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

Counting or chunking? Mathematical and heuristic abilities in patients with corticobasal syndrome and posterior cortical atrophy

A growing amount of empirical data is showing that the ability to manipulate quantities in a precise and efficient fashion is rooted in cognitive mechanisms devoted to specific aspects of numbers processing. The analog number system (ANS) has a reasonable representation of quantities up to about 4, and represents larger quantities on the basis of a numerical ratio between quantities. In order to represent the precise cardinality of a number, the ANS may be supported by external algorithms such as language, leading to a "precise number system". In the setting of limited language, other number-related systems can appear. For example the parallel individuation system (PIS) supports a "chunking mechanism" that clusters units of larger numerosities into smaller subsets. In the present study we investigated number processing in non-aphasic patients with corticobasal syndrome (CBS) and posterior cortical atrophy (PCA), two neurodegenerative conditions that are associated with progressive parietal atrophy. The present study investigated these number systems in CBS and PCA by assessing the property of the ANS associated with smaller and larger numerosities, and the chunking property of the PIS. The results revealed that CBS/PCA patients are impaired in simple calculations (e.g., addition and subtraction) and that their performance strongly correlates with the size of the numbers involved in these calculations, revealing a clear magnitude effect. This magnitude effect was correlated with gray matter atrophy in parietal regions. Moreover, a numeral-dots transcoding task showed that CBS/PCA patients were able to take advantage of clustering in the spatial distribution of the dots of the array. The relative advantage associated with chunking compared to a random spatial distribution correlated with both parietal and prefrontal regions. These results shed light on the properties of systems for representing number knowledge in non-aphasic patients with CBS and PCA.

Numerical discrimination by frogs (Bombina orientalis)

Evidence has been reported for quantity discrimination in mammals and birds and, to a lesser extent, fish and amphibians. For the latter species, however, whether quantity discrimination would reflect sensitivity to number or to the continuous physical variables that covary with number is unclear. Here we reported a series of experiments with frogs (Bombina orientalis) tested in free-choice experiments for their preferences for different amounts of preys (Tenebrio molitor larvae) with systematic controls for variables such as surface area, volume, weight, and movement. Frogs showed quantity discrimination in the range of both small (1 vs. 2, 2 vs. 3, but not 3 vs. 4) and large numerousness (3 vs. 6, 4 vs. 8, but not 4 vs. 6), with clear evidence of being able to discriminate numerousness even when continuous physical variables were controlled for in the case of small numerousness (i.e., 1 vs. 2), whereas in the case of large numerousness it remains unclear whether the number or surface areas were dominant. We suggested that task demands are likely to be responsible for the activation of different systems for small and large numerousness and for their relative susceptibility to quantitative stimulus variables.

Concurrent validity of approximate number sense tasks in adults and children

Reasoning with non-symbolic numerosities is suggested to be rooted in the Approximate Number System (ANS) and evidence pointing to a relationship between the acuity of this system and mathematics is available. In order to use the acuity of this ANS as a screening instrument to detect future math problems, it is important to model ANS acuity over development. However, whether ANS acuity and its development have been described accurately can be questioned. Namely, different tasks were used to examine the developmental trajectory of ANS acuity and studies comparing performances on these different tasks are scarce. In the present study, we examined whether different tasks designed to measure the acuity of the ANS are comparable and lead to related ANS acuity measures (i.e., the concurrent validity of these tasks). We contrasted the change detection task, which is used in infants, with tasks that are more commonly used in older children and adults (i.e., comparison and same-different tasks). Together, our results suggest that ANS acuity measures obtained with different tasks are not related. This poses serious problems for the comparison of ANS acuity measures derived from different tasks and thus for the establishment of the developmental trajectory of ANS acuity.

Visual nesting of stimuli affects rhesus monkeys' (Macaca mulatta) quantity judgments in a bisection task

Nonhuman animals are highly proficient at judging relative quantities presented in a variety of formats, including visual, auditory, and even cross-modal formats. Performance typically is constrained by the ratio between sets, as would be expected under Weber's law and as is described in the approximate number system (ANS) hypothesis. In most cases, tests are designed to avoid any perceptual confusion for animals regarding the stimulus sets, but despite this, animals show some of the perceptual biases that humans show based on organization of stimuli. Here, we demonstrate an additional perceptual bias that emerges from the illusion of nested sets. When arrays of circles were presented on a computer screen and were to be classified as larger than or smaller than an established central value, rhesus monkeys (Macaca mulatta) underestimated quantities when circles were nested within each other. This matched a previous report with adult humans (Chesney & Gelman, Attention, Perception, & Psychophysics 24:1104-1113, 2012), indicating that macaques, like humans, show the pattern of biased perception predicted by ANS estimation. Although some macaques overcame this perceptual bias, demonstrating that they could come to view nested stimuli as individual elements to be included in the estimates of quantity used for classifying arrays, the majority of the monkeys showed the bias of underestimating nested arrays throughout the experiment.

Electrophysiological evidence for the involvement of the approximate number system in preschoolers' processing of spoken number words

Little is known about the neural underpinnings of number word comprehension in young children. Here we investigated the neural processing of these words during the crucial developmental window in which children learn their meanings and asked whether such processing relies on the Approximate Number System. ERPs were recorded as 3- to 5-year-old children heard the words one, two, three, or six while looking at pictures of 1, 2, 3, or 6 objects. The auditory number word was incongruent with the number of visual objects on half the trials and congruent on the other half. Children's number word comprehension predicted their ERP incongruency effects. Specifically, children with the least number word knowledge did not show any ERP incongruency effects, whereas those with intermediate and high number word knowledge showed an enhanced, negative polarity incongruency response (N(inc)) over centroparietal sites from 200 to 500 msec after the number word onset. This negativity was followed by an enhanced, positive polarity incongruency effect (P(inc)) that emerged bilaterally over parietal sites at about 700 msec. Moreover, children with the most number word knowledge showed ratio dependence in the P(inc) (larger for greater compared with smaller numerical mismatches), a hallmark of the Approximate Number System. Importantly, a similar modulation of the P(inc) from 700 to 800 msec was found in children with intermediate number word knowledge. These results provide the first neural correlates of spoken number word comprehension in preschoolers and are consistent with the view that children map number words onto approximate number representations before they fully master the verbal count list.

Lateralized mechanisms for encoding of object. Behavioral evidence from an animal model: the domestic chick (Gallus gallus)

In our previous research we reported a leftward-asymmetry in domestic chicks required to identify a target element, on the basis of its ordinal position, in a series of identical elements. Here we re-coded behavioral data collected in previous studies from chicks tested in a task involving a different kind of numerical ability, to study lateralization in dealing with an arithmetic task. Chicks were reared with a set of identical objects representing artificial social companions. On day 4, chicks underwent a free-choice test in which two sets, each composed of a different number of identical objects (5 vs.10 or 6 vs. 9, Experiment 1), were hidden behind two opaque screens placed in front of the chick, one on the left and one on the right side. Objects disappeared, one by one, behind either screen, so that, for example, one screen occluded 5 objects and the other 10 objects. The left-right position of the larger set was counterbalanced between trials. Results show that chicks, in the attempt to rejoin the set with the higher number of social companions, performed better when this was located to the right. However, when the number of elements in the two sets was identical (2 vs. 2, in Experiment 2) and they differed only in the coloration of the objects, this bias was not observed, suggesting a predisposition to map the numerical magnitude from left to right. Future studies should be devoted to the direct investigation of this phenomenon, possibly employing an identical number of mono-chromatic imprinting stimuli in both conditions involving a numerical discrimination and conditions not involving any numerosity difference.

Common and dissociated mechanisms for estimating large and small dot arrays: value-specific fMRI adaptation

An fMRI pair-adaptation paradigm was used to identify the brain regions linked to the apprehension of small and large numbers of items. Participants classified stimuli on the basis of their numerosities (fewer or more than five dots). We manipulated the type of repetition within pairs of dot arrays. Overall processing of pairs with small as opposed to large quantities was associated with a decreased BOLD response in the midline structures and inferior parietal cortex. The opposite pattern was observed in middle cingulate cortex. Pairs in which the same numerosity category was repeated, were associated with a decreased signal in the left prefrontal and the left inferior parietal cortices, compared with when numerosities changed. Repetitions of exact numerosities irrespective of sample size were associated with decreased responses in bi-lateral prefrontal, sensory-motor regions, posterior occipital and left intraparietal sulcus (IPS). More importantly, we found value-specific adaptation specific to repeated small quantity in the left lateral occipito-temporal cortex, irrespective of whether the exact same stimulus pattern repeated. Our results indicate that a large network of regions (including the IPS) support visual quantity processing independent of the number of items present; however assimilation of small quantities is associated with additional support from regions within the left occipito-temporal cortex. We propose that processing of small quantities is aided by a subitizing-specific network. This network may account for the increased processing efficiency often reported for numerosities in the subitizing range.

Indexing the approximate number system

Much recent research attention has focused on understanding individual differences in the approximate number system, a cognitive system believed to underlie human mathematical competence. To date researchers have used four main indices of ANS acuity, and have typically assumed that they measure similar properties. Here we report a study which questions this assumption. We demonstrate that the numerical ratio effect has poor test-retest reliability and that it does not relate to either Weber fractions or accuracy on nonsymbolic comparison tasks. Furthermore, we show that Weber fractions follow a strongly skewed distribution and that they have lower test-retest reliability than a simple accuracy measure. We conclude by arguing that in the future researchers interested in indexing individual differences in ANS acuity should use accuracy figures, not Weber fractions or numerical ratio effects.

Numerical matching judgments in children with mathematical learning disabilities

Both deficits in the innate magnitude representation (i.e. representation deficit hypothesis) and deficits in accessing the magnitude representation from symbols (i.e. access deficit hypotheses) have been proposed to explain mathematical learning disabilities (MLD). Evidence for these hypotheses has mainly been accumulated through the use of numerical magnitude comparison tasks. It has been argued that the comparison distance effect might reflect decision processes on activated magnitude representations rather than number processing per se. One way to avoid such decisional processes confounding the numerical distance effect is by using a numerical matching task, in which children have to indicate whether two dot-arrays or a dot-array and a digit are numerically the same or different. Against this background, we used a numerical matching task to examined the representation deficit and access deficit hypotheses in a group children with MLD and controls matched on age, gender and IQ. The results revealed that children with MLD were slower than controls on the mixed notation trials, whereas no difference was found for the non-symbolic trials. This might be in line with the access deficit hypothesis, showing that children with MLD have difficulties in linking a symbol with its quantity representation. However, further investigation is required to exclude the possibility that children with MLD have a deficit in integrating the information from different input notations.

Open questions and a proposal: a critical review of the evidence on infant numerical abilities

Considerable research has investigated infants' numerical capacities. Studies in this domain have used procedures of habituation, head turn, violation of expectation, reaching, and crawling to ask what quantities infants discriminate and represent visually, auditorily as well as intermodally. The concensus view from these studies is that infants possess a numerical system that is amodal and applicable to the quantification of any kind of entity and that this system is fundamentally separate from other systems that represent continuous magnitude. Although there is much evidence consistent with this view, there are also inconsistencies in the data. This paper provides a broad review of what we know, including the evidence suggesting systematic early knowledge as well as the peculiarities and gaps in the empirical findings with respect to the concensus view. We argue, from these inconsistencies, that the concensus view cannot be entirely correct. In light of the evidence, we propose a new hypothesis, the Signal Clarity hypothesis, that posits a developmental role for dimensions of continuous quantity within the discrete quantity system and calls for a broader research agenda that considers the covariation of discrete and continuous quantities not simply as a problem for experimental control but as information that developing infants may use to build more precise and robust representations of number.

Estimation abilities of large numerosities in Kindergartners

The approximate number system (ANS) is thought to be a building block for the elaboration of formal mathematics. However, little is known about how this core system develops and if it can be influenced by external factors at a young age (before the child enters formal numeracy education). The purpose of this study was to examine numerical magnitude representations of 5–6 year old children at 2 different moments of Kindergarten considering children's early number competence as well as schools' socio-economic index (SEI). This study investigated estimation abilities of large numerosities using symbolic and non-symbolic output formats (8–64). In addition, we assessed symbolic and non-symbolic early number competence (1–12) at the end of the 2nd (N = 42) and the 3rd (N = 32) Kindergarten grade. By letting children freely produce estimates we observed surprising estimation abilities at a very young age (from 5 year on) extending far beyond children's symbolic explicit knowledge. Moreover, the time of testing has an impact on the ANS accuracy since 3rd Kindergarteners were more precise in both estimation tasks. Additionally, children who presented better exact symbolic knowledge were also those with the most refined ANS. However, this was true only for 3rd Kindergarteners who were a few months from receiving math instructions. In a similar vein, higher SEI positively impacted only the oldest children's estimation abilities whereas it played a role for exact early number competences already in 2nd and 3rd graders. Our results support the view that approximate numerical representations are linked to exact number competence in young children before the start of formal math education and might thus serve as building blocks for mathematical knowledge. Since this core number system was also sensitive to external components such as the SEI this implies that it can most probably be targeted and refined through specific educational strategies from preschool on.

Are the neural correlates of subitizing and estimation dissociable? An fNIRS investigation

Human performance in visual enumeration tasks typically shows two distinct patterns as a function of set size. For small sets, usually up to 4 items, numerosity judgments are extremely rapid, precise and confident, a phenomenon known as subitizing. When this limit is exceeded and serial counting is precluded, exact enumeration gives way to estimation: performance becomes error-prone and more variable. Surprisingly, despite the importance of subitizing and estimation in numerical cognition, only few neuroimaging studies have examined whether the neural activity related to these two phenomena can be dissociated. In the present work, we used multi-channel near-infrared spectroscopy (fNIRS) to measure hemodynamic activity of the bilateral parieto-occipital cortex during a visual enumeration task. Participants had to judge the numerosity of dot arrays and indicate it by means of verbal response. We observed a different hemodynamic pattern in the parietal cortex, both in terms of amplitude modulation and temporal profile, for numerosities below and beyond the subitizing range. Crucially, the neural dissociation between subitizing and estimation was strongest at the level of right IPS. The present findings confirm that fNIRS can be successfully used to detect subtle temporal differences in hemodynamic activity and to produce inferences on the neural mechanisms underlying cognitive functions.

Measuring acuity of the approximate number system reliably and validly: the evaluation of an adaptive test procedure

Two studies investigated the reliability and predictive validity of commonly used measures and models of Approximate Number System acuity (ANS). Study 1 investigated reliability by both an empirical approach and a simulation of maximum obtainable reliability under ideal conditions. Results showed that common measures of the Weber fraction (w) are reliable only when using a substantial number of trials, even under ideal conditions. Study 2 compared different purported measures of ANS acuity as for convergent and predictive validity in a within-subjects design and evaluated an adaptive test using the ZEST algorithm. Results showed that the adaptive measure can reduce the number of trials needed to reach acceptable reliability. Only direct tests with non-symbolic numerosity discriminations of stimuli presented simultaneously were related to arithmetic fluency. This correlation remained when controlling for general cognitive ability and perceptual speed. Further, the purported indirect measure of ANS acuity in terms of the Numeric Distance Effect (NDE) was not reliable and showed no sign of predictive validity. The non-symbolic NDE for reaction time was significantly related to direct w estimates in a direction contrary to the expected. Easier stimuli were found to be more reliable, but only harder (7:8 ratio) stimuli contributed to predictive validity.

A common metric magnitude system for the perception and production of numerosity, length, and duration

Numerosity, length, and duration processing may share a common functional mechanism situated within the parietal cortex. A strong parallelism between the processing of these three magnitudes has been revealed by similar behavioral signatures (e.g., Weber–Fechner's law, the distance effect) and reciprocal interference effects. Here, we extend the behavioral evidence for a common magnitude processing mechanism by exploring whether the under- and overestimation patterns observed during numerical perception and production tasks are also present in length and duration perception and production. In a first experiment, participants had to perform two estimation tasks (i.e., perception and production) on three magnitudes (i.e., numerosities, lengths, and durations). The results demonstrate similar patterns for the three magnitudes: underestimation was observed in all perception tasks, whereas overestimation was found in all production tasks. A second experiment ensured that this pattern of under- and over-estimation was not solely generated by the mere process of perceiving or producing something. Participants were required to estimate the alphabetical position of a letter (i.e., perception task) or to produce the letter corresponding to a given position (i.e., production task). No under- or overestimation were observed in this experiment, which suggests that the process of perceiving or producing something alone cannot explain the systematic pattern of estimation observed on magnitudes. Together, these findings strengthen the idea that magnitude estimations share a common metric system, requiring similar mechanisms and/or representations.

A two-minute paper-and-pencil test of symbolic and nonsymbolic numerical magnitude processing explains variability in primary school children's arithmetic competence

Recently, there has been a growing emphasis on basic number processing competencies (such as the ability to judge which of two numbers is larger) and their role in predicting individual differences in school-relevant math achievement. Children’s ability to compare both symbolic (e.g. Arabic numerals) and nonsymbolic (e.g. dot arrays) magnitudes has been found to correlate with their math achievement. The available evidence, however, has focused on computerized paradigms, which may not always be suitable for universal, quick application in the classroom. Furthermore, it is currently unclear whether both symbolic and nonsymbolic magnitude comparison are related to children’s performance on tests of arithmetic competence and whether either of these factors relate to arithmetic achievement over and above other factors such as working memory and reading ability. In order to address these outstanding issues, we designed a quick (2 minute) paper-and-pencil tool to assess children’s ability to compare symbolic and nonsymbolic numerical magnitudes and assessed the degree to which performance on this measure explains individual differences in achievement. Children were required to cross out the larger of two, single-digit numerical magnitudes under time constraints. Results from a group of 160 children from grades 1–3 revealed that both symbolic and nonsymbolic number comparison accuracy were related to individual differences in arithmetic achievement. However, only symbolic number comparison performance accounted for unique variance in arithmetic achievement. The theoretical and practical implications of these findings are discussed which include the use of this measure as a possible tool for identifying students at risk for future difficulties in mathematics.

The importance of being relevant: modulation of magnitude representations

The current study aims to answer two main questions. First, is there a difference between the representations of the numerical and the physical properties of visually presented numbers? Second, can the relevancy of the dimension change its representation? In a numerical Stroop task, participants were asked to indicate either the physically or the numerically larger value of two digits. The ratio between the physical sizes and the numerical values changed orthogonally from 0.1 (the largest difference) to 0.8. Reaction times (RT) were plotted as a function of both physical and numerical ratios. Trend analysis revealed that while the numerical dimension followed Weber's law regardless of task demands, the physical ratio deviated from linearity. Our results suggest that discrete and continuous magnitudes are represented by different yet interactive systems rather than by a shared representation.

Individual differences in inhibitory control, not non-verbal number acuity, correlate with mathematics achievement

Given the well-documented failings in mathematics education in many Western societies, there has been an increased interest in understanding the cognitive underpinnings of mathematical achievement. Recent research has proposed the existence of an Approximate Number System (ANS) which allows individuals to represent and manipulate non-verbal numerical information. Evidence has shown that performance on a measure of the ANS (a dot comparison task) is related to mathematics achievement, which has led researchers to suggest that the ANS plays a critical role in mathematics learning. Here we show that, rather than being driven by the nature of underlying numerical representations, this relationship may in fact be an artefact of the inhibitory control demands of some trials of the dot comparison task. This suggests that recent work basing mathematics assessments and interventions around dot comparison tasks may be inappropriate.

Inherently Analog Quantity Representations in Olive Baboons (Papio anubis)

Strong evidence indicates that non-human primates possess a numerical representation system, but the inherent nature of that system is still debated. Two cognitive mechanisms have been proposed to account for non-human primate numerical performance: (1) a discrete object-file system limited to quantities <4, and (2) an analog system which represents quantities comparatively but is limited by the ratio between two quantities. To test the underlying nature of non-human primate quantification, we asked eight experiment-naive olive baboons (Papio anubis) to discriminate between number pairs containing small (<4), large (>4), or span (small vs. large) numbers of food items presented simultaneously or sequentially. The prediction from the object-file hypothesis is that baboons will only accurately choose the larger quantity in small pairs, but not large or span pairs. Conversely, the analog system predicts that baboons will be successful with all numbers, and that success will be dependent on numerical ratio. We found that baboons successfully discriminated all pair types at above chance levels. In addition, performance significantly correlated with the ratio between the numerical values. Although performance was better for simultaneous trials than sequential trials, evidence favoring analog numerical representation emerged from both conditions, and was present even in the first exposure to number pairs. Together, these data favor the interpretation that a single, coherent analog representation system underlies spontaneous quantitative abilities in primates.

Non-symbolic halving in an Amazonian indigene group

Much research supports the existence of an Approximate Number System (ANS) that is recruited by infants, children, adults, and non-human animals to generate coarse, non-symbolic representations of number. This system supports simple arithmetic operations such as addition, subtraction, and ordering of amounts. The current study tests whether an intuition of a more complex calculation, division, exists in an indigene group in the Amazon, the Mundurucu, whose language includes no words for large numbers. Mundurucu children were presented with a video event depicting a division transformation of halving, in which pairs of objects turned into single objects, reducing the array's numerical magnitude. Then they were tested on their ability to calculate the outcome of this division transformation with other large-number arrays. The Mundurucu children effected this transformation even when non-numerical variables were controlled, performed above chance levels on the very first set of test trials, and exhibited performance similar to urban children who had access to precise number words and a surrounding symbolic culture. We conclude that a halving calculation is part of the suite of intuitive operations supported by the ANS.

Using potential performance theory to analyze systematic and random factors in enumeration tasks

Prior research has shown that as the number of items being enumerated increases, performance decreases, especially when the amount of time is limited. Researchers studying nonverbal enumeration have found that random noise increases as a function of the number of items presented. Over a series of 2 experiments, the authors used potential performance theory to expand these findings and discover precisely how much random noise actually influences observed performance and what performance might look like in the absence of random factors. Participants briefly viewed a visual stimulus comprising a set of 4 to 9 dots presented horizontally (Experiment 1) or randomly (Experiment 2) on a computer monitor. Findings from both experiments indicate that the decrease in performance for larger set sizes resulted almost entirely from a reduction in consistency (or an increase in random noise), whereas potential performance remained fairly constant until the maximum set size.

Numerical architecture

The idea that there is a "Number Sense" (Dehaene, 1997) or "Core Knowledge" of number ensconced in a modular processing system (Carey, 2009) has gained popularity as the study of numerical cognition has matured. However, these claims are generally made with little, if any, detailed examination of which modular properties are instantiated in numerical processing. In this article, I aim to rectify this situation by detailing the modular properties on display in numerical cognitive processing. In the process, I review literature from across the cognitive sciences and describe how the evidence reported in these works supports the hypothesis that numerical cognitive processing is modular. I outline the properties that would suffice for deeming a certain processing system a modular processing system. Subsequently, I use behavioral, neuropsychological, philosophical, and anthropological evidence to show that the number module is domain specific, informationally encapsulated, neurally localizable, subject to specific pathological breakdowns, mandatory, fast, and inaccessible at the person level; in other words, I use the evidence to demonstrate that some of our numerical capacity is housed in modular casing.

Comparing the neural distance effect derived from the non-symbolic comparison and the same-different task

As a result of the representation of numerosities, more accurate and faster discrimination between two numerosities is observed when the distance between them increases. In previous studies, the comparison and same-different task were most frequently used to investigate this distance effect. Recently, it was questioned whether the non-symbolic distance effects derived from these tasks originate at the same level. In the current study, we examined the behavioral and neural distance effects of the comparison and same-different task to assess potential differences between both tasks. Participants were first year university students. Each participant completed both tasks, while their reaction time, accuracy and brain activity on predefined components was measured. The early N1-P2p transition and the P2p component on temporo-occipital (TO) and inferior parietal (IP) electrode groups were considered, as well as the late P3 component on a central (C) electrode group. The results showed that the behavioral distance effects from both tasks were comparable, although participants' performance was worse on the same-different task. The neural results revealed similar effects of distance on the mean amplitudes for the early components for both tasks (all p′s < 0.02) and an additional effect of task difficulty on the mean amplitudes of these components. Similar as in previous studies, we found a (marginally) significant increase in mean amplitude of the later P3 component with increasing distance for the comparison (p = 0.07), but not for the same-different task. Apparently, the initial stages of number processing are comparable for both tasks, but an additional later stage is only present for the comparison task. The P3 effect would be indicative of this decisional stage, which was previously proposed to underlie the comparison distance effect (CDE).

1 < 2 and 2 < 3: non-linguistic appreciations of numerical order

Ordinal understanding is involved in understanding social hierarchies, series of actions, and everyday events. Moreover, an appreciation of numerical order is critical to understanding number at a highly abstract, conceptual level. In this paper, we review findings concerning the development and expression of ordinal numerical knowledge in preverbal human infants in light of literature about the same cognitive abilities in non-human animals. We attempt to reconcile seemingly contradictory evidence, provide new directions for prospective research, and evaluate the shared basis of ordinal knowledge among non-verbal organisms. Our review of the research leads us to conclude that both infants and non-human animals are adapted to respond to monotonic progressions in numerical order, consonant with mathematical definitions of numerical order. Further, we suggest that patterns in the way that infants and non-human animals process numerical order can be accounted for by changes across development, the conditions under which representations are generated, or both.

Infants Show Ratio-dependent Number Discrimination Regardless of Set Size

Evidence for approximate number system (ANS) representations in infancy is robust but has typically only been found when infants are presented with arrays of four or more elements. In addition, several studies have found that infants fail to discriminate between small numbers when continuous variables such as surface area and contour length are controlled. These findings suggest that under some circumstances, infants fail to recruit either the ANS or object file representations for small sets. Here, we used a numerical change detection paradigm to assess 6-month-old infants' ability to represent small values. In Experiment 1, infants were tested with 1 versus 3, 1 versus 2, and 2 versus 3 dots. Infants successfully discriminated 1 versus 3 and 1 versus 2, but failed with 2 versus 3. In Experiment 2, we tested whether infants could compare small and large values with a 2 versus 4 condition. Across both experiments, infants' performance exhibited ratio dependence, the hallmark of the ANS. Our results indicate that infants can attend to the purely numerical attributes of small sets and that the numerical change detection paradigm accesses ANS representations in infancy regardless of set size.

Visual nesting impacts approximate number system estimation

The approximate number system (ANS) allows people to quickly but inaccurately enumerate large sets without counting. One popular account of the ANS is known as the accumulator model. This model posits that the ANS acts analogously to a graduated cylinder to which one "cup" is added for each item in the set, with set numerosity read from the "height" of the cylinder. Under this model, one would predict that if all the to-be-enumerated items were not collected into the accumulator, either the sets would be underestimated, or the misses would need to be corrected by a subsequent process, leading to longer reaction times. In this experiment, we tested whether such miss effects occur. Fifty participants judged numerosities of briefly presented sets of circles. In some conditions, circles were arranged such that some were inside others. This circle nesting was expected to increase the miss rate, since previous research had indicated that items in nested configurations cannot be preattentively individuated in parallel. Logically, items in a set that cannot be simultaneously individuated cannot be simultaneously added to an accumulator. Participants' response times were longer and their estimations were lower for sets whose configurations yielded greater levels of nesting. The level of nesting in a display influenced estimation independently of the total number of items present. This indicates that miss effects, predicted by the accumulator model, are indeed seen in ANS estimation. We speculate that ANS biases might, in turn, influence cognition and behavior, perhaps by influencing which kinds of sets are spontaneously counted.

A large-scale model of the functioning brain

A central challenge for cognitive and systems neuroscience is to relate the incredibly complex behavior of animals to the equally complex activity of their brains. Recently described, large-scale neural models have not bridged this gap between neural activity and biological function. In this work, we present a 2.5-million-neuron model of the brain (called "Spaun") that bridges this gap by exhibiting many different behaviors. The model is presented only with visual image sequences, and it draws all of its responses with a physically modeled arm. Although simplified, the model captures many aspects of neuroanatomy, neurophysiology, and psychological behavior, which we demonstrate via eight diverse tasks.

Nothing to it: precursors to a zero concept in preschoolers

Do young children understand the numerical value of empty sets prior to developing a concept of symbolic zero? Are empty sets represented as mental magnitudes? In order to investigate these questions, we tested 4-year old children and adults with a numerical ordering task in which the goal was to select two stimuli in ascending numerical order with occasional empty set stimuli. Both children and adults showed distance effects for empty sets. Children who were unable to order the symbol zero (e.g., 0<1), but who successfully ordered countable integers (e.g., 2<4) nevertheless showed distance effects with empty sets. These results suggest that empty sets are represented on the same numerical continuum as non-empty sets and that children represent empty sets numerically prior to understanding symbolic zero.

A number of options: rationalist, constructivist, and Bayesian insights into the development of exact-number concepts

The question of how human beings acquire exact-number concepts has interested cognitive developmentalists since the time of Piaget. The answer will owe something to both the rationalist and constructivist traditions. On the one hand, some aspects of numerical cognition (e.g. approximate number estimation and the ability to track small sets of one to four individuals) are innate or early-developing and are shared widely among species. On the other hand, only humans create representations of exact, large numbers such as 42, as distinct from both 41 and 43. These representations seem to be constructed slowly, over a period of months or years during early childhood. The task for researchers is to distinguish the innate representational resources from those that are constructed, and to characterize the construction process. Bayesian approaches can be useful to this project in at least three ways: (1) As a way to analyze data, which may have distinct advantages over more traditional methods (e.g. making it possible to find support for a nuli hypothesis); (2) as a way of modeling children's performance on specific tasks: Peculiarities of the task are captured as a prior; the child's knowledge is captured in the way the prior is updated; and behavior is captured as a posterior distribution; and (3) as a way of modeling learning itself, by providing a formal account of how learners might choose among alternative hypotheses.

Set size, individuation, and attention to shape

Much research has demonstrated a shape bias in categorizing and naming solid objects. This research has shown that when an entity is conceptualized as an individual object, adults and children attend to the object's shape. Separate research in the domain of numerical cognition suggest that there are distinct processes for quantifying small and large sets of discrete items. This research shows that small set discrimination, comparison, and apprehension is often precise for 1-3 and sometimes 4 items; however, large numerosity representation is imprecise. Results from three experiments suggest a link between the processes for small and large number representation and the shape bias in a forced choice categorization task using naming and non-naming procedures. Experiment 1 showed that adults generalized a newly learned name for an object to new instances of the same shape only when those instances were presented in sets of less than 3 or 4. Experiment 2 showed that preschool children who were monolingual speakers of three different languages were also influenced by set size when categorizing objects in sets. Experiment 3 extended these results and showed the same effect in a non-naming task and when the novel noun was presented in a count-noun syntax frame. The results are discussed in terms of a relation between the precision of object representation and the precision of small and large number representation.

Intuitive sense of number correlates with math scores on college-entrance examination

Many educated adults possess exact mathematical abilities in addition to an approximate, intuitive sense of number, often referred to as the Approximate Number System (ANS). Here we investigate the link between ANS precision and mathematics performance in adults by testing participants on an ANS-precision test and collecting their scores on the Scholastic Aptitude Test (SAT), a standardized college-entrance exam in the USA. In two correlational studies, we found that ANS precision correlated with SAT-Quantitative (i.e., mathematics) scores. This relationship remained robust even when controlling for SAT-Verbal scores, suggesting a small but specific relationship between our primitive sense for number and formal mathematical abilities.

Coding of abstract quantity by 'number neurons' of the primate brain

Humans share with nonhuman animals a quantification system for representing the number of items as nonverbal mental magnitudes. Over the past decade, the anatomical substrates and neuronal mechanisms of this quantification system have been unraveled down to the level of single neurons. Work with behaviorally trained nonhuman primates identified a parieto-frontal cortical network with individual neurons selectively tuned to the number of items. Such 'number neurons' can track items across space, time, and modality to encode numerosity in a most abstract, supramodal way. The physiological properties of these neurons can explain fundamental psychophysical phenomena during numerosity judgments. Functionally overlapping groups of parietal neurons represent not only numerable-discrete quantity (numerosity), but also innumerable-continuous quantity (extent) and relations between quantities (proportions), supporting the idea of a generalized magnitude system in the brain. These studies establish putative homologies between the monkey and human brain and demonstrate the suitability of nonhuman primates as model system to explore the neurobiological roots of the brain's nonverbal quantification system, which may constitute the evolutionary foundation of all further, more elaborate numerical skills in humans.

Spatiotemporal dynamics of processing nonsymbolic number: an event-related potential source localization study

Coordinated studies with adults, infants, and nonhuman animals provide evidence for two distinct systems of nonverbal number representation. The "parallel individuation" (PI) system selects and retains information about one to three individual entities and the "numerical magnitude" system establishes representations of the approximate cardinal value of a group. Recent event-related potential (ERP) work has demonstrated that these systems reliably evoke functionally and temporally distinct patterns of brain response that correspond to established behavioral signatures. However, relatively little is known about the neural generators of these ERP signatures. To address this question, we targeted known ERP signatures of these systems, by contrasting processing of small versus large nonsymbolic numbers, and used a source localization algorithm (LORETA) to identify their cortical origins. Early processing of small numbers, showing the signature effects of PI on the N1 (∼150 ms), was localized primarily to extrastriate visual regions. In contrast, qualitatively and temporally distinct processing of large numbers, showing the signatures of approximate number representation on the mid-latency P2p (∼200-250 ms), was localized primarily to right intraparietal regions. In comparison, mid-latency small number processing was localized to the right temporal-parietal junction and left-lateralized intraparietal regions. These results add spatial information to the emerging ERP literature documenting the process by which we represent number. Furthermore, these results substantiate recent claims that early attentional processes determine whether a collection of objects will be represented through PI or as an approximate numerical magnitude by providing evidence that downstream processing diverges to distinct cortical regions.

A canonical model of multistability and scale-invariance in biological systems

Multistability and scale-invariant fluctuations occur in a wide variety of biological organisms from bacteria to humans as well as financial, chemical and complex physical systems. Multistability refers to noise driven switches between multiple weakly stable states. Scale-invariant fluctuations arise when there is an approximately constant ratio between the mean and standard deviation of a system's fluctuations. Both are an important property of human perception, movement, decision making and computation and they occur together in the human alpha rhythm, imparting it with complex dynamical behavior. Here, we elucidate their fundamental dynamical mechanisms in a canonical model of nonlinear bifurcations under stochastic fluctuations. We find that the co-occurrence of multistability and scale-invariant fluctuations mandates two important dynamical properties: Multistability arises in the presence of a subcritical Hopf bifurcation, which generates co-existing attractors, whilst the introduction of multiplicative (state-dependent) noise ensures that as the system jumps between these attractors, fluctuations remain in constant proportion to their mean and their temporal statistics become long-tailed. The simple algebraic construction of this model affords a systematic analysis of the contribution of stochastic and nonlinear processes to cortical rhythms, complementing a recently proposed biophysical model. Similar dynamics also occur in a kinetic model of gene regulation, suggesting universality across a broad class of biological phenomena.

Biological systems are able to adapt to rapidly and widely changing environments. Many biological organisms employ two distinct mechanisms that improve their survival in these circumstances: Firstly they exhibit rapid, qualitative changes in their internal dynamics; secondly they possess the ability to respond to change that is not absolute, but scales in proportion to the underlying intensity of the environment. In this paper, we study a simple class of noisy, dynamical systems that mathematically represent a very broad range of more complex models. We hence show how a combination of nonlinear instabilities and state-dependent noise in this model is able to unify these two apparently distinct biological phenomena. To illustrate its unifying potential, this simple model is applied to two very distinct biological processes – the spontaneous activity of the human cortex (i.e. when subjects are at rest), and genetic regulation in a bacteriophage. We also provide proof of principle that our model can be inverted from empirical data, allowing estimation of the parameters that express the nonlinear and stochastic influences at play in the underlying system.

Over-estimation in numerosity estimation tasks: more than an attentional bias?

Over- and under-estimation have been observed in numerosity estimation and approximate arithmetic tasks. Two different models have been proposed to account for these reverse patterns of performance: 1) the bi-directional mapping account (Crollen, Castronovo, & Seron, 2011); 2) the operational momentum hypothesis (McCrink, Dehaene, & Dehaene-Lambertz, 2007). Our study was designed to examine whether the operational momentum could account for the over-estimation found in numerosity estimation tasks. To this aim, a series of 3 experiments involving a symbolic to non-symbolic numerical mapping and a rightward or leftward displacement along the mental number line were designed. Over-estimation was observed in these three tasks irrespective of the direction and size of the displacement to be done on the mental number line. These results thus clearly demonstrated that overestimation was not merely due to an attentional bias, but rather relied on the cognitive operation of mapping two differently scaled numerical representations.

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.

Evidence for a shared mechanism used in multiple-object tracking and subitizing

It has been proposed that the mechanism that supports the ability to keep track of multiple moving objects also supports subitizing--the ability to quickly and accurately enumerate a small set of objects. To test this hypothesis, we investigated the effects on subitizing when human observers were required to perform a multiple object tracking task and an enumeration task simultaneously. In three experiments, participants (Exp. 1, N = 24; Exp. 2, N = 11; Exp. 3, N = 37) enumerated sets of zero to nine squares that were flashed while they tracked zero, two, or four moving discs. The results indicated that the number of items participants could subitize decreased by one for each item they tracked. No such pattern was seen when the enumeration task was paired with an equally difficult, but nonvisual, working memory task. These results suggest that a shared visual mechanism supports multiple object tracking and subitizing.

Implicit response-irrelevant number information triggers the SNARC effect: evidence using a neural overlap paradigm

There is evidence from the SNARC (spatial-numerical association of response codes) effect and NDE (numerical distance effect) that number activates spatial representations. Most of this evidence comes from tasks with explicit reference to number, whether through presentation of Arabic digits (SNARC) or through magnitude decisions to nonsymbolic representations (NDE). Here, we report four studies that use the neural overlap paradigm developed by Fias, Lauwereyns, and Lammertyn (2001) to examine whether the presentation of implicit and task-irrelevant numerosity information (nonsymbolic arrays and auditory numbers) is enough to activate a spatial representation of number. Participants were presented with either numerosity arrays (1-9 circles or triangles) to which they made colour (experiment 1) or orientation (experiment 2) judgements, or auditory numbers coupled with an on-screen stimulus to which they made a colour (experiment 3) or orientation (experiment 4) judgement. SNARC effects were observed only for the orientation tasks. Following the logic of Fias et al., we argue that this SNARC effect occurs as a result of overlap in parietal processing for number and orientation judgements irrespective of modality. Furthermore, we found stronger SNARC effects in the small number range (1-4) than in the larger number range (6-9) for both nonsymbolic displays and auditory numbers. These results suggest that quantity is extracted (and interferes with responses in the orientation task) but this is not exact for the entire number range. We discuss a number of alternative models and mechanisms of numerical processing that may account for such effects.

Evidence for two numerical systems that are similar in humans and guppies

BACKGROUND

Humans and non-human animals share an approximate non-verbal system for representing and comparing numerosities that has no upper limit and for which accuracy is dependent on the numerical ratio. Current evidence indicates that the mechanism for keeping track of individual objects can also be used for numerical purposes; if so, its accuracy will be independent of numerical ratio, but its capacity is limited to the number of items that can be tracked, about four. There is, however, growing controversy as to whether two separate number systems are present in other vertebrate species.

METHODOLOGY/PRINCIPAL FINDINGS

In this study, we compared the ability of undergraduate students and guppies to discriminate the same numerical ratios, both within and beyond the small number range. In both students and fish the performance was ratio-independent for the numbers 1-4, while it steadily increased with numerical distance when larger numbers were presented.

CONCLUSIONS/SIGNIFICANCE

Our results suggest that two distinct systems underlie quantity discrimination in both humans and fish, implying that the building blocks of uniquely human mathematical abilities may be evolutionarily ancient, dating back to before the divergence of bony fish and tetrapod lineages.