Number sense biases children's area judgments

Measuring temporal bias in sequential numerosity comparison

While several methods have been proposed to assess the influence of continuous visual cues in parallel numerosity estimation, the impact of temporal magnitudes on sequential numerosity judgments has been largely ignored. To overcome this issue, we extend a recently proposed framework that makes it possible to separate the contribution of numerical and non-numerical information in numerosity comparison by introducing a novel stimulus space designed for sequential tasks. Our method systematically varies the temporal magnitudes embedded into event sequences through the orthogonal manipulation of numerosity and two latent factors, which we designate as "duration" and "temporal spacing". This allows us to measure the contribution of finer-grained temporal features on numerosity judgments in several sensory modalities. We validate the proposed method on two different experiments in both visual and auditory modalities: results show that adult participants discriminated sequences primarily by relying on numerosity, with similar acuity in the visual and auditory modality. However, participants were similarly influenced by non-numerical cues, such as the total duration of the stimuli, suggesting that temporal cues can significantly bias numerical processing. Our findings highlight the need to carefully consider the continuous properties of numerical stimuli in a sequential mode of presentation as well, with particular relevance in multimodal and cross-modal investigations. We provide the complete code for creating sequential stimuli and analyzing participants' responses.

Developmental changes in numerosity and area perception in school-age children

Humans can quickly estimate the quantity of objects in their environment through the Approximate Number System (ANS). However, the developmental trajectories of numerical and spatial perception in school-aged children are not well understood. This study aimed to address this issue by examining the performance of 7-year-olds, 9-year-olds, and 11-year-olds in the dot-number and dot-area tasks. In each trial, participants were presented with pairs of dots array simultaneously and were asked to indicate which array had a greater quantity of dots in the dot-number task, and which array had a larger overall area of dots in the dot-area task. We manipulated number cues and area cues of the dots array in these two tasks, creating three different relationships between dot-number and dot-area: congruent, neutral, and incongruent. Our results showed school-aged children's ability to estimate numerosity improved significantly after the age of 7, with no apparent improvement observed between the ages of 9 and 11. This indicates a marked growth in acuity in the perception of numerical quantities. Conversely, the capacity to estimate area showed consistent stability across the various age groups examined. Additionally, our results demonstrated a pronounced difficulty among participants in ignoring numerical cues when assessing dot quantity or dot area, as opposed to non-numerical cues. This highlights a preferential sensitivity to numerical information in cognitive processing. These findings provide valuable insights into the cognitive development underlying quantity perception and may offer practical guidance for educational practices.

Weaker number sense accounts for impaired numerosity perception in dyscalculia: Behavioral and computational evidence

Impaired numerosity perception in developmental dyscalculia (low "number acuity") has been interpreted as evidence of reduced representational precision in the neurocognitive system supporting non-symbolic number sense. However, recent studies suggest that poor numerosity judgments might stem from stronger interference from non-numerical visual information, in line with alternative accounts that highlight impairments in executive functions and visuospatial abilities in the etiology of dyscalculia. To resolve this debate, we used a psychophysical method designed to disentangle the contribution of numerical and non-numerical features to explicit numerosity judgments in a dot comparison task and we assessed the relative saliency of numerosity in a spontaneous categorization task. Children with dyscalculia were compared to control children with average mathematical skills matched for age, IQ, and visuospatial memory. In the comparison task, the lower accuracy of dyscalculics compared to controls was linked to weaker encoding of numerosity, but not to the strength of non-numerical biases. Similarly, in the spontaneous categorization task, children with dyscalculia showed a weaker number-based categorization compared to the control group, with no evidence of a stronger influence of non-numerical information on category choice. Simulations with a neurocomputational model of numerosity perception showed that the reduction of representational resources affected the progressive refinement of number acuity, with little effect on non-numerical bias in numerosity judgments. Together, these results suggest that impaired numerosity perception in dyscalculia cannot be explained by increased interference from non-numerical visual cues, thereby supporting the hypothesis of a core number sense deficit. RESEARCH HIGHLIGHTS: A strongly debated issue is whether impaired numerosity perception in dyscalculia stems from a deficit in number sense or from poor executive and visuospatial functions. Dyscalculic children show reduced precision in visual numerosity judgments and weaker number-based spontaneous categorization, but no increasing reliance on continuous visual properties. Simulations with deep neural networks demonstrate that reduced neural/computational resources affect the developmental trajectory of number acuity and account for impaired numerosity judgments. Our findings show that weaker number acuity in developmental dyscalculia is not necessarily related to increased interference from non-numerical visual cues.

Heuristic strategy of intuitive statistical inferences in 7- to 10-year-old children

Intuitive statistical inferences refer to making inferences about uncertain events based on limited probabilistic information, which is crucial for both human and non-human species' survival and reproduction. Previous research found that 7- and 8-year-old children failed in intuitive statistical inference tasks after heuristic strategies had been controlled. However, few studies systematically explored children's heuristic strategies of intuitive statistical inferences and their potential numerical underpinnings. In the current research, Experiment 1 (N = 81) examined 7- to 10-year-olds' use of different types of heuristic strategies; results revealed that children relied more on focusing on the absolute number strategy. Experiment 2 (N = 99) and Experiment 3 (N = 94) added continuous-format stimuli to examine whether 7- and 8-year-olds could make genuine intuitive statistical inferences instead of heuristics. Results revealed that both 7- and 8-year-olds and 9- and 10-year-olds performed better in intuitive statistical inference tasks with continuous-format stimuli, even after focusing on the absolute number strategy had been controlled. The results across the three experiments preliminarily hinted that the ratio processing system might rely on the approximate number system. Future research could clarify what specific numerical processing mechanism may be used and how it might support children's statistical intuitions.

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.

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.

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.

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.

Sometimes nothing is simply nothing: Automatic processing of empty sets

Previous work using the numerical comparison task has shown that an empty set, the nonsymbolic manifestation of zero, can be represented as the smallest quantity of the numerical magnitude system. In this study, we examined whether an empty set can be represented as such under conditions of automatic processing in which deliberate processing of stimuli magnitudes is not required by the task. In Experiment 1, participants performed physical and numerical comparisons of empty sets (i.e., empty frames) and of other numerosities presented as framed arrays of 1 to 9 dots. The physical sizes of the frames varied within pairs. Both tasks revealed a size congruity effect (SCE) for comparisons of non-empty sets. In contrast, comparisons to empty sets produced an inverted SCE in the physical comparison task, whereas no SCE was found for comparisons to empty sets in the numerical comparison task. In Experiment 2, participants performed an area comparison task using the same stimuli as Experiment 1 to examine the effect of visual cues on the automatic processing of empty sets. The results replicated the findings of the physical comparison task in Experiment 1. Taken together, our findings indicate that empty sets are not perceived as "zero," but rather as "nothing," when processed automatically. Hence, the perceptual dominance of empty sets seems to play a more important role under conditions of automatic processing, making it harder to abstract the numerical meaning of zero from empty sets.

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.

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.

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.

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.