Subitizing endures in sequential rather than simultaneous comparison tasks

Subitizing is the ability to appraise a number of small quantities (up to four) rapidly and precisely. This system, however, can be impaired by distractors presented along with targets to be enumerated. To better understand whether this limitation arises in perceptual circuits or in the response selection stage, we investigated whether subitizing can endure in simultaneous comparison tasks. Participants were asked to compare the number of dots in two sets on the left and right sides of the screen, presented either simultaneously or sequentially. For comparing within the numerosity range (6-32 dots), both the error rate and reaction time increased steadily as the ratio between the two numbers compared approached "1." Namely, a phenomenon labeled the ratio effect was revealed. For comparison with small numbers (<5), the sequential comparison task was errorless despite the ratio, suggesting the feature of subitizing. Individual efficiency (measured by the inverse efficiency score [IES]) did not correlate between number ranges in sequential comparison, suggesting that distinct mechanisms were involved. However, we found that in simultaneous tasks, error rate and efficiency showed an increase as the ratios of the two numbers compared approached "1." This is similar to the ratio effect revealed in the comparison for moderate numbers. Individual efficiency within these two ranges correlated, indicating that the enumeration within these two ranges was based on a single mechanism. These results suggest that subitizing cannot process sets in parallel, and numerosity takes the job whenever subitizing fails.

EEG signature of grouping strategies in numerosity perception

The moment we see a group of objects, we can appreciate its numerosity. Our numerical estimates can be imprecise for large sets (>4 items), but they become much faster and more accurate if items are clustered into groups compared to when they are randomly displaced. This phenomenon, termed groupitizing, is thought to leverage on the capacity to quickly identify groups from 1 to 4 items (subitizing) within larger sets, however evidence in support for this hypothesis is scarce. The present study searched for an electrophysiological signature of subitizing while participants estimated grouped numerosities exceeding this range by measuring event-related potential (ERP) responses to visual arrays of different numerosities and spatial configurations. The EEG signal was recorded while 22 participants performed a numerosity estimation task on arrays with numerosities in the subitizing (3 or 4) or estimation (6 or 8) ranges. In the latter case, items could be spatially arranged into subgroups (3 or 4) or randomly scattered. In both ranges, we observed a decrease in N1 peak latency as the number of items increased. Importantly, when items were arranged to form subgroups, we showed that the N1 peak latency reflected both changes in total numerosity and changes in the number of subgroups. However, this result was mainly driven by the number of subgroups to suggest that clustered elements might trigger the recruitment of the subitizing system at a relatively early stage. At a later stage, we found that P2p was mostly modulated by the total numerosity in the set, with much less sensitivity for the number of subgroups these might be segregated in. Overall, this experiment suggests that the N1 component is sensitive to both local and global parcelling of elements in a scene suggesting that it could be crucially involved in the emergence of the groupitizing advantage. On the other hand, the later P2p component seems to be much more bounded to the global aspects of the scene coding the total number of elements while being mostly blind to the number of subgroups in which elements are parsed.

The role of neural tuning in quantity perception

Perception of quantities, such as numerosity, timing, and size, is essential for behavior and cognition. Accumulating evidence demonstrates neurons processing quantities are tuned, that is, have a preferred quantity amount, not only for numerosity, but also other quantity dimensions and sensory modalities. We argue that quantity-tuned neurons are fundamental to understanding quantity perception. We illustrate how the properties of quantity-tuned neurons can underlie a range of perceptual phenomena. Furthermore, quantity-tuned neurons are organized in distinct but overlapping topographic maps. We suggest that this overlap in tuning provides the neural basis for perceptual interactions between different quantities, without the need for a common neural representational code.

Effect of Non-canonical Spatial Symmetry on Subitizing

Subitizing refers to ability of people to accurately and effortlessly enumerate a small number of items, with a capacity around four elements. Previous research showed that "canonical" organizations, such as familiar layouts on a dice, can readily improve subitizing performance of people. However, almost all canonical shapes found in the world are also highly symmetrical; therefore, it is unclear whether previously reported facilitative effect of canonical organization is really due to canonicality, or simply driven by spatial symmetry. Here, we investigated the possible effect of symmetry on subitizing by using symmetrical, yet non-canonical, shape structures. These symmetrical layouts were compared with highly controlled random patterns (Experiment 1), as well as fully random and canonical patterns (Experiment 2). Our results showed that symmetry facilitates subitizing performance, but only at set size of 6, suggesting that the effect is insufficient to improve performance of people in the lower or upper range. This was also true, although weaker, in reaction time (RT), error distance measures, and Weber Fractions. On the other hand, canonical layouts produced faster and more accurate subitizing performances across multiple set sizes. We conclude that, although previous findings mixed symmetry in their canonical shapes, their findings on shape canonicality cannot be explained by symmetry alone. We also propose that our symmetrical and canonical results are best explained by the "groupitizing" and pattern recognition accounts, respectively.

Decoding location-specific and location-invariant stages of numerosity processing in subitizing

Extracting the number of objects in perceived scenes is a fundamental cognitive ability. Number processing is proposed to rely on two consecutive stages: an early object location map that captures individuated objects in a location‐specific way and a subsequent location‐invariant representation that captures numerosity at an abstract level. However, it is unclear whether this framework applies to small numerosities that can be individuated at once (“subitized”). Here, we reanalyzed data from two electroencephalography (EEG) experiments using multivariate pattern decoding to identify location‐specific and location‐invariant stages of numerosity processing in the subitizing range. In these experiments, one to three targets were presented in the left or right hemifield, which allowed for decoding target numerosity within each hemifield separately (location specific) or across hemifields (location invariant). Experiment 1 indicated the presence of a location‐specific stage (180–200 ms after stimulus), followed by a location‐invariant stage (300 ms after stimulus). A time‐by‐channel searchlight analysis revealed that the early location‐specific stage is most evident at occipital channels, whereas the late location‐invariant stage is most evident at parietal channels. Experiment 2 showed that both location‐specific and location‐invariant components are engaged only during tasks that explicitly require numerosity processing, ruling out automatic, and passive recording of numerosity. These results suggest that numerosity coding in subitizing is strongly grounded on an attention‐based, location‐specific stage. This stage overlaps with the subsequent activation of a location‐invariant stage, where a full representation of numerosity is finalized. Taken together, our findings provide clear evidence for a temporal and spatial segregation of location‐specific and location‐invariant numerosity coding of small object numerosities.

Groupitizing Improves Estimation of Numerosity of Auditory Sequences

Groupitizing is a recently described phenomenon of numerosity perception where clustering items of a set into smaller "subitizable" groups improves discrimination. Groupitizing is thought to be rooted on the subitizing system, with which it shares several properties: both phenomena accelerate counting and decrease estimation thresholds irrespective of stimulus format (for both simultaneous and sequential numerosity perception) and both rely on attention. As previous research on groupitizing has been almost completely limited to vision, the current study investigates whether it generalizes to other sensory modalities. Participants estimated the numerosity of a series of tones clustered either by proximity in time or by similarity in frequency. We found that compared with unstructured tone sequences, grouping lowered auditory estimation thresholds by up to 20%. The groupitizing advantage was similar across different grouping conditions, temporal proximity and tone frequency similarity. These results mirror the groupitizing effect for visual stimuli, suggesting that, like subitizing, groupitizing is an a-modal phenomenon.

Modal-independent Pattern Recognition Deficit in Developmental Dyscalculia Adults: Evidence from Tactile and Visual Enumeration

Developmental dyscalculia (DD) is characterized by lower numerical and finger-related skills. Studies of enumeration among those DD that suggested core deficiency in pattern recognition, working memory or/and attention were mostly carried out in the visual modality. In our study, we examined visual (dots) enumeration of 1-10 stimuli and tactile (vibration) enumeration of 1-10 fingers among DD and matched-control adults. We used 800-ms stimuli exposure time of either random/non-neighboring or canonical/neighboring stimuli arrangements (visual/tactile). Compared to controls, those with DD responded faster in visual random enumeration and did not differ in reaction time (RT) of canonical stimuli arrangements. However, while the control group had near perfect accuracy in random stimuli arrangements of up to five stimuli, DD participants performed accurately for only up to four stimuli, and they were less accurate in the canonical stimuli arrangements in the counting range. In the tactile task, DD participants showed less accurate tactile enumeration only for neighboring arrangements, more profoundly for finger counting (FC) patterns. The longer exposure time in the visual task enabled us to explore pattern recognition effects when working memory and attention loads were low. We discuss possible modal-independent deficits in pattern recognition and working memory on enumeration performance among those with DD and the unique role of fingers in ordinal and cardinal representation of numbers.

The relation between subitizable symbolic and non-symbolic number processing over the kindergarten school year

A long-standing debate in the field of numerical cognition concerns the degree to which symbolic and non-symbolic processing are related over the course of development. Of particular interest is the possibility that this link depends on the range of quantities in question. Behavioral and neuroimaging research with adults suggests that symbolic and non-symbolic quantities may be processed more similarly within, relative to outside of, the subitizing range. However, it remains unclear whether this unique link exists in young children at the outset of formal education. Further, no study has yet taken numerical size into account when investigating the longitudinal influence of these skills. To address these questions, we investigated the relation between symbolic and non-symbolic processing inside versus outside the subitizing range, both cross-sectionally and longitudinally, in 540 kindergarteners. Cross-sectionally, we found a consistently stronger relation between symbolic and non-symbolic number processing within versus outside the subitizing range at both the beginning and end of kindergarten. We also show evidence for a bidirectional relation over the course of kindergarten between formats within the subitizing range, and a unidirectional relation (symbolic → non-symbolic) for quantities outside of the subitizing range. These findings extend current theories on symbolic and non-symbolic magnitude development by suggesting that non-symbolic processing may in fact play a role in the development of symbolic number abilities, but that this influence may be limited to quantities within the subitizing range.

Differences in Counting Skills Between Chinese and German Children Are Accompanied by Differences in Processing of Approximate Numerical Magnitude Information

Human beings are supposed to possess an approximate number system (ANS) dedicated to extracting and representing approximate numerical magnitude information as well as an object tracking system (OTS) for the rapid and accurate enumeration of small sets. It is assumed that the OTS and the ANS independently contribute to the acquisition of more elaborate numerical concepts. Chinese children have been shown to exhibit more elaborate numerical concepts than their non-Chinese peers, but it is still an open question whether similar cross-national differences exist with regard to the underlying systems, namely the ANS and the OTS. In the present study, we investigated this question by comparing Chinese and German preschool children with regard to their performance in a non-symbolic numerical magnitude comparison task (assessing the ANS) and in an enumeration task (assessing the OTS). In addition, we compared children’s counting skills. To ensure that possible between-group differences could not be explained by differences in more general performance factors, we also assessed children’s reasoning ability and processing speed. Chinese children showed a better counting performance and a more accurate performance in the non-symbolic numerical magnitude comparison task. These differences in performance could not be ascribed to differences in reasoning abilities and processing speed. In contrast, Chinese and German children did not differ significantly in the enumeration of small sets. The superior counting performance of Chinese children was thus found to be reflected in the ANS but not in the OTS.

Comparative Judgment of Familiar Objects Is Modulated by Their Size

Perceptual decisions such as that we have more strawberries than apples left in our fruit basket seem to be made effortlessly. However, it is not examined yet whether such decisions are also biased by the size of the objects, just like numerosity comparisons with meaningless dot arrays. We presented two homogeneous sets of larger and smaller fruits (e.g., three apples and four strawberries), and participants had to indicate which set was more numerous. Although accuracy was nearly perfect, a strong congruency effect was found in reaction times, showing it is more difficult to compare the numerosities of sets of 2 apples and 3 strawberries than the opposite, that is, 3 apples and 2 strawberries. Because the stimuli were selected to simulate everyday conditions as much as possible, the present results suggest that most likely also comparative numerosity judgment in daily life is biased by nonnumerical cues such as size of the objects.

Cognitive access to numbers: the philosophical significance of empirical findings about basic number abilities

How can we acquire a grasp of cardinal numbers, even the first very small positive cardinal numbers, given that they are abstract mathematical entities? That problem of cognitive access is the main focus of this paper. All the major rival views about the nature and existence of cardinal numbers face difficulties; and the view most consonant with our normal thought and talk about numbers, the view that cardinal numbers are sizes of sets, runs into the cognitive access problem. The source of the problem is the plausible assumption that cognitive access to something requires causal contact with it. It is argued that this assumption is in fact wrong, and that in this and similar cases, we should accept that a certain recognize-and-distinguish capacity is sufficient for cognitive access. We can then go on to solve the cognitive access problem, and thereby support the set-size view of cardinal numbers, by paying attention to empirical findings about basic number abilities. To this end, some selected studies of infants, pre-school children and a trained chimpanzee are briefly discussed.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

Psychophysical evidence for the number sense

It is now clear that most animals, including humans, possess an ability to rapidly estimate number. Some have questioned whether this ability arises from dedicated numerosity mechanisms, or is derived indirectly from judgements of density or other attributes. We describe a series of psychophysical experiments, largely using adaptation techniques, which demonstrate clearly the existence of a number sense in humans. The number sense is truly general, extending over space, time and sensory modality, and is closely linked with action. We further show that when multiple cues are present, numerosity emerges as the natural dimension for discrimination. However, when element density increases past a certain level, the elements become too crowded to parse, and the scene is perceived as a texture rather than array of elements. The two different regimes are psychophysically discriminable in that they follow distinct psychophysical laws, and show different dependencies on eccentricity, luminance levels and effects of perceptual grouping. The distinction is important, as the ability to discriminate numerosity, but not texture, correlates with formal maths skills.

This article is part of the discussion meeting issue ‘The origins of numerical abilities’.

Spatial Arrangement and Set Size Influence the Coding of Non-symbolic Quantities in the Intraparietal Sulcus

Performance in visual quantification tasks shows two characteristic patterns as a function of set size. A precise subitizing process for small sets (up to four) was contrasted with an approximate estimation process for larger sets. The spatial arrangement of elements in a set also influences visual quantification performance, with frequently perceived arrangements (e.g., dice patterns) being faster enumerated than random arrangements. Neuropsychological and imaging studies identified the intraparietal sulcus (IPS), as key brain area for quantification, both within and above the subitizing range. However, it is not yet clear if and how set size and spatial arrangement of elements in a set modulate IPS activity during quantification. In an fMRI study, participants enumerated briefly presented dot patterns with random, canonical or dice arrangement within and above the subitizing range. We evaluated how activity amplitude and pattern in the IPS were influenced by size and spatial arrangement of a set. We found a discontinuity in the amplitude of IPS response between subitizing and estimation range, with steep activity increase for sets exceeding four elements. In the estimation range, random dot arrangements elicited stronger IPS response than canonical arrangements which in turn elicited stronger response than dice arrangements. Furthermore, IPS activity patterns differed systematically between arrangements. We found a signature in the IPS response for a transition between subitizing and estimation processes during quantification. Differences in amplitude and pattern of IPS activity for different spatial arrangements indicated a more precise representation of non-symbolic numerical magnitude for dice and canonical than for random arrangements. These findings challenge the idea of an abstract coding of numerosity in the IPS even within a single notation.

Simultanagnosia and object individuation

Simultanagnosic patients have difficulty in perceiving multiple objects when presented simultaneously. In this review article, I discuss how neuropsychological research on simultanagnosia has been inspirational for two interconnected lines of research related to the core mechanisms by which the visual system processes cluttered scenes. First, I review previous studies on enumeration tasks indicating that, despite their inability to identify multiple objects, simultanagnosic patients can enumerate up to 2-3 elements as efficiently as healthy individuals (the so-called "subitizing" phenomenon). This intriguing observation is one of the first results to support the existence of an "object individuation" mechanism that can spatially tag a limited set of objects simultaneously, and resonates with recent research on the brain dynamics of enumeration in healthy individuals. Second, I further develop the implications of the dissociation between object identification and object enumeration in simultanagnosia specifically for the distinction between object identification and individuation. The latter distinction has been the subject of recent neuroimaging research that has provided fine-grained information on the spatial as well as temporal aspects of object individuation and recognition. The lessons learned from neuropsychological research on exact enumeration in simultanagnosia can be generalized to the normal functioning of the human mind, and have provided insightful clues for cognitive neuroscience.

Possible Associations between Subitizing, Estimation and Visuospatial Working Memory (VSWM) in Children

Researchers have focused on identifying the mechanisms involved in subitizing and its differences with estimation. Some suggest that subitizing relies on a visual indexing system in charge of the simultaneous individuation of objects that is also used by visuospatial working memory (VSWM). In adults, studies found associations between subitizing and VSWM, in the absence of correlation between VSWM and estimation. The present study analyzed the performance of 120 4 and 6-year-old children in three tasks: dot enumeration to measure subitizing capacity, quantity discrimination for estimation, and Corsi Block-tapping task for VSWM. In the enumeration task RTs (F(9, 1062)=720.59, MSE=734394, p<.001, η2=.86) and errors (F(9, 1062)=42.15, MSE=.194, p<.001, η2=.26.) increased with the array, but this growth was statistically significant only from 4 dots onward. Each subject's subitizing range was estimated by fitting RTs with a sigmoid function of number of dots and obtaining the bend point of the curve. Data fit (age 4: R 2 = .88; SD = .08; age 6: R 2 = .91, SD = .08) showed a mean subitizing range of 2.79 (SD = .66) for 4 year-olds and of 3.11 (SD = .64) for 6 year-olds. Subitizing ranges and average RTs showed low association with storage (r = .274; p < .05; r = -.398; p < .001) and average RTs with concurrent processing (r = -.412; p < .001) in VSWM. Subitizing range and speed showed no association with estimation speed and a poor association with accuracy (r = .234, p < .01; r = -.398, p < .001), which suggests independent systems for small and large quantities. Subitizing and estimation measures correlated with VSWM (p < .01), which suggests that both processes may require VSWM resources.

Distinct Neural Signatures for Very Small and Very Large Numerosities

Behavioral studies of numerical cognition have shown that perceptual threshold for numerosity discrimination depends on the range of numerical values to be estimated. Discrimination threshold is constant when comparing very small numerosities via the mechanism called subitizing, while it increases as a function of numerosity for numbers beyond that range governed by subitizing. However, when numerosity gets so large that the individual elements start to form a cluttered ensemble, discrimination threshold increases as a function of the square root of numerosity. These behavioral patterns suggest that our sense of number is not based on a unitary mechanism and is rather based on multiple numerosity processing mechanisms depending on the absolute numerosity to be estimated. In this study, we demonstrate neurophysiological evidence for such multiple mechanisms. Participants’ electroencephalogram (EEG) was recorded while they viewed arrays containing either very small (1–4) or very large (100–400) number of dots with systematic variations in non-numerical cues. A linear model that tested the effects of numerical and non-numerical cues on the visual-evoked potentials (VEPs) revealed strong neural sensitivity to numerosity around 160–180 ms over right occipito-parietal sites irrespective of the numerical range presented. In contrast, earlier neural responses (~100 ms) showed markedly distinct patterns across the different numerical ranges tested. These results indicate that differences in behavioral response patterns in numerosity estimation across various numerical ranges may arise from the differences in the first stages of visual analysis. Collectively, the findings provide a firmer ground for the idea that there exists a brain system specifically dedicated for numerosity processing, yet they also suggest that multiple early visual cortical mechanisms converge to that numerosity processing stage later in the visual stream.

Dyscalculia and the Calculating Brain

Dyscalculia, like dyslexia, affects some 5% of school-age children but has received much less investigative attention. In two thirds of affected children, dyscalculia is associated with another developmental disorder like dyslexia, attention-deficit disorder, anxiety disorder, visual and spatial disorder, or cultural deprivation. Infants, primates, some birds, and other animals are born with the innate ability, called subitizing, to tell at a glance whether small sets of scattered dots or other items differ by one or more item. This nonverbal approximate number system extends mostly to single digit sets as visual discrimination drops logarithmically to "many" with increasing numerosity (size effect) and crowding (distance effect). Preschoolers need several years and specific teaching to learn verbal names and visual symbols for numbers and school agers to understand their cardinality and ordinality and the invariance of their sequence (arithmetic number line) that enables calculation. This arithmetic linear line differs drastically from the nonlinear approximate number system mental number line that parallels the individual number-tuned neurons in the intraparietal sulcus in monkeys and overlying scalp distribution of discrete functional magnetic resonance imaging activations by number tasks in man. Calculation is a complex skill that activates both visual and spatial and visual and verbal networks. It is less strongly left lateralized than language, with approximate number system activation somewhat more right sided and exact number and arithmetic activation more left sided. Maturation and increasing number skill decrease associated widespread non-numerical brain activations that persist in some individuals with dyscalculia, which has no single, universal neurological cause or underlying mechanism in all affected individuals.

Effects of Numerosity Range on Tactile and Visual Enumeration

Our study explores tactile enumeration using both hands and investigates the effects of numerosity range's (NR) on general enumeration. In Experiment 1, using custom-made vibro-tactile apparatus, we replicated results of Cohen, Naparstek, and Henik (2014, Acta Psychologica, 150C, 26-34) and again found a moderate increase in RT up to four stimuli and then a decrease for five stimuli. In Experiment 2, we used a within participants design and compared NR 1 to 5 and 1 to 10 in tactile and visual enumeration. The results showed that enumeration for NR 5 to 1 was faster than for NR 1 to 10, especially for numerosities four and five. Within NR 1 to 10, in the visual modality the subitizing range was 4, the counting range was from 5 to 9, and there was an end effect of 10 dots. In the tactile modality, when excluding one-hand arrangements, the subitizing range was 2, the counting range was from 3 to 5, there was an acceleration of counting from 5 and on, and there was an end effect for 10 stimuli that was stronger than for 10 visual stimuli. We suggest that NR influences enumeration and that number-hand association (i.e. resulting from finger counting) influences enumeration, resulting in faster counting.

Ratio dependence in small number discrimination is affected by the experimental procedure

Adults, infants and some non-human animals share an approximate number system (ANS) to estimate numerical quantities, and are supposed to share a second, 'object-tracking,' system (OTS) that supports the precise representation of a small number of items (up to 3 or 4). In relative numerosity judgments, accuracy depends on the ratio of the two numerosities (Weber's Law) for numerosities >4 (the typical ANS range), while for numerosities ≤4 (OTS range) there is usually no ratio effect. However, recent studies have found evidence for ratio effects for small numerosities, challenging the idea that the OTS might be involved for small number discrimination. Here we tested the hypothesis that the lack of ratio effect in the numbers 1-4 is largely dependent on the type of stimulus presentation. We investigated relative numerosity judgments in college students using three different procedures: a simultaneous presentation of intermingled and separate groups of dots in separate experiments, and a further experiment with sequential presentation. As predicted, in the large number range, ratio dependence was observed in all tasks. By contrast, in the small number range, ratio insensitivity was found in one task (sequential presentation). In a fourth experiment, we showed that the presence of intermingled distractors elicited a ratio effect, while easily distinguishable distractors did not. As the different ratio sensitivity for small and large numbers has been often interpreted in terms of the activation of the OTS and ANS, our results suggest that numbers 1-4 may be represented by both numerical systems and that the experimental context, such as the presence/absence of task-irrelevant items in the visual field, would determine which system is activated.

Number As a Primary Perceptual Attribute: A Review

Although humans are the only species to possess language-driven abstract mathematical capacities, we share with many other animals a nonverbal capacity for estimating quantities or numerosity. For some time, researchers have clearly differentiated between small numbers of items--less than about four--referred to as the subitizing range, and larger numbers, where counting or estimation is required. In this review, we examine more recent evidence suggesting a further division, between sets of items greater than the subitizing range, but sparse enough to be individuated as single items; and densely packed stimuli, where they crowd each other into what is better considered as a texture. These two different regimes are psychophysically discriminable in that they follow distinct psychophysical laws and show different dependencies on eccentricity and on luminance levels. But provided the elements are not too crowded (less than about two items per square degree in central vision, less in the periphery), there is little evidence that estimation of numerosity depends on mechanisms responsive to texture. The distinction is important, as the ability to discriminate numerosity, but not texture, correlates with formal maths skills.

Abnormalities in brain systems supporting individuation and enumeration in autism

Previous work indicates that adults with autism display a decreased capacity when rapidly enumerating small sets of elements (i.e., subitizing), compared to typically developing (TD) individuals. This ability is crucial for fundamental visual functions such as object individuation and parallel processing. Thus, the deficit in autism suggests limits in these skills. To examine the neural basis of this limitation, adults with and without high functioning autism rapidly enumerated 1 to 8 randomly located squares during a neuroimaging study. Typically, adults are thought to use parallel visual processes to quantify up to three or four elements, and serial processes to enumerate more (5+) elements. We hypothesized that parietal lobe regions associated with counting would be recruited with smaller sets of elements in adults with autism, compared to TD adults. Consistent with this hypothesis, activation in parietal regions increased with smaller set sizes in adults with autism compared to TD adults. Increased activation for three elements was evident in several regions, including those thought to underlie subitizing. In addition, regions specific to the counting range in TD adults were often equally active for set sizes in the subitizing range in the adults with autism. Finally, significant deactivation was evident in TD adults, presumably reflecting relative suppression of regions specialized for competing processes, but was not apparent in adults with autism. These differences in brain function in adults with autism on a simple enumeration task suggest atypical brain organization and function that is likely to impact most visual tasks, especially those with multiple elements.

Multiple object individuation and subitizing in enumeration: a view from electrophysiology

What are the processes involved in determining that there are exactly n objects in the visual field? The core level of representation for this process is based on a mechanism that iteratively individuates each of the set of relevant objects for exact enumeration. In support of this proposal, we review recent electrophysiological findings on enumeration-at-a-glance and consider three temporally distinct responses of the EEG signal that are modulated by object numerosity, and which have been associated respectively with perceptual modulation, attention selection, and working memory. We argue that the neural response associated with attention selection shows the hallmarks of an object individuation mechanism, including the property of simultaneous individuation of a limited number of objects thought to underlie the behavioral subitizing effect. The findings support the view that the core component of exact enumeration is an attention-based individuation mechanism that binds specific features to locations and provides a stable representation of a limited set of relevant objects. The resulting representation is made available for further cognitive operations for exact enumeration.

The limit of mental structures

The "magical" number of 4 has been demonstrated to limit much of human information processing. The relevant evidence is briefly reviewed. It is proposed that the organization of processing structures is based on interconnected bidirectional pairs, with every element paired with every other one. The limit arises because of the large increase in links among elements required beyond structures of size 3 and 4.

The role of pattern recognition in children's exact enumeration of small numbers

Enumeration can be accomplished by subitizing, counting, estimation, and combinations of these processes. We investigated whether the dissociation between subitizing and counting can be observed in 4- to 6-year-olds and studied whether the maximum number of elements that can be subitized changes with age. To detect a dissociation between subitizing and counting, it is tested whether task manipulations have different effects in the subitizing than in the counting range. Task manipulations concerned duration of presentation of elements (limited, unlimited) and configuration of elements (random, line, dice). In Study 1, forty-nine 4- and 5-year-olds were tested with a computerized enumeration task. Study 2 concerned data from 4-, 5-, and 6-year-olds, collected with Math Garden, a computer-adaptive application to practice math. Both task manipulations affected performance in the counting, but not the subitizing range, supporting the conclusion that children use two distinct enumeration processes in the two ranges. In all age groups, the maximum number of elements that could be subitized was three. The strong effect of configuration of elements suggests that subitizing might be based on a general ability of pattern recognition.

Individual differences in non-symbolic numerical abilities predict mathematical achievements but contradict ATOM

BACKGROUND

A significant debate surrounds the nature of the cognitive mechanisms involved in non-symbolic number estimation. Several studies have suggested the existence of the same cognitive system for estimation of time, space, and number, called "a theory of magnitude" (ATOM). In addition, researchers have proposed the theory that non-symbolic number abilities might support our mathematical skills. Despite the large number of studies carried out, no firm conclusions can be drawn on either topic.

METHODS

In the present study, we correlated the performance of adults on non-symbolic magnitude estimations and symbolic numerical tasks. Non-symbolic magnitude abilities were assessed by asking participants to estimate which auditory tone lasted longer (time), which line was longer (space), and which group of dots was more numerous (number). To assess symbolic numerical abilities, participants were required to perform mental calculations and mathematical reasoning.

RESULTS

We found a positive correlation between non-symbolic and symbolic numerical abilities. On the other hand, no correlation was found among non-symbolic estimations of time, space, and number.

CONCLUSIONS

Our study supports the idea that mathematical abilities rely on rudimentary numerical skills that predate verbal language. By contrast, the lack of correlation among non-symbolic estimations of time, space, and number is incompatible with the idea that these magnitudes are entirely processed by the same cognitive system.

The magical number four: A biological, historical and mythological enigma

Precise recognition of small object numbers without counting is a widespread phenomenon. It is well documented for humans and for a series of non-human vertebrates. Recently this has been confirmed for an invertebrate, the honeybee.(1) This type of inborn numerical competence has been named "subitizing", from the Latin subito = suddenly, immediately. It differs from the classical, sequential counting which has to be trained, starting with the help of our fingers. For humans it had been established since 1871 by Jevons(2) that only up to four objects are precisely recognized and memorized. Under conditions which do not allow sequential counting, mistakes start to occur in case of more than four objects. This result has been confirmed whenever the range of visual attention has been carefully tested under a variety of rigorous conditions. It provides the basis for a novel hypothesis about the evolution of counting and numbering systems in ancient civilizations.(3)  

To bee or not to bee, this is the question…: The inborn numerical competence of humans and honeybees

Human inborn numerical competence means our ability to recognize object numbers precisely under circumstances which do not allow sequential counting. This archaic process has been called "subitizing," from the Latin "subito" = suddenly, immediately, indicating that the objects in question are presented to test persons only for a fraction of a second in order to prevent counting. In contrast, however, sequential counting, an outstanding cultural achievement of mankind, means to count "1, 2, 3, 4, 5, 6, 7, 8…" without a limit. The following essay will explain how the limit of numerical competence, i.e., the recognition of object numbers without counting, has been determined for humans and how this has been achieved for the first time in case of an invertebrate, the honeybee. Finally, a hypothesis explaining the influence of our limited, inborn numerical competence on counting in our times, e.g., in the Russian language, will be presented. Subitizing versus counting by young Down syndrome infants and autistics and the Savant syndrome will be discussed.

Give me 5…The invention of number five in ancient civilizations: A consequence of our limited inborn numerical competence

Precise recognition of small numbers of objects without counting is an archaic, inborn ability of humans. Since almost 140 years it is known that we can recognize precisely only up to four objects if sequential counting is prevented. Vertebrates and invertebrates such as honeybees can recognize and remember three and up to four objects, respectively. A synopsis of counting systems in ancient civilizations reveals that our limited ability to recognize only four objects without counting influenced our counting and numbering systems and enforced the need for new symbols for numbers beyond four.

The effect of feature saliency on haptic subitizing

'Subitizing' refers to fast and error-free numerosity judgment for small (<4) sets of items. For larger sets, the slower process of 'counting' is used. Counting has a serial character, whereas subitizing is believed to have a parallel character. While subitizing was initially found in vision, it has been shown to exist in touch as well. In vision, it has been demonstrated that adding distractor items to a set of target items influences numerosity judgment of the target items. Subitizing was in this case only possible if the distractor item is highly salient among the targets. In the present study, we investigated the effect of adding a distractor item on haptic judgement of a set of target items. To this end, we asked subjects to judge the number of spheres grasped in their hand. Either a cube or an ellipsoid could be added to the set. A cube among spheres has been shown to be highly salient, while an ellipsoid among spheres is not. Our results show that adding a distractor item led to an increase in the response time slopes regardless of the distractor shape. Subitizing was, however, only possible in the case of a salient distractor. This is in agreement with results from vision.

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.

Too much to count on: impaired very small numbers in corticobasal degeneration

Patients with corticobasal degeneration (CBD) have calculation impairments. This study examined whether impaired number knowledge depends on verbal mediation. We focused particularly on knowledge of very small numbers, where there is a precise relationship between a cardinality and its number concept, but little hypothesized role for verbal mediation. We evaluated accuracy and reaction time (RT) for matching dot arrays and Arabic numerals involving smaller (2-4) and larger (5-9) cardinalities in non-aphasic patients with CBD (n=16), frontotemporal dementia (FTD; n=23), and healthy controls (n=15). CBD were less accurate and slowed at judging smaller Arabic numeral-dot array stimuli compared to FTD patients and controls. Moreover, only CBD showed longer RTs judging successively larger number-dot array pairs among the smaller cardinalities. Difficulty judging very small numbers is impaired in CBD, suggesting degraded representation of precise number knowledge that does not depend on language functioning.