No intrinsic number bias: Evaluating the role of perceptual discriminability in magnitude categorization.

Accumulating evidence suggests that there is a spontaneous preference for numerical, compared to non-numerical (e.g., cumulative surface area), information. However, given a paucity of research on the perception of non-numerical magnitudes, it is unclear whether this preference reflects a specific bias towards number, or a general bias towards the more perceptually discriminable dimension (i.e., number). Here, we found that when the number and area of visual dot displays were matched in mathematical ratio, number was more perceptually discriminable than area in both adults and children. Moreover, both adults and children preferentially categorized these ratio-matched stimuli based on number, consistent with previous work. However, when number and area were matched in perceptual discriminability, a different pattern of results emerged. In particular, children preferentially categorized stimuli based on area, suggesting that children's previously observed number bias may be due to a mismatch in the perceptual discriminability of number and area, not an intrinsic salience of number. Interestingly, adults continued to categorize the displays on the basis of number. Altogether, these findings suggest a dominant role for area during childhood, refuting the claim that number is inherently and uniquely salient. Yet they also reveal an increased salience of number that emerges over development. Potential explanations for this developmental shift are discussed. RESEARCH HIGHLIGHTS: Previous work found that children and adults spontaneously categorized dot array stimuli by number, over other magnitudes (e.g., area), suggesting number is uniquely salient. However, here we found that when number and area were matched by ratio, as in prior work, number was significantly more perceptually discriminable than area. When number and area were made equally discriminable ('perceptually-matched'), children, contra adults, spontaneously categorized stimuli by area over number (and other non-numerical magnitudes). These findings suggest that area may be uniquely salient early in childhood, with the previously-observed number bias not emerging until later in development.

Perceived number is not abstract.

To support the claim that the approximate number system (ANS) represents rational numbers, Clarke and Beck (C&B) argue that number perception is abstract and characterized by a second-order character. However, converging evidence from visual illusions and psychophysics suggests that perceived number is not abstract, but rather, is perceptually interdependent with other magnitudes. Moreover, number, as a concept, is second-order, but number, as a percept, is not.

Does training mental rotation transfer to gains in mathematical competence? Assessment of an at-home visuospatial intervention.

The current study examined whether the effect of spatial training transfers to the math domain. Sixty-two 6- and 7-year-olds completed an at-home 1-week online training intervention. The spatial-training group received mental rotation training, whereas the active control group received literacy training in a format that matched the spatial training. Results revealed near transfer of mental rotation ability in the spatial-training group. More importantly, there was also far transfer to canonical arithmetic problems, such that children in the spatial-training group performed better on these math problems than children in the control group. Such far transfer could not be attributed to general cognitive improvement, since no improvement was observed for non-symbolic quantity processing, verbal working memory (WM), or language ability following spatial training. Spatial training may have benefitted symbolic arithmetic performance by improving visualization ability, access to the mental number line, and/or increasing the capacity of visuospatial WM.

Canine sense of quantity: evidence for numerical ratio-dependent activation in parietotemporal cortex.

The approximate number system (ANS), which supports the rapid estimation of quantity, emerges early in human development and is widespread across species. Neural evidence from both human and non-human primates suggests the parietal cortex as a primary locus of numerical estimation, but it is unclear whether the numerical competencies observed across non-primate species are subserved by similar neural mechanisms. Moreover, because studies with non-human animals typically involve extensive training, little is known about the spontaneous numerical capacities of non-human animals. To address these questions, we examined the neural underpinnings of number perception using awake canine functional magnetic resonance imaging. Dogs passively viewed dot arrays that varied in ratio and, critically, received no task-relevant training or exposure prior to testing. We found evidence of ratio-dependent activation, which is a key feature of the ANS, in canine parietotemporal cortex in the majority of dogs tested. This finding is suggestive of a neural mechanism for quantity perception that has been conserved across mammalian evolution.

Cross-magnitude interactions across development: Longitudinal evidence for a general magnitude system.

There is general agreement that humans represent numerical, spatial, and temporal magnitudes from early in development. However, there is disagreement about whether different magnitudes converge within a general magnitude system and whether this system supports behavioral demonstrations of cross-magnitude interactions at different developmental time points. Using a longitudinal design, we found a relation between children's cross-magnitude interactions assessed at two developmental time points with different behavioral measures. More specifically, stronger cross-magnitude interactions in infancy (M = 9.3 months) predicted a stronger cross-magnitude congruity effect at preschool age (M = 44.2 months), even when controlling for performance on measures of inhibitory control, analogical reasoning, and verbal competence at preschool age. The results suggest a common mechanism for cross-magnitude interactions at different points in development as well as stability of the underlying individual differences. We argue that this mechanism reflects a nonverbal general magnitude system that is operational early in life and that displays continuity from infancy to preschool age.

Does 1 + 1 = 2nd? The relations between children's understanding of ordinal position and their arithmetic performance.

The current study examined the relations between 5- and 6-year-olds' understanding of ordinality and their mathematical competence. We focused specifically on "positional operations," a property of ordinality not contingent on magnitude, in an effort to better understand the unique contributions of position-based ordinality to math development. Our findings revealed that two types of positional operations-the ability to execute representational movement along letter sequences and the ability to update ordinal positions after item insertion or removal-predicted children's arithmetic performance. Nevertheless, these positional operations did not mediate the relation between magnitude processing (as measured by the acuity of the approximate number system) and arithmetic performance. Taken together, these findings suggest a unique role for positional ordinality in math development. We suggest that positional ordinality may aid children in their mental organization of number symbols, which may facilitate solving arithmetic computations and may support the development of novel numerical concepts.

Skeletal representations of shape in human vision: Evidence for a pruned medial axis model.

A representation of shape that is low dimensional and stable across minor disruptions is critical for object recognition. Computer vision research suggests that such a representation can be supported by the medial axis-a computational model for extracting a shape's internal skeleton. However, few studies have shown evidence of medial axis processing in humans, and even fewer have examined how the medial axis is extracted in the presence of disruptive contours. Here, we tested whether human skeletal representations of shape reflect the medial axis transform (MAT), a computation sensitive to all available contours, or a pruned medial axis, which ignores contours that may be considered "noise." Across three experiments, participants (N = 2062) were shown complete, perturbed, or illusory two-dimensional shapes on a tablet computer and were asked to tap the shapes anywhere once. When directly compared with another viable model of shape perception (based on principal axes), participants' collective responses were better fit by the medial axis, and a direct test of boundary avoidance suggested that this result was not likely because of a task-specific cognitive strategy (Experiment 1). Moreover, participants' responses reflected a pruned computation in shapes with small or large internal or external perturbations (Experiment 2) and under conditions of illusory contours (Experiment 3). These findings extend previous work by suggesting that humans extract a relatively stable medial axis of shapes. A relatively stable skeletal representation, reflected by a pruned model, may be well equipped to support real-world shape perception and object recognition.

Action ability modulates time-to-collision judgments.

Time-to-collision (TTC) underestimation has been interpreted as an adaptive response that allows observers to have more time to engage in a defensive behaviour. This bias seems, therefore, strongly linked to action preparation. There is evidence that the observer's physical fitness modulates the underestimation effect so that people who need more time to react (i.e. those with less physical fitness) show a stronger underestimation effect. Here we investigated whether this bias is influenced by the momentary action capability of the observers. In the first experiment, participants estimated the time-to-collision of threatening or non-threatening stimuli while being mildly immobilized (with a chin rest) or while standing freely. Having reduced the possibility of movement led participants to show more underestimation of the approaching stimuli. However, this effect was not stronger for threatening relative to non-threatening stimuli. The effect of the action capability found in the first experiment could be interpreted as an expansion of peripersonal space (PPS). In the second experiment, we thus investigated the generality of this effect using an established paradigm to measure the size of peripersonal space. Participants bisected lines from different distances while in the chin rest or standing freely. The results replicated the classic left-to-right gradient in lateral spatial attention with increasing viewing distance, but no effect of immobilization was found. The manipulation of the momentary action capability of the observers influenced the participants' performance in the TTC task but not in the line bisection task. These results are discussed in relation to the different functions of PPS.

Representations of numerical and non-numerical magnitude both contribute to mathematical competence in children.

A growing body of evidence suggests that non-symbolic representations of number, which humans share with nonhuman animals, are functionally related to uniquely human mathematical thought. Other research suggesting that numerical and non-numerical magnitudes not only share analog format but also form part of a general magnitude system raises questions about whether the non-symbolic basis of mathematical thinking is unique to numerical magnitude. Here we examined this issue in 5- and 6-year-old children using comparison tasks of non-symbolic number arrays and cumulative area as well as standardized tests of math competence. One set of findings revealed that scores on both magnitude comparison tasks were modulated by ratio, consistent with shared analog format. Moreover, scores on these tasks were moderately correlated, suggesting overlap in the precision of numerical and non-numerical magnitudes, as expected under a general magnitude system. Another set of findings revealed that the precision of both types of magnitude contributed shared and unique variance to the same math measures (e.g. calculation and geometry), after accounting for age and verbal competence. These findings argue against an exclusive role for non-symbolic number in supporting early mathematical understanding. Moreover, they suggest that mathematical understanding may be rooted in a general system of magnitude representation that is not specific to numerical magnitude but that also encompasses non-numerical magnitude.

Transitive inference of social dominance by human infants.

It is surprising that there are inconsistent findings of transitive inference (TI) in young infants given that non-linguistic species succeed on TI tests. To conclusively test for TI in infants, we developed a task within the social domain, with which infants are known to show sophistication. We familiarized 10- to 13-month-olds (M = 11.53 months) to a video of two dominance interactions between three puppets (bear > elephant; hippo > bear) consistent with a dominance hierarchy (hippo > bear > elephant; where '>' denotes greater dominance). Infants then viewed interactions between the two puppets that had not interacted during familiarization. These interactions were either congruent (hippo > elephant) or incongruent (elephant > hippo) with the inferred hierarchy. Consistent with TI, infants looked longer to incongruent than congruent displays. Control conditions ruled out the possibility that infants' expectations were based on stable behaviors specific to individual puppets rather than their inferred transitive dominance relations. We suggest that TI may be supported by phylogenetically ancient mechanisms of ordinal representation and visuospatial processing that come online early in human development.

Right idea, wrong magnitude system.

Leibovich et al. claim that number representations are non-existent early in life and that the associations between number and continuous magnitudes reside in stimulus confounds. We challenge both claims - positing, instead, that number is represented independently of continuous magnitudes already in infancy, but is nonetheless more deeply connected to other magnitudes through adulthood than acknowledged by the "sense of magnitude" theory.

Spatial Processing in Infancy Predicts Both Spatial and Mathematical Aptitude in Childhood.

Despite considerable interest in the role of spatial intelligence in science, technology, engineering, and mathematics (STEM) achievement, little is known about the ontogenetic origins of individual differences in spatial aptitude or their relation to later accomplishments in STEM disciplines. The current study provides evidence that spatial processes present in infancy predict interindividual variation in both spatial and mathematical competence later in development. Using a longitudinal design, we found that children's performance on a brief visuospatial change-detection task administered between 6 and 13 months of age was related to their spatial aptitude (i.e., mental-transformation skill) and mastery of symbolic-math concepts at 4 years of age, even when we controlled for general cognitive abilities and spatial memory. These results suggest that nascent spatial processes present in the first year of life not only act as precursors to later spatial intelligence but also predict math achievement during childhood.

The associations between space and order in numerical and non-numerical sequences.

Spatial-numerical associations have been found across different studies, yet the basis for these associations remains debated. The current study employed an order judgment task to adjudicate between two competing accounts of such associations, namely the Mental Number Line (MNL) and Working Memory (WM) models. On this task, participants judged whether number pairs were in ascending or descending order. Whereas the MNL model predicts that ascending and descending orders should map onto opposite sides of space, the WM model predicts no such mapping. Moreover, we compared the spatial-order mapping for numerical and non-numerical sequences because the WM model predicts no difference in mapping. Across two experiments, we found consistent spatial mappings for numerical order along both horizontal and vertical axes, consistent with a MNL model. In contrast, we found no consistent mappings for letter sequences. These findings are discussed in the context of conflicting extant data related to these two models.

Threat modulates neural responses to looming visual stimuli.

Objects on a collision course with an observer produce a specific pattern of optical expansion on the retina known as looming, which in theory exactly specifies the time-to-collision (TTC) of approaching objects. It was recently demonstrated that the affective content of looming stimuli influences perceived TTC, with threatening objects judged as approaching sooner than non-threatening objects. Here, the neural mechanisms by which perceived threat modulates spatiotemporal perception were investigated. Participants judged the TTC of threatening (snakes, spiders) or non-threatening (butterflies, rabbits) stimuli, which expanded in size at a rate indicating one of five TTCs. Visual-evoked potentials (VEPs) and oscillatory neural responses measured with electroencephalography were analysed. The arrival time of threatening stimuli was underestimated compared with non-threatening stimuli, though an interaction suggested that this underestimation was not constant across TTCs. Further, both speed of approach and threat modulated both VEPs and oscillatory responses. Speed of approach modulated the N1 parietal and oscillations in the beta band. Threat modulated several VEP components (P1, N1 frontal, N1 occipital, early posterior negativity and late positive potential) and oscillations in the alpha and high gamma band. The results for the high gamma band suggest an interaction between these two factors. Previous evidence suggests that looming stimuli activate sensorimotor areas, even in the absence of an intended action. The current results show that threat disrupts the synchronization over the sensorimotor areas that are likely activated by the presentation of a looming stimulus.

The potentiation of geometry by features in human children: Evidence against modularity in the domain of navigation.

Accumulating evidence demonstrates that humans and other animals use geometric information, such as the shape of a surrounding space, to recover from disorientation. Less clear is to what extent human children integrate geometry with featural cues, such as the color of walls within an enclosed space, for this purpose. One view holds that reorientation relies on a cognitive module that processes geometric information independently of features. Here we provide evidence against this position by demonstrating that prior exposure to features within a kite-shaped space facilitated the use of geometry in 3- and 4-year-old children, as has been shown with nonhuman animals. Children were tasked with localizing a hidden object within a kite space following disorientation. Their performance was compared across two blocks of trials. We found that children first exposed to features (two black walls and two white walls) within the kite space (first block) were subsequently better at relying on the space's geometry to localize the target object (second block) than children not previously exposed to features. Follow-up experiments ruled out nonspecific effects of practice and attention. Not only did featural cues interact with the processing of geometry, but also features specifically enhanced children's representations of the space's geometry, which they used for reorientation. We suggest that this potentiation of geometry was possible because the placement of wall colors highlighted the major axis of the kite space, which may be critical for aiding the encoding of global shape or for maintaining the representation of a complex geometry in memory.

Nonsymbolic number and cumulative area representations contribute shared and unique variance to symbolic math competence.

Humans and nonhuman animals share the capacity to estimate, without counting, the number of objects in a set by relying on an approximate number system (ANS). Only humans, however, learn the concepts and operations of symbolic mathematics. Despite vast differences between these two systems of quantification, neural and behavioral findings suggest functional connections. Another line of research suggests that the ANS is part of a larger, more general system of magnitude representation. Reports of cognitive interactions and common neural coding for number and other magnitudes such as spatial extent led us to ask whether, and how, nonnumerical magnitude interfaces with mathematical competence. On two magnitude comparison tasks, college students estimated (without counting or explicit calculation) which of two arrays was greater in number or cumulative area. They also completed a battery of standardized math tests. Individual differences in both number and cumulative area precision (measured by accuracy on the magnitude comparison tasks) correlated with interindividual variability in math competence, particularly advanced arithmetic and geometry, even after accounting for general aspects of intelligence. Moreover, analyses revealed that whereas number precision contributed unique variance to advanced arithmetic, cumulative area precision contributed unique variance to geometry. Taken together, these results provide evidence for shared and unique contributions of nonsymbolic number and cumulative area representations to formally taught mathematics. More broadly, they suggest that uniquely human branches of mathematics interface with an evolutionarily primitive general magnitude system, which includes partially overlapping representations of numerical and nonnumerical magnitude.

Representations of numerical sequences and the concept of middle in preschoolers.

The present study concerns preschoolers' understanding of the middle concept as it applies to numerical sequences. Previous research using implicit psychophysical assessment suggests that the numerical midpoint is embedded within numerical representations by 4 years of age. Here, we examined 3- to 5-year-olds' ability to identify the midpoint value in triplets of non-symbolic numbers when explicitly probed to do so. We found that whereas 4- and 5-year-olds were capable of explicit access to numerical midpoint values and showed ratio-dependent performance, a signature of the approximate number system (ANS), 3-year-olds performed at chance. Children's difficulty in identifying numerical midpoint values was not due to comparing multiple arrays, nor was it entirely due to a spatial association with the word "middle" used in the task. We speculate that explicit access to numerical midpoint values may be jointly supported by endogenous control of attentional mechanisms and the development of a mental number line.

Object geometry serves humans' intuitive physics of stability.

How do humans judge physical stability? A prevalent account emphasizes the mental simulation of physical events implemented by an intuitive physics engine in the mind. Here we test the extent to which the perceptual features of object geometry are sufficient for supporting judgments of falling direction. In all experiments, adults and children judged the falling direction of a tilted object and, across experiments, objects differed in the geometric features (i.e., geometric centroid, object height, base size and/or aspect ratio) relevant to the judgment. Participants' performance was compared to computational models trained on geometric features, as well as a deep convolutional neural network (ResNet-50), none of which incorporated mental simulation. Adult and child participants' performance was well fit by models of object geometry, particularly the geometric centroid. ResNet-50 also provided a good account of human performance. Altogether, our findings suggest that object geometry may be sufficient for judging the falling direction of tilted objects, independent of mental simulation.

Visual looming is a primitive for human emotion.

Looming objects afford threat of collision across the animal kingdom. Defensive responses to looming and neural computations for looming detection are strikingly conserved across species. In mammals, information about rapidly approaching threats is conveyed from the retina to the midbrain superior colliculus, where variables that indicate the position and velocity of approach are computed to enable defensive behavior. Although neuroscientific theories posit that midbrain representations contribute to emotion through connectivity with distributed brain systems, it remains unknown whether a computational system for looming detection can predict both defensive behavior and phenomenal experience in humans. Here, we show that a shallow convolutional neural network based on the Drosophila visual system predicts defensive blinking to looming objects in infants and superior colliculus responses to optical expansion in adults. Further, the responses of the convolutional network to a broad array of naturalistic video clips predict self-reported emotion largely on the basis of subjective arousal. Our findings illustrate how motor and experiential components of human emotion relate to species-general systems for survival in unpredictable environments.

Visual looming is a primitive for human emotion.

The neural computations for looming detection are strikingly similar across species. In mammals, information about approaching threats is conveyed from the retina to the midbrain superior colliculus, where approach variables are computed to enable defensive behavior. Although neuroscientific theories posit that midbrain representations contribute to emotion through connectivity with distributed brain systems, it remains unknown whether a computational system for looming detection can predict both defensive behavior and phenomenal experience in humans. Here, we show that a shallow convolutional neural network based on the Drosophila visual system predicts defensive blinking to looming objects in infants and superior colliculus responses to optical expansion in adults. Further, the neural network's responses to naturalistic video clips predict self-reported emotion largely by way of subjective arousal. These findings illustrate how a simple neural network architecture optimized for a species-general task relevant for survival explains motor and experiential components of human emotion.