Core systems of number

What representations underlie the ability to think and reason about number? Whereas certain numerical concepts, such as the real numbers, are only ever represented by a subset of human adults, other numerical abilities are widespread and can be observed in adults, infants and other animal species. We review recent behavioral and neuropsychological evidence that these ontogenetically and phylogenetically shared abilities rest on two core systems for representing number. Performance signatures common across development and across species implicate one system for representing large, approximate numerical magnitudes, and a second system for the precise representation of small numbers of individual objects. These systems account for our basic numerical intuitions, and serve as the foundation for the more sophisticated numerical concepts that are uniquely human.     

Small and large number discrimination in guppies

Non-verbal numerical behavior in human infants, human adults, and non-human primates appears to be rooted in two distinct mechanisms: a precise system for tracking and comparing small numbers of items simultaneously (up to 3 or 4 items) and an approximate system for estimating numerical magnitude of a group of objects. The most striking evidence that these two mechanisms are distinct comes from the apparent inability of young human infants and non-human primates to compare quantites across the small (<3 or 4)/large (>4) number boundary. We ask whether this distinction is present in lower animal species more distantly related to humans, guppies (Poecilia reticulata). We found that, like human infants and non-human primates, fish succeed at comparisons between large numbers only (5 vs. 10), succeed at comparisons between small numbers only (3 vs. 4), but systematically fail at comparisons that closely span the small/large boundary (3 vs. 5). Furthermore, increasing the distance between the small and large number resulted in successful discriminations (3 vs. 6, 3 vs. 7, and 3 vs. 9). This pattern of successes and failures is similar to those observed in human infants and non-human primates to suggest that the two systems are present and functionally distinct across a wide variety of animal species.     

Infants recognize counting as numerically relevant

Children do not understand the meanings of count words like “two” and “three” until the preschool years. But even before knowing the meanings of these individual words, might they recognize that counting is “about” the dimension of number? Here in five experiments, we asked whether infants already associate counting with quantities. We measured 14‐ and 18‐month olds’ ability to remember different numbers of hidden objects that either were or were not counted by an experimenter before hiding. As in previous research, we found that infants failed to differentiate four hidden objects from two when the objects were not counted—suggesting an upper limit on the number of individual objects they could represent in working memory. However, infants succeeded when the objects were simply counted aloud before hiding. We found that counting also helped infants differentiate four hidden objects from six (a 2:3 ratio), but not three hidden objects from four (a 3:4 ratio), suggesting that counting helped infants represent the arrays’ approximate cardinalities. Hence counting directs infants’ attention to numerical aspects of the world, showing that they recognize counting as numerically relevant years before acquiring the meanings of number words.     

两种数量表征系统

数量表征是人类数学能力的基础,数量表征研究中的一个争论焦点在于是否存在两种不同的数量表征系统：对小数的精确表征系统和对大数的近似表征系统。通过综述不同研究领域对数量表征的研究,总结了支持两种表征系统分离的证据：对1~3范围内小数的表征受数量大小的限制,基于指向物体本身的注意,更依赖于物体的知觉特征,对物体及其数量进行精确表征;而对4以上的数量的近似表征系统则受韦伯定律的限制,基于指向数量的模拟幅度的表征,而不依赖单个物体的知觉特征,是对数量的近似的、心理的表征。fMRI、PET和ERP的脑成像研究结果迄今尚无定论,但认知神经科学研究的深入开展将最终阐明数量表征的机制     

Memory for multiple visual ensembles in infancy

The number of individual items that can be maintained in working memory is limited. One solution to this problem is to store representations of ensembles that contain summary information about large numbers of items (e.g., the approximate number or cumulative area of a group of many items). Here we explored the developmental origins of ensemble representations by asking whether infants represent ensembles and, if so, how many at one time. We habituated 9-month-old infants to arrays containing 2, 3, or 4 spatially intermixed colored subsets of dots, then asked whether they detected a numerical change to one of the subsets or to the superset of all dots. Experiment Series 1 showed that infants detected a numerical change to 1 of the subsets when the array contained 2 subsets but not 3 or 4 subsets. Experiment Series 2 showed that infants detected a change to the superset of all dots no matter how many subsets were presented. Experiment 3 showed that infants represented both the approximate number and the cumulative surface area of these ensembles. Our results suggest that infants, like adults (Halberda, Sires, & Feigenson, 2006), can store quantitative information about 2 subsets plus the superset: a total of 3 ensembles. This converges with the known limit on the number of individual objects infants and adults can store and suggests that, throughout development, an ensemble functions much like an individual object for working memory.     

Foundational numerical capacities and the origins of dyscalculia

One important cause of very low attainment in arithmetic (dyscalculia) seems to be a core deficit in an inherited foundational capacity for numbers. According to one set of hypotheses, arithmetic ability is built on an inherited system responsible for representing approximate numerosity. One account holds that this is supported by a system for representing exactly a small number (less than or equal to four4) of individual objects. In these approaches, the core deficit in dyscalculia lies in either of these systems. An alternative proposal holds that the deficit lies in an inherited system for sets of objects and operations on them (numerosity coding) on which arithmetic is built. I argue that a deficit in numerosity coding, not in the approximate number system or the small number system, is responsible for dyscalculia. Nevertheless, critical tests should involve both longitudinal studies and intervention, and these have yet to be carried out.     

Functional brain organization for number processing in pre‐verbal infants

Humans are born with the ability to mentally represent the approximate numerosity of a set of objects, but little is known about the brain systems that sub‐serve this ability early in life and their relation to the brain systems underlying symbolic number and mathematics later in development. Here we investigate processing of numerical magnitudes before the acquisition of a symbolic numerical system or even spoken language, by measuring the brain response to numerosity changes in pre‐verbal infants using functional near‐infrared spectroscopy (fNIRS). To do this, we presented infants with two types of numerical stimulus blocks: number change blocks that presented dot arrays alternating in numerosity and no change blocks that presented dot arrays all with the same number. Images were carefully constructed to rule out the possibility that responses to number changes could be due to non‐numerical stimulus properties that tend to co‐vary with number. Interleaved with the two types of numerical blocks were audio‐visual animations designed to increase attention. We observed that number change blocks evoked an increase in oxygenated hemoglobin over a focal right parietal region that was greater than that observed during no change blocks and during audio‐visual attention blocks. The location of this effect was consistent with intra‐parietal activity seen in older children and adults for both symbolic and non‐symbolic numerical tasks. A distinct set of bilateral occipital and middle parietal channels responded more to the attention‐grabbing animations than to either of the types of numerical stimuli, further dissociating the specific right parietal response to number from a more general bilateral visual or attentional response. These results provide the strongest evidence to date that the right parietal cortex is specialized for numerical processing in infancy, as the response to number is dissociated from visual change processing and general attentional processing.     

Spontaneous analog number representations in 3‐year‐old children

When enumerating small sets of elements nonverbally, human infants often show a set‐size limitation whereby they are unable to represent sets larger than three elements. This finding has been interpreted as evidence that infants spontaneously represent small numbers with an object‐file system instead of an analog magnitude system ( Feigenson, Dehaene & Spelke, 2004 ). In contrast, non‐human animals and adult humans have been shown to rely on analog magnitudes for representing both small and large numbers ( Brannon & Terrace, 1998 ; Cantlon & Brannon, 2007 ; Cordes, Gelman, Gallistel & Whalen, 2001). Here we demonstrate that, like adults and non‐human animals, children as young as 3 years of age spontaneously employ analog magnitude representations to enumerate both small and large sets. Moreover, we show that children spontaneously attend to numerical value in lieu of cumulative surface area. These findings provide evidence of young children’s greater sensitivity to number relative to other quantities and demonstrate continuity in the process they spontaneously recruit to judge small and large values.     

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.     

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.     

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.     

Calibrating the mental number line

Human adults are thought to possess two dissociable systems to represent numbers: an approximate quantity system akin to a mental number line, and a verbal system capable of representing numbers exactly. Here, we study the interface between these two systems using an estimation task. Observers were asked to estimate the approximate numerosity of dot arrays. We show that, in the absence of calibration, estimates are largely inaccurate: responses increase monotonically with numerosity, but underestimate the actual numerosity. However, insertion of a few inducer trials, in which participants are explicitly (and sometimes misleadingly) told that a given display contains 30 dots, is sufficient to calibrate their estimates on the whole range of stimuli. Based on these empirical results, we develop a model of the mapping between the numerical symbols and the representations of numerosity on the number line.     

On the limits of infants' quantification of small object arrays

Recent work suggests that infants rely on mechanisms of object-based attention and short-term memory to represent small numbers of objects. Such work shows that infants discriminate arrays containing 1, 2, or 3 objects, but fail with arrays greater than 3 [Feigenson, L., & Carey, S. (2003). Tracking individuals via object-files: Evidence from infants' manual search. Developmental Science, 6, 568-584; Feigenson, L., Carey, S., & Hauser, M. (2002). The representations underlying infants' choice of more: Object files versus analog magnitudes. Psychological Science, 13(2), 150-156]. However, little is known about how infants represent arrays exceeding the 3-item limit of parallel representation. We explored possible formats by which infants might represent a 4-object array. Experiment 1 used a manual search paradigm to show that infants successfully discriminated between arrays of 1 vs. 2, 2 vs. 3, and 1 vs. 3 objects. However, infants failed to discriminate 1 vs. 4 despite the highly discriminable ratio, providing the strongest evidence to date for object-file representations underlying performance in this task. Experiment 2 replicated this dramatic failure to discriminate 1 from 4 in a second paradigm, a cracker choice task. We then showed that infants in the choice task succeeded at choosing the larger quantity with 0 vs. 4 crackers and with 1 small vs. 4 large crackers. These results suggest that while infants failed to represent 4 as “exactly 4”, “approximately 4”, “3”, or as even as “a plurality”, they did represent information about the array, including the existence of a cracker or cracker-material and the size of the individual objects in the array.     

Tracking individuals via object-files: evidence from infants’ manual search

In two experiments, a manual search task explored 12- to 14-month-old infants’ representations of small sets of objects. In this paradigm, patterns of searching revealed the number of objects infants represented as hidden in an opaque box. In Experiment 1, we obtained the set-size signature of object-file representations: infants succeeded at representing precisely 1, precisely 2, and precisely 3 objects in the box, but failed at representing 4 (or even that 4 is greater than 2). In Experiment 2, we showed that infants’ expectations about the contents of the box were based on number of individual objects, and not on a continuous property such as total object volume. These findings support the hypothesis that infants maintained representations of individuals, that object-files were the underlying means of representing these individuals, and that object-file models can be compared via one-to-one correspondence to establish numerical equivalence.     

Core multiplication in childhood

A dedicated, non-symbolic, system yielding imprecise representations of large quantities (approximate number system, or ANS) has been shown to support arithmetic calculations of addition and subtraction. In the present study, 5–7-year-old children without formal schooling in multiplication and division were given a task requiring a scalar transformation of large approximate numerosities, presented as arrays of objects. In different conditions, the required calculation was doubling, quadrupling, or increasing by a fractional factor (2.5). In all conditions, participants were able to represent the outcome of the transformation at above-chance levels, even on the earliest training trials. Their performance could not be explained by processes of repeated addition, and it showed the critical ratio signature of the ANS. These findings provide evidence for an untrained, intuitive process of calculating multiplicative numerical relationships, providing a further foundation for formal arithmetic instruction.     

Large-number addition and subtraction by 9-month-old infants

Do genuinely numerical computational abilities exist in infancy? It has recently been argued that previous studies putatively illustrating infants' ability to add and subtract tapped into specialized object-tracking processes that apply only with small numbers. This argument contrasts with the original interpretation that successful performance was achieved via a numerical system for estimating and calculating magnitudes. Here, we report that when continuous variables (such as area and contour length) are controlled, 9-month-old infants successfully add and subtract over numbers of items that exceed object-tracking limits. These results support the theory that infants possess a magnitude-based estimation system for representing numerosities that also supports procedures for numerical computation.     

Factors influencing infants’ ability to update object representations in memory

Remembering persisting objects over occlusion is critical to representing a stable environment. Infants remember hidden objects at multiple locations and can update their representation of a hidden array when an object is added or subtracted. However, the factors influencing these updating abilities have received little systematic exploration. Here we examined the flexibility of infants’ ability to update object representations. We tested 11-month-olds in a looking-time task in which objects were added to or subtracted from two hidden arrays. Across five experiments, infants successfully updated their representations of hidden arrays when the updating occurred successively at one array before beginning at the other. But when updating required alternating between two arrays, infants failed. However, simply connecting the two arrays with a thin strip of foam-core led infants to succeed. Our results suggest that infants’ construal of an event strongly affects their ability to update memory representations of hidden objects. When construing an event as containing multiple updates to the same array, infants succeed, but when construing the event as requiring the revisiting and updating of previously attended arrays, infants fail.     

Infants’ auditory enumeration: Evidence for analog magnitudes in the small number range

Vigorous debate surrounds the issue of whether infants use different representational mechanisms to discriminate small and large numbers. We report evidence for ratio-dependent performance in infants’ discrimination of small numbers of auditory events, suggesting that infants can use analog magnitudes to represent small values, at least in the auditory domain. Seven-month-old infants in the present study reliably discriminated two from four tones (a 1:2 ratio) in Experiment 1, when melodic and continuous temporal properties of the sequences were controlled, but failed to discriminate two from three tones (a 2:3 ratio) under the same conditions in Experiment 2. A third experiment ruled out the possibility that infants in Experiment 1 were responding to greater melodic variety in the four-tone sequences. The discrimination function obtained here is the same as that found for infants’ discrimination of large numbers of visual and auditory items at a similar age, as well as for that obtained for similar-aged infants’ duration discriminations, and thus adds to a growing body of evidence suggesting that human infants may share with adults and nonhuman animals a mechanism for representing quantities as “noisy” mental magnitudes.     

Intuitive mapping between nonsymbolic quantity and observed action across development

Adults’ concurrent processing of numerical and action information yields bidirectional interference effects consistent with a cognitive link between these two systems of representation. This link is in place early in life: infants create expectations of congruency across numerical and action-related stimuli (i.e., a small [large] hand aperture associated with a smaller [larger] numerosity). Although these studies point to a developmental continuity of this mapping, little is known about the later development and thus how experience shapes such relationships. We explored how number–action intuitions develop across early and later childhood using the same methodology as in adults. We asked 3-, 6-, and 8-year-old children, as well as adults, to relate the magnitude of an observed action (a static hand shape, open vs. closed, in Experiment 1; a dynamic hand movement, opening vs. closing, in Experiment 2) to either a small or large nonsymbolic quantity (numerosity in Experiment 1 and numerosity and/or object size in Experiment 2). From 6 years of age, children started performing in a systematic congruent way in some conditions, but only 8-year-olds (added in Experiment 2) and adults performed reliably above chance in this task. We provide initial evidence that early intuitions guiding infants’ mapping between magnitude across nonsymbolic number and observed action are used in an explicit way only from late childhood, with a mapping between action and size possibly being the most intuitive. An initial coarse mapping between number and action is likely modulated with extensive experience with grasping and related actions directed to both arrays and individual objects.     

Toddler subtraction with large sets: further evidence for an analog-magnitude representation of number

Two experiments were conducted to test the hypothesis that toddlers have access to an analog-magnitude number representation that supports numerical reasoning about relatively large numbers. Three-year-olds were presented with subtraction problems in which initial set size and proportions subtracted were systematically varied. Two sets of cookies were presented and then covered. The experimenter visibly subtracted cookies from the hidden sets, and the children were asked to choose which of the resulting sets had more. In Experiment 1, performance was above chance when high proportions of objects (3 versus 6) were subtracted from large sets (of 9) and for the subset of older participants (older than 3 years, 5 months; n = 15), performance was also above chance when high proportions (10 versus 20) were subtracted from the very large sets (of 30). In Experiment 2, which was conducted exclusively with older 3-year-olds and incorporated an important methodological control, the pattern of results for the subtraction tasks was replicated. In both experiments, success on the tasks was not related to counting ability. The results of these experiments support the hypothesis that young children have access to an analog-magnitude system for representing large approximate quantities, as performance on these subtraction tasks showed a Weber's Law signature, and was independent of conventional number knowledge.