Variability signatures distinguish verbal from nonverbal counting for both large and small numbers

The "where" and "what" in developmental dyscalculia

Developmental dyscalculia (DD) is a congenital deficit that affects the ability to acquire arithmetical skills. Individuals with DD have problems learning standard number facts and procedures. Estimates of the prevalence rate of DD are similar to those of developmental dyslexia. Recent reports and discussions suggest that those with DD suffer from specific deficits (e.g., subitizing, comparative judgment). Accordingly, DD has been described as a domain-specific disorder that involves particular brain areas (e.g., intra-parietal sulcus). However, we and others have found that DD is characterized by additional deficiencies and may be affected by domain-general (e.g., attention) factors. Hence "pure DD" might be rather rare and not as pure as one would think. We suggest that the heterogeneity of symptoms that commonly characterize learning disabilities needs to be taken into account in future research and treatment.

Sex Differences in Time Production Revisited

Individual differences in time production should indicate differences in the rate of functioning of an internal clock, assuming the existence of such a clock. And sex differences in time production should reflect a difference in the rate of functioning of that clock between men and women. One way of approaching the data is to compute individual regressions of produced duration (P) on target duration (T), after log transformation, and to derive estimates for the intercept and the slope. One could investigate a sex difference by comparing these estimates for men and women; one could also contrast them by looking at mean log(P). Using such indices, we found a sex difference in time production, female participants having a relatively faster internal clock, making shorter time productions, and having a smaller exponent. The question is whether a sex difference in time production would be found using other methods for analyzing the data: (1) the P/T ratio; (2) an absolute discrepancy (|P-T|) score; and (3) an absolute error (|P-T|/T) score. For the P/T ratio, female participants have a lower mean ratio in comparison to the male participants. In contrast, the |P-T| and |P-T|/T indices seem to be seriously compromised by wide individual differences.

Parallels in stimulus-driven oscillatory brain responses to numerosity changes in adults and seven-month-old infants

Previous studies provide indirect evidence for an ontogenetically continuous Approximate-Number System. We employed a rapid steady-state visual-presentation paradigm combined with electroencephalography to measure stimulus-driven neural oscillatory responses to numerosities in infants and adults. Steady-state repetition of the same numerosity across a 2.4-sec time block yielded an increase in the stimulus-locked neural entrainment in both groups. Entrainment changes following a numerosity switch varied by the ratio of the numerosities, consistent with Weber's Law. These similarities thus provide direct evidence for an ontogenetically continuous Approximate-Number System. Moreover, the degree of neural entrainment significantly predicted infants' number discrimination measured behaviorally two months later.

Two systems of non-symbolic numerical cognition

Studies of human adults, infants, and non-human animals demonstrate that non-symbolic numerical cognition is supported by at least two distinct cognitive systems: a “parallel individuation system” that encodes the numerical identity of individual items and an “approximate number system” that encodes the approximate numerical magnitude, or numerosity, of a set. The exact nature and role of these systems, however, have been debated for over a 100-years. Some argue that the non-symbolic representation of small numbers (<4) is carried out solely by the parallel individuation system and the non-symbolic representation of large numbers (>4) is carried out solely by the approximate number system. Others argue that all numbers are represented by the approximate number system. This debate has been fueled largely by some studies showing dissociations between small and large number processing and other studies showing similar processing of small and large numbers. Recent work has addressed this debate by showing that the two systems are present and distinct from early infancy, persist despite the acquisition of a symbolic number system, activate distinct cortical networks, and engage differentially based attentional constraints. Based on the recent discoveries, I provide a hypothesis that may explain the puzzling findings and makes testable predictions as to when each system will be engaged. In particular, when items are presented under conditions that allow selection of individuals, they will be represented as distinct mental items through parallel individuation and not as a numerical magnitude. In contrast, when items are presented outside attentional limits (e.g., too many, too close together, under high attentional load), they will be represented as a single mental numerical magnitude and not as distinct mental items. These predictions provide a basis on which researchers can further investigate the role of each system in the development of uniquely human numerical thought.

Numerical and nonnumerical estimation in children with and without mathematical learning disabilities

There are currently multiple explanations for mathematical learning disabilities (MLD). The present study focused on those assuming that MLD are due to a basic numerical deficit affecting the ability to represent and to manipulate number magnitude ( Butterworth, 1999 , 2005 ; A. J. Wilson & Dehaene, 2007 ) and/or to access that number magnitude representation from numerical symbols ( Rousselle & Noël, 2007 ). The present study provides an original contribution to this issue by testing MLD children (carefully selected on the basis of preserved abilities in other domains) on numerical estimation tasks with contrasting symbolic (Arabic numerals) and nonsymbolic (collection of dots) numbers used as input or output. MLD children performed consistently less accurately than control children on all the estimation tasks. However, MLD children were even weaker when the task involved the mapping between symbolic and nonsymbolic numbers than when the task required a mapping between two nonsymbolic numerical formats. Moreover, in the estimation of nonsymbolic numerosities, MLD children relied more than control children on perceptual cues such as the cumulative area of the dots. Finally, the task requiring a mapping from a nonsymbolic format to a symbolic format was the best predictor of MLD. In order to explain these present results, as well as those reported in the literature, we propose that the impoverished number magnitude representation of MLD children may arise from an initial mapping deficit between number symbols and that magnitude representation.

Measuring the Approximate Number System

Recent theories in numerical cognition propose the existence of an approximate number system (ANS) that supports the representation and processing of quantity information without symbols. It has been claimed that this system is present in infants, children, and adults, that it supports learning of symbolic mathematics, and that correctly harnessing the system during tuition will lead to educational benefits. Various experimental tasks have been used to investigate individuals' ANSs, and it has been assumed that these tasks measure the same system. We tested the relationship across six measures of the ANS. Surprisingly, despite typical performance on each task, adult participants' performances across the tasks were not correlated, and estimates of the acuity of individuals' ANSs from different tasks were unrelated. These results highlight methodological issues with tasks typically used to measure the ANS and call into question claims that individuals use a single system to complete all these tasks.

Human nonverbal discrimination of relative and absolute number

The nonverbal discrimination of relative and absolute number of sequential visual stimuli was investigated with humans in bisection, reproduction, and report tasks. Participants viewed a sequence of 40 red and black objects on each trial, randomly intermixed, and had to identify the number of red objects, which varied from 1 to 20. To prevent the use of a verbal-counting strategy, participants were required to name the objects as they appeared. The characteristics of human performance resembled those of pigeons in analogous procedures (Tan & Grace Learning and Behavior 38:408-417, 2010; Tan, Grace, Holland, & McLean Journal of Experimental Psychology 33:409-427, 2007): Average response number increased systematically with sample number, and bisection points were located at the arithmetic, not the geometric, mean. Additionally, in both the reproduction and report tasks, coefficients of variation decreased for values less than 6 but increased or remained constant for larger values, suggesting that different representations were used for small and large numbers.

Predicting sights from sounds: 6-month-olds' intermodal numerical abilities

Although the psychophysics of infants' nonsymbolic number representations have been well studied, less is known about other characteristics of the approximate number system (ANS) in young children. Here three experiments explored the extent to which the ANS yields abstract representations by testing infants' ability to transfer approximate number representations across sensory modalities. These experiments showed that 6-month-olds matched the approximate number of sounds they heard to the approximate number of sights they saw, looking longer at visual arrays that numerically mismatched a previously heard auditory sequence. This looking preference was observed when sights and sounds mismatched by 1:3 and 1:2 ratios but not by a 2:3 ratio. These findings suggest that infants can compare numerical information obtained in different modalities using representations stored in memory. Furthermore, the acuity of 6-month-olds' comparisons of intermodal numerical sequences appears to parallel that of their comparisons of unimodal sequences.

Developmental neuroscience of time and number: implications for autism and other neurodevelopmental disabilities

Estimations of time and number share many similarities in both non-humans and man. The primary focus of this review is on the development of time and number sense across infancy and childhood, and neuropsychological findings as they relate to time and number discrimination in infants and adults. Discussion of these findings is couched within a mode-control model of timing and counting which assumes time and number share a common magnitude representation system. A basic sense of time and number likely serves as the foundation for advanced numerical and temporal competence, and aspects of higher cognition-this will be discussed as it relates to typical childhood, and certain developmental disorders, including autism spectrum disorder. Directions for future research in the developmental neuroscience of time and number (NEUTIN) will also be highlighted.

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.

The enigma of number: why children find the meanings of even small number words hard to learn and how we can help them do better

Although number words are common in everyday speech, learning their meanings is an arduous, drawn-out process for most children, and the source of this delay has long been the subject of inquiry. Children begin by identifying the few small numerosities that can be named without counting, and this has prompted further debate over whether there is a specific, capacity-limited system for representing these small sets, or whether smaller and larger sets are both represented by the same system. Here we present a formal, computational analysis of number learning that offers a possible solution to both puzzles. This analysis indicates that once the environment and the representational demands of the task of learning to identify sets are taken into consideration, a continuous system for learning, representing and discriminating set-sizes can give rise to effective discontinuities in processing. At the same time, our simulations illustrate how typical prenominal linguistic constructions ("there are three balls") structure information in a way that is largely unhelpful for discrimination learning, while suggesting that postnominal constructions ("balls, there are three") will facilitate such learning. A training-experiment with three-year olds confirms these predictions, demonstrating that rapid, significant gains in numerical understanding and competence are possible given appropriately structured postnominal input. Our simulations and results reveal how discrimination learning tunes children's systems for representing small sets, and how its capacity-limits result naturally out of a mixture of the learning environment and the increasingly complex task of discriminating and representing ever-larger number sets. They also explain why children benefit so little from the training that parents and educators usually provide. Given the efficacy of our intervention, the ease with which it can be implemented, and the large body of research showing how early numerical ability predicts later educational outcomes, this simple discovery may have far-reaching consequences.

The effect of viewing eccentricity on enumeration

Visual acuity and contrast sensitivity progressively diminish with increasing viewing eccentricity. Here we evaluated how visual enumeration is affected by visual eccentricity, and whether subitizing capacity, the accurate enumeration of a small number (∼3) of items, decreases with more eccentric viewing. Participants enumerated gratings whose (1) stimulus size was constant across eccentricity, and (2) whose stimulus size scaled by a cortical magnification factor across eccentricity. While we found that enumeration accuracy and precision decreased with increasing eccentricity, cortical magnification scaling of size neutralized the deleterious effects of increasing eccentricity. We found that size scaling did not affect subitizing capacities, which were nearly constant across all eccentricities. We also found that size scaling modulated the variation coefficients, a normalized metric of enumeration precision, defined as the standard deviation divided by the mean response. Our results show that the inaccuracy and imprecision associated with increasing viewing eccentricity is due to limitations in spatial resolution. Moreover, our results also support the notion that the precise number system is restricted to small numerosities (represented by the subitizing limit), while the approximate number system extends across both small and large numerosities (indexed by variation coefficients) at large eccentricities.

The case for continuity

This article defends a continuity position. Infants can abstract numerosity and young preschool children do respond appropriately to tasks that tap their ability to use a count and cardinal value and/or arithmetic principles. Active use of a nonverbal domain of arithmetic serves to enable the child to find relevant data to build knowledge about the language and use rules of numerosity and quantity.

Typical development of quantity comparison in school-aged children

Although basic numerical skills have been widely studied in the last years, very few studies have undertaken a developmental approach. The present study evaluated the development of the magnitude comparison basic numerical ability, in children from first, third and sixth grades by means of the subject's response time in numerical tasks presented in symbolic and non-symbolic formats. The results showed a significant decrease on quantities processing speed as age increases, which suggests numerical skills tend to become automatic with instruction. The differences found, concerning the general achievement pattern in each school year, might express the maturational specificities of the numerical representation system through development.

Under- and Over-Estimation

Over the last 30 years, numerical estimation has been largely studied. Recently, Castronovo and Seron (2007) proposed the bi-directional mapping hypothesis in order to account for the finding that dependent on the type of estimation task (perception vs. production of numerosities), reverse patterns of performance are found (i.e., under- and over-estimation, respectively). Here, we further investigated this hypothesis by submitting adult participants to three types of numerical estimation task: (1) a perception task, in which participants had to estimate the numerosity of a non-symbolic collection; (2) a production task, in which participants had to approximately produce the numerosity of a symbolic numerical input; and (3) a reproduction task, in which participants had to reproduce the numerosity of a non-symbolic numerical input. Our results gave further support to the finding that different patterns of performance are found according to the type of estimation task: (1) under-estimation in the perception task; (2) over-estimation in the production task; and (3) accurate estimation in the reproduction task. Moreover, correlation analyses revealed that the more a participant under-estimated in the perception task, the more he/she over-estimated in the production task. We discussed these empirical data by showing how they can be accounted by the bi-directional mapping hypothesis ( Castronovo & Seron, 2007 ).

Monkeys (macaca mulatta and cebus apella) and human adults and children (homo sapiens) compare subsets of moving stimuli based on numerosity

Two monkey species (Macaca mulatta and Cebus apella) and human children and adults judged the numerousness of two subsets of moving stimuli on a computer screen. Two sets of colored dots that varied in number and size were intermixed in an array in which all dots moved in random directions and speeds. Participants had to indicate which dot color was more numerous within the array. All species performed at high and comparable levels, including on trials in which the subset with the larger number of items had a smaller total area of coloration. This indicated a similarity across species to use the number of items in the subsets, and not dimensions such as area or volume, to guide decision making. Discrimination performance was constrained by the ratio between the subsets, consistent with other reports of numerousness judgments of stationary stimuli. These results indicate a similarity in numerical estimation ability for moving stimuli across primate species, and this capacity may be necessary for naturally occurring experiences in which moving stimuli must be summed.

Spatial and numerical abilities without a complete natural language

We studied the cognitive abilities of a 13-year-old deaf child, deprived of most linguistic input from late infancy, in a battery of tests designed to reveal the nature of numerical and geometrical abilities in the absence of a full linguistic system. Tests revealed widespread proficiency in basic symbolic and non-symbolic numerical computations involving the use of both exact and approximate numbers. Tests of spatial and geometrical abilities revealed an interesting patchwork of age-typical strengths and localized deficits. In particular, the child performed extremely well on navigation tasks involving geometrical or landmark information presented in isolation, but very poorly on otherwise similar tasks that required the combination of the two types of spatial information. Tests of number- and space-specific language revealed proficiency in the use of number words and deficits in the use of spatial terms. This case suggests that a full linguistic system is not necessary to reap the benefits of linguistic vocabulary on basic numerical tasks. Furthermore, it suggests that language plays an important role in the combination of mental representations of space.

Bilateral field advantage in visual enumeration

A number of recent studies have demonstrated superior visual processing when the information is distributed across the left and right visual fields than if the information is presented in a single hemifield (the bilateral field advantage). This effect is thought to reflect independent attentional resources in the two hemifields and the capacity of the neural responses to the left and right hemifields to process visual information in parallel. Here, we examined whether a bilateral field advantage can also be observed in a high-level visual task that requires the information from both hemifields to be combined. To this end, we used a visual enumeration task—a task that requires the assimilation of separate visual items into a single quantity—where the to-be-enumerated items were either presented in one hemifield or distributed between the two visual fields. We found that enumerating large number (>4 items), but not small number (<4 items), exhibited the bilateral field advantage: enumeration was more accurate when the visual items were split between the left and right hemifields than when they were all presented within the same hemifield. Control experiments further showed that this effect could not be attributed to a horizontal alignment advantage of the items in the visual field, or to a retinal stimulation difference between the unilateral and bilateral displays. These results suggest that a bilateral field advantage can arise when the visual task involves inter-hemispheric integration. This is in line with previous research and theory indicating that, when the visual task is attentionally demanding, parallel processing by the neural responses to the left and right hemifields can expand the capacity of visual information processing.

Small Subitizing Range in People with Williams syndrome

Recent evidence suggests that the rapid apprehension of small numbers of objects-- often called subitizing-- engages a system which allows representation of up to 4 objects but is distinct from other aspects of numerical processing. We examined subitizing by studying people with Williams syndrome (WS), a genetic deficit characterized by severe visuospatial impairments, and normally developing children (4-6.5 years old). In Experiment 1, participants first explicitly counted displays of 1 to 8 squares that appeared for 5 s and reported "how many". They then reported "how many" for the same displays shown for 250 ms, a duration too brief to allow explicit counting, but sufficient for subitizing. All groups were highly accurate up to 8 objects when they explicitly counted. With the brief duration, people with WS showed almost perfect accuracy up to a limit of 3 objects, comparable to 4 year-olds but fewer than either 5 or 6.5 year-old children. In Experiment 2, participants were asked to report "how many" for displays that were presented for an unlimited duration, as rapidly as they could while remaining accurate. Individuals with WS responded as rapidly as 6.5 year-olds, and more rapidly than 4 year-olds. However, their accuracy was as in Experiment 1, comparable to 4 year-olds, and lower than older children. These results are consistent with previous results indicating that people with WS can simultaneously represent multiple objects, but that they have a smaller capacity than older children, on par with 4 year-olds. This pattern is discussed in the context of normal and abnormal development of visuospatial skills, in particular those linked to the representation of numerosity.

Neural signatures of number processing in human infants: evidence for two core systems underlying numerical cognition

Behavioral research suggests two cognitive systems are at the foundations of numerical thinking: one for representing 1-3 objects in parallel and one for representing and comparing large, approximate numerical magnitudes. We tested for dissociable neural signatures of these systems in preverbal infants, by recording event-related potentials (ERPs) as 6-7.5 month-old infants (n = 32) viewed dot arrays containing either small (1-3) or large (8-32) sets of objects in a number alternation paradigm. If small and large numbers are represented by the same neural system, then the brain response to the arrays should scale with ratio for both number ranges, a behavioral and brain signature of the approximate numerical magnitude system obtained in animals and in human adults. Contrary to this prediction, a mid-latency positivity (P500) over parietal scalp sites was modulated by the ratio between successive large, but not small, numbers. Conversely, an earlier peaking positivity (P400) over occipital-temporal sites was modulated by the absolute cardinal value of small, but not large, numbers. These results provide evidence for two early developing systems of non-verbal numerical cognition: one that responds to small quantities as individual objects and a second that responds to large quantities as approximate numerical values. These brain signatures are functionally similar to those observed in previous studies of non-symbolic number with adults, suggesting that this dissociation may persist over vast differences in experience and formal training in mathematics.

Numerosity estimation in visual stimuli in the absence of luminance-based cues

BACKGROUND

Numerosity estimation is a basic preverbal ability that humans share with many animal species and that is believed to be foundational of numeracy skills. It is notoriously difficult, however, to establish whether numerosity estimation is based on numerosity itself, or on one or more non-numerical cues like-in visual stimuli-spatial extent and density. Frequently, different non-numerical cues are held constant on different trials. This strategy, however, still allows numerosity estimation to be based on a combination of non-numerical cues rather than on any particular one by itself.

METHODOLOGY/PRINCIPAL FINDINGS

Here we introduce a novel method, based on second-order (contrast-based) visual motion, to create stimuli that exclude all first-order (luminance-based) cues to numerosity. We show that numerosities can be estimated almost as well in second-order motion as in first-order motion.

CONCLUSIONS/SIGNIFICANCE

The results show that numerosity estimation need not be based on first-order spatial filtering, first-order density perception, or any other processing of luminance-based cues to numerosity. Our method can be used as an effective tool to control non-numerical variables in studies of numerosity estimation.

Number line compression and the illusory perception of random numbers

Developmental studies indicate that children initially possess a compressed intuition of numerical distances, in which larger numbers are less discriminable than small ones. Education then "linearizes" this responding until by about age eight, children become able to map symbolic numerals onto a linear spatial scale. However, this illusion of compression of symbolic numerals may still exist in a dormant form in human adults and may be observed in appropriate experimental contexts. To investigate this issue, we asked adult participants to rate whether a random sequence of numbers contained too many small numbers or too many large ones. Participants exhibited a large bias, judging as random a geometric series that actually oversampled small numbers, consistent with a compression of large numbers. This illusion resisted training on a number-space mapping task, even though performance was linear on this task. While the illusion was moderately reduced by explicit exposure to linear sequences, responding was still significantly compressed. Thus, the illusion of compression is robust in this task, but linear and compressed responding can be exhibited in the same participants depending on the experimental context.

An association between understanding cardinality and analog magnitude representations in preschoolers

The preschool years are a time of great advances in children's numerical thinking, most notably as they master verbal counting. The present research assessed the relation between analog magnitude representations and cardinal number knowledge in preschool-aged children to ask two questions: (1) Is there a relationship between acuity in the analog magnitude system and cardinality proficiency? (2) Can evidence of the analog magnitude system be found within mappings of number words children have not successfully mastered? To address the first question, Study 1 asked three- to five-year-old children to discriminate side-by-side dot arrays with varying differences in numerical ratio, as well as to complete an assessment of cardinality. Consistent with the analog magnitude system, children became less accurate at discriminating dot arrays as the ratio between the two numbers approached one. Further, contrary to prior work with preschoolers, a significant correlation was found between cardinal number knowledge and non-symbolic numerical discrimination. Study 2 aimed to look for evidence of the analog magnitude system in mappings to the words in preschoolers' verbal counting list. Based on a modified give-a-number task (Wynn, 1990, 1992), three- to five-year-old children were asked to give quantities between 1 and 10 as many times as possible in order to assess analog magnitude variability within their developing cardinality understanding. In this task, even children who have not yet induced the cardinality principle showed signs of analog representations in their understanding of the verbal count list. Implications for the contribution of analog magnitude representations towards mastery of the verbal count list are discussed in light of the present work.

Operational Momentum in Numerosity Production Judgments of Multi-Digit Number Problems

The current study demonstrates a numerosity production task and investigates approximate mental calculations with two-digit numbers. Participants were required to produce random dot patterns to indicate the size of two-digit numbers and the results of addition and subtraction problems. The stimuli in the calculation task consisted of problems requiring a carry operation (e.g., 24 + 18) or no-carry problems (e.g., 24 + 53) or zero problems (e.g., 24 + 0). Our analysis revealed that the outcomes of additions were estimated to be larger than the outcomes of subtractions. Interestingly, this judgment bias was present for no-carry and zero problems but not for carry problems. Taken together, the presented data provide empirical support for the presence of an operational momentum effect (OM effect) in multi-digit number arithmetic. These findings and the dissociation of the OM effect for carry and no-carry problems are discussed in the context of recent models on multi-digit number processing.

Ontogeny of numerical abilities in fish

BACKGROUND

It has been hypothesised that human adults, infants, and non-human primates share two non-verbal systems for enumerating objects, one for representing precisely small quantities (up to 3-4 items) and one for representing approximately larger quantities. Recent studies exploiting fish's spontaneous tendency to join the larger group showed that their ability in numerical discrimination closely resembles that of primates but little is known as to whether these capacities are innate or acquired.

METHODOLOGY/PRINCIPAL FINDINGS

We used the spontaneous tendency to join the larger shoal to study the limits of the quantity discrimination of newborn and juvenile guppies. One-day old fish chose the larger shoal when the choice was between numbers in the small quantity range, 2 vs. 3 fish, but not when they had to choose between large numbers, 4 vs. 8 or 4 vs. 12, although the numerical ratio was larger in the latter case. To investigate the relative role of maturation and experience in large number discrimination, fish were raised in pairs (with no numerical experience) or in large social groups and tested at three ages. Forty-day old guppies from both treatments were able to discriminate 4 vs. 8 fish while at 20 days this was only observed in fish grown in groups. Control experiments showed that these capacities were maintained after guppies were prevented from using non numerical perceptual variables that co-vary with numerosity.

CONCLUSIONS/SIGNIFICANCE

Overall, our results suggest the ability of guppies to discriminate small numbers is innate and is displayed immediately at birth while discrimination of large numbers emerges later as a result of both maturation and social experience. This developmental dissociation suggests that fish like primates might have separate systems for small and large number representation.

Discrimination and representation of relative numerosity in a bisection task by pigeons

We trained 4 pigeons in a numerical bisection task to discriminate between pairs of keylight flashes with a ratio of 1:3 (2 vs. 6, 4 vs. 12, and 8 vs. 24) that were presented in a sample phase. Responses to the blue key were reinforced after a sequence of a larger number of flashes, and responses to the white key were reinforced after a sequence of a smaller number of flashes. The intervals between flashes in the sample phase were randomized to attenuate the covariation of temporal cues with flash number. Pigeons responded accurately in each of the discriminations, with typically 85%-90% correct responses. Transfer tests showed that the proportion of large responses increased with number and performance generalized to larger values outside the training ranges. Psychometric functions superposed when plotted on a relative scale, and estimates of Weber fractions were approximately constant, suggesting that variability was scalar. However, contrary to previous research in nonhumans, bisection points were located at the arithmetic, not geometric, mean. Hierarchical logistic regressions confirmed significant control over responding by number beyond that attributable to temporal cues. These results show that pigeons are able to respond accurately in a relative numerosity discrimination with successively presented visual stimuli, although the nature of the numerical representation and response rule remains unclear.

Spatial attention determines the nature of nonverbal number representation

Coordinated studies of adults, infants, and nonhuman animals provide evidence for two systems of nonverbal number representation: a "parallel individuation" system that represents individual items and a "numerical magnitude" system that represents the approximate cardinal value of a group. However, there is considerable debate about the nature and functions of these systems, due largely to the fact that some studies show a dissociation between small (1-3) and large (>3) number representation, whereas others do not. Using event-related potentials, we show that it is possible to determine which system will represent the numerical value of a small number set (1-3 items) by manipulating spatial attention. Specifically, when attention can select individual objects, an early brain response (N1) scales with the cardinal value of the display, the signature of parallel individuation. In contrast, when attention cannot select individual objects or is occupied by another task, a later brain response (P2p) scales with ratio, the signature of the approximate numerical magnitude system. These results provide neural evidence that small numbers can be represented as approximate numerical magnitudes. Further, they empirically demonstrate the importance of early attentional processes to number representation by showing that the way in which attention disperses across a scene determines which numerical system will deploy in a given context.

Range dependent processing of visual numerosity: similarities across vision and haptics

'Subitizing' refers to fast and accurate judgement of small numerosities, whereas for larger numerosities either counting or estimation are used. Counting is slow and precise, whereas estimation is fast but imprecise. In this study consisting of five experiments we investigated if and how the numerosity judgement process is affected by the relative spacing between the presented numerosities. To this end we let subjects judge the number of dots presented on a screen and recorded their response times. Our results show that subjects switch from counting to estimation if the relative differences between subsequent numerosities are large (a factor of 2), but that numerosity judgement in the subitizing range was still faster. We also show this fast performance for small numerosities only occurred when numerosity information is present. This indicates this is typical for number processing and not magnitude estimation in general. Furthermore, comparison with a previous haptic study suggests similar processing in numerosity judgement through haptics and vision.

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.

Frontal subregions mediating Elevator Counting task performance

Deficits in sustained attention may lead to action slips in everyday life as irrelevant action sequences are inappropriately triggered internally or by the environment. While deficits in sustained attention have been associated with damage to the frontal lobes of the brain, little is known about the role of the frontal lobes in the Elevator Counting subtest of the Test of Everyday Attention. In the current study, 55 frontal patients subdivided into medial, orbital and lateral subgroups, 18 patients with posterior lesions and 82 healthy controls performed the Elevator Counting task. The results revealed that patients with medial and left lateral prefrontal lesions were significantly impaired on the task compared to healthy controls. Research suggests that patients with medial lesions are susceptible to competition from task irrelevant schema; whereas the left lateral group in the current study may fail to keep track of the tones already presented.

How element visibility affects visual enumeration

The task of detection requires that at least one target component (i.e. "feature") be detected, while the task of identification requires the detection and integration of multiple features into a recognizable object. Enumeration seems to involve aspects of both feature detection and object identification. As in identification, it requires the detection of multiple features, but as in detection, it does not require the explicit encoding of a global form. Enumeration of briefly presented objects is accurate up to the "subitizing capacity" of 3-5 items. We discuss the relation of enumeration to visual detection and identification by considering the effect of target visibility on subitizing capacity. We found that while the distribution of enumeration responses changes with contrast, subitizing capacity is generally invariant with contrast until it nears detection threshold. These results suggest that component detection (associated with number estimation) and component integration (associated with subitizing) behaved differentially as contrast was manipulated. We speculate that subitizing capacity is linked to the approximate number of detected features adequate for recognizing shapes.

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.

Crossing the divide: infants discriminate small from large numerosities

Although young infants have repeatedly demonstrated successful numerosity discrimination across large sets when the number of items in the sets changes twofold (E. M. Brannon, S. Abbott, & D. J. Lutz, 2004; J. N. Wood & E. S. Spelke, 2005; F. Xu & E. S. Spelke, 2000), they consistently fail to discriminate a twofold change in number when one set is large and the other is small (<4 items; F. Feigenson, S. Carey, & M. Hauser, 2002; F. Xu, 2003). It has been theorized that this failure reflects an incompatibility in representational systems for small and large sets. The authors investigated the ability of 7-month-old infants to compare small and large sets over a variety of conditions. Results reveal that infants can successfully discriminate small from large sets when given a fourfold change, but not a twofold change, in number. The implications of these results are discussed in light of current theories of number representation.

A Model of Knower-Level Behavior in Number-Concept Development

We develop and evaluate a model of behavior on the Give-N task, a commonly-used measure of young children's number knowledge. Our model uses the knower-level theory of how children represent numbers. To produce behavior on the Give-N task, the model assumes children start out with a base-rate that make some answers more likely a priori than others, but is updated on each experimental trial in a way that depends on the interaction between the experimenter's request and the child's knower-level. We formalize this process as a generative graphical model, so that the parameters-including the base-rate distribution and each child's knower-level-can be inferred from data using Bayesian methods. Using this approach, we evaluate the model on previously published data from 82 children spanning the whole developmental range. The model provides an excellent fit to these data, and the inferences about the base-rate and knower-levels are interpretable and insightful. We discuss how our modeling approach can be extended to other developmental tasks, and can be used to help evaluate alternative theories of number representation against the knower-level theory.

The neural development of an abstract concept of number

As literate adults, we appreciate numerical values as abstract entities that can be represented by a numeral, a word, a number of lines on a scorecard, or a sequence of chimes from a clock. This abstract, notation-independent appreciation of numbers develops gradually over the first several years of life. Here, using functional magnetic resonance imaging, we examine the brain mechanisms that 6- and 7-year-old children and adults recruit to solve numerical comparisons across different notation systems. The data reveal that when young children compare numerical values in symbolic and nonsymbolic notations, they invoke the same network of brain regions as adults including occipito-temporal and parietal cortex. However, children also recruit inferior frontal cortex during these numerical tasks to a much greater degree than adults. Our data lend additional support to an emerging consensus from adult neuroimaging, nonhuman primate neurophysiology, and computational modeling studies that a core neural system integrates notation-independent numerical representations throughout development but, early in development, higher-order brain mechanisms mediate this process.

Levels of number knowledge during early childhood

Researchers have long disagreed about whether number concepts are essentially continuous (unchanging) or discontinuous over development. Among those who take the discontinuity position, there is disagreement about how development proceeds. The current study addressed these questions with new quantitative analyses of children's incorrect responses on the Give-N task. Using data from 280 children, ages 2 to 4 years, this study showed that most wrong answers were simply guesses, not counting or estimation errors. Their mean was unrelated to the target number, and they were lower-bounded by the numbers children actually knew. In addition, children learned the number-word meanings one at a time and in order; they treated the number words as mutually exclusive; and once they figured out the cardinal principle of counting, they generalized this principle to the rest of their count list. Findings support the 'discontinuity' account of number development in general and the 'knower-levels' account in particular.

Spontaneous number representation in mosquitofish

While there is convincing evidence that preverbal human infants and non-human primates can spontaneously represent number, considerable debate surrounds the possibility that such capacity is also present in other animals. Fish show a remarkable ability to discriminate between different numbers of social companions. Previous work has demonstrated that in fish the same set of signature limits that characterize non-verbal numerical systems in primates is present but yet to provide any demonstration that fish can really represent number rather than basing their discrimination on continuous attributes that co-vary with number. In the present work, using the method of 'item by item' presentation, we provide the first evidence that fish are capable of selecting the larger group of social companions relying exclusively on numerical information. In our tests subjects could choose between one large and one small group of companions when permitted to see only one fish at a time. Fish were successful when both small (3 vs. 2) and large numbers (8 vs. 4) were involved and their performance was not affected by the density of the fish or by the overall space occupied by the group.

All numbers are not equal: an electrophysiological investigation of small and large number representations

Behavioral and brain imaging research indicates that human infants, humans adults, and many nonhuman animals represent large nonsymbolic numbers approximately, discriminating between sets with a ratio limit on accuracy. Some behavioral evidence, especially with human infants, suggests that these representations differ from representations of small numbers of objects. To investigate neural signatures of this distinction, event-related potentials were recorded as adult humans passively viewed the sequential presentation of dot arrays in an adaptation paradigm. In two studies, subjects viewed successive arrays of a single number of dots interspersed with test arrays presenting the same or a different number; numerical range (small numerical quantities 1-3 vs. large numerical quantities 8-24) and ratio difference varied across blocks as continuous variables were controlled. An early-evoked component (N1), observed over widespread posterior scalp locations, was modulated by absolute number with small, but not large, number arrays. In contrast, a later component (P2p), observed over the same scalp locations, was modulated by the ratio difference between arrays for large, but not small, numbers. Despite many years of experience with symbolic systems that apply equally to all numbers, adults spontaneously process small and large numbers differently. They appear to treat small-number arrays as individual objects to be tracked through space and time, and large-number arrays as cardinal values to be compared and manipulated.

How Humans Count: Numerosity and the Parietal Cortex

Numerosity (the number of objects in a set), like color or movement, is a basic property of the environment. Animal and human brains have been endowed by evolution by mechanisms based on parietal circuitry for representing numerosity in an highly abstract, although approximate fashion. These mechanisms are functional at a very early age in humans and spontaneously deployed in the wild by animals of different species. The recent years have witnessed terrific advances in unveiling the neural code(s) underlying numerosity representations and showing similarities as well as differences across species. In humans, during development, with the introduction of symbols for numbers and the implementation of the counting routines, the parietal system undergoes profound (yet still largely mysterious) modifications, such that the neural machinery previously evolved to represent approximate numerosity gets partially “recycled” to support the representation of exact number.