Adams, 1987, Strauss and Curtis, 1984). Thus, children make ordinal numerical judgments before they are proficient at using the verbal counting system to mediate these judgments.

Even fewer studies have specifically tested for ordinal numerical knowledge in the first year of life. One relevant type of data comes from research showing that infants keep track of the number of objects behind an occluder (e.g. Wynn, 1992). These data may be interpreted as evidence that infants are capable of addition and subtraction and that infants can represent ordinal numerical relations (Wynn, 1992, Wynn, 1995, Wynn, 1998). However, the available evidence may also be explained by a non-numerical account (object-file system) whereby infants represent each of the objects behind an occluder and do not possess a symbolic representation of the numerosity of the set (see Simon, 1997, Uller et al., 1999).

In a second paradigm more directly addressing ordinal numerical knowledge in infancy, Feigenson, Carey, and Hauser (in press) found that 10- and 12-month-old infants spontaneously chose the numerically larger of two sets of food items when amount of food was confounded with number but failed to do so when amount of food was equivalent. In addition, even when amount of food could have been used as a cue infants succeeded at 1 vs. 2 and 2 vs. 3 and failed at 2 vs. 4, 3 vs. 4 and 3 vs. 6 suggesting that the numerical ratio was not what controlled performance but instead that infants were limited by the numerical size of the values being compared. Feigenson et al. interpret their results as evidence that infants use an object-file to represent each food item and that information about surface area is preserved and used in the comparison process.

In a third paradigm, Cooper (1984) habituated infants to pairs of displays that were presented successively. The displays of each pair maintained a constant ordinal relationship between the number of elements in the first and second display but the absolute values varied between trials (values ranged from 1 to 4). Thus, on habituation trials infants were always shown a small number followed by a large number or the reverse. Infants were then tested with pairs of numerical displays where the ordinal relationship between the two displays was the same as in habituation, was reversed, or was eliminated by equating the numerical value of the first and second displays. An interesting pattern of results was obtained. Ten- to 12-month-old infants dishabituated (i.e. looked longer) when tested with the novel pairs that contained two equal numerical values but failed to dishabituate to the novel pairs that reversed in ordinal direction. In contrast, 14–16-month-old-infants dishabituated to both of the novel types of test trials (change in ordinal direction and elimination of ordinal relations). This pattern of results is tantalizing and has been widely cited because it suggests a developmental trend in ordinal numerical knowledge. Infants under 12 months of age only differentiate equal and unequal numerical relations and fail to distinguish greater than from less than relations, whereas by 14 months of age infants have ordinal numerical knowledge. However, these results are difficult to interpret because surface area was not controlled.

The current research seeks to test whether infants represent ordinal numerical relations and whether there is a lower age boundary on this ability. In Experiment 1, 9- and 11-month-old infants were habituated to three-item sequences of numerical displays presented in an ascending or descending numerical order. The sequences were dynamic in that they repeated continuously and infants' looking times were measured for the whole sequence. The absolute numerical values were varied between trials and surface area was not confounded with number. Infants were then tested with new numerical values where the ordinal relations were maintained or were reversed from that of habituation. If infants represent ordinal numerical relations they should have looked longer when the ordinal direction was reversed from that of habituation compared to when it was maintained.

Section snippets

Participants

Participants were 16 healthy full-term 11-month-old infants (mean age: 10 months 23 days, range: 10 months 14 days–11 months 14 days) and 16 healthy full-term 9-month-old infants (mean age: 9 months, range: 8 months 14 days–9 months 14 days). Seven of the 11-month-old and four of the 9-month-old infants were female. Data from five additional 11-month-old and four 9-month-old infants were discarded because of fussiness resulting in failure to complete at least four test trials.

Design

Infants were

Experiment 2

The results of Experiment 1 suggest that 11-month-old infants represent the ordinal relations between numerical values and that for whatever reason 9-month-old infants do not. However, it is possible that 11-month-old infants in Experiment 1 used density rather than number to differentiate the ordinal directions of numerical sequences because density was confounded with number in the displays used in Experiment 1. In addition, in Experiment 1, element size was on average inversely correlated

Experiment 3

Experiments 1 and 2 collectively suggest that a change occurs between the 9th and 11th months of life to support the ability to make numerical comparisons. However, these results can not answer whether the failure of 9-month-old infants is numerical in nature or instead depends on a more general cognitive ability such as the ability to contrast any two rapidly and successively presented visual displays. To test this alternative possibility, 9-month-old infants were tested in another version of

General discussion

The results of Experiments 1 and 2 suggest that infants as young as 11 months of age are sensitive to the ordinal relations between numerical values. Such a finding would be important because it would demonstrate that preverbal infants within the first year of life appreciate that 16 is numerically more than 8 rather than simply discriminating that 16 is different from 8. It could also be argued that such a finding would be evidence that infants have numerical concepts because it would

Acknowledgements

The experimental design benefited from discussions with Susan Hespos who was conducting similar experiments in collaboration with Elizabeth Spelke. Thanks also to Susan Carey, Lisa Feigenson, Amy Needham, and Gretchen Van de Walle for helpful advice on experimental design and procedures and to Susan Carey, Amy Needham, Michael Platt, Gretchen Van de Walle, and two anonymous reviewers for their comments on the paper. Finally, thanks to Stan Dehaene for suggesting the stimulus controls used in

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