Search

As adults, we represent and think about number, space, and time in at least two ways: our intuitive-but imprecise-perceptual representations, and the slowly learned-but precise-number words. With development, these representational formats interface, allowing us to use precise number words to estimate imprecise perceptual experiences. We test two accounts of this developmental milestone. Either slowly learned associations are required for the interface to form, predicting that deviations from typical experiences (e.g., presentation of a novel unit or unpracticed dimension) will disrupt children's ability to map number words to their perceptual experiences or children's understanding of the logical similarity between number words and perceptual representations allows them to flexibly extend this interface to novel experiences (e.g., units and dimensions they have not yet learned how to formally measure). 5-11-year-olds completed verbal estimation and perceptual sensitivity tasks across three dimensions: Number, Length, and Area. For verbal estimation, they were given novel units (i.e., a three-dot unit called one "toma" for Number, a 44 px long line called one "blicket" for Length, a 111 px 2 blob called one "modi" for Area) and asked to estimate how many tomas/blickets/modies they saw when shown a larger set of dots, lines, and blobs. Children could flexibly link number words to novel units across dimensions, demonstrating positive estimation slopes, even for Length and Area, which younger children had limited experience with. This suggests that the logic of structure mapping can be dynamically utilized across perceptual dimensions, even without extensive experience., (© 2023 The Authors. Developmental Science published by John Wiley & Sons Ltd.)

The mind represents abstract magnitude information, including time, space, and number, but in what format is this information stored? We show support for the bipartite format of perceptual magnitudes, in which the measured value on a dimension is scaled to the dynamic range of the input, leading to a privileged status for values at the lowest and highest end of the range. In six experiments with college undergraduates, we show that observers are faster and more accurate to find the endpoints (i.e., the minimum and maximum) than any of the inner values, even as the number of items increases beyond visual short-term memory limits. Our results show that length, size, and number are represented in a dynamic format that allows for comparison-free sorting, with endpoints represented with an immediately accessible status, consistent with the bipartite model of perceptual magnitudes. We discuss the implications for theories of visual search and ensemble perception.

When observing a simple visual scene such as an array of dots, observers can easily and automatically extract their number. How does our visual system accomplish this? We investigate the role of specific spatial frequencies to the encoding of number through cross-adaptation. In two experiments, observers were peripherally adapted to six randomly generated sinusoidal gratings varying from relatively low-spatial frequency (M = 0.44 c/deg) to relatively high-spatial frequency (M = 5.88 c/deg). Subsequently, observers judged which side of the screen had a higher number of dots. We found a strong number-adaptation effect to low-spatial frequency gratings (i.e., participants significantly underestimated the number of dots on the adapted side) but a significantly reduced adaptation effect for high-spatial frequency gratings. Various control conditions demonstrate that these effects are not due to a generic response bias for the adapted side, nor moderated by dot size or spacing effects. In a third experiment, we observed no cross-adaptation for centrally presented gratings. Our results show that observers' peripheral number perception can be adapted even with stimuli lacking any numeric or segmented object information and that low spatial frequencies adapt peripheral number perception more than high ones. Together, our results are consistent with recent number perception models that suggest a key role for spatial frequency in the extraction of number from the visual signal (e.g., Paul, Ackooij, Ten Cate, & Harvey, 2022), but additionally suggest that some spatial frequencies - especially in the low range and in the periphery - may be weighted more by the visual system when estimating number. We argue that the cross-adaptation paradigm is also a useful methodology for discovering the primitives of visual number encoding., (© 2023. The Psychonomic Society, Inc.)

While the human visual system is sensitive to numerosity, the mechanisms that allow perception to extract and represent the number of objects in a scene remains unknown. Prominent theoretical approaches posit that numerosity perception emerges from passive experience with visual scenes throughout development, and that unsupervised deep neural network models mirror all characteristic behavioral features observed in participants. Here, we derive and test a novel prediction: if the visual number sense emerges from exposure to real-world scenes, then the closer a stimulus aligns with the natural statistics of the real world, the better number perception should be. But - in contrast to this prediction - we observe no such advantage (and sometimes even a notable impairment) in number perception for natural scenes compared to artificial dot displays in college-aged adults. These findings are not accounted for by the difficulty in object identification, visual clutter, the parsability of objects from the rest of the scene, or increased occlusion. This pattern of results represents a fundamental challenge to recent models of numerosity perception based in experiential learning of statistical regularities, and instead suggests that the visual number sense is attuned to abstract number of objects, independent of their underlying correlation with non-numeric features. We discuss our results in the context of recent proposals that suggest that object complexity and entropy may play a role in number perception., (Copyright © 2022 Elsevier B.V. All rights reserved.)

The emerging field of robotics education (RE) is a new and rapidly growing subject area worldwide. It may provide a playful and novel learning environment for children to engage with all aspects of science, technology, engineering, and mathematics (STEM) learning. The purpose of this research is to examine how robotics learning activities may affect the cognitive abilities and cognitive processes of 6-8 years old children. The study adopted the mixed methods approach with a repeated measures design; three waves of data collection over 6 months, including quantitative data obtained from cognitive assessments and eye-tracking, and qualitative data from the interviews. A total of 31 children were recruited from an afterschool robotics program. To the best of our knowledge, this study is the first RE research that used a combination of eye-tracking, cognitive assessments, and interviews for examining the effect of RE on children. Using linear growth models, the results of cognitive assessments showed that children's visuospatial working memory as well as logical and abstract reasoning skills improved over time. The interview data were analyzed by a thematic analysis. The results revealed that children perceived RE activities as game play, which made children more engaged in their study; parents found their children to be more focused on activities comparing to six months ago. Additionally, the visualization of the eye-tracking data suggested that children became more focused on RE activities and got faster to process the information across six months in general, which echoed the findings in assessments and interviews. Our findings may help educators and policymakers better understand the benefits of RE for young children., Competing Interests: Competing InterestsThe authors declare no competing interests., (© The Author(s), under exclusive licence to Springer Nature B.V. 2023, Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.)

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., Competing Interests: Declaration of competing interest The authors declare no conflict of interest., (Copyright © 2022 The Authors. Published by Elsevier B.V. All rights reserved.)

Strategic collaboration according to the law of comparative advantage involves dividing tasks based on the relative capabilities of group members. Three experiments (N = 405, primarily White and Asian, 45% female, collected 2016-2019 in Canada) examined how this strategy develops in children when dividing cognitive labor. Children divided questions about numbers between two partners. By 7 years, children allocated difficult questions to the skilled partner (Experiment 1, d = 1.42; Experiment 2, d = 0.87). However, younger children demonstrated a self-serving bias, choosing the easiest questions for themselves. Only when engaging in a third-party collaborative task did 5-year-olds assign harder questions to the more skilled individual (Experiment 3, d = 0.55). These findings demonstrate early understanding of strategic collaboration subject to a self-serving bias., (© 2021 The Authors. Child Development © 2021 Society for Research in Child Development.)

Clarke and Beck use behavioural evidence to argue that (1) approximate ratio computations are sufficient for claiming that the approximate number system (ANS) represents the rationals, and (2) the ANS does not represent the reals. We argue that pure behaviour is a poor litmus test for this problem, and that we should trust the psychophysical models that place ANS representations within the reals.

Despite the global importance of science, engineering, and math-related fields, women are consistently underrepresented in these areas. One source of this disparity is likely the prevalence of gender stereotypes that constrain girls' and women's math performance and interest. The current research explores the developmental roots of these effects by examining the impact of stereotypes on young girls' intuitive number sense, a universal skill that predicts later math ability. Across four studies, 762 children ages 3-6 were presented with a task measuring their Approximate Number System accuracy. Instructions given before the task varied by condition. In the two control conditions, the task was described to children either as a game or a test of eyesight ability. In the experimental condition, the task was described as a test of math ability and that researchers were interested in whether boys or girls were better at math and counting. Separately, we measured children's explicit beliefs about math and gender. Results conducted on the combined dataset indicated that while only a small number of girls in the sample had stereotypes associating math with boys, these girls performed significantly worse on a test of Approximate Number System accuracy when it was framed as a math test rather than a game or an eyesight test. These results provide novel evidence that for young girls who do endorse stereotypes about math and gender, contextual activation of these stereotypes may impair their intuitive number sense, potentially affecting their acquisition of formal mathematics concepts and developing interest in math-related fields., Competing Interests: The authors have declared that no competing interests exist.

Natural languages like English connect pronunciations with meanings. Linguistic pronunciations can be described in ways that relate them to our motor system (e.g., to the movement of our lips and tongue). But how do linguistic meanings relate to our nonlinguistic cognitive systems? As a case study, we defend an explicit proposal about the meaning of most by comparing it to the closely related more: whereas more expresses a comparison between two independent subsets, most expresses a subset-superset comparison. Six experiments with adults and children demonstrate that these subtle differences between their meanings influence how participants organize and interrogate their visual world. In otherwise identical situations, changing the word from most to more affects preferences for picture-sentence matching (experiments 1-2), scene creation (experiments 3-4), memory for visual features (experiment 5), and accuracy on speeded truth judgments (experiment 6). These effects support the idea that the meanings of more and most are mental representations that provide detailed instructions to conceptual systems., (© 2021 New York Academy of Sciences.)

The world can be a confusing place, which leads to a significant challenge: how do we figure out what is true? To accomplish this, children possess two relevant skills: reasoning about the likelihood of their own accuracy (metacognitive confidence) and reasoning about the likelihood of others' accuracy (mindreading). Guided by Signal Detection Theory and Simulation Theory, we examine whether these two self- and other-oriented skills are one in the same, relying on a single cognitive process. Specifically, Signal Detection Theory proposes that confidence in a decision is purely derived from the imprecision of that decision, predicting a tight correlation between decision accuracy and confidence. Simulation Theory further proposes that children attribute their own cognitive experience to others when reasoning socially. Together, these theories predict that children's self and other reasoning should be highly correlated and dependent on decision accuracy. In four studies ( N  = 374), children aged 4-7 completed a confidence reasoning task and selective social learning task each designed to eliminate confounding language and response biases, enabling us to isolate the unique correlation between self and other reasoning. However, in three of the four studies, we did not find that individual differences on the two tasks correlated, nor that decision accuracy explained performance. These findings suggest self and other reasoning are either independent in childhood, or the result of a single process that operates differently for self and others., Supplementary Information: The online version contains supplementary material available at 10.1007/s11409-021-09263-x., Competing Interests: Conflicts of InterestThe authors declare that they have no conflict of interest., (© The Author(s) 2021.)

Research over the past 20 years has suggested that our intuitive sense of number-the Approximate Number System (ANS)-is associated with individual differences in symbolic math performance. The mechanism supporting this relationship, however, remains unknown. Here, we test whether the ANS contributes to how well adult observers judge the direction and magnitude of symbolic math equation errors. We developed a novel task in which participants view symbolic equations with incorrect answers (e.g., 47 + 21 = 102), and indicate whether the provided answer was too high or too low. By varying the ratio between the correct and the provided answers, we measured individual differences in how well participants detect the magnitude and direction of symbolic equation errors. We find that individual differences in equation error detection were uniquely predicted by ANS acuity-that is, the precision of each participant's intuitive number representations-even when controlling for differences in surface area perception, working memory span, and operational span. This suggests that the ANS can act as a unique source of error detection variability for formal mathematics, providing a plausible mechanism for how our universally shared number sense might link with human-specific symbolic math abilities. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

How does a person make decisions across perceptual boundaries? Here, we test the account that confidence constitutes a common currency for perceptual decisions even in childhood by examining whether confidence can be compared across distinct perceptual dimensions. We conducted a strict test of domain-generality in confidence reasoning by asking 6- to 7-year-olds to compare their confidence in 2 decisions, either from the same perceptual dimension (e.g., number vs. number) or from two different perceptual dimensions (e.g., area vs. emotion). Not only could children compare their confidence across and within domains but there were no differences in their abilities to make within- and across-domain comparisons. Our findings support the idea that confidence is represented in a common format even in childhood, which could provide an account for perceptual integration in childhood that doesn't necessitate the use of language. (PsycInfo Database Record (c) 2020 APA, all rights reserved).

Research with children and adults suggests that people's math performance is predicted by individual differences in an evolutionarily ancient ability to estimate and compare numerical quantities without counting (the approximate number system or ANS). However, previous work has almost exclusively used visual stimuli to measure ANS precision, leaving open the possibility that the observed link might be driven by aspects of visuospatial competence, rather than the amodal ANS. We addressed this possibility in an ANS training study. Sixty-eight 6-year-old children participated in a 5-week study that either trained their visual ANS ability or their phonological awareness (an active control group). Immediately before and after training, we assessed children's visual and auditory ANS precision, as well as their symbolic math ability and phonological awareness. We found that, prior to training, children's precision in a visual ANS task related to their math performance - replicating recent studies. Importantly, precision in an auditory ANS task also related to math performance. Furthermore, we found that children who completed visual ANS training showed greater improvements in auditory ANS precision than children who completed phonological awareness training. Finally, children in the ANS training group showed significant improvements in math ability but not phonological awareness. These results suggest that the link between ANS precision and school math ability goes beyond visuospatial abilities and that the modality-independent ANS is causally linked to math ability in early childhood., (Copyright © 2020 Libertus, Odic, Feigenson and Halberda.)

Women are notably underrepresented in the academic sciences. Psychology is a pertinent case study of gender inequality in science, because women make up over three quarters of undergraduate and graduate students but only a third of all full professors. Here, publication records from 125 high-impact, peer-reviewed psychology journals are analyzed to describe nuanced patterns about how men and women contribute to research psychology. To determine gender, we classified over 750,000 authors on 200,000 unique publications by comparing the 1st name of each author to openly available census data. The data replicate previous reports of publication and citation gender gaps in psychology and significantly extend these results by showing that these gaps are persistent across subdiscipline and time but are mediated by various contextual factors. For example, although the size of the publication and citation gaps are not explained by the university affiliation of the authors' and frequency of coauthorship, the gaps are larger in high-impact journals and at the last-author position. These patterns have remained largely unchanged since at least 2003. These results provide a detailed look at the variety of factors contributing to the differences in how men and women publish in research psychology and provide free and openly available tools for assessing publication and citation differences across time, journals, and other academic disciplines. (PsycINFO Database Record (c) 2020 APA, all rights reserved).

The human perceptual system is responsive to numerical information within visual and auditory scenes. For example, when shown 2 displays of dots, observers can instantly, albeit approximately, identify the set that is more numerous. Theories in perceptual and cognitive psychology have focused on 2 mechanisms for how vision accomplishes such a feat: Under the domain-specific encoding theory, number is represented as a primary visual feature of perception, much like motion or color, while under the domain-general theory, the visual system represents number indirectly, through a complex combination of features such as the size of the dots, their total cluster, and so forth. Evidence for the latter theory often comes from "congruency effects:" the finding that participants frequently select the side where the dots on the screen are denser, larger, or brighter, rather than the side that is actually more numerous. However, such effects could also stem from response conflicts between otherwise independent dimensions. Here, we test these 2 competing accounts by embedding numerical displays within visual illusions that create large conflicts between number and other non-numeric dimensions-including contour length, convex hull, and density-and contrast participants' performance on a number discrimination task (i.e., "Which side has more dots?") against a number estimation task (i.e., "How many dots are there?"), which should eliminate response conflicts. Across 3 experiments, we find that while contour length illusions only affect number perception in discrimination tasks, the influences of convex hull and density on number perception persist in both discrimination and estimation tasks, supporting a more domain-general account of number encoding. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

What are young children's first intuitions about numbers and what role do these play in their later understanding of mathematics? Traditionally, number has been viewed as a culturally derived breakthrough occurring relatively recently in human history that requires years of education to master. Contrary to this view, research in cognitive development indicates that our minds come equipped with a rich and flexible sense of number-the Approximate Number System (ANS). Recently, several major challenges have been mounted to the existence of the ANS and its value as a domain-specific system for representing number. In this article, we review five questions related to the ANS (what, who, why, where, and how) to argue that the ANS is defined by key behavioral and neural signatures, operates independently from nonnumeric dimensions such as time and space, and is used for a variety of functions (including formal mathematics) throughout life. We identify research questions that help elucidate the nature of the ANS and the role it plays in shaping children's earliest understanding of the world around them.

Young children can quickly and intuitively represent the number of objects in a visual scene through the Approximate Number System (ANS). The precision of the ANS - indexed as the most difficult ratio of two numbers that children can reliably discriminate - is well known to improve with development: whereas infants require relatively large ratios to discriminate number, children can discriminate finer and finer changes in number between toddlerhood and early adulthood. Which factors drive the developmental improvements in ANS precision? Here, we investigate the influence of four non-numeric dimensions - area, density, line length, and time - on ANS development, exploring the degree to which the ANS develops independently from these other dimensions, from inhibitory control, and from domain-general factors such as attention and working memory that are shared between these tasks. A sample of 185 children between the ages of 2 and 12 years completed five discrimination tasks: approximate number, area, density, length, and time. We report three main findings. First, logistic growth models applied to both accuracy and Weber fractions (w; an index of ANS precision) across age reveal distinct developmental trajectories across the five dimensions: while area and length develop by adolescence, time and density do not develop fully until early adulthood, with ANS precision developing at an intermediate rate. Second, we find that ANS precision develops independently of the other four dimensions, which in turn develop independently of the ANS. Third, we find that ANS precision also develops independently from individual differences in inhibitory control (indexed as the difference in accuracy and w between Congruent and Incongruent ANS trials). Together, these results are the first to provide evidence for domain-specific improvements in ANS precision, and place children's maturing perception of number, space, and time into a broader developmental context., (© 2017 John Wiley & Sons Ltd.)

Leibovich et al. suggest that congruency effects in number perception (biases towards smaller, denser, etc., dots) are evidence for the number's dependence on these dimensions. I argue that they fail to differentiate between effects at three distinct levels of number perception - encoding, representations, and decision making - and that differentiating between these allows the number to be independent from, but correlated with, non-numeric dimensions.

The results of our recent experiments suggest that temporarily modulating children's approximate number system (ANS) precision leads to a domain-specific change in their symbolic math performance (Journal of Experimental Child Psychology, 2016, Vol. 147, pp. 82-99). We interpreted these results as evidence for a causal relationship between ANS precision and symbolic math. In a commentary on our work, Merkley, Matejko, and Ansari argue that our methodology limits the interpretation of our results, primarily because our experiments did not meet the criteria for an intervention study as set out by What Works Clearinghouse and others. Here, we clarify the goals and limitations of our study and emphasize the variety of approaches to demonstrating causality. We argue that our goal was not to design and test an intervention or to compare the effectiveness of different treatments. Instead, we aimed to experimentally manipulate one variable (i.e., ANS acuity) and, in a randomized sample of children, observe whether this manipulation had any statistically significant effect on a dependent variable (i.e., performance on a set of symbolic math questions). We provide further analyses to support our assertion that a temporary manipulation of ANS performance does lead to a change in math performance. These results point to a causal relationship between ANS precision and math, and they suggest that further investigation of this relationship will be fruitful., (Copyright © 2016 Elsevier Inc. All rights reserved.)

Children can represent number in at least two ways: by using their non-verbal, intuitive approximate number system (ANS) and by using words and symbols to count and represent numbers exactly. Furthermore, by the time they are 5years old, children can map between the ANS and number words, as evidenced by their ability to verbally estimate numbers of items without counting. How does the quality of the mapping between approximate and exact numbers relate to children's math abilities? The role of the ANS-number word mapping in math competence remains controversial for at least two reasons. First, previous work has not examined the relation between verbal estimation and distinct subtypes of math abilities. Second, previous work has not addressed how distinct components of verbal estimation-mapping accuracy and variability-might each relate to math performance. Here, we addressed these gaps by measuring individual differences in ANS precision, verbal number estimation, and formal and informal math abilities in 5- to 7-year-old children. We found that verbal estimation variability, but not estimation accuracy, predicted formal math abilities, even when controlling for age, expressive vocabulary, and ANS precision, and that it mediated the link between ANS precision and overall math ability. These findings suggest that variability in the ANS-number word mapping may be especially important for formal math abilities., (Copyright © 2016 Elsevier Inc. All rights reserved.)

From early in life, humans have access to an approximate number system (ANS) that supports an intuitive sense of numerical quantity. Previous work in both children and adults suggests that individual differences in the precision of ANS representations correlate with symbolic math performance. However, this work has been almost entirely correlational in nature. Here we tested for a causal link between ANS precision and symbolic math performance by asking whether a temporary modulation of ANS precision changes symbolic math performance. First, we replicated a recent finding that 5-year-old children make more precise ANS discriminations when starting with easier trials and gradually progressing to harder ones, compared with the reverse. Next, we show that this brief modulation of ANS precision influenced children's performance on a subsequent symbolic math task but not a vocabulary task. In a supplemental experiment, we present evidence that children who performed ANS discriminations in a random trial order showed intermediate performance on both the ANS task and the symbolic math task, compared with children who made ordered discriminations. Thus, our results point to a specific causal link from the ANS to symbolic math performance., (Copyright © 2016 Elsevier Inc. All rights reserved.)

A simple and popular psychophysical model-usually described as overlapping Gaussian tuning curves arranged along an ordered internal scale-is capable of accurately describing both human and nonhuman behavioral performance and neural coding in magnitude estimation, production, and reproduction tasks for most psychological dimensions (e.g., time, space, number, or brightness). This model traditionally includes two parameters that determine how a physical stimulus is transformed into a psychological magnitude: (1) an exponent that describes the compression or expansion of the physical signal into the relevant psychological scale (β), and (2) an estimate of the amount of inherent variability (often called internal noise) in the Gaussian activations along the psychological scale (σ). To date, linear slopes on log-log plots have traditionally been used to estimate β, and a completely separate method of averaging coefficients of variance has been used to estimate σ. We provide a respectful, yet critical, review of these traditional methods, and offer a tutorial on a maximum-likelihood estimation (MLE) and a Bayesian estimation method for estimating both β and σ [PsiMLE(β,σ)], coupled with free software that researchers can use to implement it without a background in MLE or Bayesian statistics (R-PsiMLE). We demonstrate the validity, reliability, efficiency, and flexibility of this method through a series of simulations and behavioral experiments, and find the new method to be superior to the traditional methods in all respects.

Human mathematical abilities comprise both learned, symbolic representations of number and unlearned, non-symbolic evolutionarily primitive cognitive systems for representing quantities. However, the mechanisms by which our symbolic (verbal) number system becomes integrated with the non-symbolic (non-verbal) representations of approximate magnitude (supported by the Approximate Number System, or ANS) are not well understood. To explore this connection, forty-six children participated in a 6-month longitudinal study assessing verbal number knowledge and non-verbal numerical acuity. Cross-sectional analyses revealed a strong relationship between verbal number knowledge and ANS acuity. Longitudinal analyses suggested that increases in ANS acuity were most strongly related to the acquisition of the cardinal principle, but not to other milestones of verbal number acquisition. These findings suggest that experience with culture and language is intimately linked to changes in the properties of a core cognitive system.

What is the relationship between our intuitive sense of number (e.g., when estimating how many marbles are in a jar), and our intuitive sense of other quantities, including time (e.g., when estimating how long it has been since we last ate breakfast)? Recent work in cognitive, developmental, comparative psychology, and computational neuroscience has suggested that our representations of approximate number, time, and spatial extent are fundamentally linked and constitute a "generalized magnitude system". But, the shared behavioral and neural signatures between number, time, and space may alternatively be due to similar encoding and decision-making processes, rather than due to shared domain-general representations. In this study, we investigate the relationship between approximate number and time in a large sample of 6-8 year-old children in Uruguay by examining how individual differences in the precision of number and time estimation correlate with school mathematics performance. Over four testing days, each child completed an approximate number discrimination task, an approximate time discrimination task, a digit span task, and a large battery of symbolic math tests. We replicate previous reports showing that symbolic math abilities correlate with approximate number precision and extend those findings by showing that math abilities also correlate with approximate time precision. But, contrary to approximate number and time sharing common representations, we find that each of these dimensions uniquely correlates with formal math: approximate number correlates more strongly with formal math compared to time and continues to correlate with math even when precision in time and individual differences in working memory are controlled for. These results suggest that there are important differences in the mental representations of approximate number and approximate time and further clarify the relationship between quantity representations and mathematics., (Copyright © 2015 Elsevier B.V. All rights reserved.)

Humans can represent number either exactly--using their knowledge of exact numbers as supported by language, or approximately--using their approximate number system (ANS). Adults can map between these two systems--they can both translate from an approximate sense of the number of items in a brief visual display to a discrete number word estimate (i.e., ANS-to-Word), and can generate an approximation, for example by rapidly tapping, when provided with an exact verbal number (i.e., Word-to-ANS). Here we ask how these mappings are initially formed and whether one mapping direction may become functional before the other during development. In two experiments, we gave 2-5 year old children both an ANS-to-Word task, where they had to give a verbal number response to an approximate presentation (i.e., after seeing rapidly flashed dots, or watching rapid hand taps), and a Word-to-ANS task, where they had to generate an approximate response to a verbal number request (i.e., rapidly tapping after hearing a number word). Replicating previous results, children did not successfully generate numerically appropriate verbal responses in the ANS-to-Word task until after 4 years of age--well after they had acquired the Cardinality Principle of verbal counting. In contrast, children successfully generated numerically appropriate tapping sequences in the Word-to-ANS task before 4 years of age--well before many understood the Cardinality Principle. We further found that the accuracy of the mapping between the ANS and number words, as captured by error rates, continues to develop after this initial formation of the interface. These results suggest that the mapping between the ANS and verbal number representations is not functionally bidirectional in early development, and that the mapping direction from number representations to the ANS is established before the reverse., (Copyright © 2015 Elsevier B.V. All rights reserved.)

There exists a conspicuous gap of knowledge about the organization of life at mesoscopic levels. Ultra-fast coherent diffractive imaging with X-ray free-electron lasers can probe structures at the relevant length scales and may reach sub-nanometer resolution on micron-sized living cells. Here we show that we can introduce a beam of aerosolised cyanobacteria into the focus of the Linac Coherent Light Source and record diffraction patterns from individual living cells at very low noise levels and at high hit ratios. We obtain two-dimensional projection images directly from the diffraction patterns, and present the results as synthetic X-ray Nomarski images calculated from the complex-valued reconstructions. We further demonstrate that it is possible to record diffraction data to nanometer resolution on live cells with X-ray lasers. Extension to sub-nanometer resolution is within reach, although improvements in pulse parameters and X-ray area detectors will be necessary to unlock this potential.

Humans can quickly and intuitively represent the number of objects in a scene using visual evidence through the Approximate Number System (ANS). But the computations that support the encoding of visual number-the transformation from the retinal input into ANS representations-remain controversial. Two types of number encoding theories have been proposed: those arguing that number is encoded through a dedicated, enumeration computation, and those arguing that visual number is inferred from nonnumber specific visual features, such as surface area, density, convex hull, etc. Here, we attempt to adjudicate between these two theories by testing participants on both a number and a cumulative area task while also tracking their eye-movements. We hypothesize that if approximate number and surface area depend on distinct encoding computations, saccadic signatures should be distinct for the two tasks, even if the visual stimuli are identical. Consistent with this hypothesis, we find that discriminating number versus cumulative area modulates both where participants look (i.e., participants spend more time looking at the more numerous set in the number task and the larger set in the cumulative area task), and how participants look (i.e., cumulative area encoding shows fewer, longer saccades, while number encoding shows many short saccades and many switches between targets). We further identify several saccadic signatures that are associated with task difficulty and correct versus incorrect trials for both dimensions. These results suggest distinct encoding algorithms for number and cumulative area extraction, and thereby distinct representations of these dimensions.

Perceptual decisions are often affected not only by the evidence gathered during a trial but also by the history of preceding trials. This effect--termed perceptual hysteresis--provides evidence for how perceptual information is represented and how it is used. The present research focuses on how the difficulty of preceding trials affects subsequent ones--we find that how well 5-year-old children perform in a 2-alternative forced-choice numerical discrimination task depends on whether they have had a prior history of easier discriminations or a prior history of harder discriminations. Furthermore, this effect is modulated by the feedback children receive. In 3 experiments, we demonstrate that these effects are not related to practice or loss of interest due to negative feedback, or simply to trial difficulty or discriminability. Instead, children appear to have state-dependent confidence states such that prolonged experience making low-confidence decisions degrades performance, whereas prolonged experience making high-confidence decisions improves it. These results are discussed in the context of dynamical psychophysics, representations of confidence, and work on children's and adults' number perception abilities.

From very early in life, humans can approximate the number and surface area of objects in a scene. The ability to discriminate between 2 approximate quantities, whether number or area, critically depends on the ratio between the quantities, with the most difficult ratio that a participant can reliably discriminate known as the Weber fraction. While developmental improvements in the Weber fraction have been demonstrated for number, the developmental trajectory of improvement in area discrimination remains unknown. Here we investigated whether the development of area discrimination parallels that of number discrimination. We tested forty 3- to 6-year-old children and adults in both a number and an area discrimination task in which participants selected the greater of 2 quantities across a range of ratios. We used formal psychophysical models to derive, for each participant and each age group, the Weber fraction for both number and area discrimination. We found that, like number acuity, area acuity steadily improves during childhood. However, we also found area acuity to be consistently higher than number acuity, suggesting a potential difference in the underlying mechanisms that encode and/or represent approximate area and approximate number. We discuss these findings in the context of quantity processing and its development., (PsycINFO Database Record (c) 2013 APA, all rights reserved)

The psychology supporting the use of quantifier words (e.g., "some," "most," "more") is of interest to both scientists studying quantity representation (e.g., number, area) and to scientists and linguists studying the syntax and semantics of these terms. Understanding quantifiers requires both a mastery of the linguistic representations and a connection with cognitive representations of quantity. Some words (e.g., "many") refer to only a single dimension, whereas others, like the comparative "more," refer to comparison by numeric ("more dots") or nonnumeric dimensions ("more goo"). In the present work, we ask 2 questions. First, when do children begin to understand the word "more" as used to compare nonnumeric substances and collections of discrete objects? Second, what is the underlying psychophysical character of the cognitive representations children utilize to verify such sentences? We find that children can understand and verify sentences including "more goo" and "more dots" at around 3.3 years-younger than some previous studies have suggested-and that children employ the Approximate Number System and an Approximate Area System in verification. These systems share a common underlying format (i.e., Gaussian representations with scalar variability). The similarity in the age of onset we find for understanding "more" in number and area contexts, along with the similar psychophysical character we demonstrate for these underlying cognitive representations, suggests that children may learn "more" as a domain-neutral comparative term., ((c) 2013 APA, all rights reserved.)

Many educated adults possess exact mathematical abilities in addition to an approximate, intuitive sense of number, often referred to as the Approximate Number System (ANS). Here we investigate the link between ANS precision and mathematics performance in adults by testing participants on an ANS-precision test and collecting their scores on the Scholastic Aptitude Test (SAT), a standardized college-entrance exam in the USA. In two correlational studies, we found that ANS precision correlated with SAT-Quantitative (i.e., mathematics) scores. This relationship remained robust even when controlling for SAT-Verbal scores, suggesting a small but specific relationship between our primitive sense for number and formal mathematical abilities., (Copyright © 2012 Elsevier B.V. All rights reserved.)

The Ternus display produces a bistable illusion of motion: at very short interstimulus intervals (ISIs; < 30 ms) observers perceive element motion while at longer ISIs (> 30 ms) observers perceive group motion. In experiment 1, however, we find that, when the Ternus display's ISI contains an occluding box, group motion is mostly eliminated. These results do not fit the predictions made by the short-range/long-range two-process theory [Braddick and Adlard, 1978, in Visual Psychophysics and Psychology (New York: Academic Press)]. We propose that the differential-activation theory (Gilroy et al, 2001 Perception & Psychophysics 63 847-861) accounts for our results. We then extend the differential-activation theory as an explanatory mechanism for the Ternus display in experiment 2 by selectively placing an occluder over the first, second, or third Ternus display element. As predicted by the differential-activation theory, the occlusion of the far-left element produced a normal distribution of group motion increasing with ISI, while the occlusion of the other two elements produced an illusion of occluded elements remaining stationary throughout the display. Furthermore, as predicted by the differential-activation theory, each moving element was assigned to its nearest neighbour, producing, in the case of second and third element occlusion, a novel Ternus display motion illusion where only two out of three elements are perceived as moving.