Numerosity but not texture-density discrimination correlates with math ability in children

Auditory time perception impairment in children with developmental dyscalculia

Developmental dyscalculia (DD) is a specific learning disability which prevents children from acquiring adequate numerical and arithmetical competences. We investigated whether difficulties in children with DD spread beyond the numerical domain and impact also their ability to perceive time. A group of 37 children/adolescent with and without DD were tested with an auditory categorization task measuring time perception thresholds in the sub-second (0.25-1 s) and supra-second (0.75-3 s) ranges. Results showed that auditory time perception was strongly impaired in children with DD at both time scales. The impairment remained even when age, non-verbal reasoning, and gender were regressed out. Overall, our results show that the difficulties of DD can affect magnitudes other than numerical and contribute to the increasing evidence that frames dyscalculia as a disorder affecting multiple neurocognitive and perceptual systems.

Inter-individual, hemispheric and sex variability of brain activations during numerosity processing

Numerosity perception is a fundamental and innate cognitive function shared by both humans and many animal species. Previous research has primarily focused on exploring the spatial and functional consistency of neural activations that were associated with the processing of numerosity information. However, the inter-individual variability of brain activations of numerosity perception remains unclear. In the present study, with a large-sample functional magnetic resonance imaging (fMRI) dataset (n = 460), we aimed to localize the functional regions related to numerosity perceptions and explore the inter-individual, hemispheric, and sex differences within these brain regions. Fifteen subject-specific activated regions, including the anterior intraparietal sulcus (aIPS), posterior intraparietal sulcus (pIPS), insula, inferior frontal gyrus (IFG), inferior temporal gyrus (ITG), premotor area (PM), middle occipital gyrus (MOG) and anterior cingulate cortex (ACC), were delineated in each individual and then used to create a functional probabilistic atlas to quantify individual variability in brain activations of numerosity processing. Though the activation percentages of most regions were higher than 60%, the intersections of most regions across individuals were considerably lower, falling below 50%, indicating substantial variations in brain activations related to numerosity processing among individuals. Furthermore, significant hemispheric and sex differences in activation location, extent, and magnitude were also found in these regions. Most activated regions in the right hemisphere had larger activation volumes and activation magnitudes, and were located more lateral and anterior than their counterparts in the left hemisphere. In addition, in most of these regions, males displayed stronger activations than females. Our findings demonstrate large inter-individual, hemispheric, and sex differences in brain activations related to numerosity processing, and our probabilistic atlas can serve as a robust functional and spatial reference for mapping the numerosity-related neural networks.

Neurocognitive Assessment of Mathematics-Related Capacities in Neurosurgical Patients

A precise neuropsychological assessment is of the utmost importance for neurosurgical patients undergoing the surgical excision of cerebral lesions. The assessment of mathematical abilities is usually limited to arithmetical operations while other fundamental visuo-spatial aspects closely linked to mathematics proficiency, such as the perception of numerical quantities and geometrical reasoning, are completely neglected. We evaluated these abilities with two objective and reproducible psychophysical tests, measuring numerosity perception and non-symbolic geometry, respectively. We tested sixteen neuro-oncological patients before the operation and six after the operation with classical neuropsychological tests and with two psychophysical tests. The scores of the classical neuropsychological tests were very heterogeneous, possibly due to the distinct location and histology of the tumors that might have spared (or not) brain areas subserving these abilities or allowed for plastic reorganization. Performance in the two non-symbolic tests reflected, on average, the presumed functional role of the lesioned areas, with participants with parietal and frontal lesions performing worse on these tests than patients with occipital and temporal lesions. Single-case analyses not only revealed some interesting exceptions to the group-level results (e.g., patients with parietal lesions performing well in the numerosity test), but also indicated that performance in the two tests was independent of non-verbal reasoning and visuo-spatial working memory. Our results highlight the importance of assessing non-symbolic numerical and geometrical abilities to complement typical neuropsychological batteries. However, they also suggest an avoidance of reliance on an excessively rigid localizationist approach when evaluating the neuropsychological profile of oncological patients.

Omit needless words: Sentence length perception

Short sentences improve readability. Short sentences also promote social justice through accessibility and inclusiveness. Despite this, much remains unknown about sentence length perception-an important factor in producing readable writing. Accordingly, we conducted a psychophysical study using procedures from Signal Detection Theory to examine sentence length perception in naive adults. Participants viewed real-world full-page text samples and judged whether a bolded target sentence contained more or fewer than 17 words. The experiment yielded four findings. First, naïve adults perceived sentence length in real-world text samples quickly (median = 300-400 ms) and precisely (median = ~90% correct). Second, flipping real-world text samples upside-down generated no reaction-time cost and nearly no loss in the precision of sentence length perception. This differs from the large inversion effects that characterize other highly practiced, real-world perceptual tasks involving canonically oriented stimuli, most notably face perception and reading. Third, participants significantly underestimated the length of mirror-reversed sentences-but not upside-down, nor standard sentences. This finding parallels participants' familiarity with commonly occurring left-justified right-ragged text, and suggests a novel demonstration of left-lateralized anchoring in scene syntax. Fourth, error patterns demonstrated that participants achieved their high speed, high precision sentence-length judgments by heuristically counting text lines, not by explicitly counting words. This suggests practical advice for writing instructors to offer students. When copy editing, students can quickly and precisely identify their long sentences via a line-counting heuristic, e.g., "a 17-word sentence spans about 1.5 text lines". Students can subsequently improve a long sentence's readability and inclusiveness by omitting needless words.

Quantum spin models for numerosity perception

Humans share with animals, both vertebrates and invertebrates, the capacity to sense the number of items in their environment already at birth. The pervasiveness of this skill across the animal kingdom suggests that it should emerge in very simple populations of neurons. Current modelling literature, however, has struggled to provide a simple architecture carrying out this task, with most proposals suggesting the emergence of number sense in multi-layered complex neural networks, and typically requiring supervised learning; while simple accumulator models fail to predict Weber's Law, a common trait of human and animal numerosity processing. We present a simple quantum spin model with all-to-all connectivity, where numerosity is encoded in the spectrum after stimulation with a number of transient signals occurring in a random or orderly temporal sequence. We use a paradigmatic simulational approach borrowed from the theory and methods of open quantum systems out of equilibrium, as a possible way to describe information processing in neural systems. Our method is able to capture many of the perceptual characteristics of numerosity in such systems. The frequency components of the magnetization spectra at harmonics of the system's tunneling frequency increase with the number of stimuli presented. The amplitude decoding of each spectrum, performed with an ideal-observer model, reveals that the system follows Weber's law. This contrasts with the well-known failure to reproduce Weber's law with linear system or accumulators models.

Math difficulties in attention deficit hyperactivity disorder do not originate from the visual number sense

There is ample evidence from literature and clinical practice indicating mathematical difficulties in individuals with ADHD, even when there is no concomitant diagnosis of developmental dyscalculia. What factors underlie these difficulties is still an open question. Research on dyscalculia and neurotypical development suggests visual perception of numerosity (the number sense) as a building block for math learning. Participants with lower numerosity estimation thresholds (higher precision) are often those with higher math capabilities. Strangely, the role of numerosity perception in math skills in ADHD has been neglected, leaving open the question whether math difficulties in ADHD also originate from a deficitary visual number sense. In the current study we psychophysically measured numerosity thresholds and accuracy in a sample of children/adolescents with ADHD, but not concomitant dyscalculia (N = 20, 8–16 years). Math abilities were also measured by tasks indexing different mathematical competences. Numerosity performance and math scores were then compared to those obtained from an age-matched control group (N = 20). Bayesian statistics indicated no difference between ADHD and controls on numerosity perception, despite many of the symbolic math tasks being impaired in participants with ADHD. Moreover, the math deficits showed by the group with ADHD remained substantial even when numerosity thresholds were statistically regressed out. Overall, these results indicate that math difficulties in ADHD are unlikely to originate from an impaired visual number sense.

Assessing the ecological validity of numerosity-selective neuronal populations with real-world natural scenes

Animals and humans are able to quickly and effortlessly estimate the number of items in a set: their numerosity. Numerosity perception is thought to be critical to behavior, from feeding to escaping predators to human mathematical cognition. Virtually, all scientific studies on numerosity mechanisms use well controlled but artificial stimuli to isolate the numerosity dimension from other physical quantities. Here, we probed the ecological validity of these artificial stimuli and evaluate whether an important component in numerosity processing, the numerosity-selective neural populations, also respond to numerosity of items in real-world natural scenes. Using 7T MRI and natural images from a wide range of categories, we provide evidence that the numerosity-tuned neuronal populations show numerosity-selective responses when viewing images from a real-world natural scene. Our findings strengthen the role of numerosity-selective neurons in numerosity perception and provide an important link to their function in numerosity perception in real-world settings.

Auditory time thresholds in the range of milliseconds but not seconds are impaired in ADHD

The literature on time perception in individuals with ADHD is extensive but inconsistent, probably reflecting the use of different tasks and performances indexes. A sample of 40 children/adolescents (20 with ADHD, 20 neurotypical) was engaged in two identical psychophysical tasks measuring auditory time thresholds in the milliseconds (0.25–1 s) and seconds (0.75–3 s) ranges. Results showed a severe impairment in ADHD for milliseconds thresholds (Log10BF = 1.9). The deficit remained strong even when non-verbal IQ was regressed out and correlation with age suggests a developmental delay. In the seconds range, thresholds were indistinguishable between the two groups (Log10BF = − 0.5) and not correlated with milliseconds thresholds. Our results largely confirm previous evidence suggesting partially separate mechanisms for time perception in the ranges of milliseconds and seconds. Moreover, since the evidence suggests that time perception of milliseconds stimuli might load relatively less on cognitive control and working memory, compared to longer durations, the current results are consistent with a pure timing deficit in individuals with ADHD.

The relationship between numerosity perception and mathematics ability in adults: the moderating role of dots number

Background

It has been proposed that numerosity perception is the cognitive underpinning of mathematics ability. However, the existence of the association between numerosity perception and mathematics ability is still under debate, especially in adults. The present study examined the relationship between numerosity perception and mathematics ability and the moderating role of dots number (i.e., the numerosity of items in dot set) in adults.

Methods

Sixty-four adult participants from Anshun University completed behavioral measures that tested numerosity perception of small numbers and large numbers, mathematics ability, inhibition ability, visual-spatial memory, and set-switching ability.

Results

We found that numerosity perception of small numbers correlated significantly with mathematics ability after controlling the influence of inhibition ability, visual-spatial memory, and set-switching ability, but numerosity perception of large numbers was not related to mathematics ability in adults.

Conclusions

These findings suggest that the dots number moderates the relationship between numerosity perception and mathematics ability in adults and may contribute to explaining the contradictory findings in the previous literature about the link between numerosity perception and mathematics ability.

Perception of geometric sequences and numerosity both predict formal geometric competence in primary school children

While most animals have a sense of number, only humans have developed symbolic systems to describe and organize mathematical knowledge. Some studies suggest that human arithmetical knowledge may be rooted in an ancient mechanism dedicated to perceiving numerosity, but it is not known if formal geometry also relies on basic, non-symbolic mechanisms. Here we show that primary-school children who spontaneously detect and predict geometrical sequences (non-symbolic geometry) perform better in school-based geometry tests indexing formal geometric knowledge. Interestingly, numerosity discrimination thresholds also predicted and explained a specific portion of variance of formal geometrical scores. The relation between these two non-symbolic systems and formal geometry was not explained by age or verbal reasoning skills. Overall, the results are in line with the hypothesis that some human-specific, symbolic systems are rooted in non-symbolic mechanisms.

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.

Numerosity adaptation partly depends on the allocation of implicit numerosity-contingent visuo-spatial attention

Like other perceptual attributes, numerosity is susceptible to adaptation. Nevertheless, it has never been fully investigated whether adaptation to numerosity is fully perceptual in nature or if it stems from the mixed influence of perception and attention. In the present work, we addressed this point throughout three separate experiments aiming at investigating the potential role played by visuo-spatial attentional mechanisms in shaping numerosity perception and adaptation. In Experiments 1 and 2, we showed that the magnitude of numerosity adaptation can be strongly influenced by the distribution of numerosity-contingent visuo-spatial attentional resources during the adaptation period. Results from Experiment 1 revealed a robust reduction of adaptation magnitude whenever a second numerical stimulus was presented in a diametrically opposite location from that of the adaptor, despite this second adapter being neutral as matched in numerosity with the following stimulus displayed in that location. In Experiment 2, we showed that this reduction in adaptation did not occur in cases where the second stimulus was not numerical, suggesting that attentional resources specifically related to numerosity information accounts for the results of Experiment 1. Finally, in Experiment 3, we showed that uninformative visuo-spatial cues shape numerosity discrimination judgments both at baseline and during adaptation. Taken together, our results seem to indicate that visuo-spatial attention plays a relevant role in numerosity perception and that adaptation to numerosity is actively influenced by this cognitive process.

Developmental Changes in ANS Precision Across Grades 1-9: Different Patterns of Accuracy and Reaction Time

The main aim of this study was to analyze the patterns of changes in Approximate Number Sense (ANS) precision from grade 1 (mean age: 7.84 years) to grade 9 (mean age: 15.82 years) in a sample of Russian schoolchildren. To fulfill this aim, the data from a longitudinal study of two cohorts of children were used. The first cohort was assessed at grades 1-5 (elementary school education plus the first year of secondary education), and the second cohort was assessed at grades 5-9 (secondary school education). ANS precision was assessed by accuracy and reaction time (RT) in a non-symbolic comparison test ("blue-yellow dots" test). The patterns of change were estimated via mixed-effect growth models. The results revealed that in the first cohort, the average accuracy increased from grade 1 to grade 5 following a non-linear pattern and that the rate of growth slowed after grade 3 (7-9 years old). The non-linear pattern of changes in the second cohort indicated that accuracy started to increase from grade 7 to grade 9 (13-15 years old), while there were no changes from grade 5 to grade 7. However, the RT in the non-symbolic comparison test decreased evenly from grade 1 to grade 7 (7-13 years old), and the rate of processing non-symbolic information tended to stabilize from grade 7 to grade 9. Moreover, the changes in the rate of processing non-symbolic information were not explained by the changes in general processing speed. The results also demonstrated that accuracy and RT were positively correlated across all grades. These results indicate that accuracy and the rate of non-symbolic processing reflect two different processes, namely, the maturation and development of a non-symbolic representation system.

Longitudinal Effects of the Home Learning Environment and Parental Difficulties on Reading and Math Development Across Grades 1-9

This study focuses on parental reading and mathematical difficulties, the home literacy environment, and the home numeracy environment as well as their predictive role in Finnish children’s reading and mathematical development through Grades 1–9. We examined if parental reading and mathematical difficulties directly predict children’s academic performance and/or if they are mediated by the home learning environment. Mothers (n = 1590) and fathers (n = 1507) reported on their reading and mathematical difficulties as well as on the home environment (shared reading, teaching literacy, and numeracy) when their children were in kindergarten. Tests for reading fluency, reading comprehension, and arithmetic fluency were administered to children in Grades 1, 2, 3, 4, 7, and 9. Parental reading difficulties predicted children’s reading fluency, whereas parental mathematical difficulties predicted their reading comprehension and arithmetic fluency. Familial risk was associated with neither formal nor informal home environment factors, whereas maternal education had a significant relationship with both, with higher levels of education among mothers predicting less time spent on teaching activities and more time spent on shared reading. In addition, shared reading was significantly associated with the development of reading comprehension up to Grades 3 and 4, whereas other components of the home learning environment were not associated with any assessed skills. Our study highlights that taken together, familial risk, parental education, and the home learning environment form a complex pattern of associations with children’s mathematical and reading skills.

Fast saccadic eye-movements in humans suggest that numerosity perception is automatic and direct

Fast saccades are rapid automatic oculomotor responses to salient and ecologically important visual stimuli such as animals and faces. Discriminating the number of friends, foe, or prey may also have an evolutionary advantage. In this study, participants were asked to saccade rapidly towards the more numerous of two arrays. Participants could discriminate numerosities with high accuracy and great speed, as fast as 190 ms. Intermediate numerosities were more likely to elicit fast saccades than very low or very high numerosities. Reaction-times for vocal responses (collected in a separate experiment) were slower, did not depend on numerical range, and correlated only with the slow not the fast saccades, pointing to different systems. The short saccadic reaction-times we observe are surprising given that discrimination using numerosity estimation is thought to require a relatively complex neural circuit, with several relays of information through the parietal and prefrontal cortex. Our results suggest that fast numerosity-driven saccades may be generated on a single feed-forward pass of information recruiting a primitive system that cuts through the cortical hierarchy and rapidly transforms the numerosity information into a saccade command.

"Groupitizing": a strategy for numerosity estimation

Previous work has shown that when arrays of objects are grouped within clusters, participants can enumerate their numerosity more rapidly than when objects are randomly scattered, a phenomenon termed "groupitizing". Importantly, the magnitude of the grouping advantage correlates with math abilities in children. Here we show that sensory precision of numerosity estimation is also improved when grouping cues are available, by up to 20%. The grouping can be induced by color and/or spatial proximity, and occurs in temporal sequences as well as spatial arrays. The improvement is strongest for participants with the highest thresholds in the random, ungrouped conditions. Taken together with previous research, our data suggest that measurements correlations between numerosity estimation and formal math skills may be driven by grouping strategies, which require a minimal level of basic arithmetic.

Excessive visual crowding effects in developmental dyscalculia

Visual crowding refers to the inability to identify objects when surrounded by other similar items. Crowding-like mechanisms are thought to play a key role in numerical perception by determining the sensory mechanisms through which ensembles are perceived. Enhanced visual crowding might hence prevent the normal development of a system involved in segregating and perceiving discrete numbers of items and ultimately the acquisition of more abstract numerical skills. Here, we investigated whether excessive crowding occurs in developmental dyscalculia (DD), a neurodevelopmental disorder characterized by difficulty in learning the most basic numerical and arithmetical concepts, and whether it is found independently of associated major reading and attentional difficulties. We measured spatial crowding in two groups of adult individuals with DD and control subjects. In separate experiments, participants were asked to discriminate the orientation of a Gabor patch either in isolation or under spatial crowding. Orientation discrimination thresholds were comparable across groups when stimuli were shown in isolation, yet they were much higher for the DD group with respect to the control group when the target was crowded by closely neighbouring flanking gratings. The difficulty in discriminating orientation (as reflected by the combination of accuracy and reaction times) in the DD compared to the control group persisted over several larger target flanker distances. Finally, we found that the degree of such spatial crowding correlated with impairments in mathematical abilities even when controlling for visual attention and reading skills. These results suggest that excessive crowding effects might be a characteristic of DD, independent of other associated neurodevelopmental disorders.

Math Anxiety Mediates the Link Between Number Sense and Math Achievements in High Math Anxiety Young Adults

In the past few years, many studies have suggested that subjects with high sensory precision in the processing of non-symbolic numerical quantities (approximate number system; ANS) also have higher math abilities. At the same time, there has been interest in another non-cognitive factor affecting mathematical learning: mathematical anxiety (MA). MA is defined as a debilitating emotional reaction to mathematics that interferes with the manipulation of numbers and the solving of mathematical problems. Few studies have been dedicated to uncovering the interplay between ANS and MA and those have provided conflicting evidence. Here we measured ANS precision (numerosity discrimination thresholds) in a cohort of university students with either a high (>75th percentile; n = 49) or low (<25th percentile; n = 39) score on the Abbreviate Math Anxiety Scale (AMAS). We also assessed math proficiency using a standardized test (MPP: Mathematics Prerequisites for Psychometrics), visuo-spatial attention capacity by means of a Multiple Objects Tracking task (MOT) and sensory precision for non-numerical quantities (disk size). Our results confirmed previous studies showing that math abilities and ANS precision correlate in subjects with high math anxiety. Neither precision in size-discrimination nor visuo-spatial attentional capacity were found to correlate with math capacities. Interestingly, within the group with high MA, our data also revealed a relationship between ANS precision and MA, with MA playing a key role in mediating the correlation between ANS and math achievement. Taken together, our results suggest an interplay between extreme levels of MA and the sensory precision in the processing of non-symbolic numerosity.

Interaction of disparity size and depth structure on perceived numerosity in a three-dimensional space

The number of elements in two stereo-surfaces parallelly overlapped in depth is overestimated compared to that in a single flat surface, even when both have the same number of elements. Using stereoscopic pairs of elements, we evaluated two hypotheses on the overestimation: one that a higher-order process, forming a background surface, increases the number of perceived elements, and the other that the number of elements potentially occluded by the elements on a front surface is taken accounted for. The data from four experiments showed that (a) when binocular disparity between (or among) stereoscopic elements was small, the overestimation occurred for the stimuli we used-a two-surface-overlapping stimulus, where the likelihood for the process to operate was manipulated by changing the averaged luminance of each surface, a volumetric stimulus, where the likelihood for the background surface to be formed would decrease, and a two-non-overlapping-surface stimulus, where the surfaces in depth were not overlapped-, and (b) when binocular disparity was large, the overestimation occurred for the two-surfaces-overlapping stimulus, when the averaged luminance of the two surfaces were the same, and for the volumetric stimulus, but diminished for the surface-overlapping stimulus, when the averaged luminance differed between the surfaces and for the surfaces-non-overlapping stimulus. These results cannot be explained either hypothesis only. We explain the results by postulating that the sensory system processing disparities of elements interferes with that estimating the number of elements, resulting in an overestimation of the elements in a stereo-stimulus, and the disparity range within which the interference occurs may depend on the stimulus depth structure.

Higher attentional costs for numerosity estimation at high densities

Humans can estimate numerosity over a large range, but the precision with which they do so varies considerably over that range. For very small sets, within the subitizing range of up to about four items, estimation is rapid and errorless. For intermediate numerosities, errors vary directly with the numerosity, following Weber’s law, but for very high numerosities, with very dense patterns, thresholds continue to rise with the square root of numerosity. This suggests that three different mechanisms operate over the number range. In this study we provide further evidence for three distinct numerosity mechanisms, by studying their dependence on attentional resources. We measured discrimination thresholds over a wide range of numerosities, while manipulating attentional load with both visual and auditory dual tasks. The results show that attentional effects on thresholds vary over the number range. Both visual and auditory attentional loads strongly affect subitizing, much more than for larger numerosities. Attentional costs remain stable over the estimation range, then rise again for very dense patterns. These results reinforce the idea that numerosity is processed by three separates but probably overlapping systems.

Learning disabilities: Developmental dyscalculia

Developmental dyscalculia (DD) is a developmental learning disability that manifests as a persistent difficulty in comprehending even the most basic numeric and arithmetic concepts, despite normal intelligence and schooling opportunities. Given the predominant use of numbers in modern society, this condition can pose major challenges in the sufferer's everyday life, both in personal and professional development. Since, to date, we still lack a universally recognized and psychometrically driven definition of DD, its diagnosis has been applied to a wide variety of cognitive profiles. In this chapter, we review the behavioral and neural characterization of DD as well as the different neurocognitive and etiologic accounts of this neurodevelopmental disorder. We underline the multicomponential nature of this heterogeneous disability: different aspects of mathematical competence can be affected by both the suboptimal recruitment of general cognitive functions supporting mathematical cognition (such as attention, memory, and cognitive control) and specific deficits in mastering numeric concepts and operations. Accordingly, both intervention paradigms focused on core numeric abilities and more comprehensive protocols targeting multiple neurocognitive systems have provided evidence for effective positive outcomes.

Three-systems for visual numerosity: A single case study

Humans possess the remarkable capacity to assess the numerosity of a set of items over a wide range of conditions, from a handful of items to hundreds of them. Recent evidence is starting to show that judgments over such a large range is possible because of the presence of three mechanisms, each tailored to specific stimulation conditions. Previous evidence in favour of this theory comes from the fact that discrimination thresholds and estimation reaction times are not constants across numerosity levels. Likewise, attention is capable of dissociating the three mechanisms: when healthy adult observers are asked to perform concurrently a taxing task, the judgments of low numerosities (<4 dots) or of high numerosities is affected greatly, not so however for intermediate numerosities. Here we bring evidence from a neuropsychological perspective. To this end we measured perceptual performance in PA, a 41 year-old patient who suffers simultanagnosia after a hypoxic brain injury. PA showed a profound deficit in attentively tracking objects over space and time (multiple object tracking), even in very simple conditions where controls made no errors. PA also showed a massive deficit on sensory thresholds when comparing dot-arrays containing extremely low (3 dots) or extremely high (64, 128 dots) numerosities as well as in comparing dot-distances. Surprisingly, PA discrimination thresholds were relatively spared for intermediate numerosity (12 and 16 dots). Overall his deficit on the numerosity task results in a U-shape function across numerosity which, combined with the attentional deficit and the inability to judge dot-distances, confirms previously suggested three-systems for numerosity judgments.

Small numerosity advantage for sequential enumeration on RSVP stimuli: an object individuation-based account

Although there is a large literature demonstrating rapid and accurate enumeration of small sets of simultaneously presented items (i.e., subitizing), it is unclear whether this small numerosity advantage (SNA) can also manifest in sequential enumeration. The present study thus has two aims: to establish a robust processing advantage for small numerosities during sequential enumeration using a rapid serial visual presentation (RSVP) paradigm, and to examine the underlying mechanism for a SNA in sequential enumeration. The results indicate that a small set of items presented in fast sequences can be enumerated accurately with a high precision and a SOA (stimulus onset asynchrony)-sensitive capacity limit, essentially generalizing the large literature on small numerosity advantage from spatial domain to temporal domain. A resource competition hypothesis was proposed and confirmed in further experiments. Specifically, sequential enumeration and other cognitive process, such as visual working memory (VWM), compete for a shared resource of object individuation by which items are segregated as individual entities. These results implied that the limited resource of object individuation can be allocated within time windows of flexible temporal scales during simultaneous and sequential enumerations. Taken together, the present study calls for attention to the dynamic aspect of the enumeration process and highlights the pivotal role of object individuation in underlying a wide range of mental operations, such as enumeration and VWM.

Attentional amplification of neural codes for number independent of other quantities along the dorsal visual stream

Humans and other animals base important decisions on estimates of number, and intraparietal cortex is thought to provide a crucial substrate of this ability. However, it remains debated whether an independent neuronal processing mechanism underlies this ‘number sense’, or whether number is instead judged indirectly on the basis of other quantitative features. We performed high-resolution 7 Tesla fMRI while adult human volunteers attended either to the numerosity or an orthogonal dimension (average item size) of visual dot arrays. Along the dorsal visual stream, numerosity explained a significant amount of variance in activation patterns, above and beyond non-numerical dimensions. Its representation was selectively amplified and progressively enhanced across the hierarchy when task relevant. Our results reveal a sensory extraction mechanism yielding information on numerosity separable from other dimensions already at early visual stages and suggest that later regions along the dorsal stream are most important for explicit manipulation of numerical quantity.

Numbers and the ability to count and calculate are an essential part of human culture. They are part of everyday life, featuring in calendars, computers or the weekly shop, but also in some of humanity’s biggest achievements: without them the pyramids or space travel would not exist. A precursor of sophisticated mathematical skill could reside in a simpler mental ability: the capacity to assess numerical quantities at a glance. This ‘number sense’ appears in humans in early childhood and it is also present in other animals, but it is still poorly understood.

Brain imaging techniques have identified the parts of the brain that are active when perceiving numbers or making calculations. As techniques have advanced, it has become possible to resolve fine differences in brain activity that occur when people switch their attention between different visual tasks. But how exactly does the human brain process visual information to make sense of numbers? One theory suggests that humans use visual cues, such as the size of a group of objects or how densely packed objects are, to estimate numbers. On the other hand, it is also possible that humans can sense number directly, without reference to other properties of the group being observed.

Castaldi et al. presented twenty adult volunteers with groups of dots and asked them to focus either on the number of dots or on the size of the dots during a brain scan. This approach allowed the separation of brain signals specific to number from signals corresponding to other visual cues, such as size or density of the group. The experiment revealed that brain activity changed depending on the number of dots displayed. The signal related to number became stronger when people focused on the number of dots, while signals related to other properties of the group remained unchanged. Moreover, brain signals for number were observed at the very early stages of visual processing, in the parts of the brain that receive input from the eyes first.

These results suggest that the human visual system perceives number directly, and not by processing information about the size or density of a group of objects. This finding provides insights into how human brains encode numbers, which could be important to understand disorders where number sense can be impaired leading to difficulties learning math and operating with numbers.

Spontaneous perception of numerosity in pre-school children

There is strong evidence that humans can make rough estimates of the numerosity of a set of items, almost from birth. However, as numerosity covaries with many non-numerical variables, the idea of a direct number sense has been challenged. Here we applied two different psychophysical paradigms to demonstrate the spontaneous perception of numerosity in a cohort of young pre-school children. The results of both tasks showed that even at that early developmental stage, humans spontaneously base the perceptual choice on numerosity, rather than on area or density. Precision in one of these tasks predicted mathematical abilities. The results reinforce strongly the idea of a primary number sense and provide further evidence linking mathematical skills to the sensory precision of the spontaneous number sense, rather than to mechanisms involved in handling explicit numerosity judgements or extensive exposure to mathematical teaching.

Different reaction-times for subitizing, estimation, and texture

Humans can estimate and encode numerosity over a large range, from very few items to several hundreds. Two distinct mechanisms have been proposed: subitizing, for numbers up to four and estimation for larger numerosities. We have recently extended this idea by suggesting that for very densely packed arrays, when items are less segregable, a third "texture" mechanism comes into play. In this study, we provide further evidence for the existence of a third regime for numerosity. Reaction times were very low in the subitizing range, rising rapidly for numerosities greater than four. However, for tightly packed displays of very high numerosities, reaction times became faster. These results reinforce the idea of three regimes in the processing of numerosity, subitizing, estimation, and texture.

Inhibitory surrounds of motion mechanisms revealed by continuous tracking

Continuous psychophysics is a newly developed technique that allows rapid estimation of visual thresholds by asking subjects to track a moving object, then deriving the integration window underlying tracking behavior (Bonnen, Burge, Yates, Pillow, & Cormack, 2015). Leveraging the continuous flow of stimuli and responses, continuous psychophysics allows for estimation of psychophysical thresholds in as little as 1 min. To date this technique has been applied only to tracking visual objects, where it has been used to measure localization thresholds. Here we adapt the technique to visual motion discrimination, by displaying a drifting grating that changes direction on a binary random walk and asking participants to continuously report drift direction by alternate key press. This technique replicates and confirms well-known findings of the motion-perception system. It also proves particularly valuable in demonstrating induced motion, reinforcing evidence for the existence of antagonistic surround fields. At low contrasts, the surround summates with the center, rather than opposing it, again consistent with existing evidence on classical techniques. The user-friendliness and efficiency of the method may lend it to clinical and developmental work.

Perceiving numerosity from birth

Leibovich et al. opened up an important discussion on the nature and origins of numerosity perception. The authors rightly point out that non-numerical features of stimuli influence this ability. Despite these biases, there is evidence that from birth, humans perceive and represent numerosities, and not just non-numerical quantitative features such as item size, density, and convex hull.

Spontaneous representation of numerosity in typical and dyscalculic development

Animals including humans are endowed with a remarkable capacity to estimate rapidly the number of items in a scene. Some have questioned whether this ability reflects a genuine sense of number, or whether numerosity is derived indirectly from other covarying attributes, such as density and area. In previous work we have demonstrated that adult observers are more sensitive to changes in numerosity than to area or density, particularly changes that leave numerosity constant, pointing to a spontaneous sensitivity to numerosity, not attributable to area and density. Here we extend this line of research with a novel technique where participants reproduce the size and density of a dot-array. They were given no explicit instructions of what to match, but could regulate freely all combinations of area and density by trackpad. If the task is mediated by matching separately area and texture-density, the errors in the two attributes have to be independent. Contrarily to this prediction, we found that errors in area and density were negatively correlated, suggesting that subjects matched numerosity, rather than area and density. We employed this technique to investigate processing of number in adolescents with typical and low math abilities (dyscalculia). Interestingly, we found that dyscalculics also reproduced numerosity rather than area or density. However, compared to typicals, dyscalculics had longer reaction times, a tendency to rely also on area, and their performance did not improve over sessions. Taken together, the data demonstrate that numerosity emerges as the most spontaneous and sensitive dimension, supporting the existence of a dedicated number sense and confirm numerosity atypicalities in dyscalculia.

Numerical Processing Impairment in 22q11.2 (LCR22-4 to LCR22-5) Microdeletion: A Cognitive-Neuropsychological Case Study

Although progress has been made, the cognitive, biological and, particularly, the genetic underpinnings of math learning difficulties (MD) remain largely unknown. This difficulty stems from the heterogeneity of MD and from the large contribution of environmental factors to its etiology. Understanding endophenotypes, e.g., the role of the Approximate Number System (ANS), may help understanding the nature of MD. MD associated with ANS impairments has been described in some genetic conditions, e.g., 22q11.2 deletion syndrome (22q11.2DS or Velocardiofacial syndrome, VCFS). Recently, a girl with MD was identified in a school population screening. She has a new syndrome resulting from a microdeletion in 22q11.2 (LCR22-4 to LCR22-5), a region adjacent to but not overlapping with region 22q11.2 (LCR22-2 to LCR22-4), typically deleted in VCFS. Here, we describe her cognitive-neuropsychological and numerical-cognitive profiles. The girl was assessed twice, at 8 and 11 years. Her numerical-cognitive performance at both times was compared to demographically similar girls with normal intelligence in a single-case, quasi-experimental study. Neuropsychological assessment was normal, except for relatively minor impairments in executive functions. She presented severe and persistent difficulties in the simplest single-digit calculations. Difficulties in commutative operations improved from the first to the second assessment. Difficulties in subtraction persisted and were severe. No difficulties were observed in Arabic number writing. Difficulties in single-digit calculation co-occurred with basic numerical processing impairments in symbolic and non-symbolic (single-digit comparison, dot sets size comparison and estimation) tasks. Her difficulties suggest ANS impairment. No difficulties were detected in visuospatial/visuoconstructional and in phonological processing tasks. The main contributions of the present study are: (a) this is the first characterization of the neuropsychological phenotype in 22q11.2DS (LCR22-4 to LCR22.5) with normal intelligence; (b) mild forms of specific genetic conditions contribute to persistent MD in otherwise typical persons; (c) heterogeneity of neurogenetic underpinnings of MD is suggested by poor performance in non-symbolic numerical processing, dissociated from visuospatial/visuoconstructional and phonological impairments; (d) similar to what happens in 22q11.2DS (LCR22-2 to LCR22-4), ANS impairments may also characterize 22q11.2DS (LCR22-4 to LCR22-5).

Asymmetrical interference between number and item size perception provides evidence for a domain specific impairment in dyscalculia

Dyscalculia, a specific learning disability that impacts arithmetical skills, has previously been associated to a deficit in the precision of the system that estimates the approximate number of objects in visual scenes (the so called 'number sense' system). However, because in tasks involving numerosity comparisons dyscalculics' judgements appears disproportionally affected by continuous quantitative dimensions (such as the size of the items), an alternative view linked dyscalculia to a domain-general difficulty in inhibiting task-irrelevant responses. To arbitrate between these views, we evaluated the degree of reciprocal interference between numerical and non-numerical quantitative dimensions in adult dyscalculics and matched controls. We used a novel stimulus set orthogonally varying in mean item size and numerosity, putting particular attention into matching both features' perceptual discriminability. Participants compared those stimuli based on each of the two dimensions. While control subjects showed no significant size interference when judging numerosity, dyscalculics' numerosity judgments were strongly biased by the unattended size dimension. Importantly however, both groups showed the same degree of interference from the unattended dimension when judging mean size. Moreover, only the ability to discard the irrelevant size information when comparing numerosity (but not the reverse) significantly predicted calculation ability across subjects. Overall, our results show that numerosity discrimination is less prone to interference than discrimination of another quantitative feature (mean item size) when the perceptual discriminability of these features is matched, as here in control subjects. By quantifying, for the first time, dyscalculic subjects' degree of interference on another orthogonal dimension of the same stimuli, we are able to exclude a domain-general inhibition deficit as explanation for their poor / biased numerical judgement. We suggest that enhanced reliance on non-numerical cues during numerosity discrimination can represent a strategy to cope with a less precise number sense.

Independent adaptation mechanisms for numerosity and size perception provide evidence against a common sense of magnitude

How numerical quantity is processed is a central issue for cognition. On the one hand the "number sense theory" claims that numerosity is perceived directly, and may represent an early precursor for acquisition of mathematical skills. On the other, the "theory of magnitude" notes that numerosity correlates with many continuous properties such as size and density, and may therefore not exist as an independent feature, but be part of a more general system of magnitude. In this study we examined interactions in sensitivity between numerosity and size perception. In a group of children, we measured psychophysically two sensory parameters: perceptual adaptation and discrimination thresholds for both size and numerosity. Neither discrimination thresholds nor adaptation strength for numerosity and size correlated across participants. This clear lack of correlation (confirmed by Bayesian analyses) suggests that numerosity and size interference effects are unlikely to reflect a shared sensory representation. We suggest these small interference effects may rather result from top-down phenomena occurring at late decisional levels rather than a primary "sense of magnitude".

Variability in Single Digit Addition Problem-Solving Speed Over Time Identifies Typical, Delay and Deficit Math Pathways

We assessed the degree to which the variability in the time children took to solve single digit addition (SDA) problems longitudinally, predicted their ability to solve more complex mental addition problems. Beginning at 5 years, 164 children completed a 12-item SDA test on four occasions over 6 years. We also assessed their (1) digit span, visuospatial working memory, and non-verbal IQ, and (2) the speed with which they named single numbers and letters, as well the speed enumerating one to three dots as a measure of subitizing ability. Children completed a double-digit mental addition test at the end of the study. We conducted a latent profile analysis to determine if there were different SDA problem solving response time (PRT) variability patterns across the four test occasions, which yielded three distinct PRT variability patterns. In one pattern, labeled a typical acquisition pathway, mean PRTs were relatively low and PRT variability diminished over time. In a second pattern, label a delayed pathway, mean PRT and variability was high initially but diminished over time. In a third pattern, labeled a deficit pathway, mean PRT and variability remained relatively high throughout the study. We investigated the degree to which the three SDA PRT variability pathways were associated with (1) different cognitive ability measures, and (2) double-digit mental addition abilities. The deficit pathway differed from the typical and delayed pathway on the subitizing measure only, but not other measures; and the latter two pathways also differed from each other on the subitizing but not other measures. Double-digit mental addition problem solving success differed between each of the three pathways, and mean PRT variability differed between the typical and the delayed and deficit pathways. The latter two pathways did not differ from each other. The findings emphasize the value of examining individual differences in problem-solving PRT variability longitudinally as an index of math ability, and highlight the important of subitizing ability as a diagnostic index of math ability/difficulties.

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’.

The Developing Mental Number Line: Does Its Directionality Relate to 5- to 7-Year-Old Children's Mathematical Abilities?

Spatial representations of number, such as a left-to-right oriented mental number line, are well documented in Western culture. Yet, the functional significance of such a representation remains unclear. To test the prominent hypothesis that a mental number line may support mathematical development, we examined the relation between spatial-numerical associations (SNAs) and math proficiency in 5- to 7-year-old children. We found evidence of SNAs with two tasks: a non-symbolic magnitude comparison task, and a symbolic "Where was the number?" (WTN) task. Further, we found a significant correlation between these two tasks, demonstrating convergent validity of the directional mental number line across numerical format. Although there were no significant correlations between children's SNAs on the WTN task and math ability, children's SNAs on the magnitude comparison task were negatively correlated with their performance on a measure of cross-modal arithmetic, suggesting that children with a stronger left-to-right oriented mental number line were less competent at cross-modal arithmetic, an effect that held when controlling for age and a set of general cognitive abilities. Despite some evidence for a negative relation between SNAs and math ability in adulthood, we argue that the effect here may reflect task demands specific to the magnitude comparison task, not necessarily an impediment of the mental number line to math performance. We conclude with a discussion of the different properties that characterize a mental number line and how these different properties may relate to mathematical ability.

Numerical encoding in early visual cortex

The ability to estimate numerosity in a visual array arose early in evolution, develops early in human development, and is correlated with mathematical ability. Previous work with visually presented arrays indicates that the intraparietal sulcus (IPS) represents number. However, it is not clear if the number signal originates in IPS or is propagated from earlier visual areas. Previous work from our group has demonstrated a rapidly instantiated representation of number in low-level regions of visual cortex using the high temporal resolution of event-related electro-encephalography (EEG). Here, we use a rapid event-related functional magnetic resonance imaging (fMRI) paradigm and find convergent evidence for a number signal in low-level visual cortex (areas V1, V2, and V3). Employing a stringent set of stimulus controls, we demonstrate that this signal cannot be explained by the total extent of the array, the density of the items in the array, the aggregate visual area of the items, the size of individual items, the proportion of the array covered by items, nor the overall scale of the array and items. Our findings thus provide strong support for the hypothesis that number is rapidly and directly encoded early in the visual processing stream.

Discriminability of numerosity-evoked fMRI activity patterns in human intra-parietal cortex reflects behavioral numerical acuity

Areas of the primate intraparietal cortex have been identified as an important substrate of numerical cognition. In human fMRI studies, activity patterns in these and other areas have allowed researchers to read out the numerosity a subject is viewing, but the relation of such decodable information with behavioral numerical proficiency remains unknown. Here, we estimated the precision of behavioral numerosity discrimination (internal Weber fraction) in twelve adult subjects based on psychophysical testing in a delayed numerosity comparison task outside the scanner. FMRI data were then recorded during a similar task, to obtain the accuracy with which the same sample numerosities could be read out from evoked brain activity patterns, as a measure of the precision of the neuronal representation. Sample numerosities were decodable in both early visual and intra-parietal cortex with approximately equal accuracy on average. In parietal cortex, smaller numerosities were better discriminated than larger numerosities of the same ratio, paralleling smaller behavioral Weber fractions for smaller numerosities. Furthermore, in parietal but not early visual cortex, fMRI decoding performance was correlated with behavioral number discrimination acuity across subjects (subjects with a more precise behavioral Weber fraction measured prior to scanning showed greater discriminability of fMRI activity patterns in intraparietal cortex, and more specifically, the right LIP region). These results suggest a crucial role for intra-parietal cortex in supporting a numerical representation which is explicitly read out for numerical decisions and behavior.

Evidence for a number sense

Numerosity is inherently confounded by related stimulus attributes such as density and area, and many studies have reported interactions of various strengths between area, density, and numerosity. However, direct measurements of sensitivity within the area-density-numerosity space show that numerosity emerges as the most spontaneous and sensitive dimension, strongly supporting the existence of a dedicated number sense.

Approximate number sense correlates with math performance in gifted adolescents

Nonhuman animals, human infants, and human adults all share an Approximate Number System (ANS) that allows them to imprecisely represent number without counting. Among humans, people differ in the precision of their ANS representations, and these individual differences have been shown to correlate with symbolic mathematics performance in both children and adults. For example, children with specific math impairment (dyscalculia) have notably poor ANS precision. However, it remains unknown whether ANS precision contributes to individual differences only in populations of people with lower or average mathematical abilities, or whether this link also is present in people who excel in math. Here we tested non-symbolic numerical approximation in 13- to 16-year old gifted children enrolled in a program for talented adolescents (the Center for Talented Youth). We found that in this high achieving population, ANS precision significantly correlated with performance on the symbolic math portion of two common standardized tests (SAT and ACT) that typically are administered to much older students. This relationship was robust even when controlling for age, verbal performance, and reaction times in the approximate number task. These results suggest that the Approximate Number System is linked to symbolic math performance even at the top levels of math performance.

Effects of adaptation on numerosity decoding in the human brain

Psychophysical studies have shown that numerosity is a sensory attribute susceptible to adaptation. Neuroimaging studies have reported that, at least for relatively low numbers, numerosity can be accurately discriminated in the intra-parietal sulcus. Here we developed a novel rapid adaptation paradigm where adapting and test stimuli are separated by pauses sufficient to dissociate their BOLD activity. We used multivariate pattern recognition to classify brain activity evoked by non-symbolic numbers over a wide range (20-80), both before and after psychophysical adaptation to the highest numerosity. Adaptation caused underestimation of all lower numerosities, and decreased slightly the average BOLD responses in V1 and IPS. Using support vector machine, we showed that the BOLD response of IPS, but not in V1, classified numerosity well, both when tested before and after adaptation. However, there was no transfer from training pre-adaptation responses to testing post-adaptation, and vice versa, indicating that adaptation changes the neuronal representation of the numerosity. Interestingly, decoding was more accurate after adaptation, and the amount of improvement correlated with the amount of perceptual underestimation of numerosity across subjects. These results suggest that numerosity adaptation acts directly on IPS, rather than indirectly via other low-level stimulus parameters analysis, and that adaptation improves the capacity to discriminate numerosity.

Spontaneous perception of numerosity in humans

Humans, including infants, and many other species have a capacity for rapid, nonverbal estimation of numerosity. However, the mechanisms for number perception are still not clear; some maintain that the system calculates numerosity via density estimates-similar to those involved in texture-while others maintain that more direct, dedicated mechanisms are involved. Here we show that provided that items are not packed too densely, human subjects are far more sensitive to numerosity than to either density or area. In a two-dimensional space spanning density, area and numerosity, subjects spontaneously react with far greater sensitivity to changes in numerosity, than either area or density. Even in tasks where they were explicitly instructed to make density or area judgments, they responded spontaneously to number. We conclude, that humans extract number information, directly and spontaneously, via dedicated mechanisms.