Modeling the approximate number system to quantify the contribution of visual stimulus features

Adaptation to numerosity affects the pupillary light response

We recently showed that the gain of the pupillary light response depends on numerosity, with weaker responses to fewer items. Here we show that this effect holds when the stimuli are physically identical but are perceived as less numerous due to numerosity adaptation. Twenty-eight participants adapted to low (10 dots) or high (160 dots) numerosities and subsequently watched arrays of 10-40 dots, with variable or homogeneous dot size. Luminance was constant across all stimuli. Pupil size was measured with passive viewing, and the effects of adaptation were checked in a separate psychophysical session. We found that perceived numerosity was systematically lower, and pupillary light responses correspondingly smaller, following adaptation to high rather than low numerosities. This is consistent with numerosity being a primary visual feature, spontaneously encoded even when task irrelevant, and affecting automatic and unconscious behaviours like the pupillary light response.

Disparate processing of numerosity and associated continuous magnitudes in rats

The studies of number sense in different species are severely hampered by the inevitable entanglement of non-numerical attributes inherent in nonsymbolic stimuli representing numerosity, resulting in contrasting theories of numerosity processing. Here, we developed an algorithm and associated analytical methods to generate stimuli that not only minimized the impact of non-numerical magnitudes in numerosity perception but also allowed their quantification. We trained number-naïve rats with these stimuli as sound pulses representing two or three numbers and demonstrated that their numerical discrimination ability mainly relied on numerosity. Also, studying the learning process revealed that rats used numerosity before using magnitudes for choices. This numerical processing could be impaired specifically by silencing the posterior parietal cortex. Furthermore, modeling this capacity by neural networks shed light on the separation of numerosity and magnitudes extraction. Our study helps dissect the relationship between magnitude and numerosity processing, and the above different findings together affirm the independent existence of innate number and magnitudes sense in rats.

Modeling Magnitude Discrimination: Effects of Internal Precision and Attentional Weighting of Feature Dimensions

Given a rich environment, how do we decide on what information to use? A view of a single entity (e.g., a group of birds) affords many distinct interpretations, including their number, average size, and spatial extent. An enduring challenge for cognition, therefore, is to focus resources on the most relevant evidence for any particular decision. In the present study, subjects completed three tasks-number discrimination, surface area discrimination, and convex hull discrimination-with the same stimulus set, where these three features were orthogonalized. Therefore, only the relevant feature provided consistent evidence for decisions in each task. This allowed us to determine how well humans discriminate each feature dimension and what evidence they relied on to do so. We introduce a novel computational approach that fits both feature precision and feature use. We found that the most relevant feature for each decision is extracted and relied on, with minor contributions from competing features. These results suggest that multiple feature dimensions are separately represented for each attended ensemble of many items and that cognition is efficient at selecting the appropriate evidence for a decision.

Effects of spatial frequency cross-adaptation on the visual number sense

When observing a simple visual scene such as an array of dots, observers can easily and automatically extract their number. How does our visual system accomplish this? We investigate the role of specific spatial frequencies to the encoding of number through cross-adaptation. In two experiments, observers were peripherally adapted to six randomly generated sinusoidal gratings varying from relatively low-spatial frequency (M = 0.44 c/deg) to relatively high-spatial frequency (M = 5.88 c/deg). Subsequently, observers judged which side of the screen had a higher number of dots. We found a strong number-adaptation effect to low-spatial frequency gratings (i.e., participants significantly underestimated the number of dots on the adapted side) but a significantly reduced adaptation effect for high-spatial frequency gratings. Various control conditions demonstrate that these effects are not due to a generic response bias for the adapted side, nor moderated by dot size or spacing effects. In a third experiment, we observed no cross-adaptation for centrally presented gratings. Our results show that observers' peripheral number perception can be adapted even with stimuli lacking any numeric or segmented object information and that low spatial frequencies adapt peripheral number perception more than high ones. Together, our results are consistent with recent number perception models that suggest a key role for spatial frequency in the extraction of number from the visual signal (e.g., Paul, Ackooij, Ten Cate, & Harvey, 2022), but additionally suggest that some spatial frequencies - especially in the low range and in the periphery - may be weighted more by the visual system when estimating number. We argue that the cross-adaptation paradigm is also a useful methodology for discovering the primitives of visual number encoding.

The Interaction between Congruency and Numerical Ratio Effects in the Nonsymbolic Comparison Test

The nonsymbolic comparison task is used to investigate the precision of the Approximate Number Sense, the ability to process discrete numerosity without counting and symbols. There is an ongoing debate regarding the extent to which the ANS is influenced by the processing of non-numerical visual cues. To address this question, we assessed the congruency effect in a nonsymbolic comparison task, examining its variability across different stimulus presentation formats and numerical proportions. Additionally, we examined the variability of the numerical ratio effect with the format and congruency. Utilizing generalized linear mixed-effects models with a sample of 290 students (89% female, mean age 19.33 years), we estimated the congruency effect and numerical ratio effect for separated and intermixed formats of stimulus presentation, and for small and large numerical proportions. The findings indicated that the congruency effect increased in large numerical proportion conditions, but this pattern was observed only in the separated format. In the intermixed format, the congruency effect was insignificant for both types of numerical proportion. Notably, the numerical ratio effect varied for congruent and incongruent trials in different formats. The results may suggest that the processing of visual non-numerical parameters may be crucial when numerosity processing becomes noisier, specifically when numerical proportion becomes larger. The implications of these findings for refining the ANS theory are discussed.

Development of precision of non-symbolic arithmetic operations in 4-6-year-old children

Children can represent the approximate quantity of sets of items using the Approximate Number System (ANS), and can perform arithmetic-like operations over ANS representations. Previous work has shown that the representational precision of the ANS develops substantially during childhood. However, less is known about the development of the operational precision of the ANS. We examined developmental change in the precision of the solutions to two non-symbolic arithmetic operations in 4-6-year-old U.S. children. We asked children to represent the quantity of an occluded set (Baseline condition), to compute the sum of two sequentially occluded arrays (Addition condition), or to infer the quantity of an addend after observing an initial array and then the array incremented by the unknown addend (Unknown-addend condition). We measured the precision of the solutions of these operations by asking children to compare their solutions to visible arrays, manipulating the ratio between the true quantity of the solution and the comparison array. We found that the precision of ANS representations that were not the result of operations (in the Baseline condition) was higher than the precision of solutions to ANS operations (in the Addition and Unknown-addend conditions). Further, we found that precision in the Baseline and Addition conditions improved significantly between 4 and 6 years, while precision in the Unknown-Addend condition did not. Our results suggest that ANS operations may inject "noise" into the representations they operate over, and that the development of the precision of different operations may follow different trajectories in childhood.

Studying the neural representations of uncertainty

The study of the brain's representations of uncertainty is a central topic in neuroscience. Unlike most quantities of which the neural representation is studied, uncertainty is a property of an observer's beliefs about the world, which poses specific methodological challenges. We analyze how the literature on the neural representations of uncertainty addresses those challenges and distinguish between 'code-driven' and 'correlational' approaches. Code-driven approaches make assumptions about the neural code for representing world states and the associated uncertainty. By contrast, correlational approaches search for relationships between uncertainty and neural activity without constraints on the neural representation of the world state that this uncertainty accompanies. To compare these two approaches, we apply several criteria for neural representations: sensitivity, specificity, invariance and functionality. Our analysis reveals that the two approaches lead to different but complementary findings, shaping new research questions and guiding future experiments.

Electrophysiological Comparison of Cumulative Area and Non-Symbolic Number Judgments

Despite the importance of representing different magnitudes (i.e., number and cumulative area) for action planning and formal mathematics, there is much debate about the nature of these representations, particularly the extent to which magnitudes interact in the mind and brain. Early interaction views suggest that there are shared perceptual processes that form overlapping magnitude representations. However, late interaction views hold that representations of different magnitudes remain distinct, interacting only when preparing a motor response. The present study sheds light on this debate by examining the temporal onset of ratio and congruity effects as participants made ordinal judgments about number and cumulative area. Event-related potentials (ERPs) were recorded to identify whether the onset of such effects aligned with early versus late views. Ratio effects for both magnitudes were observed starting in the P100. Moreover, a congruity effect emerged within the P100. That interactions were observed early in processing, at the same time that initial ratio effects occurred, suggests that number and cumulative area processes interacted when magnitude representations were being formed, prior to preparing a decision response. Our findings are consistent with an early interaction view of magnitude processing, in which number and cumulative area may rely on shared perceptual mechanisms.

Numerosity Comparison in Three Dimensions in the Case of Low Numerical Values

This study investigated the perception of numbers in humans in 3D stimuli. Recent research has shown that number processing relies on "number sense" for small values, in line with Weber's law. While previous studies have reported 3D numerosity overestimation mainly in higher numerical values, our experiment examined whether this phenomenon occurs at lower numerical values. We also explored whether the Weber ratio follows Weber's law when comparing 2D and 3D stimuli in terms of the number of elements. Observers were presented with pairs of stimuli on a monitor and were asked to identify the stimulus with a larger number of elements. Using the constant method, we calculated the point of subjective equality (PSE), just noticeable difference (JND), and Weber ratios from the collected data. As a result, it was confirmed that the phenomenon of over-estimation of 3D numerical values occurs even when the numerical values are small. Additionally, we observed that the Weber fraction adhered to Weber's law within the measured range. These findings contribute to the existing body of research, supporting the existence of distinct mechanisms for perceiving numerosity and density.

Neurophysiological signatures of approximate number system acuity in preschoolers

BACKGROUND

A hallmark of the approximate number system (ANS) is ratio dependence. Previous work identified specific event-related potentials (ERPs) that are modulated by numerical ratio throughout the lifespan. In adults, ERP ratio dependence was correlated with the precision of the numerical judgments with individuals who make more precise judgments showing larger ratio-dependent ERP effects. The current study evaluated if this relationship generalizes to preschoolers.

METHOD

ERPs were recorded from 56 4.5 to 5.5-year-olds while they compared the numerosity of two sequentially presented dot arrays. Nonverbal numerical precision, often called ANS acuity, was assessed using a similar behavioral task.

RESULTS

Only children with high ANS acuity exhibited a P2p ratio-dependent effect onsetting ∼250 ms after the presentation of the comparison dot array. Furthermore, P2p amplitude positively correlated with ANS acuity across tasks.

CONCLUSION

Results demonstrate developmental continuity between preschool years and adulthood in the neural basis of the ANS.

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.

The role of spatial information in an approximate cross-modal number matching task

The approximate number system (ANS) is thought to be an innate cognitive system that allows humans to perceive numbers (>4) in a fuzzy manner. One assumption of the ANS is that numerosity is represented amodally due to a mechanism, which filters out nonnumerical information from stimulus material. However, some studies show that nonnumerical information (e.g., spatial parameters) influence the numerosity percept as well. Here, we investigated whether there is a cross-modal transfer of spatial information between the haptic and visual modality in an approximate cross-modal number matching task. We presented different arrays of dowels (haptic stimuli) to 50 undergraduates and asked them to compare haptically perceived numerosity to two visually presented dot arrays. Participants chose which visually presented array matched the numerosity of the haptic stimulus. The distractor varied in number and displayed a random pattern, whereas the matching (target) dot array was either spatially identical or spatially randomized (to the haptic stimulus). We hypothesized that if a "numerosity" percept is based solely on number, neither spatially identical nor spatial congruence between the haptic and the visual target arrays would affect the accuracy in the task. However, results show significant processing advantages for targets with spatially identical patterns and, furthermore, that spatial congruency between haptic source and visual target facilitates performance. Our results show that spatial information was extracted from the haptic stimuli and influenced participants' responses, which challenges the assumption that numerosity is represented in a truly abstract manner by filtering out any other stimulus features.

Attention to number requires magnitude-specific inhibition

Recent studies have shown that the ability to process number in the face of conflicting dimensions of magnitude is a crucial aspect of numerosity judgments, relying in part on the inhibition of the non-numerical dimensions. Here we report, for the first time, that these inhibitory control processes are specific to the conflicting dimension of magnitude. Using a non-symbolic numerical comparison task adapted to a conflict adaptation paradigm on a group of 82 adults, we show that congruency effects between numerical and non-numerical information were reduced only when the conflicting dimension was the same in the preceding incongruent trial. Attention to number thus involves inhibitory control processes acting at a specific level of information. These results contribute to better characterize the domain general abilities involved in numerical cognition, and provide evidence for a specific interaction between numerosity perception and inhibitory control.

Visual numerosity perception shows no advantage in real-world scenes compared to artificial displays

While the human visual system is sensitive to numerosity, the mechanisms that allow perception to extract and represent the number of objects in a scene remains unknown. Prominent theoretical approaches posit that numerosity perception emerges from passive experience with visual scenes throughout development, and that unsupervised deep neural network models mirror all characteristic behavioral features observed in participants. Here, we derive and test a novel prediction: if the visual number sense emerges from exposure to real-world scenes, then the closer a stimulus aligns with the natural statistics of the real world, the better number perception should be. But - in contrast to this prediction - we observe no such advantage (and sometimes even a notable impairment) in number perception for natural scenes compared to artificial dot displays in college-aged adults. These findings are not accounted for by the difficulty in object identification, visual clutter, the parsability of objects from the rest of the scene, or increased occlusion. This pattern of results represents a fundamental challenge to recent models of numerosity perception based in experiential learning of statistical regularities, and instead suggests that the visual number sense is attuned to abstract number of objects, independent of their underlying correlation with non-numeric features. We discuss our results in the context of recent proposals that suggest that object complexity and entropy may play a role in number perception.

Exploring spatiotemporal interactions: On the superiority of time over space

Space and time mutually influence each other such that space affects time estimation (space-on-time effect), and conversely (time-on-space effect). These reciprocal interferences suggest that space and time are intrinsically linked in the human mind. Yet, recent evidence for an asymmetrical advantage for space over time challenges the classical theoretical interpretation. In the present study, we tested whether the superiority of space over time in magnitude interference depends on the cognitive resources engaged in the spatial task. We conducted three experiments in which participants performed judgments on temporal intervals and spatial distances in separate blocks. In each trial, two dots were successively flashed at various locations, and participants were to judge whether the duration or distance between the dots was short or long. To manipulate cognitive demands in the spatial task, distances varied across experiments (highly discriminable for the non-demanding spatial task in Experiment 1 and scarcely discriminable for the demanding spatial task in Experiment 2). Importantly, this manipulation tended to enhance perceptual sensitivity (as indexed by Weber Ratios) but slowed down the decision process (as indexed by response times) in the demanding experiment. Our results provide evidence for robust space-on-time and time-on-space effects (Experiments 1 and 2). More crucially, the involvement of cognitive resources in a demanding spatial task causes a massive time-on-space effect: Spatial judgments are indeed more influenced by irrelevant temporal information than the reverse (Experiments 2 and 3). Overall, the flexibility of spatiotemporal interferences has direct theoretical implications and questions the origins of space-time interaction.

The malleable impact of non-numeric features in visual number perception

Non-numeric stimulus features frequently influence observers' number judgments: when judging the number of items in a display, we will often (mis)perceive the set with a larger cumulative surface area as more numerous. These "congruency effects" are often used as evidence for how vision extracts numeric information and have been invoked in arguments surrounding whether non-numeric cues (e.g., cumulative area, density, etc.) are combined for number perception. We test whether congruency effects for one such cue - cumulative area - provide evidence that it is necessarily used and integrated in number perception, or if its influence on number is malleable. In Experiment 1, we replicate and extend prior work showing that the presence of feedback eliminates congruency effects between number and cumulative area, suggesting that the role of cumulative area in number perception is malleable rather than obligatory. In Experiment 2, we test whether this malleable influence is because of use of prior experiences about how number naturalistically correlates with cumulative area, or the result of response competition, with number and cumulative area actively competing for the same behavioral decision. We preserve cumulative area as a visual cue but eliminate response competition with number by replacing one side of the dot array with its corresponding Hindu-Arabic numeral. Independent of the presence or absence of feedback, we do not observe congruency effects in Experiment 2. These experiments suggest that cumulative area is not necessarily integrated in number perception nor a reflection of a rational use of naturalistic correlations, but rather congruency effects between cumulative area and number emerge as a consequence of response competition. Our findings help to elucidate the mechanism through which non-numeric cues and number interact, and provide an explanation for why congruency effects are only sometimes observed across studies.

The effect of abstract representation and response feedback on serial dependence in numerosity perception

Serial dependence entails an attractive bias based on the recent history of stimulation, making the current stimulus appear more similar to the preceding one. Although serial dependence is ubiquitous in perception, its nature and mechanisms remain unclear. Here, in two independent experiments, we test the hypothesis that this bias originates from high-level processing stages at the level of abstract information processing (Exp. 1) or at the level of judgment (Exp. 2). In Experiment 1, serial dependence was induced by a task-irrelevant "inducer" stimulus in a numerosity discrimination task, similarly to previous studies. Importantly, in this experiment, the inducers were either arrays of dots similar to the task-relevant stimuli (e.g., 12 dots), or symbolic numbers (e.g., the numeral "12"). Both dots and symbol inducers successfully yielded attractive serial dependence biases, suggesting that abstract information about an image is sufficient to bias the perception of the current stimulus. In Experiment 2, participants received feedback about their responses in each trial of a numerosity estimation task, which was designed to assess whether providing external information about the accuracy of judgments would modulate serial dependence. Providing feedback significantly increased the attractive serial dependence effect, suggesting that external information at the level of judgment may modulate the weight of past perceptual information during the processing of the current image. Overall, our results support the idea that, although serial dependence may operate at a perceptual level, it originates from high-level processing stages at the level of abstract information processing and at the level of judgment.

Attention drives human numerosity-selective responses

Numerosity, the set size of a group of items, helps guide behavior and decisions. Previous studies have shown that neural populations respond selectively to numerosities. How numerosity is extracted from the visual scene is a longstanding debate, often contrasting low-level visual with high-level cognitive processes. Here, we investigate how attention influences numerosity-selective responses. The stimuli consisted of black and white dots within the same display. Participants' attention was focused on either black or white dots, while we systematically changed the numerosity of black, white, and total dots. Using 7 T fMRI, we show that the numerosity-tuned neural populations respond only when attention is focused on their preferred numerosity, irrespective of the unattended or total numerosities. Without attention, responses to preferred numerosity are suppressed. Unlike traditional effects of attention in the visual cortex, where attention enhances already existing responses, these results suggest that attention is required to drive numerosity-selective responses.

Incentive motivation improves numerosity discrimination in children and adolescents

We recently showed that incentive motivation improves the precision of the Approximate Number System (ANS) in young adults. To shed light on the development of incentive motivation, the present study investigated whether this effect and its underlying mechanisms may also be observed in younger samples. Specifically, seven-year-old children (n = 23; 12 girls) and 14-year-old adolescents (n = 30; 15 girls) performed a dot comparison task with monetary reward incentives. Both age groups showed higher accuracy in a reward compared to a neutral condition and, similarly, higher processing efficiency as revealed by the drift rate parameter of the EZ-diffusion model. Furthermore, in line with the Incentive Salience Hypothesis, phasic pupil dilations—indicating the activation of the brain’s salience network—were greater in incentivized trials in both age groups. Together these finding suggest that incentive modulation improves numerosity discrimination in children and adolescents by enhancing the perceptual saliency of numerosity information. However, the observed reward anticipation effects were less pronounced in children relative to adolescents. Furthermore, unlike previous findings regarding young adults, the decision thresholds of children and adolescents were not raised by the monetary reward, which may indicate a more protracted development of incentive regulation of response caution than perceptual evidence accumulation.

Get in touch with numbers - an approximate number comparison task in the haptic modality

The Approximate Number System (ANS) is conceptualized as an innate cognitive system that allows humans to perceive numbers of objects or events (>4) in a fuzzy, imprecise manner. The representation of numbers is assumed to be abstract and not bound to a particular sense. In the present study, we test the assumption of a shared cross-sensory system. We investigated approximate number processing in the haptic modality and compared performance to that of the visual modality. We used a dot comparison task (DCT), in which participants compare two dot arrays and decide which one contains more dots. In the haptic DCT, 67 participants had to compare two simultaneously presented dot arrays with the palms of their hands; in the visual DCT, participants inspected and compared dot arrays on a screen. Tested ratios ranged from 2.0 (larger/smaller number) to 1.1. As expected, in both the haptic and the visual DCT responses similarly depended on the ratio of the numbers of dots in the two arrays. However, on an individual level, we found evidence against medium or stronger positive correlations between “ANS acuity” in the visual and haptic DCTs. A regression model furthermore revealed that besides number, spacing-related features of dot patterns (e.g., the pattern’s convex hull) contribute to the percept of numerosity in both modalities. Our results contradict the strong theory of the ANS solely processing number and being independent of a modality. According to our regression and response prediction model, our results rather point towards a modality-specific integration of number and number-related features.

Quantity perception: The forest and the trees

Park (2021) has described "flawed stimulus design(s)" in our recent studies on area perception. Here, we briefly respond to those critiques. While the rigorous, computational approaches taken by Park (and others) certainly have value, we believe that our approach - one that focuses the perceptual reality of quantity rather than the physical reality - is essential. We emphasize again (as we have many times in our work) that the study of quantity perception benefits from both approaches. To further illustrate our point, we collected additional data and show that some of Park's arguments, while sensible in principle, further support our view in practice.

Numerosity tuning in human association cortices and local image contrast representations in early visual cortex

Human early visual cortex response amplitudes monotonically increase with numerosity (object number), regardless of object size and spacing. However, numerosity is typically considered a high-level visual or cognitive feature, while early visual responses follow image contrast in the spatial frequency domain. We find that, at fixed contrast, aggregate Fourier power (at all orientations and spatial frequencies) follows numerosity closely but nonlinearly with little effect of object size, spacing or shape. This would allow straightforward numerosity estimation from spatial frequency domain image representations. Using 7T fMRI, we show monotonic responses originate in primary visual cortex (V1) at the stimulus’s retinotopic location. Responses here and in neural network models follow aggregate Fourier power more closely than numerosity. Truly numerosity tuned responses emerge after lateral occipital cortex and are independent of retinotopic location. We propose numerosity’s straightforward perception and neural responses may result from the pervasive spatial frequency analyses of early visual processing.

The authors show that spatial frequency domain Fourier power closely but nonlinearly follows numerosity, simplifying computing numerosity from early visual responses. Monotonic early visual cortex and neural network responses follow Fourier power, while later tuned responses follow numerosity.

Young Children Intuitively Divide Before They Recognize the Division Symbol

Children bring intuitive arithmetic knowledge to the classroom before formal instruction in mathematics begins. For example, children can use their number sense to add, subtract, compare ratios, and even perform scaling operations that increase or decrease a set of dots by a factor of 2 or 4. However, it is currently unknown whether children can engage in a true division operation before formal mathematical instruction. Here we examined the ability of 6- to 9-year-old children and college students to perform symbolic and non-symbolic approximate division. Subjects were presented with non-symbolic (dot array) or symbolic (Arabic numeral) dividends ranging from 32 to 185, and non-symbolic divisors ranging from 2 to 8. Subjects compared their imagined quotient to a visible target quantity. Both children (Experiment 1 N = 89, Experiment 2 N = 42) and adults (Experiment 3 N = 87) were successful at the approximate division tasks in both dots and numeral formats. This was true even among the subset of children that could not recognize the division symbol or solve simple division equations, suggesting intuitive division ability precedes formal division instruction. For both children and adults, the ability to divide non-symbolically mediated the relation between Approximate Number System (ANS) acuity and symbolic math performance, suggesting that the ability to calculate non-symbolically may be a mechanism of the relation between ANS acuity and symbolic math. Our findings highlight the intuitive arithmetic abilities children possess before formal math instruction.

Numbers, numerosities, and new directions

In our target article, we argued that the number sense represents natural and rational numbers. Here, we respond to the 26 commentaries we received, highlighting new directions for empirical and theoretical research. We discuss two background assumptions, arguments against the number sense, whether the approximate number system (ANS) represents numbers or numerosities, and why the ANS represents rational (but not irrational) numbers.

Sharpening, focusing, and developing: A study of change in nonsymbolic number comparison skills and math achievement in 1st grade

Children's ability to discriminate nonsymbolic number (e.g., the number of items in a set) is a commonly studied predictor of later math skills. Number discrimination improves throughout development, but what drives this improvement is unclear. Competing theories suggest that it may be due to a sharpening numerical representation or an improved ability to pay attention to number and filter out non-numerical information. We investigate this issue by studying change in children's performance (N = 65) on a nonsymbolic number comparison task, where children decide which of two dot arrays has more dots, from the middle to the end of 1st grade (mean age at time 1 = 6.85 years old). In this task, visual properties of the dot arrays such as surface area are either congruent (the more numerous array has more surface area) or incongruent. Children rely more on executive functions during incongruent trials, so improvements in each congruency condition provide information about the underlying cognitive mechanisms. We found that accuracy rates increased similarly for both conditions, indicating a sharpening sense of numerical magnitude, not simply improved attention to the numerical task dimension. Symbolic number skills predicted change in congruent trials, but executive function did not predict change in either condition. No factor predicted change in math achievement. Together, these findings suggest that nonsymbolic number processing undergoes development related to existing symbolic number skills, development that appears not to be driving math gains during this period. Children's ability to discriminate nonsymbolic number improves throughout development. Competing theories suggest improvement due to sharpening magnitude representations or changes in attention and inhibition. The current study investigates change in nonsymbolic number comparison performance during first grade and whether symbolic number skills, math skills, or executive function predict change. Children's performance increased across visual control conditions (i.e., congruent or incongruent with number) suggesting an overall sharpening of number processing. Symbolic number skills predicted change in nonsymbolic number comparison performance.

Resources Underlying Visuo-Spatial Working Memory Enable Veridical Large Numerosity Perception

Humans can quickly approximate how many objects are in a visual image, but no clear consensus has been achieved on the cognitive resources underlying this ability. Previous work has lent support to the notion that mechanisms which explicitly represent the locations of multiple objects in the visual scene within a mental map are critical for both visuo-spatial working memory and enumeration (at least for relatively small numbers of items). Regarding the cognitive underpinnings of large numerosity perception, an issue currently subject to much controversy is why numerosity estimates are often non-veridical (i.e., susceptible to biases from non-numerical quantities). Such biases have been found to be particularly pronounced in individuals with developmental dyscalculia (DD), a learning disability affecting the acquisition of arithmetic skills. Motivated by findings showing that DD individuals are also often impaired in visuo-spatial working memory, we hypothesized that resources supporting this type of working memory, which allow for the simultaneous identification of multiple objects, might also be critical for precise and unbiased perception of larger numerosities. We therefore tested whether loading working memory of healthy adult participants during discrimination of large numerosities would lead to increased interference from non-numerical quantities. Participants performed a numerosity discrimination task on multi-item arrays in which numerical and non-numerical stimulus dimensions varied congruently or incongruently relative to each other, either in isolation or in the context of a concurrent visuo-spatial or verbal working memory task. During performance of the visuo-spatial, but not verbal, working memory task, precision in numerosity discrimination decreased, participants' choices became strongly biased by item size, and the strength of this bias correlated with measures of arithmetical skills. Moreover, the interference between numerosity and working memory tasks was bidirectional, with number discrimination impacting visuo-spatial (but not verbal) performance. Overall, these results suggest that representing visual numerosity in a way that is unbiased by non-numerical quantities relies on processes which explicitly segregate/identify the locations of multiple objects that are shared with visuo-spatial (but not verbal) working memory. This shared resource may potentially be impaired in DD, explaining the observed co-occurrence of working memory and numerosity discrimination deficits in this clinical population.

Context-Dependent Modulation of Early Visual Cortical Responses to Numerical and Nonnumerical Magnitudes

Whether and how the brain encodes discrete numerical magnitude differently from continuous nonnumerical magnitude is hotly debated. In a previous set of studies, we orthogonally varied numerical (numerosity) and nonnumerical (size and spacing) dimensions of dot arrays and demonstrated a strong modulation of early visual evoked potentials (VEPs) by numerosity and not by nonnumerical dimensions. Although very little is known about the brain's response to systematic changes in continuous dimensions of a dot array, some authors intuit that the visual processing stream must be more sensitive to continuous magnitude information than to numerosity. To address this possibility, we measured VEPs of participants viewing dot arrays that changed exclusively in one nonnumerical magnitude dimension at a time (size or spacing) while holding numerosity constant and compared this to a condition where numerosity was changed while holding size and spacing constant. We found reliable but small neural sensitivity to exclusive changes in size and spacing; however, exclusively changing numerosity elicited a much more robust modulation of the VEPs. Together with previous work, these findings suggest that sensitivity to magnitude dimensions in early visual cortex is context dependent: The brain is moderately sensitive to changes in size and spacing when numerosity is held constant, but sensitivity to these continuous variables diminishes to a negligible level when numerosity is allowed to vary at the same time. Neurophysiological explanations for the encoding and context dependency of numerical and nonnumerical magnitudes are proposed within the framework of neuronal normalization.

The role of neural tuning in quantity perception

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

Flawed stimulus design in additive-area heuristic studies

In a series of recently published studies purportedly on the "additive-area heuristic," Yousif & Keil (2019, 2020) argue for a systematic distortion in the perception of the cumulative area of an item array and further claim that previous findings of numerical cognition and magnitude perception in general are "at risk" (Yousif & Keil, 2021). This commentary describes serious stimulus design flaws present in all of Yousif and colleagues experiments that prevent from making such conclusions. Specifically, item arrays used in those studies demonstrate a skewed correlational structure between selected magnitude dimensions and exhibit unbalanced ranges across different magnitude dimensions of interest. Because the perception of magnitude dimensions interferes one another and because our perceptual system is sensitive to the statistical regularities of the sensory input, such a biased design makes it difficult, if not impossible, to interpret the choice behavior of an observer making magnitude judgments. By re-introducing the mathematical framework for a systematic construction of dot array stimuli (DeWind et al., 2015) and by re-analyzing the data from another recent study on area perception (Tomlinson et al., 2020), this paper explains-and introduces a MATLAB code for-an optimal method for designing and constructing dot arrays for magnitude perception studies.

Shaping the way from the unknown to the known: The role of convex hull shape in numerical comparisons

Various studies have shown that numerical processing is modulated by non-numerical physical properties. One such physical property is the convex hull - the smallest convex polygon surrounding all items in an array. The convex hull is usually discussed only in terms of its area. However, our group has shown that observers use the convex hull shape, as defined according to the number of vertices of the convex hull, to make numerical estimations (Katzin, Katzin, Rosén, Henik, & Salti, 2020). Yet, it is still unknown if and how the convex hull shape affects comparison tasks, and how it interacts with its counterpart, convex hull area. Here we re-examine the data collected by Katzin, Salti, and Henik (2019). Using image processing, we extracted the information on the convex hull shape and showed that the shape affects latency and accuracy of numerical comparisons. We found that both the convex hull shape and other physical properties (i.e., convex hull area, average diameter, density, total circumference, and total surface area) have distinct effects on performance. Finally, the convex hull shape effect was found in counting and estimation ranges, however its effect decreased with numerosities above the counting range. Our results indicate that the interplay between convex hull shape and other physical properties, including convex hull area and numerosity, plays an important role in numerical decisions. We suggest that the convex hull shape should be controlled for when designing non-symbolic numerical tasks.

The specious interaction of time and numerosity perception

Magnitude information is essential to create a representation of the external environment and successfully interact with it. Duration and numerosity, for example, can shape our predictions and bias each other (i.e. the greater the number of people queuing, the longer we expect to wait). While these biases suggest the existence of a generalized magnitude system, asymmetric effects (i.e. numerosity affecting duration but not vice versa) challenged this idea. Here, we propose that such asymmetric integration depends on the stimuli used and the neural processing dynamics they entail. Across multiple behavioural experiments employing different stimulus presentation displays (static versus dynamic) and experimental manipulations known to bias numerosity and duration perceptions (i.e. connectedness and multisensory integration), we show that the integration between numerosity and time can be symmetrical if the stimuli entail a similar neural time-course and numerosity unfolds over time. Overall, these findings support the idea of a generalized magnitude system, but also highlight the role of early sensory processing in magnitude representation and integration.

"BreaThink": breathing affects production and perception of quantities

Cognition is shaped by signals from outside and within the body. Following recent evidence of interoceptive signals modulating higher-level cognition, we examined whether breathing changes the production and perception of quantities. In Experiment 1, 22 adults verbally produced on average larger random numbers after inhaling than after exhaling. In Experiment 2, 24 further adults estimated the numerosity of dot patterns that were briefly shown after either inhaling or exhaling. Again, we obtained on average larger responses following inhalation than exhalation. These converging results extend models of situated cognition according to which higher-level cognition is sensitive to transient interoceptive states.

Towards a standardization of non-symbolic numerical experiments: GeNEsIS, a flexible and user-friendly tool to generate controlled stimuli

Several studies have suggested that vertebrate and invertebrate species may possess a number sense, i.e. an ability to process in a non-symbolic and non-verbal way the numerousness of a set of items. However, this hypothesis has been challenged by the presence of other non-numerical continuous physical variables, which vary along with numerosity (i.e., any change in the number of visual physical elements in a set naturally involves a related change in visual features such as area, density, contour length and convex hull of the stimulus). It is therefore necessary to control and manipulate the continuous physical information when investigating the ability of humans and other animals to perceive numerousness. During decades of research, different methods have been implemented in order to address this issue, which has implications for experiment replicability and inter-species comparisons, since no general standardized procedure is currently being used. Here we present the ‘Generation of Numerical Elements Images Software’ (GeNEsIS) for the creation of non-symbolic numerical arrays in a standardized and user-friendly environment. The main aim of this tool is to provide researchers in the field of numerical cognition a manageable and precise instrument to produce visual numerical arrays controlled for all the continuous variables. Additionally, we implemented the ability to actively guide stimuli presentation during habituation/dishabituation and dual-choice comparison tasks used in human and comparative research.

Approximate Number Sense in Students With Severe Hearing Loss: A Modality-Neutral Cognitive Ability

The Approximate Number System (ANS) allows humans and non-human animals to estimate large quantities without counting. It is most commonly studied in visual contexts (i.e., with displays containing different numbers of dots), although the ANS may operate on all approximate quantities regardless of modality (e.g., estimating the number of a series of auditory tones). Previous research has shown that there is a link between ANS and mathematics abilities, and that this link is resilient to differences in visual experience (Kanjlia et al., 2018). However, little is known about the function of the ANS and its relationship to mathematics abilities in the absence of other types of sensory input. Here, we investigated the acuity of the ANS and its relationship with mathematics abilities in a group of students from the Sichuan Province in China, half of whom were deaf. We found, consistent with previous research, that ANS acuity improves with age. We found that mathematics ability was predicted by Non-verbal IQ and Inhibitory Control, but not visual working memory capacity or Attention Network efficiencies. Even above and beyond these predictors, ANS ability still accounted for unique variance in mathematics ability. Notably, there was no interaction with hearing, which indicates that the role played by the ANS in explaining mathematics competence is not modulated by hearing capacity. Finally, we found that age, Non-verbal IQ and Visual Working Memory capacity were predictive of ANS performance when controlling for other factors. In fact, although students with hearing loss performed slightly worse than students with normal hearing on the ANS task, hearing was no longer significantly predictive of ANS performance once other factors were taken into account. These results indicate that the ANS is able to develop at a consistent pace with other cognitive abilities in the absence of auditory experience, and that its relationship with mathematics ability is not contingent on sensory input from hearing.

A Ubiquitous Illusion of Volume: Are Impressions of 3D Volume Captured by an "Additive Heuristic"?

Several empirical approaches have attempted to explain perception of 2D and 3D size. While these approaches have documented interesting perceptual effects, they fail to offer a compelling, general explanation of everyday size perception. Here, we offer one. Building on prior work documenting an "Additive Area Heuristic" by which observers estimate perceived area by summing objects' dimensions, we show that this same principle-an "additive heuristic"-explains impressions of 3D volume. Observers consistently discriminate sets that vary in "additive volume," even when there is no true difference; they also fail to discriminate sets that truly differ (even by amounts as much as 30%) when they are equated in "additive volume." These results suggest a failure to properly integrate multiple spatial dimensions, and frequent reliance on a perceptual heuristic instead.

Serial dependence in time and numerosity perception is dimension-specific

The perception of a visual event (e.g., a flock of birds) at the present moment can be biased by a previous perceptual experience (e.g., the perception of an earlier flock). Serial dependence is a perceptual bias whereby a current stimulus appears more similar to a previous one than it actually is. Whereas serial dependence emerges within several visual stimulus dimensions, whether it could simultaneously operate across different dimensions of the same stimulus (e.g., the numerosity and the duration of a visual pattern) remains unclear. Here we address this question by assessing the presence of serial dependence across duration and numerosity, two stimulus dimensions that are often associated and can bias each other. Participants performed either a duration or a numerosity discrimination task, in which they compared a constant reference with a variable test stimulus, varying along the task-relevant dimension (either duration or numerosity). Serial dependence was induced by a task-irrelevant inducer, that is, a stimulus presented before the reference and always varying in both duration and numerosity. The results show systematic serial dependencies only within the task-relevant stimulus dimension, that is, stimulus numerosity affects numerosity perception only, and duration affects duration perception only. Additionally, at least in the numerosity condition, the task-irrelevant dimension of the inducer (duration) had an opposite, repulsive effect. These findings thus show that attractive serial dependence operates in a highly specific fashion and does not transfer across different stimulus dimensions. Instead, the repulsive influence, possibly reflecting perceptual adaptation, can transfer from one dimension to another.

How We See Area and Why It Matters

A large and growing literature examines how we see the visual quantities of number, area, and density. The literature rests on an untested assumption: that our perception of area is veridical. Here, we discuss a systematic distortion of perceived area and its implications for quantity perception more broadly.

Numerosities and Other Magnitudes in the Brains: A Comparative View

The ability to represent, discriminate, and perform arithmetic operations on discrete quantities (numerosities) has been documented in a variety of species of different taxonomic groups, both vertebrates and invertebrates. We do not know, however, to what extent similarity in behavioral data corresponds to basic similarity in underlying neural mechanisms. Here, we review evidence for magnitude representation, both discrete (countable) and continuous, following the sensory input path from primary sensory systems to associative pallial territories in the vertebrate brains. We also speculate on possible underlying mechanisms in invertebrate brains and on the role played by modeling with artificial neural networks. This may provide a general overview on the nervous system involvement in approximating quantity in different animal species, and a general theoretical framework to future comparative studies on the neurobiology of number cognition.

The Number Sense Represents (Rational) Numbers

On a now orthodox view, humans and many other animals possess a "number sense," or approximate number system (ANS), that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique-the arguments from congruency, confounds, and imprecision-and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds or exotic substitutes for number, such as "numerosities" or "quanticals," as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind(s) of number being represented. In response, we propose that the ANS represents not only natural numbers (e.g. 7), but also non-natural rational numbers (e.g. 3.5). It does not represent irrational numbers (e.g. √2), however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research.

Time and numerosity estimation in peripersonal and extrapersonal space

The representation of space, time and number is believed to rely on a common encoding system developed to support action guidance. While the ecological advantage of such a shared system is evident when objects are located within the region of space we can act on (known as peri-personal space), it is less obvious in the case of objects located beyond our arms' reach. In the current study we investigated whether and to what extent the distance of the stimuli from the observer affects the perception of duration and numerosity. We first replicated Anelli et al.'s (2015) experiment by asking adult participants to perform a duration reproduction task with stimuli of different sizes displayed in the peri- or extra-personal space, and then applied the same paradigm to a non-symbolic numerosity estimation task. Results show that, independently of size, duration estimates were overestimated when visual stimuli were presented in the extra-personal space, replicating previous findings. A similar effect was also found for numerosity perception, however overestimation for far stimuli was much smaller in magnitude and was accounted by the difference in perceived size between stimuli presented in peripersonal or extrapersonal space. Overall, these results suggest that, while the processing of temporal information is robustly affected by the position of the stimuli in either the peri- or extra-personal space, numerosity perception is independent from stimulus distance. We speculate that, while time and numerosity may be encoded by a shared system in the peri-personal space (to optimize action execution), different and partially independent mechanisms may underlie the representation of time and numerosity in extra-personal space. Furthermore, these results suggest that investigating magnitude perception across spatial planes (where it is or is not possible to act) may unveil processing differences that would otherwise pass unnoticed.

Approximate multiplication in young children prior to multiplication instruction

Prior work indicates that children have an untrained ability to approximately calculate using their approximate number system (ANS). For example, children can mentally double or halve a large array of discrete objects. Here, we asked whether children can perform a true multiplication operation, flexibly attending to both the multiplier and multiplicand, prior to formal multiplication instruction. We presented 5- to 8-year-olds with nonsymbolic multiplicands (dot arrays) or symbolic multiplicands (Arabic numerals) ranging from 2 to 12 and with nonsymbolic multipliers ranging from 2 to 8. Children compared each imagined product with a visible comparison quantity. Children performed with above-chance accuracy on both nonsymbolic and symbolic approximate multiplication, and their performance was dependent on the ratio between the imagined product and the comparison target. Children who could not solve any single-digit symbolic multiplication equations (e.g., 2 × 3) on a basic math test were nevertheless successful on both our approximate multiplication tasks, indicating that children have an intuitive sense of multiplication that emerges independent of formal instruction about symbolic multiplication. Nonsymbolic multiplication performance mediated the relation between children's Weber fraction and symbolic math abilities, suggesting a pathway by which the ANS contributes to children's emerging symbolic math competence. These findings may inform future educational interventions that allow children to use their basic arithmetic intuition as a scaffold to facilitate symbolic math learning.

The ratio effect in visual numerosity comparisons is preserved despite spatial frequency equalisation

How non-symbolic numerosity is visually extracted remains a matter of intense debate. Most evidence suggests that numerosity is directly extracted on individual objects following Weber's law, at least for a moderate numerical range. Alternative accounts propose that, whatever the range, numerosity is indirectly derived from summary texture-statistics of the raw image such as spatial frequency (SF). Here, to disentangle these accounts, we tested whether the well-known behavioural signature of numerosity encoding (ratio effect) is preserved despite the equalisation of the SF content. In Experiment 1, participants had to select the numerically larger of two briefly presented moderate-range numerical sets (i.e., 8-18 dots) carefully matched for SF; the ratio between numerosities was manipulated by levels of increasing difficulty (e.g., 0.66, 0.75, 0.8). In Experiment 2, participants performed the same task, but they were presented with both the original and SF equalised stimuli. In both experiments, the results clearly showed a ratio-dependence of the performance: numerosity discrimination became harder and slower as the ratio between numerosities increased. Moreover, this effect was found to be independent of the stimulus type, although the overall performance was better with the original rather than the SF equalised stimuli (Experiment 2). Taken together, these findings indicate that the power spectrum per se cannot explain the main behavioural signature of Weber-like encoding of numerosities (the ratio effect), at least over the tested numerical range, partially challenging alternative indirect accounts of numerosity processing.