Connecting Visual Objects Reduces Perceived Numerosity and Density for Sparse but not Dense Patterns

How is numerosity encoded by the visual system? – directly, or derived indirectly from texture density? We recently suggested that the numerosity of sparse patterns is encoded directly by dedicated mechanisms (which have been described as the “Approximate Number System” ANS). However, at high dot densities, where items become “crowded” and difficult to segregate, “texture-density” mechanisms come into play. Here we tested the importance of item segmentation on numerosity and density perception at various stimulus densities, by measuring the effect of connecting visual objects with thin lines. The results confirmed many previous studies showing that connecting items robustly reduces the apparent numerosity of patterns of moderate density. We further showed that the apparent density of moderate-density patterns is also reduced by connecting the dots. Crucially, we found that both these effects are strongly reduced at higher numerosities. Indeed for density judgments, the effect reverses, so connecting dots in dense patterns increases the apparent density (as expected from the physical characteristics). The results provide clear support for the three-regime framework of number perception, and suggest that for moderately sparse stimuli, numerosity – but not texture-density – is perceived directly.

Related papers:

title: Separate Mechanisms for Perception of Numerosity and Density
abstract: Despite the existence of much evidence for a number sense in humans, several researchers have questioned whether number is sensed directly or derived indirectly from texture density. Here, we provide clear evidence that numerosity and density judgments are subserved by distinct mechanisms with different psychophysical characteristics. We measured sensitivity for numerosity discrimination over a wide range of numerosities: For low densities (less than 0.25 dots/deg2), thresholds increased directly with numerosity, following Weber’s law; for higher densities, thresholds increased with the square root of texture density, a steady decrease in the Weber fraction. The existence of two different psychophysical systems is inconsistent with a model in which number is derived indirectly from noisy estimates of density and area; rather, it points to the existence of separate mechanisms for estimating density and number. These results provide strong confirmation for the existence of neural mechanisms that sense number directly, rather than indirectly from texture density.

title: Number As a Primary Perceptual Attribute: A Review
abstract: 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.

title: Mechanisms for perception of numerosity or texture-density are governed by crowding-like effects.
abstract: We have recently provided evidence that the perception of number and texture density is mediated by two independent mechanisms: numerosity mechanisms at relatively low numbers, obeying Weber's law, and texture-density mechanisms at higher numerosities, following a square root law. In this study we investigated whether the switch between the two mechanisms depends on the capacity to segregate individual dots, and therefore follows similar laws to those governing visual crowding. We measured numerosity discrimination for a wide range of numerosities at three eccentricities. We found that the point where the numerosity regime (Weber's law) gave way to the density regime (square root law) depended on eccentricity. In central vision, the regime changed at 2.3 dots/°2, while at 15° eccentricity, it changed at 0.5 dots/°2, three times less dense. As a consequence, thresholds for low numerosities increased with eccentricity, while at higher numerosities thresholds remained constant. We further showed that like crowding, the regime change was independent of dot size, depending on distance between dot centers, not distance between dot edges or ink coverage. Performance was not affected by stimulus contrast or blur, indicating that the transition does not depend on low-level stimulus properties. Our results reinforce the notion that numerosity and texture are mediated by two distinct processes, depending on whether the individual elements are perceptually segregable. Which mechanism is engaged follows laws that determine crowding.

title: Emergence of a 'visual number sense' in hierarchical generative models
abstract: Numerosity estimation is phylogenetically ancient and foundational to human mathematical learning, but its computational bases remain controversial. Here we show that visual numerosity emerges as a statistical property of images in 'deep networks' that learn a hierarchical generative model of the sensory input. Emergent numerosity detectors had response profiles resembling those of monkey parietal neurons and supported numerosity estimation with the same behavioral signature shown by humans and animals.

title: A common visual metric for approximate number and density
abstract: There is considerable interest in how humans estimate the number of objects in a scene in the context of an extensive literature on how we estimate the density (i.e., spacing) of objects. Here, we show that our sense of number and our sense of density are intertwined. Presented with two patches, observers found it more difficult to spot differences in either density or numerosity when those patches were mismatched in overall size, and their errors were consistent with larger patches appearing both denser and more numerous. We propose that density is estimated using the relative response of mechanisms tuned to low and high spatial frequencies (SFs), because energy at high SFs is largely determined by the number of objects, whereas low SF energy depends more on the area occupied by elements. This measure is biased by overall stimulus size in the same way as human observers, and by estimating number using the same measure scaled by relative stimulus size, we can explain all of our results. This model is a simple, biologically plausible common metric for perceptual number and density.

title: Texture density adaptation and visual number revisited
abstract: Burr and Ross  have recently proposed that the visual dimension of number is itself directly adaptable. The aftereffect they describe is one that my colleagues and I previously used to investigate the perception of texture density [2–4]. Burr and Ross  argue that the effect is new because it concerns visual number, rather than texture density, but they did not report critical tests to support this claim. Here, I shall briefly describe the striking similarity between our prior work and that of Burr and Ross , and discuss how some of our results rule out Burr and Ross's  interpretation that numerosity, and not density, is at play. I shall also provide a new demonstration that confirms that these effects are based on density, using a display that explicitly dissociates density from numerosity. Taken together, this line of arguments suggests that Burr and Ross's  recent study may best be thought of as replicating support within a well-established line of work on texture density.

title: A texture-processing model of the ‘visual sense of number’
abstract: It has been suggested that numerosity is an elementary quality of perception, similar to colour. If so (and despite considerable investigation), its mechanism remains unknown. Here, we show that observers require on average a massive difference of approximately 40% to detect a change in the number of objects that vary irrelevantly in blur, contrast and spatial separation, and that some naive observers require even more than this. We suggest that relative numerosity is a type of texture discrimination and that a simple model computing the contrast energy at fine spatial scales in the image can perform at least as well as human observers. Like some human observers, this mechanism finds it harder to discriminate relative numerosity in two patterns with different degrees of blur, but it still outpaces the human. We propose energy discrimination as a benchmark model against which more complex models and new data can be tested.

title: Number and density discrimination rely on a common metric: Similar psychophysical effects of size, contrast, and divided attention.
abstract: While observers are adept at judging the density of elements (e.g., in a random-dot image), it has recently been proposed that they also have an independent visual sense of number. To test the independence of number and density discrimination, we examined the effects of manipulating stimulus structure (patch size, element size, contrast, and contrast-polarity) and available attentional resources on both judgments. Five observers made a series of two-alternative, forced-choice discriminations based on the relative numerosity/density of two simultaneously presented patches containing 16-1,024 Gaussian blobs. Mismatches of patch size and element size (across reference and test) led to bias and reduced sensitivity in both tasks, whereas manipulations of contrast and contrast-polarity had varied effects on observers, implying differing strategies. Nonetheless, the effects reported were consistent across density and number judgments, the only exception being when luminance cues were made available. Finally, density and number judgment were similarly impaired by attentional load in a dual-task experiment. These results are consistent with a common underlying metric to density and number judgments, with the caveat that additional cues may be exploited when they are available.

title: Modeling the approximate number system to quantify the contribution of visual stimulus features
abstract: The approximate number system (ANS) subserves estimation of the number of items in a set. Typically, ANS function is assessed by requiring participants to compare the number of dots in two arrays. Accuracy is determined by the numerical ratio of the sets being compared, and each participant's Weber fraction (w) provides a quantitative index of ANS acuity. When making numerical comparisons, however, performance is also influenced by non-numerical features of the stimuli, such as the size and spacing of dots. Current models of numerosity comparison do not account for these effects and consequently lead to different estimates of w depending on the methods used to control for non-numerical features. Here we proffer a new model that teases apart the effects of ANS acuity from the effects of non-numerical stimulus features. The result is an estimate of w that is a more theoretically valid representation of numerical acuity and novel terms that denote the degree to which a participant's perception of number is affected by non-numerical features. We tested this model in a sample of 20 adults and found that, by correctly attributing errors due to non-numerical stimulus features, the w obtained was more reliable across different stimulus conditions. We found that although non-numerical features biased numerosity discriminations in all participants, number was the primary feature driving discriminations in most of them. Our findings support the idea that, while numerosity is a distinct visual quantity, the internal representation of number is tightly bound to the representation of other magnitudes. This tool for identifying the different effects of the numerical and non-numerical features of a stimulus has important implications not only for the behavioral investigation of the ANS, but also for the collection and analyses of neural data sets associated with ANS function.

title: The interplay between nonsymbolic number and its continuous visual properties.
abstract: To date, researchers investigating nonsymbolic number processes devoted little attention to the visual properties of their stimuli. This is unexpected, as nonsymbolic number is defined by its visual characteristics. When number changes, its visual properties change accordingly. In this study, we investigated the influence of different visual properties on nonsymbolic number processes and show that the current assumptions about the relation between number and its visual characteristics are incorrect. Similar to previous studies, we controlled the visual cues: Each visual cue was not predictive of number. Nevertheless, participants showed congruency effects induced by the visual properties of the stimuli. These congruency effects scaled with the number of visual cues manipulated, implicating that people do not extract number from a visual scene independent of its visual cues. Instead, number judgments are based on the integration of information from multiple visual cues. Consequently, current ways to control the visual cues of the number stimuli are insufficient, as they control only a single variable at the time. And, more important, the existence of an approximate number system that can extract number independent of the visual cues appears unlikely. We therefore propose that number judgment is the result of the weighing of several distinct visual cues.

title: The Role of Visual Information in Numerosity Estimation
abstract: Mainstream theory suggests that the approximate number system supports our non-symbolic number abilities (e.g. estimating or comparing different sets of items). It is argued that this system can extract number independently of the visual cues present in the stimulus (diameter, aggregate surface, etc.). However, in a recent report we argue that this might not be the case. We showed that participants combined information from different visual cues to derive their answers. While numerosity comparison requires a rough comparison of two sets of items (smaller versus larger), numerosity estimation requires a more precise mechanism. It could therefore be that numerosity estimation, in contrast to numerosity comparison, might rely on the approximate number system. To test this hypothesis, we conducted a numerosity estimation experiment. We controlled for the visual cues according to current standards: each single visual property was not informative about numerosity. Nevertheless, the results reveal that participants were influenced by the visual properties of the dot arrays. They gave a larger estimate when the dot arrays consisted of dots with, on average, a smaller diameter, aggregate surface or density but a larger convex hull. The reliance on visual cues to estimate numerosity suggests that the existence of an approximate number system that can extract numerosity independently of the visual cues is unlikely. Instead, we propose that humans estimate numerosity by weighing the different visual cues present in the stimuli.

title: Sometimes area counts more than number
abstract: Using an interference paradigm, we show across three experiments that adults' order judgments of numbers, sizes, or combined area of dots in pairs of arrays occur spontaneously and automatically, but at different speeds and levels of accuracy. Experiment 1 used circles whose sizes varied between but not within arrays. Variation in circle size interfered with judgments of which array had more circles. Experiment 2 used displays in which circle size varied within and between arrays. Between-array differences in the amount of “circle stuff” (area occupied by circles) interfered with judgments of number. Experiment 3 examined whether variation in number also interferes with judgments of area. Interference between discrete and continuous stimulus dimensions occurred in both directions, although it was stronger from the continuous to the discrete than vice versa. These results bear on interpretations of studies with infants and preschoolers wherein subjects respond on the basis of continuous quantity rather than discrete quantity. In light of our results with adults, these findings do not license the conclusion that young children cannot represent discrete quantity. Absent data on attentional hierarchies and speed of processing, it is premature to conclude that infant and child quantity processes are fundamentally different from that of adults.

title: Judgement of discrete and continuous quantity in adults: Number counts!
abstract: Three experiments involving a Stroop-like paradigm were conducted. In Experiment 1, adults received a number comparison task in which large sets of dots, orthogonally varying along a discrete dimension (number of dots) and a continuous dimension (cumulative area), were presented. Incongruent trials were processed more slowly and with less accuracy than congruent trials, suggesting that continuous dimensions such as cumulative area are automatically processed and integrated during a discrete quantity judgement task. Experiment 2, in which adults were asked to perform area comparison on the same stimuli, revealed the reciprocal interference from number on the continuous quantity judgements. Experiment 3, in which participants received both the number and area comparison tasks, confirmed the results of Experiments 1 and 2. Contrasting with earlier statements, the results support the view that number acts as a more salient cue than continuous dimensions in adults. Furthermore, the individual predisposition to automatically access approximate number representations was found to correlate significantly with adults' exact arithmetical skills.

title: A generalized sense of number
abstract: Much evidence has accumulated to suggest that many animals, including young human infants, possess an abstract sense of approximate quantity, a number sense. Most research has concentrated on apparent numerosity of spatial arrays of dots or other objects, but a truly abstract sense of number should be capable of encoding the numerosity of any set of discrete elements, however displayed and in whatever sensory modality. Here, we use the psychophysical technique of adaptation to study the sense of number for serially presented items. We show that numerosity of both auditory and visual sequences is greatly affected by prior adaptation to slow or rapid sequences of events. The adaptation to visual stimuli was spatially selective (in external, not retinal coordinates), pointing to a sensory rather than cognitive process. However, adaptation generalized across modalities, from auditory to visual and vice versa. Adaptation also generalized across formats: adapting to sequential streams of flashes affected the perceived numerosity of spatial arrays. All these results point to a perceptual system that transcends vision and audition to encode an abstract sense of number in space and in time.

title: Visual aftereffects
abstract: interacting neuronal networks make adaptive behavioral decisions. Further new avenues opened up by recent work on Lymnaea may lead to an understanding of how learning and homeostatic processes interact, to the building of sophisticated computer models of functional connectivity and interacting networks, and even to the development of applications to artificial complex systems. At a different level, the discovery and structural analysis of a glia-derived acetylcholine-binding protein that modulates synaptic transmission between identified neurons of the Lymnaea respiratory CPG has provided important cues to the structural basis of ligand binding in nicotinic acethylcholine receptors in vertebrates. This pioneering work in Lymnaea opened the way for new research that might lead to the development of new drugs involving nicotinic acetylcholine receptorassociated diseases. A recently initiated pilot sequencing project is expected to result in a cDNA sequence database from the CNS, and in turn, will potentially lead to a full genomic sequencing project of Lymnaea stagnalis.

title: Adaptation affects both high and low (subitized) numbers under conditions of high attentional load.
abstract: It has recently been reported that, like most sensory systems, numerosity is subject to adaptation. However, the effect seemed to be limited to numerosity estimation outside the subitizing range. In this study we show that low numbers, clearly in the subitizing range, are adaptable under conditions of high attentional load. These results support the idea that numerosity is detected by a perceptual mechanism that operates over the entire range of numbers, supplemented by an attention-based system for small numbers (subitizing).

title: Development of Elementary Numerical Abilities: A Neuronal Model
abstract: Despite their lack of language, human infants and several animal species possess some elementary abilities for numerical processing. These include the ability to recognize that a given numerosity is being presented visually or auditorily, and, at a later stage of development, the ability to compare two nume-rosities and to decide which is larger. We propose a model for the development of these abilities in a formal neuronal network. Initially, the model is equipped only with unordered numerosity detectors. It can therefore detect the numerosity of an input set and can be conditioned to react accordingly. In a later stage, the addition of a short-term memory network is shown to be sufficient for number comparison abilities to develop. Our computer simulations account for several phenomena in the numerical domain, including the distance effect and Fechner's law for numbers. They also demonstrate that infants' numerosity detection abilities may be explained without assuming that infants can count. The neurobiological bases of the critical components of the model are discussed.

title: Number estimation relies on a set of segmented objects
abstract: How do we estimate the number of objects in a set? Two types of visual representations might underlie this ability - an unsegmented visual image or a segmented collection of discrete objects. We manipulated whether individual objects were isolated from each other or grouped into pairs by irrelevant lines. If number estimation operates over an unsegmented image, then this manipulation should not affect estimates. But if number estimation relies on a segmented image, then grouping pairs of objects into single units should lead to lower estimates. In Experiment 1 participants underestimated the number of grouped objects, relative to disconnected objects in which the connecting lines were 'broken'. Experiment 2 presents evidence that this segmentation process occurred broadly across the entire set of objects. In Experiment 3, a staircase procedure provides a quantitative measure of the underestimation effect. Experiment 4 shows that the strength of the grouping effect was equally strong for a single thin line, and the effect can be eliminated by a small break in the line. These results provide direct evidence that number estimation relies on a segmented input.

title: Connectedness affects dot numerosity judgment: Implications for configural processing
abstract: Participants judged the number of dots in visual displays with brief presentations (200 msec), such that the numerosity judgment was based on an instantaneous impression without counting. In some displays, pairs of adjacent dots were connected by line segments, whereas, in others, line segments were freely hanging without touching the dots. In Experiments 1, 2A, and 2B, connecting pairs of dots by line segments led to underestimation of dot numbers in those patterns. In Experiment 3, we controlled for the number of freely hanging line segments, whereas Experiment 4 showed that line segments that were merely attached to dots without actually connecting them did not produce a considerable underestimation effect. Experiment 5 showed that a connectedness effect existed when stimulus duration was reduced (50 msec) or extended (1,000 msec). We conclude that connectivity affects dot numerosity judgments, consistent with earlier findings of a configural effect in numerosity processing. Implications of the role of connectedness in object representation are discussed.

title: Topology-defined units in numerosity perception
abstract: What is a number? The answer to this age-old and fundamental question of philosophy has increasingly benefited from recent scientific investigation using psychology and neuroscience. To verify the invariant nature of numerosity perception, we manipulated the numbers of items connected/enclosed in arbitrary and irregular forms while controlling for various low-level visual features in different tasks and across small and large numbers. Results were consistent with the topological account, namely that numbers were strongly influenced by topological invariants (connectivity and the inside/outside relationship): connecting/enclosing items led to robust numerosity underestimation, with the extent of underestimation increasing monotonically with the number of connected/enclosed items. Brain image results also provided evidence that numbers represented in the intraparietal sulcus were influenced by topology. What is a number? The number sense hypothesis suggests that numerosity is “a primary visual property” like color, contrast, or orientation. However, exactly what attribute of a stimulus is the primary visual property and determines numbers in the number sense? To verify the invariant nature of numerosity perception, we manipulated the numbers of items connected/enclosed in arbitrary and irregular forms while controlling for low-level features (e.g., orientation, color, and size). Subjects performed discrimination, estimation, and equality judgment tasks in a wide range of presentation durations and across small and large numbers. Results consistently show that connecting/enclosing items led to robust numerosity underestimation, with the extent of underestimation increasing monotonically with the number of connected/enclosed items. In contrast, grouping based on color similarity had no effect on numerosity judgment. We propose that numbers or the primitive units counted in numerosity perception are influenced by topological invariants, such as connectivity and the inside/outside relationship. Beyond the behavioral measures, neural tuning curves to numerosity in the intraparietal sulcus were obtained using functional MRI adaptation, and the tuning curves showed that numbers represented in the intraparietal sulcus were strongly influenced by topology.

title: Numerosity underestimation in sets with illusory contours
abstract: People underestimate the numerosity of collections in which a few dots are connected in pairs by task-irrelevant lines. Such configural processing suggests that visual numerosity depends on the perceived scene segments, rather than on the perceived total area occupied by a collection. However, a methodology that uses irrelevant line connections may also introduce unnecessary distraction and variety, or obscure the perception of task-relevant items, given the saliency of the lines. To avoid such potentially confounding variables, we conducted four experiments where the line-connected dots were replaced with collinear inducers of Kanizsa-type illusory contours. Our participants had to compare two simultaneously presented collections and choose the more numerous one. Displays comprised c-shaped inducers and disks (Experiment 1), c-shaped inducers only (Experiments 2 and 4), or closed inducers (Experiment 3). One display always showed a 12- (Experiments 1-3) or 48-item reference pattern (Experiment 4); the other was a test pattern with numerosity varying between 9 and 15 (Experiments 1-3) or 36-60 items (Experiment 4). By manipulating the number of illusory contours in the test patterns, the level of connectedness increased or decreased respectively. The fitted psychometric functions revealed an underestimation that increased with the number of illusory contours in Experiments 1 and 2, but was absent in Experiments 3 and 4, where illusory contours were more difficult to perceive or larger numerosities were used. Results corroborate claims that visual numerosity estimation depends on segmented inputs, but only within moderate numerical ranges.

title: Adaptation to number operates on perceived rather than physical numerosity
abstract: Humans share with many animals a number sense, the ability to estimate rapidly the approximate number of items in a scene. Recent work has shown that like many other perceptual attributes, numerosity is susceptible to adaptation. It is not clear, however, whether adaptation works directly on mechanisms selective to numerosity, or via related mechanisms, such as those tuned to texture density. To disentangle this issue we measured adaptation of numerosity of 10 pairs of connected dots, as connecting dots makes them appear to be less numerous than unconnected dots. Adaptation to a 20-dot pattern (same number of dots as the test) caused robust reduction in apparent numerosity of the connected-dot pattern, but not of the unconnected dot-pattern. This suggests that adaptation to numerosity, at least for relatively sparse dot-pattern, occurs at neural levels encoding perceived numerosity, rather than at lower levels responding to the number of elements in the scene.

title: Quest: A Bayesian adaptive psychometric method
abstract: An adaptive psychometric procedure that places each trial at the current most probable Bayesian estimate of threshold is described. The procedure takes advantage of the common finding that the human psychometric function is invariant in form when expressed as a function of log intensity. The procedure is simple, fast, and efficient, and may be easily implemented on any computer.

title: What The Statistics Of Natural Images Tell Us About Visual Coding
abstract: Why does the mammalian visual system represent information as it does? If we assume that visual systems have evolved to cope with the natural environment then we might expect the coding properties of the visual system to be related to the statistical structure of our environment. Indeed, images of the natural environment do not have random statistics. The first-order statistics (e.g., distribution of pixel values) and second-order statistics (e.g., power spectra) of natural images have been discussed previously and they bear important relations to visual coding. Statistics higher than second-order are difficult to measure but provide crucial information about the image. For example, it can be shown that the lines and edges found in natural images are a function of these higher-order statistics. In this paper, these higher-order statistics will be discussed in relation to the coding properties of the mammalian visual system. It is suggested that the spatial parameters of the cortical 'filters' (e.g., bandwidths of simple and complex cells) are closely related to these higher-order statistics. In particular, it will be shown that the spatial non-linearities shown by cortical complex cells provide the early visual system with the information required to learn about these statistics.

title: Amplitude spectra of natural images.
abstract: Several studies have suggested that the amplitude spectra of photographs of natural scenes are remarkably similar and have the form: amplitude varies; is directly proportional to spatial frequency-1.0. This is, of course, a straight line with slope of -1.0 when plotted on double logarithmic coordinates. We have examined the amplitude spectra of 135 digitized photographs of natural scenes and have found that relatively few images conform exactly to the suggestion. About 25% of the images in our sample have spectra which show significant curvature when plotted on log-log coordinates. The best-fitting regression lines have slopes that range from -0.8 to -1.5; the average slope is -1.2, rather steeper than previously suggested.

title: Numerosity but not texture-density discrimination correlates with math ability in children.
abstract: Considerable recent work suggests that mathematical abilities in children correlate with the ability to estimate numerosity. Does math correlate only with numerosity estimation, or also with other similar tasks? We measured discrimination thresholds of school-age (6- to 12.5-years-old) children in 3 tasks: numerosity of patterns of relatively sparse, segregatable items (24 dots); numerosity of very dense textured patterns (250 dots); and discrimination of direction of motion. Thresholds in all tasks improved with age, but at different rates, implying the action of different mechanisms: In particular, in young children, thresholds were lower for sparse than textured patterns (the opposite of adults), suggesting earlier maturation of numerosity mechanisms. Importantly, numerosity thresholds for sparse stimuli correlated strongly with math skills, even after controlling for the influence of age, gender and nonverbal IQ. However, neither motion-direction discrimination nor numerosity discrimination of texture patterns showed a significant correlation with math abilities. These results provide further evidence that numerosity and texture-density are perceived by independent neural mechanisms, which develop at different rates; and importantly, only numerosity mechanisms are related to math. As developmental dyscalculia is characterized by a profound deficit in discriminating numerosity, it is fundamental to understand the mechanism behind the discrimination.

title: Numerosity Estimation in Visual Stimuli in the Absence of Luminance-Based Cues
abstract: Numerosity estimation is a basic preverbal ability that humans share with many animal species and that is believed to be foundational of numeracy skills. It is notoriously difficult, however, to establish whether numerosity estimation is based on numerosity itself, or on one or more non-numerical cues like—in visual stimuli—spatial extent and density. Frequently, different non-numerical cues are held constant on different trials. This strategy, however, still allows numerosity estimation to be based on a combination of non-numerical cues rather than on any particular one by itself.  Here we introduce a novel method, based on second-order (contrast-based) visual motion, to create stimuli that exclude all first-order (luminance-based) cues to numerosity. We show that numerosities can be estimated almost as well in second-order motion as in first-order motion.  The results show that numerosity estimation need not be based on first-order spatial filtering, first-order density perception, or any other processing of luminance-based cues to numerosity. Our method can be used as an effective tool to control non-numerical variables in studies of numerosity estimation.

title: Vision senses number directly.
abstract: We have recently suggested that numerosity is a primary sensory attribute, and shown that it is strongly susceptible to adaptation. Here we use the Method of Single Stimuli to show that observers can extract a running average of numerosity of a succession of stimuli to use as a standard of comparison for subsequent stimuli. On separate sessions observers judged whether the perceived numerosity or density of a particular trial was greater or less than the average of previous stimuli. Thresholds were as precise for this task as for explicit comparisons of test with standard stimuli. Importantly, we found no evidence that numerosity judgments are mediated by density. Under all conditions, judgements of numerosity were as precise as those of density. Thresholds in intermingled conditions, where numerosity varied unpredictably with density, were as precise as the blocked thresholds. Judgments in constant-density conditions were more precise thresholds than those in variable-density conditions, and numerosity judgements in conditions of constant-numerosity showed no tendency to follow density. We further report the novel finding that perceived numerosity increases with decreasing luminance, whereas texture density does not, further evidence for independent processing of the two attributes. All these measurements suggest that numerosity judgments can be, and are, made independently of judgments of the density of texture.

title: Neurocognitive start-up tools for symbolic number representations
abstract: Attaching meaning to arbitrary symbols (i.e. words) is a complex and lengthy process. In the case of numbers, it was previously suggested that this process is grounded on two early pre-verbal systems for numerical quantification: the approximate number system (ANS or 'analogue magnitude'), and the object tracking system (OTS or 'parallel individuation'), which children are equipped with before symbolic learning. Each system is based on dedicated neural circuits, characterized by specific computational limits, and each undergoes a separate developmental trajectory. Here, I review the available cognitive and neuroscientific data and argue that the available evidence is more consistent with a crucial role for the ANS, rather than for the OTS, in the acquisition of abstract numerical concepts that are uniquely human.

title: Visual sustained attention and numerosity sensitivity correlate with math achievement in children.
abstract: In this study, we investigated in school-age children the relationship among mathematical performance, the perception of numerosity (discrimination and mapping to number line), and sustained visual attention. The results (on 68 children between 8 and 11 years of age) show that attention and numerosity perception predict math scores but not reading performance. Even after controlling for several variables, including age, gender, nonverbal IQ, and reading accuracy, attention remained correlated with math skills and numerosity discrimination. These findings support previous reports showing the interrelationship between visual attention and both numerosity perception and math performance. It also suggests that attentional deficits may be implicated in disturbances such as developmental dyscalculia.

title: Association between individual differences in non-symbolic number acuity and math performance: a meta-analysis.
abstract: Many recent studies have examined the association between number acuity, which is the ability to rapidly and non-symbolically estimate the quantity of items appearing in a scene, and symbolic math performance. However, various contradictory results have been reported. To comprehensively evaluate the association between number acuity and symbolic math performance, we conduct a meta-analysis to synthesize the results observed in previous studies. First, a meta-analysis of cross-sectional studies (36 samples, N = 4705) revealed a significant positive correlation between these skills (r = 0.20, 95% CI = [0.14, 0.26]); the association remained after considering other potential moderators (e.g., whether general cognitive abilities were controlled). Moreover, a meta-analysis of longitudinal studies revealed 1) that number acuity may prospectively predict later math performance (r = 0.24, 95% CI = [0.11, 0.37]; 6 samples) and 2) that number acuity is retrospectively correlated to early math performance as well (r = 0.17, 95% CI = [0.07, 0.26]; 5 samples). In summary, these pieces of evidence demonstrate a moderate but statistically significant association between number acuity and math performance. Based on the estimated effect sizes, power analyses were conducted, which suggested that many previous studies were underpowered due to small sample sizes. This may account for the disparity between findings in the literature, at least in part. Finally, the theoretical and practical implications of our meta-analytic findings are presented, and future research questions are discussed.

title: Individual differences in non-verbal number acuity correlate with maths achievement
abstract: Human mathematical competence emerges from two representational systems. Competence in some domains of mathematics, such as calculus, relies on symbolic representations that are unique to humans who have undergone explicit teaching. More basic numerical intuitions are supported by an evolutionarily ancient approximate number system that is shared by adults, infants and non-human animals—these groups can all represent the approximate number of items in visual or auditory arrays without verbally counting, and use this capacity to guide everyday behaviour such as foraging. Despite the widespread nature of the approximate number system both across species and across development, it is not known whether some individuals have a more precise non-verbal ‘number sense’ than others. Furthermore, the extent to which this system interfaces with the formal, symbolic maths abilities that humans acquire by explicit instruction remains unknown. Here we show that there are large individual differences in the non-verbal approximation abilities of 14-year-old children, and that these individual differences in the present correlate with children’s past scores on standardized maths achievement tests, extending all the way back to kindergarten. Moreover, this correlation remains significant when controlling for individual differences in other cognitive and performance factors. Our results show that individual differences in achievement in school mathematics are related to individual differences in the acuity of an evolutionarily ancient, unlearned approximate number sense. Further research will determine whether early differences in number sense acuity affect later maths learning, whether maths education enhances number sense acuity, and the extent to which tertiary factors can affect both.

title: Preschool acuity of the approximate number system correlates with school math ability.
abstract: Previous research shows a correlation between individual differences in people's school math abilities and the accuracy with which they rapidly and nonverbally approximate how many items are in a scene. This finding is surprising because the Approximate Number System (ANS) underlying numerical estimation is shared with infants and with non-human animals who never acquire formal mathematics. However, it remains unclear whether the link between individual differences in math ability and the ANS depends on formal mathematics instruction. Earlier studies demonstrating this link tested participants only after they had received many years of mathematics education, or assessed participants' ANS acuity using tasks that required additional symbolic or arithmetic processing similar to that required in standardized math tests. To ask whether the ANS and math ability are linked early in life, we measured the ANS acuity of 200 3- to 5-year-old children using a task that did not also require symbol use or arithmetic calculation. We also measured children's math ability and vocabulary size prior to the onset of formal math instruction. We found that children's ANS acuity correlated with their math ability, even when age and verbal skills were controlled for. These findings provide evidence for a relationship between the primitive sense of number and math ability starting early in life.

title: Number sense in infancy predicts mathematical abilities in childhood
abstract: The uniquely human mathematical mind sets us apart from all other animals. How does this powerful capacity emerge over development? It is uncontroversial that education and environment shape mathematical ability, yet an untested assumption is that number sense in infants is a conceptual precursor that seeds human mathematical development. Our results provide the first support for this hypothesis. We found that preverbal number sense in 6-month-old infants predicted standardized math scores in the same children 3 years later. This discovery shows that number sense in infancy is a building block for later mathematical ability and invites educational interventions to improve number sense even before children learn to count. Human infants in the first year of life possess an intuitive sense of number. This preverbal number sense may serve as a developmental building block for the uniquely human capacity for mathematics. In support of this idea, several studies have demonstrated that nonverbal number sense is correlated with mathematical abilities in children and adults. However, there has been no direct evidence that infant numerical abilities are related to mathematical abilities later in childhood. Here, we provide evidence that preverbal number sense in infancy predicts mathematical abilities in preschool-aged children. Numerical preference scores at 6 months of age correlated with both standardized math test scores and nonsymbolic number comparison scores at 3.5 years of age, suggesting that preverbal number sense facilitates the acquisition of numerical symbols and mathematical abilities. This relationship held even after controlling for general intelligence, indicating that preverbal number sense imparts a unique contribution to mathematical ability. These results validate the many prior studies purporting to show number sense in infancy and support the hypothesis that mathematics is built upon an intuitive sense of number that predates language.

title: The Approximate Number System is not Predictive for Symbolic Number Processing in Kindergarteners
abstract: The relation between the approximate number system (ANS) and symbolic number processing skills remains unclear. Some theories assume that children acquire the numerical meaning of symbols by mapping them onto the preexisting ANS. Others suggest that in addition to the ANS, children also develop a separate, exact representational system for symbolic number processing. In the current study, we contribute to this debate by investigating whether the nonsymbolic number processing of kindergarteners is predictive for symbolic number processing. Results revealed no association between the accuracy of the kindergarteners on a nonsymbolic number comparison task and their performance on the symbolic comparison task six months later, suggesting that there are two distinct representational systems for the ANS and numerical symbols.

title: Impaired acuity of the approximate number system underlies mathematical learning disability (dyscalculia).
abstract: Many children have significant mathematical learning disabilities (MLD, or dyscalculia) despite adequate schooling. The current study hypothesizes that MLD partly results from a deficiency in the Approximate Number System (ANS) that supports nonverbal numerical representations across species and throughout development. In this study of 71 ninth graders, it is shown that students with MLD have significantly poorer ANS precision than students in all other mathematics achievement groups (low, typically, and high achieving), as measured by psychophysical assessments of ANS acuity (w) and of the mappings between ANS representations and number words (cv). This relation persists even when controlling for domain-general abilities. Furthermore, this ANS precision does not differentiate low-achieving from typically achieving students, suggesting an ANS deficit that is specific to MLD.

title: Developmental trajectory of number acuity reveals a severe impairment in developmental dyscalculia
abstract: Developmental dyscalculia is a learning disability that affects the acquisition of knowledge about numbers and arithmetic. It is widely assumed that numeracy is rooted on the "number sense", a core ability to grasp numerical quantities that humans share with other animals and deploy spontaneously at birth. To probe the links between number sense and dyscalculia, we used a psychophysical test to measure the Weber fraction for the numerosity of sets of dots, hereafter called number acuity. We show that number acuity improves with age in typically developing children. In dyscalculics, numerical acuity is severely impaired, with 10-year-old dyscalculics scoring at the level of 5-year-old normally developing children. Moreover, the severity of the number acuity impairment predicts the defective performance on tasks involving the manipulation of symbolic numbers. These results establish for the first time a clear association between dyscalculia and impaired "number sense", and they may open up new horizons for the early diagnosis and rehabilitation of mathematical learning deficits.

title: Basic numerical skills in children with mathematics learning disabilities: A comparison of symbolic vs non-symbolic number magnitude processing
abstract: Forty-five children with mathematics learning disabilities, with and without comorbid reading disabilities, were compared to 45 normally achieving peers in tasks assessing basic numerical skills. Children with mathematics disabilities were only impaired when comparing Arabic digits (i.e., symbolic number magnitude) but not when comparing collections (i.e., non-symbolic number magnitude). Moreover, they automatically processed number magnitude when comparing the physical size of Arabic digits in an Stroop paradigm adapted for processing speed differences. Finally, no evidence was found for differential patterns of performance between MD and MD/RD children in these tasks. These findings suggest that children with mathematics learning disabilities have difficulty in accessing number magnitude from symbols rather than in processing numerosity per se.

title: Mathematical difficulties in developmental coordination disorder: Symbolic and nonsymbolic number processing.
abstract: At school, children with Developmental Coordination Disorder (DCD) struggle with mathematics. However, little attention has been paid to their numerical cognition abilities. The goal of this study was to better understand the cognitive basis for mathematical difficulties in children with DCD. Twenty 7-to-10 years-old children with DCD were compared to twenty age-matched typically developing children using dot and digit comparison tasks to assess symbolic and nonsymbolic number processing and in a task of single digits additions. Results showed that children with DCD had lower performance in nonsymbolic and symbolic number comparison tasks than typically developing children. They were also slower to solve simple addition problems. Moreover, correlational analyses showed that children with DCD who experienced greater impairments in the nonsymbolic task also performed more poorly in the symbolic tasks. These findings suggest that DCD impairs both nonsymbolic and symbolic number processing. A systematic assessment of numerical cognition in children with DCD could provide a more comprehensive picture of their deficits and help in proposing specific remediation.

title: Compromised approximate number system acuity in extremely preterm school‐aged children
abstract: The aim of this study was to compare the approximate number system acuity in children born extremely preterm aged 6 years 6 months and typically developing, age‐matched peers.

title: Time, number and attention in very low birth weight children
abstract: Premature birth has been associated with damage in many regions of the cerebral cortex, although there is a particularly strong susceptibility for damage within the parieto-occipital lobes (Volpe, 2009). As these areas have been shown to be critical for both visual attention and magnitudes perception (time, space, and number), it is important to investigate the impact of prematurity on both the magnitude and attentional systems, particularly for children without overt white matter injuries, where the lack of obvious injury may cause their difficulties to remain unnoticed. In this study, we investigated the ability to judge time intervals (visual, audio and audio-visual temporal bisection), discriminate between numerical quantities (numerosity comparison), map numbers onto space (numberline task) and to maintain visuo-spatial attention (multiple-object-tracking) in school-age preterm children (N29). The results show that various parietal functions may be more or less robust to prematurity-related difficulties, with strong impairments found on time estimation and attentional task, while numerical discrimination or mapping tasks remained relatively unimpaired. Thus while our study generally supports the hypothesis of a dorsal stream vulnerability in children born preterm relative to other cortical locations, it further suggests that particular cognitive processes, as highlighted by performance on different tasks, are far more susceptible than others.

title: Discrete and analogue quantity processing in the parietal lobe: a functional MRI study.
abstract: The human intraparietal sulcus (IPS) is implicated in processing symbolic number information and possibly in nonsymbolic number information. Specific IPS activity for discrete quantities (numerosities) as compared with continuous, analogue quantity has not been demonstrated. Here we use a stimulus-driven paradigm to distinguish automatic estimation of "how many things" from "how much" and "how long." The discrete analogue response task (DART) uses the perception of hues which can change either abruptly (discrete, numerous stimuli) or smoothly (analogue, nonnumerous stimuli) in space or in time. Subjects decide whether they saw more green or more blue. A conjunction analysis of spatial and temporal conditions revealed that bilateral IPS was significantly more active during the processing of discrete stimuli than during analogue stimuli, as was a parietal-occipital transition zone. We suggest that processing numerosity is a distinct process from processing analogue quantity, whether extended in space or time, and that an intraparietal network connects objects' segmentation to the estimation of their numerosity.