Variability signatures distinguish verbal from nonverbal counting for both large and small numbers

No intrinsic gender differences in children's earliest numerical abilities

Recent public discussions have suggested that the under-representation of women in science and mathematics careers can be traced back to intrinsic differences in aptitude. However, true gender differences are difficult to assess because sociocultural influences enter at an early point in childhood. If these claims of intrinsic differences are true, then gender differences in quantitative and mathematical abilities should emerge early in human development. We examined cross-sectional gender differences in mathematical cognition from over 500 children aged 6 months to 8 years by compiling data from five published studies with unpublished data from longitudinal records. We targeted three key milestones of numerical development: numerosity perception, culturally trained counting, and formal and informal elementary mathematics concepts. In addition to testing for statistical differences between boys' and girls' mean performance and variability, we also tested for statistical equivalence between boys' and girls' performance. Across all stages of numerical development, analyses consistently revealed that boys and girls do not differ in early quantitative and mathematical ability. These findings indicate that boys and girls are equally equipped to reason about mathematics during early childhood.

Numerical distance effect size is a poor metric of approximate number system acuity

Individual differences in the ability to compare and evaluate nonsymbolic numerical magnitudes-approximate number system (ANS) acuity-are emerging as an important predictor in many research areas. Unfortunately, recent empirical studies have called into question whether a historically common ANS-acuity metric-the size of the numerical distance effect (NDE size)-is an effective measure of ANS acuity. NDE size has been shown to frequently yield divergent results from other ANS-acuity metrics. Given these concerns and the measure's past popularity, it behooves us to question whether the use of NDE size as an ANS-acuity metric is theoretically supported. This study seeks to address this gap in the literature by using modeling to test the basic assumption underpinning use of NDE size as an ANS-acuity metric: that larger NDE size indicates poorer ANS acuity. This assumption did not hold up under test. Results demonstrate that the theoretically ideal relationship between NDE size and ANS acuity is not linear, but rather resembles an inverted J-shaped distribution, with the inflection points varying based on precise NDE task methodology. Thus, depending on specific methodology and the distribution of ANS acuity in the tested population, positive, negative, or null correlations between NDE size and ANS acuity could be predicted. Moreover, peak NDE sizes would be found for near-average ANS acuities on common NDE tasks. This indicates that NDE size has limited and inconsistent utility as an ANS-acuity metric. Past results should be interpreted on a case-by-case basis, considering both specifics of the NDE task and expected ANS acuity of the sampled population.

An Introduction to the Approximate Number System

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.

A threshold-free model of numerosity comparisons

A dominant mechanism in the Judgment and Decision Making literature states that information is accumulated about each choice option until a decision threshold is met. Only after that threshold does a subject start to execute a motor response to indicate their choice. However, recent research has revealed spatial gradients in motor responses as a function of comparison difficulty as well as changes-of-mind in the middle of an action, both suggesting continued accumulation and processing of decision-related signals after the decision boundary. Here we present a formal model and supporting data from a number comparison task that a continuous motor planner, combined with a simple statistical inference scheme, can model detailed behavioral effects without assuming a threshold. This threshold-free model reproduces subjects’ sensitivity to numerical distance in reaching, accuracy, reaction time, and changes of mind. We argue that the motor system positions the effectors using an optimal biomechanical feedback controller, and continuous statistical inference on outputs from cognitive processes.

Effects of non-symbolic arithmetic training on symbolic arithmetic and the approximate number system

The approximate number system (ANS) is an innate cognitive template that allows for the mental representation of approximate magnitude, and has been controversially linked to symbolic number knowledge and math ability. A series of recent studies found that an approximate arithmetic training (AAT) task that draws upon the ANS can improve math skills, which not only supports the existence of this link, but suggests it may be causal. However, no direct transfer effects to any measure of the ANS have yet been reported, calling into question the mechanisms by which math improvements may emerge. The present study investigated the effects of a 7-day AAT and successfully replicated previously reported transfer effects to math. Furthermore, our exploratory analyses provide preliminary evidence that certain ANS-related skills may also be susceptible to training. We conclude that AAT has reproducible effects on math performance, and provide avenues for future studies to further explore underlying mechanisms - specifically, the link between improvements in math and improvements in ANS skills.

The integration between nonsymbolic and symbolic numbers: Evidence from an EEG study

INTRODUCTION

Adults can represent numerical information in nonsymbolic and symbolic formats and flexibly switch between the two. While some studies suggest a strong link between the two number representation systems (e.g., Piazza, Izard, Pinel, Le Bihan, & Dehaene, 2004 Neuron, 44(3), 547), other studies show evidence against the strong-link hypothesis (e.g., Lyons, Ansari, & Beilock, 2012 Journal of Experimental Psychology: General, 141(4), 635). This inconsistency could arise from the relation between task demands and the closeness of the link between the two number systems.

METHODS

We used a passive viewing task and event-related potentials (ERP) to examine the temporal dynamics of the implicit integration between the nonsymbolic and symbolic systems. We focused on two ERP components over posterior scalp sites that were found to be sensitive to numerical distances and ratio differences in both numerical formats: a negative component that peaks around 170 ms poststimulus (N1) and a positive component that peaks around 200 ms poststimulus (P2p). We examined adults' (n = 55) ERPs when they were passively viewing simultaneously presented dot quantities and Arabic numerals (i.e., nonsymbolic and symbolic numerical information) in the double-digit range. For each stimulus, the nonsymbolic and symbolic content either matched or mismatched in number. We also asked each participant to estimate dot quantities in a separate behavioral task and observed that they tended to underestimate the actual dot quantities, suggesting a need to adjust the match between nonsymbolic and symbolic information to reflect the perceived quantity of the nonsymbolic information.

RESULTS

Using this adjustment, participants showed greater N1 and P2p amplitudes when perceived dot quantities matched Arabic numerals than when there was a mismatch. However, no differences were found between the unadjusted match and mismatch conditions.

CONCLUSION

Our findings suggest that adults rapidly integrate nonsymbolic and symbolic formats of double-digit numbers, but evidence of such integration is best observed when the perceived (rather than veridical) dot quantity is considered.

Capuchin monkeys (Cebus apella) treat small and large numbers of items similarly during a relative quantity judgment task

A key issue in understanding the evolutionary and developmental emergence of numerical cognition is to learn what mechanism(s) support perception and representation of quantitative information. Two such systems have been proposed, one for dealing with approximate representation of sets of items across an extended numerical range and another for highly precise representation of only small numbers of items. Evidence for the first system is abundant across species and in many tests with human adults and children, whereas the second system is primarily evident in research with children and in some tests with non-human animals. A recent paper (Choo & Franconeri, Psychonomic Bulletin & Review, 21, 93-99, 2014) with adult humans also reported "superprecise" representation of small sets of items in comparison to large sets of items, which would provide more support for the presence of a second system in human adults. We first presented capuchin monkeys with a test similar to that of Choo and Franconeri in which small or large sets with the same ratios had to be discriminated. We then presented the same monkeys with an expanded range of comparisons in the small number range (all comparisons of 1-9 items) and the large number range (all comparisons of 10-90 items in 10-item increments). Capuchin monkeys showed no increased precision for small over large sets in making these discriminations in either experiment. These data indicate a difference in the performance of monkeys to that of adult humans, and specifically that monkeys do not show improved discrimination performance for small sets relative to large sets when the relative numerical differences are held constant.

How Evolution Constrains Human Numerical Concepts

The types of cognitive and neural mechanisms available to children for making concepts depend on the problems their brains evolved to solve over the past millions of years. Comparative research on numerical cognition with humans and nonhuman primates has revealed a system for quantity representation that lays the foundation for quantitative development. Nonhuman primates in particular share many human abilities to compute quantities, and are likely to exhibit evolutionary continuity with humans. While humans conceive of quantity in ways that are similar to other primates, they are unique in their capacity for symbolic counting and logic. These uniquely human constructs interact with primitive systems of numerical reasoning. In this article, I discuss how evolution shapes human numerical concepts through evolutionary constraints on human object-based perception and cognition, neural homologies among primates, and interactions between uniquely human concepts and primitive logic.

Temporal causal inference with stochastic audiovisual sequences

Integration of sensory information across multiple senses is most likely to occur when signals are spatiotemporally coupled. Yet, recent research on audiovisual rate discrimination indicates that random sequences of light flashes and auditory clicks are integrated optimally regardless of temporal correlation. This may be due to 1) temporal averaging rendering temporal cues less effective; 2) difficulty extracting causal-inference cues from rapidly presented stimuli; or 3) task demands prompting integration without concern for the spatiotemporal relationship between the signals. We conducted a rate-discrimination task (Exp 1), using slower, more random sequences than previous studies, and a separate causal-judgement task (Exp 2). Unisensory and multisensory rate-discrimination thresholds were measured in Exp 1 to assess the effects of temporal correlation and spatial congruence on integration. The performance of most subjects was indistinguishable from optimal for spatiotemporally coupled stimuli, and generally sub-optimal in other conditions, suggesting observers used a multisensory mechanism that is sensitive to both temporal and spatial causal-inference cues. In Exp 2, subjects reported whether temporally uncorrelated (but spatially co-located) sequences were perceived as sharing a common source. A unified percept was affected by click-flash pattern similarity and the maximum temporal offset between individual clicks and flashes, but not on the proportion of synchronous click-flash pairs. A simulation analysis revealed that the stimulus-generation algorithms of previous studies is likely responsible for the observed integration of temporally independent sequences. By combining results from Exps 1 and 2, we found better rate-discrimination performance for sequences that are more likely to be integrated than those that are not. Our results support the principle that multisensory stimuli are optimally integrated when spatiotemporally coupled, and provide insight into the temporal features used for coupling in causal inference.

The impact of emotion on numerical estimation: A developmental perspective

Recent literature has revealed underestimation effects in numerical judgments when adult participants are presented with emotional stimuli (as opposed to neutral). Whether these numerical biases emerge early in development however, or instead reflect overt, learned responses to emotional stimuli across development are unclear. Moreover, reported links between numerical acuity and mathematics achievement point to the importance of exploring how numerical approximation abilities in childhood may be influenced in real-world affective contexts. In this study, children (aged 6-10 years) and adults were presented with happy and neutral facial stimuli in the context of a numerical bisection task. Results reveal that children, like adults, underestimate number following emotional (i.e., happy) faces (relative to neutral). However, children's, but not adult's, responses were also significantly more precise following emotional stimuli. In a second experiment, adult judgments revealed a similar increase in precision following emotional stimuli when numerical discriminations were more challenging (involving larger sets). Together, results are the first to reveal children, like adults, underestimate number in the context of emotional stimuli and this underestimation bias is accompanied with enhanced response precision.

The relationship between non-symbolic multiplication and division in childhood

Children without formal education in addition and subtraction are able to perform multi-step operations over an approximate number of objects. Further, their performance improves when solving approximate (but not exact) addition and subtraction problems that allow for inversion as a shortcut (e.g., a + b - b = a). The current study examines children's ability to perform multi-step operations, and the potential for an inversion benefit, for the operations of approximate, non-symbolic multiplication and division. Children were trained to compute a multiplication and division scaling factor (*2 or /2, *4 or /4), and were then tested on problems that combined two of these factors in a way that either allowed for an inversion shortcut (e.g., 8*4/4) or did not (e.g., 8*4/2). Children's performance was significantly better than chance for all scaling factors during training, and they successfully computed the outcomes of the multi-step testing problems. They did not exhibit a performance benefit for problems with the a*b/b structure, suggesting that they did not draw upon inversion reasoning as a logical shortcut to help them solve the multi-step test problems.

Clinical Applications of Stochastic Dynamic Models of the Brain, Part I: A Primer

Biological phenomena arise through interactions between an organism's intrinsic dynamics and stochastic forces-random fluctuations due to external inputs, thermal energy, or other exogenous influences. Dynamic processes in the brain derive from neurophysiology and anatomical connectivity; stochastic effects arise through sensory fluctuations, brainstem discharges, and random microscopic states such as thermal noise. The dynamic evolution of systems composed of both dynamic and random effects can be studied with stochastic dynamic models (SDMs). This article, Part I of a two-part series, offers a primer of SDMs and their application to large-scale neural systems in health and disease. The companion article, Part II, reviews the application of SDMs to brain disorders. SDMs generate a distribution of dynamic states, which (we argue) represent ideal candidates for modeling how the brain represents states of the world. When augmented with variational methods for model inversion, SDMs represent a powerful means of inferring neuronal dynamics from functional neuroimaging data in health and disease. Together with deeper theoretical considerations, this work suggests that SDMs will play a unique and influential role in computational psychiatry, unifying empirical observations with models of perception and behavior.

Primitive Concepts of Number and the Developing Human Brain

Counting is an evolutionarily recent cultural invention of the human species. In order for humans to have conceived of counting in the first place, certain representational and logical abilities must have already been in place. The focus of this review is the origins and nature of those fundamental mechanisms that promoted the emergence of the human number concept. Five claims are presented that support an evolutionary view of numerical development: 1) number is an abstract concept with an innate basis in humans, 2) maturational processes constrain the development of humans' numerical representations between infancy and adulthood, 3) there is evolutionary continuity in the neural processes of numerical cognition in primates, 4) primitive logical abilities support verbal counting development in humans, and 5) primitive neural processes provide the foundation for symbolic numerical development in the human brain. We support these claims by examining current evidence from animal cognition, child development, and human brain function. The data show that at the basis of human numerical concepts are primitive perceptual and logical mechanisms that have evolutionary homologs in other primates and form the basis of numerical development in the human brain. In the final section of the review, we discuss some hypotheses for what makes human numerical reasoning unique by drawing on evidence from human and non-human primate neuroimaging research.

Universal and uniquely human factors in spontaneous number perception

A capacity for nonverbal numerical estimation is widespread among humans and animals. However, it is currently unclear whether numerical percepts are spontaneously extracted from the environment and whether nonverbal perception is influenced by human exposure to formal mathematics. We tested US adults and children, non-human primates, and numerate and innumerate Tsimane' adults on a quantity task in which they could choose to categorize sets of dots on the basis of number alone, surface area alone or a combination of the two. Despite differences in age, species and education, subjects are universally biased to base their judgments on number as opposed to the alternatives. Numerical biases are uniquely enhanced in humans compared to non-human primates, and correlated with degree of mathematics experience in both the US and Tsimane' groups. We conclude that humans universally and spontaneously extract numerical information, and that human nonverbal numerical perception is enhanced by symbolic numeracy.

Attaching meaning to the number words: contributions of the object tracking and approximate number systems

Children's understanding of the quantities represented by number words (i.e., cardinality) is a surprisingly protracted but foundational step in their learning of formal mathematics. The development of cardinal knowledge is related to one or two core, inherent systems - the approximate number system (ANS) and the object tracking system (OTS) - but whether these systems act alone, in concert, or antagonistically is debated. Longitudinal assessments of 198 preschool children on OTS, ANS, and cardinality tasks enabled testing of two single-mechanism (ANS-only and OTS-only) and two dual-mechanism models, controlling for intelligence, executive functions, preliteracy skills, and demographic factors. Measures of both OTS and ANS predicted cardinal knowledge in concert early in the school year, inconsistent with single-mechanism models. The ANS but not the OTS predicted cardinal knowledge later in the school year as well the acquisition of the cardinal principle, a critical shift in cardinal understanding. The results support a Merge model, whereby both systems initially contribute to children's early mapping of number words to cardinal value, but the role of the OTS diminishes over time while that of the ANS continues to support cardinal knowledge as children come to understand the counting principles.

The precision of mapping between number words and the approximate number system predicts children's formal math abilities

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.

Common Representations of Abstract Quantities

Representations of abstract quantities such as time and number are essential for survival. A number of studies have revealed that both humans and nonhuman animals are able to nonverbally estimate time and number; striking similarities in the behavioral data suggest a common magnitude-representation system shared across species. It is unclear, however, whether these representations provide animals with a true concept of time and number, as posited by Gallistel and Gelman (2000) . In this article, we review the prominent cognitive and neurobiological models of timing and counting and explore the current evidence suggesting that nonhuman animals represent these quantities in a modality-independent (i.e., abstract) and ordered manner. Avenues for future research in the area of temporal and mathematical cognition are also discussed.

What counts in estimation? The nature of the preverbal system

It has been proposed that the development of verbal counting is supported by a more ancient preverbal system of estimation, the most widely canvassed candidates being the accumulator originally proposed by Gibbon and colleagues and the analogue magnitude system proposed by Dehaene and colleagues. The aim of this chapter is to assess the strengths and weaknesses of these models in terms of their capacity to emulate the statistical properties of verbal counting. The emphasis is put on the emergence of exact representations, autoscaling, and commensurability of noise characteristics. We also outline the modified architectures that may help improve models' power to meet these criteria. We propose that architectures considered in this chapter can be used to generate predictions for experimental testing and provide an example where we test the hypothesis whether the visual sense of number, ie, ability to discriminate numerosity without counting, entails enumeration of objects.

Cognitive Aging and Time Perception: Roles of Bayesian Optimization and Degeneracy

This review outlines the basic psychological and neurobiological processes associated with age-related distortions in timing and time perception in the hundredths of milliseconds-to-minutes range. The difficulty in separating indirect effects of impairments in attention and memory from direct effects on timing mechanisms is addressed. The main premise is that normal aging is commonly associated with increased noise and temporal uncertainty as a result of impairments in attention and memory as well as the possible reduction in the accuracy and precision of a central timing mechanism supported by dopamine-glutamate interactions in cortico-striatal circuits. Pertinent to these findings, potential interventions that may reduce the likelihood of observing age-related declines in timing are discussed. Bayesian optimization models are able to account for the adaptive changes observed in time perception by assuming that older adults are more likely to base their temporal judgments on statistical inferences derived from multiple trials than on a single trial's clock reading, which is more susceptible to distortion. We propose that the timing functions assigned to the age-sensitive fronto-striatal network can be subserved by other neural networks typically associated with finely-tuned perceptuo-motor adjustments, through degeneracy principles (different structures serving a common function).

Multiple Spatially Overlapping Sets Can Be Enumerated in Parallel

A system for nonverbally representing the approximate number of items in visual and auditory arrays has been documented in multiple species, including humans. Although many aspects of this approximate number system are well characterized, fundamental questions remain unanswered: How does attention select which items in a scene to enumerate, and how many enumerations can be computed simultaneously? Here we show that when presented an array containing different numbers of spatially overlapping dots of many colors, human adults can select and enumerate items on the basis of shared color and can enumerate approximately three color subsets from a single glance. This three-set limit converges with previously observed three-item limits of parallel attention and visual short-term memory. This suggests that participants can select a subset of items from a complex array as a single individual set, which then serves as the input to the approximate number system.

Symbolic, Nonsymbolic and Conceptual: An Across-Notation Study on the Space Mapping of Numerals

Previous studies suggested that there are interconnections between two numeral modalities of symbolic notation and nonsymbolic notation (array of dots), differences and similarities of the processing, and representation of the two modalities have both been found in previous research. However, whether there are differences between the spatial representation and numeral-space mapping of the two numeral modalities of symbolic notation and nonsymbolic notation is still uninvestigated. The present study aims to examine whether there are differences between the spatial representation and numeral-space mapping of the two numeral modalities of symbolic notation and nonsymbolic notation; especially how zero, as both a symbolic magnitude numeral and a nonsymbolic conceptual numeral, mapping onto space; and if the mapping happens automatically at an early stage of the numeral information processing. Results of the two experiments demonstrate that the low-level processing of symbolic numerals including zero and nonsymbolic numerals except zero can mapping onto space, whereas the low-level processing of nonsymbolic zero as a semantic conceptual numeral cannot mapping onto space, which indicating the specialty of zero in the numeral domain. The present study indicates that the processing of non-semantic numerals can mapping onto space, whereas semantic conceptual numerals cannot mapping onto space.

Non-symbolic division in childhood

The approximate number system (ANS) underlies representations of large numbers of objects as well as the additive, subtractive, and multiplicative relationships between them. In this set of studies, 5- and 6-year-old children were shown a series of video-based events that conveyed a transformation of a large number of objects into one-half or one-quarter of the original number. Children were able to estimate correctly the outcomes to these halving and quartering problems, and they based their responses on scaling by number, not on continuous quantities or guessing strategies. Children's performance exhibited the ratio signature of the ANS. Moreover, children performed above chance on relatively early trials, suggesting that this scaling operation is easily conveyed and readily performed. The results support the existence of a flexible and substantially untrained capacity to scale numerical amounts.

Developmental bias for number words in the intraparietal sulcus

Children and adults show behavioral evidence of psychological overlap between their early, non-symbolic numerical concepts and their later-developing symbolic numerical concepts. An open question is to what extent the common cognitive signatures observed between different numerical notations are coupled with physical overlap in neural processes. We show that from 8 years of age, regions of the intraparietal sulcus (IPS) that exhibit a numerical ratio effect during non-symbolic numerical judgments also show a semantic distance effect for symbolic number words. In both children and adults, the IPS showed a semantic distance effect during magnitude judgments of number words (i.e. larger/smaller number) but not for magnitude judgments of object words (i.e. larger/smaller object size). The results provide novel evidence of conceptual overlap between neural representations of symbolic and non-symbolic numerical values that cannot be explained by a general process, and present the first demonstration of an early-developing dissociation between number words and object words in the human brain.

Evidence for distinct magnitude systems for symbolic and non-symbolic number

Cognitive models of magnitude representation are mostly based on the results of studies that use a magnitude comparison task. These studies show similar distance or ratio effects in symbolic (Arabic numerals) and non-symbolic (dot arrays) variants of the comparison task, suggesting a common abstract magnitude representation system for processing both symbolic and non-symbolic numerosities. Recently, however, it has been questioned whether the comparison task really indexes a magnitude representation. Alternatively, it has been hypothesized that there might be different representations of magnitude: an exact representation for symbolic magnitudes and an approximate representation for non-symbolic numerosities. To address the question whether distinct magnitude systems exist, we used an audio-visual matching paradigm in two experiments to explore the relationship between symbolic and non-symbolic magnitude processing. In Experiment 1, participants had to match visually and auditory presented numerical stimuli in different formats (digits, number words, dot arrays, tone sequences). In Experiment 2, they were instructed only to match the stimuli after processing the magnitude first. The data of our experiments show different results for non-symbolic and symbolic number and are difficult to reconcile with the existence of one abstract magnitude representation. Rather, they suggest the existence of two different systems for processing magnitude, i.e., an exact symbolic system next to an approximate non-symbolic system.

Monkeys display classic signatures of human symbolic arithmetic

Non-human primates compare quantities in a crude manner, by approximating their values. Less is known about the mental transformations that non-humans can perform over approximate quantities, such as arithmetic transformations. There is evidence that human symbolic arithmetic has a deep psychological connection with the primitive, approximate forms of quantification of non-human animals. Here, we ask whether the subtle performance signatures that humans exhibit during symbolic arithmetic also bear a connection to primitive arithmetic. Specifically, we examined the problem size effect, the tie effect, and the practice effect-effects which are commonly observed in children's math performance in school. We show that, like humans, monkeys exhibited the problem size and tie effects, indicating commonalities in arithmetic algorithms with humans. Unlike humans, however, monkeys did not exhibit a practice effect. Together, these findings provide new evidence for a cognitive relation between non-symbolic and symbolic arithmetic.

Analog Magnitudes Support Large Number Ordinal Judgments in Infancy

Few studies have explored the source of infants' ordinal knowledge, and those that have are equivocal regarding the underlying representational system. The present study sought clear evidence that the approximate number system, which underlies children's cardinal knowledge, may also support ordinal knowledge in infancy; 10 - to 12-month-old infants' were tested with large sets (>3) in an ordinal choice task in which they were asked to choose between two hidden sets of food items. The difficulty of the comparison varied as a function of the ratio between the sets. Infants reliably chose the greater quantity when the sets differed by a 2:3 ratio (4v6 and 6v9), but not when they differed by a 3:4 ratio (6v8) or a 7:8 ratio (7v8). This discrimination function is consistent with previous studies testing the precision of number and time representations in infants of roughly this same age, thus providing evidence that the approximate number system can support ordinal judgments in infancy. The findings are discussed in light of recent proposals that different mechanisms underlie infants' reasoning about small and large numbers.

Precision and Bias in Approximate Numerical Judgment in Auditory, Tactile, and Cross-modal Presentation

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.

Children’s Approximate Number System in Haptic Modality

The approximate number system (ANS) is a primitive system used to estimate quantities. It can process quantities in visual and auditory modalities. The aim of the present study was to examine whether ANS can process quantities presented haptically. Moreover, to assess age-related changes, two groups of children (5- and 7-year-olds) were compared. In a newly designed haptic task, children compared two arrays of dots by touching them simultaneously using both hands, without seeing them, and for limited duration to prevent counting. Using Panamath, a frequently used visual ANS task, we verified that our population exhibited the typical pattern of approximation with visual arrays: Older children outperformed younger children, and an increased ratio between the two quantities to be compared led to more accurate responses. Performance in the haptic task revealed that children, in both age-groups, were able to haptically compare two quantities above chance level, with improved performance in older compared with younger children. Moreover, our results revealed a ratio effect, a well-known signature of the ANS. These findings suggest that haptic numerical discrimination in children is dictated by the ANS, and that ANS acuity measured with a haptic task improves with age, as commonly observed with the visual task.

Ratio dependence in small number discrimination is affected by the experimental procedure

Adults, infants and some non-human animals share an approximate number system (ANS) to estimate numerical quantities, and are supposed to share a second, 'object-tracking,' system (OTS) that supports the precise representation of a small number of items (up to 3 or 4). In relative numerosity judgments, accuracy depends on the ratio of the two numerosities (Weber's Law) for numerosities >4 (the typical ANS range), while for numerosities ≤4 (OTS range) there is usually no ratio effect. However, recent studies have found evidence for ratio effects for small numerosities, challenging the idea that the OTS might be involved for small number discrimination. Here we tested the hypothesis that the lack of ratio effect in the numbers 1-4 is largely dependent on the type of stimulus presentation. We investigated relative numerosity judgments in college students using three different procedures: a simultaneous presentation of intermingled and separate groups of dots in separate experiments, and a further experiment with sequential presentation. As predicted, in the large number range, ratio dependence was observed in all tasks. By contrast, in the small number range, ratio insensitivity was found in one task (sequential presentation). In a fourth experiment, we showed that the presence of intermingled distractors elicited a ratio effect, while easily distinguishable distractors did not. As the different ratio sensitivity for small and large numbers has been often interpreted in terms of the activation of the OTS and ANS, our results suggest that numbers 1-4 may be represented by both numerical systems and that the experimental context, such as the presence/absence of task-irrelevant items in the visual field, would determine which system is activated.

Grouping by proximity and the visual impression of approximate number in random dot arrays

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

Is 9 louder than 1? Audiovisual cross-modal interactions between number magnitude and judged sound loudness

The cross-modal impact of number magnitude (i.e. Arabic digits) on perceived sound loudness was examined. Participants compared a target sound's intensity level against a previously heard reference sound (which they judged as quieter or louder). Paired with each target sound was a task irrelevant Arabic digit that varied in magnitude, being either small (1, 2, 3) or large (7, 8, 9). The degree to which the sound and the digit were synchronized was manipulated, with the digit and sound occurring simultaneously in Experiment 1, and the digit preceding the sound in Experiment 2. Firstly, when target sounds and digits occurred simultaneously, sounds paired with large digits were categorized as loud more frequently than sounds paired with small digits. Secondly, when the events were separated, number magnitude ceased to bias sound intensity judgments. In Experiment 3, the events were still separated, however the participants held the number in short-term memory. In this instance the bias returned.

Children's mappings between number words and the approximate number system

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.

Approximate number word knowledge before the cardinal principle

Approximate number word knowledge-understanding the relation between the count words and the approximate magnitudes of sets-is a critical piece of knowledge that predicts later math achievement. However, researchers disagree about when children first show evidence of approximate number word knowledge-before, or only after, they have learned the cardinal principle. In two studies, children who had not yet learned the cardinal principle (subset-knowers) produced sets in response to number words (verbal comprehension task) and produced number words in response to set sizes (verbal production task). As evidence of approximate number word knowledge, we examined whether children's numerical responses increased with increasing numerosity of the stimulus. In Study 1, subset-knowers (ages 3.0-4.2 years) showed approximate number word knowledge above their knower-level on both tasks, but this effect did not extend to numbers above 4. In Study 2, we collected data from a broader age range of subset-knowers (ages 3.1-5.6 years). In this sample, children showed approximate number word knowledge on the verbal production task even when only examining set sizes above 4. Across studies, children's age predicted approximate number word knowledge (above 4) on the verbal production task when controlling for their knower-level, study (1 or 2), and parents' education, none of which predicted approximation ability. Thus, children can develop approximate knowledge of number words up to 10 before learning the cardinal principle. Furthermore, approximate number word knowledge increases with age and might not be closely related to the development of exact number word knowledge.

A temporal basis for Weber's law in value perception

Weber's law—the observation that the ability to perceive changes in magnitudes of stimuli is proportional to the magnitude—is a widely observed psychophysical phenomenon. It is also believed to underlie the perception of reward magnitudes and the passage of time. Since many ecological theories state that animals attempt to maximize reward rates, errors in the perception of reward magnitudes and delays must affect decision-making. Using an ecological theory of decision-making (TIMERR), we analyze the effect of multiple sources of noise (sensory noise, time estimation noise, and integration noise) on reward magnitude and subjective value perception. We show that the precision of reward magnitude perception is correlated with the precision of time perception and that Weber's law in time estimation can lead to Weber's law in value perception. The strength of this correlation is predicted to depend on the reward history of the animal. Subsequently, we show that sensory integration noise (either alone or in combination with time estimation noise) also leads to Weber's law in reward magnitude perception in an accumulator model, if it has balanced Poisson feedback. We then demonstrate that the noise in subjective value of a delayed reward, due to the combined effect of noise in both the perception of reward magnitude and delay, also abides by Weber's law. Thus, in our theory we prove analytically that the perception of reward magnitude, time, and subjective value change all approximately obey Weber's law.

On the Internal Representation of Numerical Magnitude and Physical Size

A nascent idea in the numerical cognition literature – the analogical hypothesis ( Pinel, Piazza, Bihan, & Dehaene, 2004 ) – assumes a common noisy code for the representation of symbolic (e.g., numerals) and nonsymbolic (e.g., numerosity, physical size, luminance) magnitudes. The present work subjected this assumption to various tests from the perspective of General Recognition Theory (GRT; Ashby & Townsend, 1986 ) – a multidimensional extension of Signal Detection Theory ( Green & Swets, 1966 ). The GRT was applied to the dimensions of numerical magnitude and physical size with the following goals: (a) characterizing the internal representation of these dimensions in the psychological space, and (b) assessing various types of (in)dependence and separability governing the perception of these dimensions. The results revealed various violations of independence and separability with Stroop incongruent, but not with Stroop congruent stimuli. The outcome suggests that there are deep differences in architecture between Stroop congruent and incongruent stimuli that reach well beyond the semantic relationship involved.