Congruency effects in dot comparison tasks: convex hull is more important than dot area

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.

Factors influencing the role of inhibitory control in non-symbolic numerical processing

Previous studies have found that inhibitory control plays an important role in non-symbolic numerical processing. However, this role may be influenced by the visual cue control method or the stimulus' presentation time. We investigated these questions by conducting three experiments using a priming paradigm to compare the level of inhibitory control in a sequential dot comparison task with single-dimensional and multi-dimensional control of visual cues under two presentation time conditions (300 ms and 1500 ms). We found that neither the method of visual cue control nor the presentation time of dot arrays affected the level of inhibitory control in the dot comparison task. These results reveal a stable role of inhibitory control in non-symbolic numerical processing, providing further evidence for integrating numerical and visual information during non-symbolic numerical processing.

The radial-tangential anisotropy of numerosity perception

Humans can estimate the number of visually presented items without counting. In most studies on numerosity perception, items are uniformly distributed across displays, with identical distributions in central and eccentric parts. However, the neural and perceptual representation of the human visual field differs between the fovea and the periphery. For example, in peripheral vision, there are strong asymmetries with regard to perceptual interferences between visual items. In particular, items arranged radially usually interfere more strongly with each other than items arranged tangentially (the radial-tangential anisotropy). This has been shown for crowding (the deleterious effect of clutter on target identification) and redundancy masking (the reduction of the number of perceived items in repeating patterns). In the present study, we tested how the radial-tangential anisotropy of peripheral vision impacts numerosity perception. In four experiments, we presented displays with varying numbers of discs that were predominantly arranged radially or tangentially, forming strong and weak interference conditions, respectively. Participants were asked to report the number of discs. We found that radial displays were reported as less numerous than tangential displays for all radial and tangential manipulations: weak (Experiment 1), strong (Experiment 2), and when using displays with mixed contrast polarity discs (Experiments 3 and 4). We propose that numerosity perception exhibits a significant radial-tangential anisotropy, resulting from local spatial interactions between items.

Developmental changes of the impact of visual cues on ANS acuity across grades 1-5: Different patterns of visual cues on numerosity processing

Numerous studies have consistently demonstrated the presence of the approximate number system (ANS) throughout development. Research has also revealed that visual cues may influence the ANS acuity, which may change with age. However, most studies have drawn conclusions based on performance differences between incongruent and congruent trials, which may be confounded by an individual's ability to inhibit interference. Therefore, to examine the developmental changes of the impact of visual cues on ANS acuity, we utilized congruent trials with varying visual cues. Our sample comprised Chinese children from grade one to grade five. We manipulated the salience of numerical cues (numerical ratio) and visual cues (dot size) in a non-symbolic numerosity comparison task. The results revealed a discernible leap in development from first to third grade and first to fifth grade; however, this upward trajectory did not persist into the transition from third to fifth grade, where no appreciable advancement was observed. Moreover, we observed different effects of visual cues on the dot-comparison task depending on the numerical cues and age. Specifically, visual cues (i.e., dot size) only facilitated ANS acuity in older school-aged children when numerical cues were weakened. The results indicate the presence of two distinct magnitude representational systems-one for the numerical dimension and another for the non-numerical dimension-during development.

Numerosity comparison, cognitive strategies, and general cognitive functioning in older people

Introduction

Studies have shown age-related differences in numerical cognition, for example, in the level of numerosity comparison ability. Moreover, some studies point out individual differences in the cognitive strategies employed during the performance of numerosity comparison tasks and reveal that they are related to the aging process. One probable cause of these differences is the level of cognitive functioning. The aim of our study was to determine the relationships among numerosity comparison ability, the cognitive strategies utilized in the performance of numerosity comparison tasks and the general cognitive functioning in older people.

Methods

Forty-seven elderly people participated in the study. The participants were examined using overall cognitive functioning scales and computerized numerosity comparison task.

Results

The results showed many correlations between the participants' level of cognitive functioning and the percent of correct responses (PCR) and response time (RT) during numerosity comparison, as well as with the cognitive strategies applied by the participants. Task correctness was positively related to the level of performance in the attention and executive function tasks. In contrast, the long-term memory resources index and visuospatial skills level were negatively correlated with RT regarding numerosity comparison task performance. The level of long-term memory resources was also positively associated with the frequency of use of more complex cognitive strategies. Series of regression analyses showed that both the level of general cognitive functioning and the cognitive strategies employed by participants in numerosity comparison can explain 9-21 percent of the variance in the obtained results.

Discussion

In summary, these results showed significant relationships between the level of cognitive functioning and proficiency in numerosity comparison measured in older people. Moreover, it has been shown that cognitive resources level is related to the strategies utilized by older people, which indicates the potential application for cognitive strategy examinations in the development of new diagnostic tools.

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.

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.

Are they equivalent? An examination of task variants of approximate number comparison

Nonverbal numerical ability supports individuals' numerical information processing in everyday life and is also correlated with their learning of mathematics. This ability is typically measured with an approximate number comparison paradigm, in which participants are presented with two sets of objects and instructed to choose the numerically larger set. This paradigm has multiple task variants, where the two sets are presented in different ways (e.g., two sets are presented either simultaneously or sequentially, or two sets are presented either intermixed or separately). Despite the fact that different task variants have often been used interchangeably, it remains unclear whether these variants measure the same aspects of nonverbal numerical ability. Using a latent variable modeling approach with 270 participants (Mage = 20.75 years, SDage = 2.03, 94 males), this study examined the degree to which three commonly used task variants tapped into the same construct. The results showed that a bi-factor model corresponding to the hypothesis that task variants had both commonalities and uniqueness was a better fit for the data than a single-factor model, corresponding to the hypothesis that task variants were construct equivalent. These findings suggested that task variants of approximate number comparison did not measure the same construct and cannot be used interchangeably. This study also quantified the extent to which general cognitive abilities were involved in both common and unique parts of these task variants.

Large-scale study of the precision of the approximate number system: Differences between formats, heterogeneity and congruency effects

The study used a large sample of elementary schoolchildren in Russia (N = 3,448, 51.6% were girls, with a mean age of 8.70 years, ranging 6-11 years) to investigate the congruency, format and heterogeneity effects in a nonsymbolic comparison test and between-individual differences in these effects with generalized linear mixed effects models (GLMMs). The participants were asked to compare two arrays of figures of different colours in spatially separated or spatially intermixed formats. In addition, the figures could be similar or different for the two arrays. The results revealed that congruency (difference between congruent and incongruent items), format (difference between mixed and separated formats) and heterogeneity (difference between homogeneous and heterogeneous conditions) interacted. The heterogeneity effect was higher in the separated format, while the format effect was higher for the homogeneous condition. The separated format produced a greater congruency effect than the mixed format. In addition, the congruency effect was lower in the heterogeneous condition than in the homogeneous condition. Analysis of between-individual differences revealed that there was significant between-individual variance in the format and congruency effects. Analysis of between-grade differences revealed that accuracy improved from grade 1 to grade 4 only for congruent trials in separated formats. Consequently, the congruency effect increased in separated/homogeneous and separated/heterogeneous conditions. In general, the study demonstrated that the test format and heterogeneity affected accuracy and that this effect varied for congruent and incongruent items.

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.

To what extent is dot comparison an appropriate measure of approximate number system?

Introduction

Number sense has been systematically measured using dot comparison tasks. However, recent studies have reported that performance on dot comparison might be influenced inhibitory control and visual properties of dot arrays. In the present study, we analysed the influence of continuous magnitude, inhibitory control, and numerical ratio on the dot comparison performance of preschool children.

Methods

Participants were 517 preschool children from 13 different schools in Chile. Children completed a dot comparison and two inhibitory control tasks. Gebuis and Reynvoet method was used to create well-controlled dot arrays for use in the dot comparison task. A logistic mixed effects model was conducted to predict participants' dot comparison accuracy. Continuous magnitude and ratio were entered as level-1 predictors and inhibitory control as level-2 predictors.

Results

The results showed that all predictors made a significant contribution to dot comparison accuracy. Furthermore, a significant double interaction (inhibitory control x continuous magnitude) and a triple interaction (inhibitory control x continuous magnitude x ratio) showed that the contribution of inhibitory control skills in dot comparison accuracy depends on the continuous properties of dot arrays and ratio.

Discussion

These findings suggest that preschool children rely more on continuous magnitudes than numerosity in dot comparison tasks. They also indicate that the greater children's inhibitory control, the more able they are to respond based on numerosity in fully incongruent trials, particularly when ratio is low (easiest items). Taken together, the above findings support the competing processes account provided that both ANS and inhibitory control skills influence performance on dot comparison tasks.

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.

Anisotropic representations of visual space modulate visual numerosity estimation

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.

On the flexibility of strategies for center estimation

When subjects are asked to indicate the center of a spatially distributed stimulus, the features that control their responses tend to vary (1) across subjects and (2) as stimulus properties are altered. Here we ask: can subjects bring these different response tendencies under top-down control? In each of three tasks, all using briefly displayed, Gaussian dot-clouds, subjects were trained to perform different center-estimation responses. In the "mass task," the target was the centroid of the dots. In the "hull task," the target was the centroid of the region circumscribed by the convex hull of the dot-cloud. In the "hull-vertex task," the target was the centroid of the vertices of the convex hull. Subjects were able to perform each of the mass- and hull-tasks accurately and reliably. However, they found the hull-vertex task more difficult; errors were substantially larger in this task, and responses tended to be closer to both of the hull- and mass-task centers. The finding that subjects can intentionally target either the centroid of the dots in the stimulus or the centroid of the stimulus convex hull suggests that individual differences in feedback-free experiments may reflect idiosyncratic decisions by different subjects about what combination of these statistics to use in responding.

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.

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.

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.

Developmental dyscalculia: the progress of cognitive modeling in the field of numerical cognition deficits for children

The study of dyscalculia requires an analysis of the current developed hypotheses which describe the cognitive mechanisms involved in this neurodevelopmental disorder. The objective of our review is to determine any progress in modeling developmental dyscalculia. The first hypothesis suggests that dyscalculia is the consequence of a specific deficit level number on the precise number system and the approximate system. Then, the second hypothesis states that developmental dyscalculia is linked to a failure to process non-symbolic representations of numbers. On the other hand, the third suggests that dyscalculia is caused by a lack of access to numerical quantities from symbols. However, the last hypothesis asserts that developmental dyscalculia is linked to general deficits. All these hypotheses are compatible with recent neuroimaging results and raise new horizons for experimentation, which will allow the development of precise diagnostic tools and the improvement of intervention strategies and the remediation of developmental dyscalculia.

Developmental Changes in ANS Precision Across Grades 1-9: Different Patterns of Accuracy and Reaction Time

The main aim of this study was to analyze the patterns of changes in Approximate Number Sense (ANS) precision from grade 1 (mean age: 7.84 years) to grade 9 (mean age: 15.82 years) in a sample of Russian schoolchildren. To fulfill this aim, the data from a longitudinal study of two cohorts of children were used. The first cohort was assessed at grades 1-5 (elementary school education plus the first year of secondary education), and the second cohort was assessed at grades 5-9 (secondary school education). ANS precision was assessed by accuracy and reaction time (RT) in a non-symbolic comparison test ("blue-yellow dots" test). The patterns of change were estimated via mixed-effect growth models. The results revealed that in the first cohort, the average accuracy increased from grade 1 to grade 5 following a non-linear pattern and that the rate of growth slowed after grade 3 (7-9 years old). The non-linear pattern of changes in the second cohort indicated that accuracy started to increase from grade 7 to grade 9 (13-15 years old), while there were no changes from grade 5 to grade 7. However, the RT in the non-symbolic comparison test decreased evenly from grade 1 to grade 7 (7-13 years old), and the rate of processing non-symbolic information tended to stabilize from grade 7 to grade 9. Moreover, the changes in the rate of processing non-symbolic information were not explained by the changes in general processing speed. The results also demonstrated that accuracy and RT were positively correlated across all grades. These results indicate that accuracy and the rate of non-symbolic processing reflect two different processes, namely, the maturation and development of a non-symbolic representation system.

Alzheimer's disease disrupts domain-specific and domain-general processes in numerosity estimation

INTRODUCTION

This study investigated how Alzheimer's Disease (AD) affects numerosity estimation abilities (e.g., finding the approximate number of items in a collection).

METHOD

Across two experiments, performance from HOA (i.e., Healthy Older Adults; N = 48) and AD patients (N = 50) was compared on dot comparison tasks. Participants were presented with two dot arrays and had to select the more numerous dot array in comparison tasks. They also took a Simon task and a number-line tasks (i.e., number-line tasks in which they had to indicate the position of a number on a line 0 to 100 or on a line 0 to 1,000 in the number-line task).

RESULTS

In Experiment 1, (a) AD patients obtained significantly poorer performance while comparing collections of dots, especially harder (small-ratio) collections, (b) these deficits correlated with poorer performance on the number-line task for larger numerosities (i.e., 0 to 1,000), and (c) AD patients showed poorer performance on incongruent (where numerosity and area occupied by dots mismatched) than on congruent items (where both features matched), while HOA showed no congruency effects. Experiment 2 showed (a) congruency effects in both groups when convex hull was tested as an incongruent feature, and (b) comparable sequential modulations of congruency effects in both groups.

CONCLUSIONS

Our findings showed that numerosity abilities decline in AD patients, and that this decline results from impaired domain-specific processes (i.e., numerosity processing) and domain-general processes (i.e., inhibition). These findings have important implications to further our understanding of how specific and general cognitive processes contribute to numerosity estimation/comparison performance, and how such contributions change during Alzheimer's disease.

Number sense biases children's area judgments

Humans are thought to use the approximate number system (ANS) to make quick approximations based on quantity even before learning to count. However, there has long been controversy regarding the salience of number versus other stimulus dimensions throughout development, including a recent proposal that number sense is derived from a sense of general magnitude. Here, we used a regression approach to disentangle numerical acuity from sensitivity to total surface area in both 5-year-old children and adults. We found that both children and adults displayed higher acuity when making numerosity judgments than total surface area judgments. Adults were largely able to ignore irrelevant stimulus features when making numerosity or total area judgments. Children were more biased by numerosity when making total area judgments than by total area when making numerosity judgments. These results provide evidence that number is more salient than total surface area even before the start of formal education and are inconsistent with the Sense of Magnitude proposal.

Food Quantity Discrimination in Angelfish (Pterophyllum scalare): The Role of Number, Density, Size and Area Occupied by the Food Items

Quantity discrimination, the ability to identify, process, and respond to differences in number, has been shown in a variety of animal species and may have fitness value. In fish, the ability to distinguish between numerically different shoals has been well studied. However, little work has been devoted to the investigation of such ability in a foraging context. Nevertheless, angelfish (Pterophyllum scalare) have been previously shown to be able to discriminate numerically different sets of food items, with variables such as size and density of the food items playing important roles in making the choice. Here, we examine the possible role of other numerical and non-numerical variables. Using a spontaneous binary choice task, we contrasted sets of food items differing in specifically controlled ways: (1) different numerical size but equal inter-item distance; (2) different numerical size and different inter-item distance; and (3) identical total contour length and area occupied but different individual food size and inter-food distance between the contrasted food sets. In Experiment 1, angelfish were found to prefer the sets with a large number of food items. In Experiment 2, they preferred the numerically smaller sets with clustered items to the numerically larger sets with scattered items, but only when the sets were in the large number range (10 vs. 5 food items). Finally, in Experiment 3 fish preferred numerically smaller sets with large-sized and scattered food items in the large number range sets. We conclude that food item number, density, and size may not be considered individually by angelfish, but instead, the fish respond to all these factors attempting to maximize energy gained from eating the food while minimizing energy expenditure collecting and/or protecting the food.

Numerosity and cumulative surface area are perceived holistically as integral dimensions

Human and nonhuman animals have a remarkable capacity to rapidly estimate the quantity of objects in the environment. The dominant view of this ability posits an abstract numerosity code, uncontaminated by nonnumerical visual information. The present study provides novel evidence in contradiction to this view by demonstrating that number and cumulative surface area are perceived holistically, classically known as integral dimensions. Whether assessed explicitly (Experiment 1) or implicitly (Experiment 2), perceived similarity for dot arrays that varied parametrically in number and cumulative area was best modeled by Euclidean, as opposed to city-block, distance within the stimulus space, comparable to other integral dimensions (brightness/saturation and radial frequency components) but different from separable dimensions (shape/color and brightness/size). Moreover, Euclidean distance remained the best-performing model, even when compared to models that controlled for other magnitude properties (e.g., density) or image similarity. These findings suggest that numerosity perception entails the obligatory processing of nonnumerical magnitude. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

The Relationship Between Non-symbolic and Symbolic Numerosity Representations in Elementary School: The Role of Intelligence

This study aimed to estimate the extent to which the development of symbolic numerosity representations relies on pre-existing non-symbolic numerosity representations that refer to the Approximate Number System. To achieve this aim, we estimated the longitudinal relationships between accuracy in the Number Line (NL) test and “blue–yellow dots” test across elementary school children. Data from a four-wave longitudinal study involving schoolchildren in grades 1–4 in Russia and Kyrgyzstan (N = 490, mean age 7.65 years in grade 1) were analyzed. We applied structural equation modeling and tested several competing models. The results revealed that at the start of schooling, the accuracy in the NL test predicted subsequent accuracy in the “blue–yellow dots” test, whereas subsequently, non-symbolic representation in grades 2 and 3 predicted subsequent symbolic representation. These results indicate that the effect of non-symbolic representation on symbolic representation emerges after a child masters the basics of symbolic number knowledge, such as counting in the range of twenty and simple arithmetic. We also examined the extent to which the relationships between non-symbolic and symbolic representations might be explained by fluid intelligence, which was measured by Raven’s Standard Progressive Matrices test. The results revealed that the effect of symbolic representation on non-symbolic representation was explained by fluid intelligence, whereas at the end of elementary school, non-symbolic representation predicted subsequent symbolic representation independently of fluid intelligence.

The interplay between non-symbolic number and its continuous visual properties revisited: Effects of mixing trials of different types

In the last few years, the existence of a pure number sense has been challenged. Recent studies suggest that numerosity processing is influenced not only by the number of elements in a display but also by continuous magnitudes, such as the size of the elements. The aim of our study was to replicate and extend the findings by Gebuis and Reynvoet, who systematically manipulated different continuous magnitudes either congruently or incongruently with discrete numerosity. We were particularly interested in finding the same pattern of congruency effects and assess its stability and robustness as this pattern indicates a complex influence of continuous magnitudes on numerosity judgements. We did so by showing stimuli of different conditions either in separate blocks or mixed together while participants solved a dot comparison task. Our results are in line with the notion that discrete number and continuous magnitudes are integrated in numerosity judgements by means of a weighing process. Moreover, our findings suggest that this integration is modified by the mode of presentation (blocked vs. mixed).

The progressive 6-year-old conserver: Numerical saliency and sensitivity as core mechanisms of numerical abstraction in a Piaget-like estimation task

In Piaget's theory of number development, children do not possess a true concept of number until they are able to reason on numerical quantity regardless of changes in other nonnumerical magnitudes, such as length. Recent studies have echoed this result by arguing that abstracting number from nonnumerical dimensions of magnitude is a developmental milestone and a strong predictor of mathematics achievement. However, the mechanisms supporting such abstraction remain largely underspecified. We aimed to study how identification of the numerical equivalence in a Piaget-like estimation task by 6-year-old children is affected by (a) the degree of interference between number and nonnumerical magnitudes and (b) children's spontaneous orientation to numerosity. Six-year-old children first performed a card sorting task assessing their spontaneous orientation towards numerosity, spacing, or item size in a set of dots. Then, they completed a Piaget-like same/different numerical estimation task using two rows of dots in which the length ratio between the two rows varied systematically. Children were less likely to accept the numerical equivalence in the Piaget-like estimation task (a) as the difference in spacing between the dots increased and (b) as the children were more spontaneously oriented towards spacing over number in the card sorting task. Our results suggest that abstracting number depends on its saliency, which varies both as a function of the context (i.e., length ratio between the two rows) and of individual differences in children's sensitivity to the numerical aspects of their environment. These factors could be at the root of the observed development of performance in the seminal number-conservation task, which appears as a progressive abstraction of number rather than a conceptual shift, as Piaget hypothesized.

Visuo-spatial (but not verbal) executive working memory capacity modulates susceptibility to non-numerical visual magnitudes during numerosity comparison

The present study tested whether visuo-spatial vs. verbal executive working memory capacity (hereafter EWM) modulates the degree to which non-numerical visual magnitudes influence numerosity comparison using pairs of dot arrays. We hypothesized that visuo-spatial (rather than verbal) EWM capacity would influence one’s ability to selectively focus on numerical as opposed to non-numerical visual properties (such as dot size, cumulative area, density) of the dot arrays during numerosity comparison. Participants’ performance was better on trials in which non-numerical visual magnitudes were negatively (vs. positively) correlated with numerosity (i.e., reverse congruency effect). The Low visuo-spatial EWM group manifested greater reverse congruency effect compared to the High visuo-spatial EWM group. A trial-based hierarchical regression on the accuracy of each trial using the ratio of (numerical and non-numerical) visual magnitudes as predictors revealed that the ratio of numerical vs. non-numerical visual magnitudes explained the greatest variance in the performance of the High vs. Low visuo-spatial EWM groups, respectively. In contrast, there was no difference between the High vs. Low verbal EWM groups from the same analysis. These results reveal differential susceptibility to numerical vs. non-numerical visual information depending on the capacity of visuo-spatial (but not verbal) EWM. Taken together, numerosity comparison performance measured with the dot comparison paradigm seems to reflect not only one’s acuity for numerosity discrimination but also visuo-spatial EWM capacity likely required during integration of visual magnitudes during numerosity comparison.

Toward an integrative approach to numerical cognition

In response to the commentaries, we have refined our suggested model and discussed ways in which the model could be further expanded. In this context, we have elaborated on the role of specific continuous magnitudes. We have also found it important to devote a section to evidence considered the "smoking gun" of the approximate number system theory, including cross-modal studies, animal studies, and so forth. Lastly, we suggested some ways in which the scientific community can promote more transparent and collaborative research by using an open science approach, sharing both raw data and stimuli. We thank the contributors for their enlightening comments and look forward to future developments in the field.

Attention to number: The convergence of numerical magnitude processing, attention, and mathematics in the inferior frontal gyrus

Research indicates that the neurocognitive system representing nonsymbolic numerical magnitudes is foundational for the development of mathematical competence. However, recent studies found that the most common task used to measure numerical acuity, the nonsymbolic number comparison task, is heavily influenced by non-numerical visual parameters of stimuli that increase executive function demands. Further, this influence may be a confound invalidating theoretical accounts of the relation between number comparison performance and mathematical competence. Instead of acuity, the relation may depend on one's ability to attend to numerical information in the face of competing, non-numerical cues. The current study investigated this issue by measuring neural activity associated with numerical magnitude processing acuity, domain-general attention, and selective attention to number via functional magnetic resonance imaging while children 8-11 years old completed a nonsymbolic number comparison task and a flanker task. Results showed that activation in the right inferior frontal gyrus during incongruent versus congruent trials of the comparison task, our construct for attention to number, predicted mathematics achievement after controlling for verbal IQ, flanker accuracy rate, and the neural congruency effect from the flanker task. In contrast, activity in frontal and parietal regions responding to differences in difficulty of numerical comparisons, our construct for numerical magnitude processing acuity, did not correlate with achievement. Together, these findings suggest a need to reframe existing models of the relation between number processing and math competence to include the interaction between attention and use of numerical information, or in other words "attention to number."

Using Hierarchical Linear Models to Examine Approximate Number System Acuity: The Role of Trial-Level and Participant-Level Characteristics

The ability to intuitively and quickly compare the number of items in collections without counting is thought to rely on the Approximate Number System (ANS). To assess individual differences in the precision of peoples' ANS representations, researchers often use non-symbolic number comparison tasks in which participants quickly choose the numerically larger of two arrays of dots. However, some researchers debate whether this task actually measures the ability to discriminate approximate numbers or instead measures the ability to discriminate other continuous magnitude dimensions that are often confounded with number (e.g., the total surface area of the dots or the convex hull of the dot arrays). In this study, we used hierarchical linear models (HLMs) to predict 132 adults' accuracy on each trial of a non-symbolic number comparison task from a comprehensive set of trial-level characteristics (including numerosity ratio, surface area, convex hull, and temporal and spatial variations in presentation format) and participant-level controls (including cognitive abilities such as visual-short term memory, working memory, and math ability) in order to gain a more nuanced understanding of how individuals complete this task. Our results indicate that certain trial-level characteristics of the dot arrays contribute to our ability to compare numerosities, yet numerosity ratio, the critical marker of the ANS, remains a highly significant predictor of accuracy above and beyond trial-level characteristics and across individuals with varying levels of math ability and domain-general cognitive abilities.

The measurement of approximate number system acuity across the lifespan is compromised by congruency effects

Recent studies have highlighted the influence of visual cues such as dot size and cumulative surface area on the measurement of the approximate number system (ANS). Previous studies assessing ANS acuity in ageing have all applied stimuli generated by the Panamath protocol, which does not control nor measure the influence of convex hull. Crucially, convex hull has recently been identified as an influential visual cue present in dot arrays, with its impact on older adults’ ANS acuity yet to be investigated. The current study therefore investigated the manipulation of convex hull by the Panamath protocol, and its effect on the measurement of ANS acuity in younger and older participants. First, analyses of the stimuli generated by Panamath revealed a confound between numerosity ratio and convex hull ratio. Second, although older adults were somewhat less accurate than younger adults on convex hull incongruent trials, ANS acuity was broadly similar between the groups. These findings have implications for the valid measurement of ANS acuity across all ages, and suggest that the Panamath protocol produces stimuli that do not adequately control for the influence of convex hull on numerosity discrimination.

Deficits in Approximate Number System Acuity and Mathematical Abilities in 6.5-Year-Old Children Born Extremely Preterm

Preterm children are at increased risk for poor academic achievement, especially in math. In the present study, we examined whether preterm children differ from term-born children in their intuitive sense of number that relies on an unlearned, approximate number system (ANS) and whether there is a link between preterm children’s ANS acuity and their math abilities. To this end, 6.5-year-old extremely preterm (i.e., <27 weeks gestation, n = 82) and term-born children (n = 89) completed a non-symbolic number comparison (ANS acuity) task and a standardized math test. We found that extremely preterm children had significantly lower ANS acuity than term-born children and that these differences could not be fully explained by differences in verbal IQ, perceptual reasoning skills, working memory, or attention. Differences in ANS acuity persisted even when demands on visuo-spatial skills and attention were reduced in the ANS task. Finally, we found that ANS acuity and math ability are linked in extremely preterm children, similar to previous results from term-born children. These results suggest that deficits in the ANS may be at least partly responsible for the deficits in math abilities often observed in extremely preterm children.