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Humans can approximately enumerate a large number of objects at a single glance. While several mechanisms have been proposed to account for this ability, the fundamental units over which they operate remain unclear. Previous studies have argued that estimation mechanisms act only on topologically distinct units or on units formed by spatial grouping cues such as proximity and connectivity, but not on units grouped by similarity. Over four experiments, we tested this claim by systematically assessing and demonstrating that similarity grouping leads to underestimation, just as spatial grouping does. Ungrouped objects with the same low-level properties as grouped objects did not cause underestimation. Further, the underestimation caused by spatial and similarity grouping was additive, suggesting that these grouping processes operate independently. These findings argue against the proposal that estimation mechanisms operate solely on topological units. Instead, we conclude that estimation processes act on representations constructed after Gestalt grouping principles, whether similarity based or spatial, have organised incoming visual input.

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Estimation is a primary activity in everyday life, so getting it "right" in primary school provides a foundational basis in mathematical reasoning. This study focuses on numerosity estimation in primary mathematics, which is one of four types of estimation reported in literature. In numerosity estimation, a non-numerical quantitative representation is typically translated into a number. While it is assumed that fostering numerosity estimation has a great impact on the development of mathematical skills, research indicates that math achievement is influenced by non-cognitive aspects such as students' math interests. So, math interest could also influence the accuracy in numerosity estimation. In this study, we investigate the relationship between accuracy in numerosity estimation, math achievement, and math interest in third-grade students. For capturing accuracy in numerosity estimation in a standardized way, we developed an online numerosity estimation test. For assessing the construct of math interest, we used an existing questionnaire. Math achievement was assessed by a standardized math test that includes two subtests focusing on arithmetic and application tasks. The sample was comprised of 185 third-grade students. We analyzed the data using correlation and multiple linear regression analysis. The results showed a significant positive correlation between math interest and math achievement. However, no relationship was found between accuracy in numerosity estimation and math interest nor between accuracy in numerosity estimation and math achievement. These partly unexpected findings suggest further studies dedicated to numerosity estimation and its relationship to other constructs.

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Humans can estimate the number of visually displayed items without counting. This capacity of numerosity perception has often been attributed to a dedicated system to estimate numerosity, or alternatively to the exploitation of various stimulus features, such as density, convex hull, the size of items, and occupancy area. The distribution of the presented items is usually not varied with eccentricity in the visual field. However, our visual fields are highly asymmetric. To date, it is unclear how inhomogeneities of the visual field impact numerosity perception. Besides eccentricity, a pronounced asymmetry is the radial-tangential anisotropy. For example, in crowding, radially placed flankers interfere more strongly with target perception than tangentially placed flankers. Similarly, in redundancy masking, the number of perceived items in repeating patterns is reduced when the items are arranged radially but not when they are arranged tangentially. Here, we investigated whether numerosity perception is subject to the radial-tangential anisotropy of spatial vision to shed light on the underlying topology of numerosity perception. In Experiment 1, observers were presented with varying numbers of discs, predominantly arranged radially or tangentially, and asked to report their perceived number. In Experiment 2, observers were presented with the same displays as in Experiment 1, and were asked to encircle items that were perceived as a group. We found that numerosity estimation depended on the arrangement of discs, suggesting a radial-tangential anisotropy of numerosity perception. Grouping among discs did not seem to explain our results. We suggest that the topology of spatial vision modulates numerosity estimation and that asymmetries of visual space should be taken into account when investigating numerosity estimation.

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This study explores whether and how different tasks associated with approximate number system (ANS) ability are related to numeracy and cognitive reflection in adults. We conducted an online experiment using a sample of 300 Japanese adults aged 20-39. Participants were given three ANS tasks (numerosity comparison, numerosity estimation, and proportion estimation) as well as Rasch-based numeracy scale and cognitive reflection test, and we tested the correlation among the measures of these tasks. We explored the hypothesis that the typical measures used to gauge ANS ability, numerosity comparison and numerosity estimation may mediate different cognitive mechanisms in adults. We also introduced a task measuring proportion estimation, added because such estimation requires numerosity perception and the ability to map symbolic numerals. Our findings suggest that there is a weak, but significant correlation among the three ANS-related tasks. Moreover, there is a significant relationship between each of these measures and the numeracy and CRT score, suggesting that the ANS-related ability may be associated with higher cognitive abilities such as numeracy and cognitive reflection. In addition, we found that performances on the numerosity and proportion estimation are more clearly related to CRT score than the numerosity comparison task.

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Both humans and nonhuman animals can exhibit sensitivity to the approximate number of items in a visual array or events in a sequence, and across various paradigms, uncertainty in numerosity judgments increases with the number estimated or produced. The pattern of increase is usually described as exhibiting approximate adherence to Weber's law, such that uncertainty increases proportionally to the mean estimate, resulting in a constant coefficient of variation. Such a pattern has been proposed to be a signature characteristic of an innate "number sense." We reexamine published behavioral data from two studies that have been cited as prototypical evidence of adherence to Weber's law and observe that in both cases variability increases less than this account would predict, as indicated by a decreasing coefficient of variation with an increase in number. We also consider evidence from numerosity discrimination studies that show deviations from the constant coefficient of variation pattern. Though behavioral data can sometimes exhibit approximate adherence to Weber's law, our findings suggest that such adherence is not a fixed characteristic of the mechanisms whereby humans and animals estimate numerosity. We suggest instead that the observed pattern of increase in variability with number depends on the circumstances of the task and stimuli, and reflects an adaptive ensemble of mechanisms composed to optimize performance under these circumstances.

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Humans tend to be inaccurate and inconsistent when estimating a large number of objects. Furthermore, we modify our estimates when feedback or a reference array is provided, indicating that the mappings between perceived numerosity and their corresponding numerals are largely malleable in response to calibration. However, there is great variability in response to calibration across individuals. Using uncalibrated and calibrated numerosity estimation conditions, the current study explored the factors underlying individual differences in the extent and nature of the malleability of numerosity estimation performance as a result of calibration in a sample of 71 undergraduate students. We found that individual differences in performance were reliable across conditions, and participants' responses to calibration varied greatly. Participants who were less consistent or had more proportionally spaced (i.e., linear) estimates before calibration tended to shift the distributions of their estimates to a greater extent. Higher calculation competence also predicted an increase in how linear participants' estimates were after calibration. Moreover, the effect of calibration was not continuous across numerosities within participants. This suggests that the mechanisms underlying numeral-numerosity mappings may be less systematic than previously thought and likely depend on cognitive mechanisms beyond representation of numerosities. Taken together, the mappings between numerosities and numerical symbols may not be stable and direct, but transient and mediated by task-related (e.g., strategic) mechanisms. Rather than estimation skills being foundational for math competence, math competence may also influence estimation skills. Therefore, numerosity estimation tasks are not a pure measure of number representations.

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

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Numerical information can be perceived at multiple levels (e.g., one bird, or a flock of birds). The level of input has typically been defined by explicit grouping cues, such as contours or connecting lines. Here we examine how regularities of object co-occurrences shape numerosity perception in the absence of explicit grouping cues. Participants estimated the number of colored circles in an array. We found that estimates were lower in arrays containing colors that consistently appeared next to each other across the experiment, even though participants were not explicitly aware of the color pairs (Experiments 1a and 1b). To provide support for grouping, we introduced color duplicates and found that estimates were lower in arrays with two identical colors (Experiment 2). The underestimation could not be explained by increased attention to individual objects (Experiment 3). These results suggest that statistical regularities reduce perceived numerosity consistent with a grouping mechanism.

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We address the challenges of how to model human perceptual grouping in random dot arrays and how perceptual grouping affects human number estimation in these arrays. We introduce a modeling approach relying on a modified k-means clustering algorithm to formally describe human observers' grouping behavior. We found that a default grouping window size of approximately 4° of visual angle describes human grouping judgments across a range of random dot arrays (i.e., items within 4° are grouped together). This window size was highly consistent across observers and images, and was also stable across stimulus durations, suggesting that the k-means model captured a robust signature of perceptual grouping. Further, the k-means model outperformed other models (e.g., CODE) at describing human grouping behavior. Next, we found that the more the dots in a display are clustered together, the more human observers tend to underestimate the numerosity of the dots. We demonstrate that this effect is independent of density, and the modified k-means model can predict human observers' numerosity judgments and underestimation. Finally, we explored the robustness of the relationship between clustering and dot number underestimation and found that the effects of clustering remain, but are greatly reduced, when participants receive feedback on every trial. Together, this work suggests some promising avenues for formal models of human grouping behavior, and it highlights the importance of a 4° window of perceptual grouping. Lastly, it reveals a robust, somewhat plastic, relationship between perceptual grouping and number estimation.

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We investigated whether the size and number of objects mentioned in digit-word expressions influenced participants' performance in covert numerosity estimations (i.e., property probability ratings). Participants read descriptions of big or small animals standing in short, medium, and long rows (e.g., There are 8 elephants/ants in a row) and subsequently estimated the probability that a health statement about them was true (e.g., All elephants/ants are healthy). Statements about large animals scored lower than statements about small animals, confirming classical findings that humans perceive groups of large objects as being more numerous than groups of small objects (Binet, 1890) and suggesting that object size effects in covert numerosity estimations are particularly robust. Also, statements about longer rows scored lower than statements about shorter rows (cf. Sears, 1983) but no interaction between factors obtained, suggesting that quantity information is not fully retrieved in digit-word expressions or that their values are processed separately.

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Visual skills, including numerosity estimation are reported to be superior in autism spectrum disorders (ASD). This phenomenon is attributed to individuals with ASD processing local features, rather than the Gestalt. We examined the neural correlates of numerosity estimation in adults with and without ASD, to disentangle perceptual atypicalities from numerosity processing. Fourteen adults with ASD and matched typically developed (TD) controls estimated the number of dots (80-150) arranged either randomly (local information) or in meaningful patterns (global information) while brain activity was recorded with magnetoencephalography (MEG). Behavioral results showed no significant group difference in the errors of estimation. However, numerical estimation in ASD was more variable across numerosities than TD and was not affected by the global arrangement of the dots. At 80-120 ms, MEG analyses revealed early significant differences (TD > ASD) in source amplitudes in visual areas, followed from 120 to 400 ms by group differences in temporal, and then parietal regions. After 400 ms, a source was found in the superior frontal gyrus in TD only. Activation in temporal areas was differently sensitive to the global arrangement of dots in TD and ASD. MEG data show that individuals with autism exhibit widespread functional abnormalities. Differences in temporal regions could be linked to atypical global perception. Occipital followed by parietal and frontal differences might be driven by abnormalities in the processing and conversion of visual input into a number-selective neural code and complex cognitive decisional stages. These results suggest overlapping atypicalities in sensory, perceptual and number-related processing during numerosity estimation in ASD.

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