Dissociation between small and large numerosities in newborn infants

In the first year of life, infants possess two cognitive systems encoding numerical information: one for processing the numerosity of sets of 4 or more items, and the second for tracking up to 3 objects in parallel. While a previous study showed the former system to be already present a few hours after birth, it is unknown whether the latter system is functional at this age. Here, we adapt the auditory‐visual matching paradigm that previously revealed sensitivity to large numerosities to test sensitivity to numerosities spanning the range from 2 to 12. Across studies, newborns discriminated pairs of large numerosities in a 3:1 ratio, even when the smaller numerosity was 3 (3 vs. 9). In contrast, newborn infants failed to discriminate pairs including the numerosity 2, even at the same ratio (2 vs. 6). These findings mirror the dissociation that has been reported with older infants, albeit with a discontinuity situated between numerosities 2 and 3. Two alternative explanations are compatible with our results: either newborn infants have a separate system for processing small sets, and the capacity of this system is limited to 2 objects; or newborn infants possess only one system to represent numerosities, and this system either is not functional or is extremely imprecise when it is applied to small numerosities.     

Baby arithmetic: one object plus one tone

Recent studies using a violation-of-expectation task suggest that preverbal infants are capable of recognizing basic arithmetical operations involving visual objects. There is still debate, however, over whether their performance is based on any expectation of the arithmetical operations, or on a general perceptual tendency to prefer visually familiar and complex displays. Here we provide new evidence that 5-month-old infants recognize basic arithmetic operations across sensory modalities. Using a violation-of-expectation task that eliminated the possibility of the familiarity and complexity preference, 5-month-old infants were presented alternatively with two types of arithmetical events: the expected, correct outcomes of operations (1 object+1 tone=2 objects and 1 object+2 tones=3 objects) and the unexpected, incorrect ones (1 object+2 tones=2 objects and 1 object+1 tone=3 objects). Results showed that subjects looked significantly longer at the unexpected events than at the expected events, suggesting that infants are able to recognize basic arithmetic operations across sensory modalities.     

Numerical abstraction by human infants

Across several experiments, 6- to 8-month-old human infants were found to detect numerical correspondences between sets of entities presented in different sensory modalities and bearing no natural relation to one another. At the basis of this ability, we argue, is a sensitivity to numerosity, an abstract property of collections of objects and events. Our findings provide evidence that the emergence of the earliest numerical abilities does not depend upon the development of language or complex actions, or upon cultural experience with number.     

Multisensory information boosts numerical matching abilities in young children

This study presents the first evidence that preschool children perform more accurately in a numerical matching task when given multisensory rather than unisensory information about number. Three- to 5-year-old children learned to play a numerical matching game on a touchscreen computer, which asked them to match a sample numerosity with a numerically equivalent choice numerosity. Samples consisted of a series of visual squares on some trials, a series of auditory tones on other trials, and synchronized squares and tones on still other trials. Children performed at chance on this matching task when provided with either type of unisensory sample, but improved significantly when provided with multisensory samples. There was no speed–accuracy tradeoff between unisensory and multisensory trial types. Thus, these findings suggest that intersensory redundancy may improve young children’s abilities to match numerosities.     

Individuation of pairs of objects in infancy

Looking-time studies examined whether 11-month-old infants can individuate two pairs of objects using only shape information. In order to test individuation, the object pairs were presented sequentially. Infants were familiarized either with the sequential pairs, disk-triangle/disk-triangle (XY/XY), whose shapes differed within but not across pairs, or with the sequential pairs, disk-disk/triangle-triangle (XX/YY), whose shapes differed across but not within pairs. The XY/XY presentation looked to adults like a single pair of objects presented repeatedly, whereas the XX/YY presentation looked like different pairs of objects. Following familiarization to these displays, infants were given a series of test trials in which the screen was removed, revealing two pairs of objects in one of two outcomes, XYXY or XXYY. On the first test trial, infants familiarized with the identical pairs (XY/XY) apparently expected a single pair to be revealed because they looked longer than infants familiarized with the distinct pairs (XX/YY). Infants who had seen the distinct pairs apparently expected a double pair outcome. A second experiment showed outcomes of a single XY pair. This outcome is unexpected for XX/YY-familiarized infants but expected for XY/XY-familiarized infants, the reverse of Experiment 1. This time looking times were longer for XX/YY infants. Eleven-month-olds appear to be able to represent not just individual objects but also pairs of objects. These results suggest that if they can group the objects into sets, infants may be able to track more objects than their numerosity limit or available working memory slots would normally allow. We suggest possible small exact numerosity representations that would allow tracking of such sets.     

Imprinted numbers: newborn chicks’ sensitivity to number vs. continuous extent of objects they have been reared with

Newborn chicks were tested for their sensitivity to number vs. continuous physical extent of artificial objects they had been reared with soon after hatching. Because of the imprinting process, such objects were treated by chicks as social companions. We found that when the objects were similar, chicks faced with choices between 1 vs. 2 or 2 vs. 3 objects chose the set of objects of larger numerosity, irrespective of the number of objects they had been reared with. Moreover, when volume, surface or contour length were controlled for using sets of 1 vs. 4, 1 vs. 6 or 1 vs. 3 objects, chicks resorted to choosing the larger object, rather than the familiar numerosity. When, however, chicks were reared with objects differing in their aspect (colour, size, and shape) and then tested with completely novel objects (of different colour and shape but controlled for continuous extent), they chose to associate with the same number of objects they had been reared with. These results suggest that identification of objects as different and separate individuals is crucial for the computation of number rather than continuous extent in numerical representation of small numerosities and provide a striking parallel with results obtained in human infants. Early availability of small numerosity discrimination by chicks strongly suggests that these abilities are in place at birth.     

Evidence of amodal representation of small numbers across visuo-tactile modalities in 5-month-old infants

Two experiments investigated 5-month-old infants’ amodal sensitivity to numerical correspondences between sets of objects presented in the tactile and visual modes. A classical cross-modal transfer task from touch to vision was adopted. Infants were first tactually familiarized with two or three different objects presented one by one in their right hand. Then, they were presented with visual displays containing two or three objects. Visual displays were presented successively (Experiment 1) or simultaneously (Experiment 2). In both experiments, results showed that infants looked longer at the visual display which contained a different number of objects from the tactile familiarization phase. Taken together, the results revealed that infants can detect numerical correspondences between a sequence of tactile and visual stimulation, and they strengthen the hypothesis of amodal and abstract representation of small numbers of objects (two or three) across sensory modalities in 5-month-old infants.     

Number sense in human infants

Four experiments used a preferential looking method to investigate 6-month-old infants' capacity to represent numerosity in visual-spatial displays. Building on previous findings that such infants discriminate between arrays of eight versus 16 discs, but not eight versus 12 discs (Xu & Spelke, 2000), Experiments 1 and 2 investigated whether infants' numerosity discrimination depends on the ratio of the two set sizes with even larger numerosities. Infants successfully discriminated between arrays of 16 versus 32 discs, but not 16 versus 24 discs, providing evidence that their discrimination shows the set-size ratio signature of numerosity discrimination in human adults, children and many non-human animals. Experiments 3 and 4 addressed a controversy concerning infants' ability to discriminate large numerosities (observed under conditions that control for total filled area, array size and density, item size and correlated properties such as brightness: Brannon, 2002; Xu, 2003b; Xu & Spelke, 2000) versus small numerosities (not observed under conditions that control for total contour length: Clearfield & Mix, 1999). To investigate the sources of these differing findings, Experiment 3 tested infants' large-number discrimination with controls for contour length, and Experiment 4 tested small-number discrimination with controls for total filled area. Infants successfully discriminated the large-number displays but showed no evidence of discriminating the small-number displays. These findings provide evidence that infants have robust abilities to represent large numerosities. In contrast, infants may fail to represent small numerosities in visual-spatial arrays with continuous quantity controls, consistent with the thesis that separate systems serve to represent large versus small numerosities.     

Auditory–visual intermodal matching based on individual recognition in a chimpanzee (Pan troglodytes)

The ability to recognize familiar individuals with different sensory modalities plays an important role in animals living in complex physical and social environments. Individual recognition of familiar individuals was studied in a female chimpanzee named Pan. In previous studies, Pan learned an auditory–visual intermodal matching task (AVIM) consisting of matching vocal samples with the facial pictures of corresponding vocalizers (humans and chimpanzees). The goal of this study was to test whether Pan was able to generalize her AVIM ability to new sets of voice and face stimuli, including those of three infant chimpanzees. Experiment 1 showed that Pan performed intermodal individual recognition of familiar adult chimpanzees and humans very well. However, individual recognition of infant chimpanzees was poorer relative to recognition of adults. A transfer test with new auditory samples (Experiment 2) confirmed the difficulty in recognizing infants. A remaining question was what kind of cues were crucial for the intermodal matching. We tested the effect of visual cues (Experiment 3) by introducing new photographs representing the same chimpanzees in different visual perspectives. Results showed that only the back view was difficult to recognize, suggesting that facial cues can be critical. We also tested the effect of auditory cues (Experiment 4) by shortening the length of auditory stimuli, and results showed that 200 ms vocal segments were the limit for correct recognition. Together, these data demonstrate that auditory–visual intermodal recognition in chimpanzees might be constrained by the degree of exposure to different modalities and limited to specific visual cues and thresholds of auditory cues.     

Origins of number sense. Large-number discrimination in human infants

Four experiments investigated infants' sensitivity to large, approximate numerosities in auditory sequences. Prior studies provided evidence that 6-month-old infants discriminate large numerosities that differ by a ratio of 2.0, but not 1.5, when presented with arrays of visual forms in which many continuous variables are controlled. The present studies used a head-turn preference procedure to test for infants' numerosity discrimination with auditory sequences designed to control for element duration, sequence duration, interelement interval, and amount of acoustic energy. Six-month-old infants discriminated 16 from 8 sounds but failed to discriminate 12 from 8 sounds, providing evidence that the same 2.0 ratio limits numerosity discrimination in auditory-temporal sequences and visual-spatial arrays. Nine-month-old infants, in contrast, successfully discriminated 12 from 8 sounds, but not 10 from 8 sounds, providing evidence that numerosity discrimination increases in precision over development, prior to the emergence of language or symbolic counting.     

Core information processing deficits in developmental dyscalculia and low numeracy

There are two different conceptions of the innate basis for numerical abilities. On the one hand, it is claimed that infants possess a 'number module' that enables them to construct concepts of the exact numerosities of sets upon which arithmetic develops (e.g. Butterworth, 1999; Gelman & Gallistel, 1978). On the other hand, it has been proposed that infants are equipped only with a sense of approximate numerosities (e.g. Feigenson, Dehaene & Spelke, 2004), upon which the concepts of exact numerosities are constructed with the aid of language (Carey, 2004) and which forms the basis of arithmetic (Lemer, Dehaene, Spelke & Cohen, 2003). These competing proposals were tested by assessing whether performance on approximate numerosity tasks is related to performance on exact numerosity tasks. Moreover, performance on an analogue magnitude task was tested, since it has been claimed that approximate numerosities are represented as analogue magnitudes. In 8-9-year-olds, no relationship was found between exact tasks and either approximate or analogue tasks in normally achieving children, in children with low numeracy or in children with developmental dyscalculia. Low numeracy was related not to a poor grasp of exact numerosities, but to a poor understanding of symbolic numerals.     

Pigeons (Columba livia) learn to link numerosities with symbols

After responding to each element in varying, successive numerosity displays, pigeons (Columba livia) had to choose, out of an array of symbols, the symbol designated to correspond to the preceding number of elements. After extensive training, 5 pigeons responded with significant accuracy to the numerosities 1 to 4, and 2 pigeons to the numerosities 1 to 5. Several tests showed that feedback tones accompanying element pecks, the familiarity of element configurations, and the shape of the elements were not crucial to this performance. One test, however, indicated that the number of pecks issued to the elements was important for numerosities above 2. An additional test confirmed that the birds chose the symbol that corresponded to a particular numerosity rather than the positions that the symbols had held during training.     

Do infants have a sense of numerosity? A p‐curve analysis of infant numerosity discrimination studies

Research demonstrating that infants discriminate between small (e.g., 1 vs. 3 dots) and large numerosities (e.g., 8 vs. 16 dots) is central to theories concerning the origins of human numerical abilities. To date, there has been no quantitative meta‐analysis of the infant numerical competency data. Here, we quantitatively synthesize the evidential value of the available literature on infant numerosity discrimination using a meta‐analytic tool called p‐curve. In p‐curve the distribution of available p‐values is analyzed to determine whether the published literature examining particular hypotheses contains evidential value. p‐curves demonstrated evidential value for the hypotheses that infants can discriminate between both small and large unimodal and cross‐modal numerosities. However, the analyses also revealed that the published data on infants’ ability to discriminate between large numerosities is less robust and statistically powered than the data on their ability to discriminate small numerosities. We argue there is a need for adequately powered replication studies to enable stronger inferences in order to use infant data to ground theories concerning the ontogenesis of numerical cognition.     

Enumeration of small and large numerosities: the effect of element visibility

Precise enumeration is associated with small numerosities within the subitizing range (<4 items), while approximate enumeration is associated with large numerosities (>4 items). To date, there is still debate on whether a single continuous process or dual mutually exclusive processes mediate enumeration of small and large numerosities. Here, we evaluated a compromise between these two notions: that the precise representation of number is limited to small numerosities, but that the approximate representation of numerosity spans across both small and large numerosities. We assessed the independence of precise and approximate enumeration by looking at how luminance contrast affected enumeration of elements that differ by ones (1-8) or by tens (10-80). We found that enumeration functions of ones and tens have different characteristics, which is consistent with the presence of two number systems. Subitizing was preserved for small numerosities. However, simply decreasing element visibility changed the variability signatures of small numerosities to match those of large numerosities. Together, our results suggest that small numerosities are mediated by both precise and approximate representations of numerosity.     

Small and large number processing in infants and toddlers with Williams syndrome

Previous studies have suggested that typically developing 6‐month‐old infants are able to discriminate between small and large numerosities. However, discrimination between small numerosities in young infants is only possible when variables continuous with number (e.g. area or circumference) are confounded. In contrast, large number discrimination is successful even when variables continuous with number are systematically controlled for. These findings suggest the existence of different systems underlying small and large number processing in infancy. How do these develop in atypical syndromes? Williams syndrome (WS) is a rare neurocognitive developmental disorder in which numerical cognition has been found to be impaired in older children and adults. Do impairments of number processing have their origins in infancy? Here this question is investigated by testing the small and large number discrimination abilities of infants and toddlers with WS. While infants with WS were able to discriminate between 2 and 3 elements when total area was confounded with numerosity, the same infants did not discriminate between 8 and 16 elements, when number was not confounded with continuous variables. These findings suggest that a system for tracking the features of small numbers of object (object‐file representation) may be functional in WS, while large number discrimination is impaired from an early age onwards. Finally, we argue that individual differences in large number processing in infancy are more likely than small number processing to be predictive of later development of numerical cognition.     

Discrimination of small numerosities in young chicks

Chicks were trained to discriminate small sets of identical elements. They were then tested for choices (unrewarded) between sets of similar numerosities, when continuous physical variables such as spatial distribution, contour length, and overall surface were equalized. In all conditions chicks discriminated one versus two and two versus three stimulus sets. Similar results were obtained when elements were presented under conditions of partial occlusion. In contrast, with sets of four versus five, four versus six, and three versus four elements chicks seemed unable to discriminate on the basis of number, although nonnumerical discrimination based on perceptual cues was observed. This adds to increasing evidence for discrimination of small numerosities of up to three elements in human infants and nonhuman animals.     

Subitizing and similarity: Toward a pattern-matching theory of enumeration

Pattern-matching theories of subitizing claim that subjects enumerate displays with small numerosities by retrieving numerical responses associated with similar displays experienced in the past. Such retrieval implies that displays with small numerosities are similar to other displays of the same numerosity and dissimilar to other displays of different numerosities. These hypotheses were tested by having subjects rate the similarities of displays of dot patterns with numerosities in the range of 1-10. One group of subjects rated patterns of the same numerosity. Their ratings were higher for patterns in the subitizing range (numerosities of 1-3) than for patterns beyond the subitizing range (numerosities of 4-10). Another group rated patterns of different numerosities. Their ratings were lower in the subitizing range than beyond the subitizing range. An analysis based on multidimensional scaling suggested that numerosity could be retrieved accurately for displays of 1-3 dots, but not for displays of 4-10 dots.     

Enumeration of small and large numerosities: The effect of element visibility

Precise enumeration is associated with small numerosities within the subitizing range (<4 items), while approximate enumeration is associated with large numerosities (>4 items). To date, there is still debate on whether a single continuous process or dual mutually exclusive processes mediate enumeration of small and large numerosities. Here, we evaluated a compromise between these two notions: that the precise representation of number is limited to small numerosities, but that the approximate representation of numerosity spans across both small and large numerosities. We assessed the independence of precise and approximate enumeration by looking at how luminance contrast affected enumeration of elements that differ by ones (1–8) or by tens (10–80). We found that enumeration functions of ones and tens have different characteristics, which is consistent with the presence of two number systems. Subitizing was preserved for small numerosities. However, simply decreasing element visibility changed the variability signatures of small numerosities to match those of large numerosities. Together, our results suggest that small numerosities are mediated by both precise and approximate representations of numerosity.     

A double-dissociation in infants' representations of object arrays

Previous studies show that infants can compute either the total continuous extent (e.g. Clearfield, M.W., & Mix, K.S. (1999). Number versus contour length in infants' discrimination of small visual sets. Psychological Science, 10(5), 408-411; Feigenson, L., & Carey, S. (2003). Tracking individuals via object-files: evidence from infants' manual search. Developmental Science, 6, 568-584) or the numerosity (Feigenson, L., & Carey, S. (2003). Tracking individuals via object-files: evidence from infants' manual search. Developmental Science, 6, 568-584) of small object arrays. The present experiments asked whether infants can compute both extent and number over a given array. Experiment 1 used a habituation procedure to show that 7-month-old infants can compute numerosity when the objects in the array contrast in color, pattern, and texture. Experiment 2 revealed that, with these heterogeneous arrays, infants no longer represent the array's total continuous extent. Since previous work shows that infants compute continuous extent but not numerosity when objects have identical rather than contrasting properties, these results form a double dissociation. Infants computed number but not extent over representations of contrasting objects, and computed extent but not number over representations of identical objects.