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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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A number of studies revealed that our visual system can extract different types of summary statistics, such as the mean and variance, from sets of items. Although the extraction of such summary statistics has been studied well in isolation, the relationship between these statistics remains unclear. In this study, we explored this issue using an individual differences approach. Observers viewed illustrations of strawberries and lollypops varying in size or orientation and performed four tasks in a within-subject design, namely mean and variance discrimination tasks with size and orientation domains. We found that the performances in the mean and variance discrimination tasks were not correlated with each other and demonstrated that extractions of the mean and variance are mediated by different representation mechanisms. In addition, we tested the relationship between performances in size and orientation domains for each summary statistic (i.e. mean and variance) and examined whether each summary statistic has distinct processes across perceptual domains. The results illustrated that statistical summary representations of size and orientation may share a common mechanism for representing the mean and possibly for representing variance. Introspections for each observer performing the tasks were also examined and discussed.

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Several studies have shown that our visual system may construct a "summary statistical representation" over groups of visual objects. Although there is a general understanding that human observers can accurately represent sets of a variety of features, many questions on how summary statistics, such as an average, are computed remain unanswered. This study investigated sampling properties of visual information used by human observers to extract two types of summary statistics of item sets, average and variance. We presented three models of ideal observers to extract the summary statistics: a global sampling model without sampling noise, global sampling model with sampling noise, and limited sampling model. We compared the performance of an ideal observer of each model with that of human observers using statistical efficiency analysis. Results suggest that summary statistics of items in a set may be computed without representing individual items, which makes it possible to discard the limited sampling account. Moreover, the extraction of summary statistics may not necessarily require the representation of individual objects with focused attention when the sets of items are larger than 4.

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Many studies have claimed that the numerosity of any set of discrete elements can be depicted by a genuinely abstract number representation, irrespective of whether they are presented in a visual, auditory, or tactile modality. However, in behavioral studies, some inconsistencies have been observed in the performance of number comparisons among different modalities. In this study, we have tested whether numerical comparisons of auditory, tactile, and cross-modal presentations would differ under adequate control of stimulus presentation, and, if so, how they would differ. The unimodal and cross-modal stimuli pairs were presented in sequential manner. We measured the Weber fractions (i.e., precision) and points of subjective equality (i.e., accuracy) of numerical discriminations in auditory, tactile, and crossmodal conditions. The results showed that the Weber fractions are constant over standard stimuli, indicating that the Weber's law holds for the range of numerical values that was tested. Furthermore, the Weber fractions are consistent over unimodal and cross-modal comparisons, and this indicates that there is no additional noise involved in the cross-modal comparisons. Interestingly, the bias measure showed that the number of auditory stimuli is systematically overestimated compared with that of tactile stimuli.

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Most research on the multimodal perception of material properties has investigated the perception of material properties of two modalities such as vision-touch, vision-audition, audition-touch, and vision-action. Here, we investigated whether the same affective classifications of materials can be found in three different modalities of vision, audition, and touch, using wood as the target object. Fifty participants took part in an experiment involving the three modalities of vision, audition, and touch, in isolation. Twenty-two different wood types including genuine, processed, and fake were perceptually evaluated using a questionnaire consisting of twenty-three items (12 perceptual and 11 affective). The results demonstrated that evaluations of the affective properties of wood were similar in all three modalities. The elements of "expensiveness, sturdiness, rareness, interestingness, and sophisticatedness" and "pleasantness, relaxed feelings, and liked-disliked" were separately grouped for all three senses. Our results suggest that the affective material properties of wood are at least partly represented in a supramodal fashion. Our results also suggest an association between perceptual and affective properties, which will be a useful tool not only in science, but also in applied fields.

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The numerosity of any set of discrete elements can be depicted by a genuinely abstract number representation, irrespective of whether they are presented in the visual or auditory modality. The accumulator model predicts that no cost should apply for comparing numerosities within and across modalities. However, in behavioral studies, some inconsistencies have been apparent in the performance of number comparisons among different modalities. In this study, we tested whether and how numerical comparisons of visual, auditory, and cross-modal presentations would differ under adequate control of stimulus presentation. We measured the Weber fractions and points of subjective equality of numerical discrimination in visual, auditory, and cross-modal conditions. The results demonstrated differences between the performances in visual and auditory conditions, such that numerical discrimination of an auditory sequence was more precise than that of a visual sequence. The performance of cross-modal trials lay between performance levels in the visual and auditory conditions. Moreover, the number of visual stimuli was overestimated as compared to that of auditory stimuli. Our findings imply that the process of approximate numerical representation is complex and involves multiple stages, including accumulation and decision processes.

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Although a critical issue in the debate over the existence of abstract numerical representation, it remains unclear whether and how perceptual variables affect numerosity judgment and how they change across development stages. In this research, we examine the effects of perceptual variables on approximate numerosity comparison in 5-6-year-olds and adults using the identical experimental procedure. In the assessment of the effect of the perceptual variables, we measured precision (i.e., Weber fraction) and accuracy (i.e., point of subjective equality; PSE) of the numerosity comparison. In Study 1, we tested how the manipulation of the cumulative element area would affect approximate numerosity comparison. The results showed that Weber fractions increased and the size of bias enlarged in the large element condition in both adults and 5-6-year-olds. In study 2, we tested how the manipulation of the array area would affect the precision and accuracy of approximate numerosity comparison. The results demonstrated that Weber fractions increased and the size of bias enlarged in the large array condition in both adults and 5-6-year-olds. Overall, our results suggest that the effect of perceptual variables on 5-6-year-olds is qualitatively similar to that on adults. In addition, we also tested whether the performance of approximate comparison correlated with the initial numerical skill in 5-6-year-olds to reveal least relationship between them.

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A genuinely abstract number representation is thought to be capable of representing the numerosity of any set of discrete elements, whether they are sequentially or simultaneously presented. Recent neuroimaging studies, however, have demonstrated that different areas of intraparietal sulcus play a role in extracting numerosity across simultaneous or sequential presentation during a quantification process, suggesting the existence of a format-dependent numerical system. To test whether behavioral evidence exists for format-dependent numerical processing in adult humans, we measured the Weber fractions of numerosity discrimination for sequential stimuli, simultaneous stimuli, and cross-format stimuli with a carefully controlled experimental procedure. The results showed distinct differences between the performance in the simultaneous and sequential conditions, supporting the existence of format-dependent processes for numerosity representation. Moreover, the performance on cross-format trials differed among participants, with the exception that performance was always worse than in the simultaneous condition. Taken together, our findings suggest that numerical representation may involve a complex set of multiple stages.

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In this study, we tested whether and how temporal information would affect the performance of numerosity discrimination in sequential events, in an attempt to make the relationship clear between temporal and numerosity processing. We manipulated the duration of event presentation (i.e., the stimulus duration) and the duration of the sequence (i.e., the total interval). We also employed three levels of standard event numbers (i.e., 5, 10, and 20) to test whether and how the effect would differ among event numbers. The results showed that temporal information affected the performance of numerosity discrimination; precision deteriorated when stimulus duration and the total interval were manipulated, and the number of events in the longer total interval were judged as less numerous than those in the shorter total interval across standard numbers. Accordingly, our results suggested the existence of a shared system between time and numerosity processing.

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It has been claimed that a genuinely abstract numerical representation would be capable of representing the numerosity of any set of discrete elements independently of the attributes of the individual elements. In practice, however, perceptual variables, such as element size, density, and others, affect numerosity judgment. In this study, we examined how perceptual variables affect the performance of numerosity judgments under the control of factors such as amount of practice and set size. In Experiment 1, we introduced the numerosity comparison task to less experienced observers in order to examine the effect of element size and array area. In Experiment 2, we examined whether and how practice would influence precision and accuracy in numerosity comparison and demonstrated that the effects of perceptual variables mostly disappeared. In Experiment 3, we examined whether the effect of practice could transfer to the performance in different stimulus conditions. Taking the results together, we demonstrate that differences in practice might be the source of inconsistent results for perceptual variables.

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We investigated effects of element features on statistical description of relative frequency. In Experiment 1, we manipulated the proportion of two elements types forming a set and the set size and measured response time and the accuracy of relative numerosity discrimination. We did this with element pairs that had been shown to produce search symmetries or asymmetries in visual search task. We found that pop-out elements in the search asymmetry pair were numerically overestimated. In Experiment 2, we used sets of circles and circles with gaps to eliminate the possibility that the overestimation found in Experiment 1 was due to larger contour length of pop-out elements. In Experiment 3, we manipulated proportion of two elements types and the set size to measure point of subjective equality (PSE) and the slopes of z-score functions in relative numerosity discrimination to support the results of Experiments 1 and 2. The results generally showed that the proportion of pop-out elements is likely to be overestimated and that set size had no effect, suggesting that the types of features characterized by visual search could influence the accuracy and precision in discrimination of relative numerosity.

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