Set size influences the relationship between ANS acuity and math performance: a result of different strategies?

Children's comparison of different-length numbers: Managing different attributes in multidigit number processing

In everyday life the comparison of numbers usually occurs between numbers with different numbers of digits. However, experimental research here is scarce. Recent research has shown that adults respond faster to congruent pairs (the initial digit in the number with more digits is larger, e.g., 2384 vs. 107) than to incongruent pairs (the initial digit is larger in the number with fewer digits, e.g., 2675 vs. 398). This has been interpreted as support for the processing of multiple attributes in parallel and against serial accounts. The current research asked whether there is a change in the relevance of these attributes as school grades increase. School-age children from the second to sixth grades (N = 206) were presented with pairs of numbers that had either the same number of digits (3 vs. 3 or 4 vs. 4) or a different number of digits (3 vs. 4). In this latter condition, the stimuli, matched by distance, could be either length/digit congruent (e.g., 2384 vs. 107) or length/digit incongruent (e.g., 2675 vs. 398). Linear mixed models showed a length/digit congruity effect from second graders. Interestingly, in the response time measure, congruity interacted with school grade and the side in which the larger number of the pair was presented. Whereas these results support a model that considers number comparison as a process that weighs different attributes in parallel, it is also argued that developmental changes are associated with differences in the level of automatization of the componential skills involved in the comparison.

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.

The relationship between numerosity perception and mathematics ability in adults: the moderating role of dots number

Background

It has been proposed that numerosity perception is the cognitive underpinning of mathematics ability. However, the existence of the association between numerosity perception and mathematics ability is still under debate, especially in adults. The present study examined the relationship between numerosity perception and mathematics ability and the moderating role of dots number (i.e., the numerosity of items in dot set) in adults.

Methods

Sixty-four adult participants from Anshun University completed behavioral measures that tested numerosity perception of small numbers and large numbers, mathematics ability, inhibition ability, visual-spatial memory, and set-switching ability.

Results

We found that numerosity perception of small numbers correlated significantly with mathematics ability after controlling the influence of inhibition ability, visual-spatial memory, and set-switching ability, but numerosity perception of large numbers was not related to mathematics ability in adults.

Conclusions

These findings suggest that the dots number moderates the relationship between numerosity perception and mathematics ability in adults and may contribute to explaining the contradictory findings in the previous literature about the link between numerosity perception and mathematics ability.

Visual Perception Supports Adults in Numerosity Processing and Arithmetical Performance

Previous studies have found a correlation between numerosity processing and arithmetical performance. Visual perception has already been indicated as the shared cognitive mechanism between these two; however, these studies mostly focused on children. It is not clear whether the association between numerosity processing and arithmetical performance still existed following the development of individual arithmetical performance. Consequently, the underlying role of visual perception in numerosity processing and arithmetical performance has not been sufficiently studied in adults. For this study, researchers selected a total of 205 adult participants with an average age of 22years. The adults were administered arithmetic tests, numerosity comparison, and visual figure matching. Mental rotation, choice reaction time, and nonverbal intelligence were used as cognitive covariates. Results showed that numerosity comparison of adults correlated with their arithmetical performance, even after controlling for age and gender differences as well as general cognitive processing. However, after controlled for visual figure matching, the well-established association between numerosity comparison and arithmetic performance disappeared. These results supported the visual perception hypothesis, that visual perception measured by visual figure matching can account for the correlation between numerosity comparison and arithmetic performance. This indicated that even for adult populations, visual perceptual ability was the underlying component of numerosity processing and arithmetic performance.

The relation between working memory, number sense, and mathematics throughout primary education in children with and without mathematical difficulties

Number sense and working memory contribute to mathematical development throughout primary school. However, it is still unclear how the contributions of each of these predictors may change across development and whether the cognitive contribution is the same for children with and without mathematical difficulties. The aim of the two studies in this paper was to shed light on these topics. In a cross-sectional design, a typically developing group of children (study 1; N = 459, Grades 1-4) and a group with mathematical difficulties (study 2; N = 61, Grades 4-6) completed a battery of number sense and working memory tests, as well as a measure of arithmetic competence. Results of study 1 indicated that number sense was important in first grade, while working memory gained importance in second grade, before predictive value of both predictors waned. Number sense and working memory supported mathematics development independently from one another from Grade 1. Analysis of task demands showed that typically developing children rely on comprehension and visualization of quantity-to-number associations in early development. Later in development, pupils rely on comparing larger numerals and working memory until automatization. Children with mathematical difficulties were less able to employ number sense during mathematical operations, and thus might remain dependent on their working memory resources during arithmetic tasks. This suggests that children with mathematical difficulties need aid to employ working memory for mathematics from an early age to be able to automatize mathematical abilities later in development.

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

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

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.