Infants' discrimination of number vs. continuous extent

Crossing the Boundary: No Catastrophic Limits on Infants' Capacity to Represent Linguistic Sequences

The boundary effect, namely the infants' failures to compare small and large numerosities, is well documented in studies using visual stimuli. The prevailing explanation is that the numerical system used to process sets up to 3 is incompatible with the system employed for numbers >3. This study investigates the boundary effect in 10-month-old infants presented with linguistic sequences. In Condition 1 (2 vs. 3), infants can differentiate small syllable sequences (2 vs. 3), with better performance for the 2-syllable sequence, which imposes a lower memory load. Condition 2 (2 vs. 4) revealed that infants are capable of discriminating across bounds, with relatively higher performance for the 4-syllable sequence, possibly encoded as one large ensemble. This study offers evidence that, when processing linguistic sounds, infants flexibly deal with small and large numerical representations with no boundaries or incompatibilities between them. Simultaneously encoding units of different magnitudes might aid early speech processing beyond memory limits.

Estimating the distribution of numerosity and non-numerical visual magnitudes in natural scenes using computer vision

Humans share with many animal species the ability to perceive and approximately represent the number of objects in visual scenes. This ability improves throughout childhood, suggesting that learning and development play a key role in shaping our number sense. This hypothesis is further supported by computational investigations based on deep learning, which have shown that numerosity perception can spontaneously emerge in neural networks that learn the statistical structure of images with a varying number of items. However, neural network models are usually trained using synthetic datasets that might not faithfully reflect the statistical structure of natural environments, and there is also growing interest in using more ecological visual stimuli to investigate numerosity perception in humans. In this work, we exploit recent advances in computer vision algorithms to design and implement an original pipeline that can be used to estimate the distribution of numerosity and non-numerical magnitudes in large-scale datasets containing thousands of real images depicting objects in daily life situations. We show that in natural visual scenes the frequency of appearance of different numerosities follows a power law distribution. Moreover, we show that the correlational structure for numerosity and continuous magnitudes is stable across datasets and scene types (homogeneous vs. heterogeneous object sets). We suggest that considering such "ecological" pattern of covariance is important to understand the influence of non-numerical visual cues on numerosity judgements.

From developmental theory to effective training: long-term and transfer effects of promoting the quantity-to-number word linkage in first-graders at risk for mathematical difficulties

Introduction

The model of quantity-to-number word linkage (QNL model) identifies relevant milestones in the process of early numerical acquisition and describes a developmental sequence that can guide the fostering of foundational mathematical abilities in at-risk children. While there is substantial evidence for the predictive value of the quantity-number competencies (QNC) described by the model, evidence supporting the preventive potential of interventions targeting these QNC is so far largely restricted to short-term effects. Findings regarding their long-term preventive impact, especially in terms of transfer to mathematical school achievement, are still limited. This quasi-experimental study aimed to address this gap by evaluating the long-term transfer effects of an intervention program that is strictly derived from the QNL model of mathematical development [QNL training; in German "Mengen, zählen, Zahlen" (MZZ)].

Methods

We assessed the quantity-number competencies of 575 first-graders and identified 119 of them as being at risk for mathematical learning difficulties, who were then assigned to three experimental conditions. Sixty one children received 12 sessions of the QNL training, while 30 underwent training in inductive reasoning. Another 28 children served as a control group, receiving no specific intervention.

Results and Discussion

Multi-level analyses confirmed both significant short-and long-term effects in the specifically trained quantity-number competencies as well as transfer effects on subsequent mathematical school achievement. In accordance with previous findings, transfer effects of the QNL training on mathematical school achievement were not yet evident immediately after the intervention but turned out to be significant after a delay of 6 months and remained stable even 15 months after training. Effect sizes ranged from d = 0.32 to d = 1.12. These findings both underscore the preventive potential of interventions that are strictly driven by developmental theory and, conversely, support the theoretical assumptions of the QNL model.

Understanding the social-emotional components of our "number sense": insights from a novel non-symbolic numerical comparison task

Introduction

A large body of work has identified a core sense of number supported by the Approximate Number System (ANS) that is present in infancy and across species. Although it is commonly assumed that the ANS directly processes perceptual input and is relatively independent from affective factors, some evidence points at a correlation between ANS performance and math anxiety. However, the evidence is mixed. We tested whether giving participants active control in completing a numerical task would change the relationship between math anxiety on performance.

Methods

Adult participants (N = 103) completed a novel four-alternative-forced-choice non-symbolic numerical comparison task. In a repeated-measures design, participants either passively viewed different dot arrays or actively chose to view each array (i.e., active information-seeking) before deciding on the largest quantity. Participants also provided confidence judgments during the passive version of the task.

Results

We replicated the ratio-dependent signature in participants' accuracy in both the passive and active versions of the task using this novel paradigm, as well as in trial-level confidence judgments and information-seeking behavior. Participants' self-reported math anxiety significantly correlated with their accuracy on the passive version of the task. Critically, the correlation disappeared in the active version of the task. Gender also emerged as a predictor of confidence judgments and a moderator of the effect of task on overall accuracy and the effect of active information seeking on accuracy in the active version of the task. Exploratory analysis of estimated Weber Fraction suggests that these results may be driven by auxiliary factors instead of changes in ANS acuity.

Conclusion

These findings have implications for understanding the relationship between math anxiety and performance on numerical tasks.

tDCS effects in basic symbolic number magnitude processing are not significantly lateralized

Functional lateralization was previously established for various cognitive domains-but not for number processing. Although numbers are considered to be bilaterally represented in the intraparietal sulcus (IPS), there are some indications of different functional roles of the left vs. right IPS in processing number pairs with small vs. large distance, respectively. This raises the question whether number size plays a distinct role in the lateralization within the IPS. In our preregistered study, we applied anodal transcranial direct current stimulation (tDCS) over the left vs. right IPS to investigate the effect of stimulation as compared to sham on small vs. large distance, in both single-digit and two-digit number comparison. We expected that anodal tDCS over the left IPS facilitates number comparison with small distance, while anodal tDCS over the right IPS facilitates number comparison with large distance. Results indicated no effect of stimulation; however, exploratory analyses revealed that tDCS over the right IPS slowed down single-digit number processing after controlling for the training effect. In conclusion, number magnitude processing might be bilaterally represented in the IPS, however, our exploratory analyses emphasise the need for further investigation on functional lateralization of number processing.

The database for extracting numerical and visual properties of numerosity processing in the Chinese population

The ability to handle non-symbolic numerosity has been recurrently linked to mathematical abilities. The accumulated data provide a rich resource that can reflect the underlying properties (i.e., dot ratio, area, convex hull, perimeters, distance, and hash) of numerosity processing. This article reports a database of numerosity processing in the Chinese population. The database contains five independent datasets with 7459, 4902, 415, 671, 414 participants respectively. For each dataset, all data were collected in the same online computerized test, examination room, professorial tester, and using the same protocols. Computational modeling method could be used to extract the dot ratio and visual properties of numerosity from five types of dot stimuli. This database enables researchers to test the theoretical hypotheses regarding numerosity processing using a large sample population. The database can also indicate the individual difference of non-symbolic numerosity in mathematical abilities.

Abstract representations of small sets in newborns

From the very first days of life, newborns are not tied to represent narrow, modality- and object-specific aspects of their environment. Rather, they sometimes react to abstract properties shared by stimuli of very different nature, such as approximate numerosity or magnitude. As of now, however, there is no evidence that newborns possess abstract representations that apply to small sets: in particular, while newborns can match large approximate numerosities across senses, this ability does not extend to small numerosities. In two experiments, we presented newborn infants (N = 64, age 17 to 98 h) with patterned sets AB or ABB simultaneously in the auditory and visual modalities. Auditory patterns were presented as periodic sequences of sounds (AB: triangle-drum-triangle-drum-triangle-drum …; ABB: triangle-drum-drum-triangle-drum-drum-triangle-drum-drum …), and visual patterns as arrays of 2 or 3 shapes (AB: circle-diamond; ABB: circle-diamond-diamond). In both experiments, we found that participants reacted and looked longer when the patterns matched across the auditory and visual modalities – provided that the first stimulus they received was congruent. These findings uncover the existence of yet another type of abstract representations at birth, applying to small sets. As such, they bolster the hypothesis that newborns are endowed with the capacity to represent their environment in broad strokes, in terms of its most abstract properties. This capacity for abstraction could later serve as a scaffold for infants to learn about the particular entities surrounding them.

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Newborns were presented with auditory and visual patterns (AB vs. ABB).

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Participants reacted when the patterns presented were congruent across modalities.

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Newborns possess abstract representations applying to small sets.

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These representations may encode numerosity and/or repetitions.

The complexity of simple counting: ERP findings reveal early perceptual and late numerical processes in different arrangements

The counting process can only be fully understood when taking into account the visual characteristics of the sets counted. Comparing behavioral data as well as event-related brain potentials (ERPs) evoked by different task-irrelevant arrangements of dots during an exact enumeration task, we aimed to investigate the effect of illusory contour detection on the counting process while other grouping cues like proximity were controlled and dot sparsity did not provide a cue to the numerosity of sets. Adult participants (N = 37) enumerated dots (8-12) in irregular and two different types of regular arrangements which differed in the shape of their illusory dot lattices. Enumeration speed was affected by both arrangement and magnitude. The type of arrangement influenced an early ERP negativity peaking at about 270 ms after stimulus onset, whereas numerosity only affected later ERP components (> 300 ms). We also observed that without perceptual cues, magnitude was constructed at a later stage of cognitive processing. We suggest that chunking is a prerequisite for more fluent counting which influences automatic processing (< 300 ms) during enumeration. We conclude that the procedure of exact enumeration depends on the interaction of several perceptual and numerical processes that are influenced by magnitude and arrangement.

Numerosity tuning in human association cortices and local image contrast representations in early visual cortex

Human early visual cortex response amplitudes monotonically increase with numerosity (object number), regardless of object size and spacing. However, numerosity is typically considered a high-level visual or cognitive feature, while early visual responses follow image contrast in the spatial frequency domain. We find that, at fixed contrast, aggregate Fourier power (at all orientations and spatial frequencies) follows numerosity closely but nonlinearly with little effect of object size, spacing or shape. This would allow straightforward numerosity estimation from spatial frequency domain image representations. Using 7T fMRI, we show monotonic responses originate in primary visual cortex (V1) at the stimulus's retinotopic location. Responses here and in neural network models follow aggregate Fourier power more closely than numerosity. Truly numerosity tuned responses emerge after lateral occipital cortex and are independent of retinotopic location. We propose numerosity's straightforward perception and neural responses may result from the pervasive spatial frequency analyses of early visual processing.

Constructing rationals through conjoint measurement of numerator and denominator as approximate integer magnitudes in tradeoff relations

To investigate mechanisms of rational representation, I consider (1) construction of an ordered continuum of psychophysical scale of magnitude of sensation; (2) counting mechanism leading to an approximate numerosity scale for integers; and (3) conjoint measurement structure pitting the denominator against the numerator in tradeoff positions. Number sense of resulting rationals is neither intuitive nor expedient in their manipulation.

Visual Perception Supports Adults in Numerosity Processing and Arithmetical Performance

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

Towards a standardization of non-symbolic numerical experiments: GeNEsIS, a flexible and user-friendly tool to generate controlled stimuli

Several studies have suggested that vertebrate and invertebrate species may possess a number sense, i.e. an ability to process in a non-symbolic and non-verbal way the numerousness of a set of items. However, this hypothesis has been challenged by the presence of other non-numerical continuous physical variables, which vary along with numerosity (i.e., any change in the number of visual physical elements in a set naturally involves a related change in visual features such as area, density, contour length and convex hull of the stimulus). It is therefore necessary to control and manipulate the continuous physical information when investigating the ability of humans and other animals to perceive numerousness. During decades of research, different methods have been implemented in order to address this issue, which has implications for experiment replicability and inter-species comparisons, since no general standardized procedure is currently being used. Here we present the ‘Generation of Numerical Elements Images Software’ (GeNEsIS) for the creation of non-symbolic numerical arrays in a standardized and user-friendly environment. The main aim of this tool is to provide researchers in the field of numerical cognition a manageable and precise instrument to produce visual numerical arrays controlled for all the continuous variables. Additionally, we implemented the ability to actively guide stimuli presentation during habituation/dishabituation and dual-choice comparison tasks used in human and comparative research.

Small-range numerical representations of linguistic sounds in 9- to 10-month-old infants

Coordinated studies provide evidence that very young infants, like human adults and nonhuman animals, readily discriminate small and large number of visual displays on the basis of numerical information. This capacity has been considerably less studied in the auditory modality. Surprisingly, the available studies yielded mixed evidence concerning whether numerical representations of auditory items in the small number range (1 to 3) are present early in human development. Specifically, while newborns discriminate 2- from 3-syllable sequences, older infants at 6 and 9 months of age fail to differentiate 2 from 3 tones. This study tested the hypothesis that infants can represent small sets more precisely when listening to ecologically relevant linguistic sounds. The aim was to probe 9- to 10-month-olds' (N = 74) ability to represent sound sets in a working memory test. In experiments 1 and 2, infants successfully discriminated 2- and 3-syllable sequences on the basis of their numerosity, when continuous variables, such as individual item duration, inter-stimulus duration, pitch, intensity, and total duration, were controlled for. In experiment 3, however, infants failed to discriminate 3- from 4-syllable sequences under similar conditions. Finally, in experiment 4, infants were tested on their ability to distinguish 2 and 3 tone sequences. The results showed no evidence that infants discriminated these non-linguistic stimuli. These findings indicate that, by means of linguistic sounds, infants can access a numerical system that yields precise auditory representations in the small number range.

Macaque species with varying social tolerance show no differences in understanding what other agents perceive

A growing body of work demonstrates that a species' socioecology can impact its cognitive abilities. Indeed, even closely related species with different socioecological pressures often show different patterns of cognitive performance on the same task. Here, we explore whether major differences in social tolerance in two closely related macaque species can impact a core sociocognitive ability, the capacity to recognize what others see. Specifically, we compared the performance of Barbary macaques (Macaca sylvanus, n = 80) and rhesus macaques (Macaca mulatta, n = 62) on a standard test of visual perspective understanding. In contrast to the difference in performance, one might expect from these species' divergent socioecologies that our results show similar performance across Barbary and rhesus macaques, with both species forming expectations about how another agent will act based on that agent's visual perspective. These results suggest that differences in socioecology may not play as big of a role in the evolution of some theory of mind capacities as they do in other decision-making or foraging contexts.

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

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

Visual sense of number vs. sense of magnitude in humans and machines

Numerosity perception is thought to be foundational to mathematical learning, but its computational bases are strongly debated. Some investigators argue that humans are endowed with a specialized system supporting numerical representations; others argue that visual numerosity is estimated using continuous magnitudes, such as density or area, which usually co-vary with number. Here we reconcile these contrasting perspectives by testing deep neural networks on the same numerosity comparison task that was administered to human participants, using a stimulus space that allows the precise measurement of the contribution of non-numerical features. Our model accurately simulates the psychophysics of numerosity perception and the associated developmental changes: discrimination is driven by numerosity, but non-numerical features also have a significant impact, especially early during development. Representational similarity analysis further highlights that both numerosity and continuous magnitudes are spontaneously encoded in deep networks even when no task has to be carried out, suggesting that numerosity is a major, salient property of our visual environment.

The ontogeny of continuous quantity discrimination in zebrafish larvae (Danio rerio)

Several studies have investigated the ontogeny of the capacity to discriminate between discrete numerical information in human and non-human animals. Contrarily, less attention has been devoted to the development of the capacity to discriminate continuous quantities. Recently, we set up a fast procedure for screening continuous quantity abilities in adult individuals of an animal model in neurodevelopmental research, the zebrafish. Two different sized holes are presented in a wall that divides the home tank in two halves and the spontaneous preference of fish for passing through the larger hole is exploited to measure their discrimination ability. We tested zebrafish larvae in the first, second and third week of life varying the relative size of the smaller circle (0.60, 0.75, 0.86, 0.91 area ratio). We found that the number of passages increased across the age. The capacity to discriminate the larger hole decreased as the ratio between the areas increased. No difference in accuracy was found as a function of age. The accuracy of larval zebrafish almost overlaps that found in adults in a previous study, suggesting a limited role of maturation and experience on the ability to estimate areas in this species.

Food density and preferred quantity: discrimination of small and large numbers in angelfish (Pterophyllum scalare)

Many animal species share the ability to discriminate between sets with different quantity of food items. In fish, this ability has rarely been investigated, although findings have been obtained do indicate a preference, as in other animals, for sets with large over small quantities. The role played by food item size has also been found to be important in the discrimination. However, another potentially important non-numerical variable, food density, has not been investigated. In this study, we examined the influence of density (inter-item distance) in the decision-making process of food discrimination in angelfish (Pterophyllum scalare). In a binary choice task, we kept the number and size of food items constant, but contrasted a set containing food items spaced further apart (sparse set) to another set with food items spaced more closely (dense set). We conducted this analysis with sets in the small (3 vs 3 food items) and in the large number range (5 vs 5 food items) and also varied the specific spatial arrangements of the food items in the sets. Contrary to expectations, angelfish showed a preference for the sparse sets over the dense sets in the five vs five contrasts irrespective of the specific spatial arrangement, but exhibited no preference in case of the three vs three contrasts. Subsequently, we slightly lengthened the inter-item distance in the dense sets, and found preference for the dense over the sparse sets. Last, we further examined the potential effect of spatial configuration of the items in the sets, but found no effect of this latter factor. Overall, these results indicate that higher density of the contrasted food item sets significantly influences choice in angelfish, which prefer denser sets if a clear discriminability of each individual item within the sets is provided.

The role of item size on choosing contrasted food quantities in angelfish (Pterophyllum scalare)

Comparative studies on quantity discrimination in animals are important for understanding potential evolutionary roots of numerical competence. A previous study with angelfish has shown that they discriminate numerically different sets of same-sized food items and prefer the larger set. However, variables that covary with number were not controlled and choice could have been influenced by variables such as size or density of the food items rather than numerical attributes. Here using a recently developed approach, we examined whether contour length of the food items affects choice in a spontaneous binary choice task. In Experiment 1, a contrast of 1 vs. 1 food item was presented, but the ratio between the size (diameter) of the food items was varied. In Experiment 2, numerically different food sets were equated in overall size by increasing the size (diameter) of the items in the numerically small sets. In both Experiments, subjects showed a preference for the larger sized food items with a discrimination limit. These results show that item size plays a prominent role in foraging decisions in angelfish. Experiment 3 placed numerical and size attributes of the sets in conflict by presenting one larger-sized food item in the numerically smaller set that also had smaller overall size (diameter) of food items. Angelfish showed no preference in any of the contrasts, suggesting that they could not make optimal foraging decisions when these attributes were in conflict. Maximization of energy return is central to optimal foraging. Accordingly, here item size was also found to be a key feature of the sets, although the numerical attributes of the sets also influenced the choice.

Do infants have a sense of numerosity? A p-curve analysis of infant numerosity discrimination studies

Research demonstrating that infants discriminate between small (e.g., 1 vs. 3 dots) and large numerosities (e.g., 8 vs. 16 dots) is central to theories concerning the origins of human numerical abilities. To date, there has been no quantitative meta-analysis of the infant numerical competency data. Here, we quantitatively synthesize the evidential value of the available literature on infant numerosity discrimination using a meta-analytic tool called p-curve. In p-curve the distribution of available p-values is analyzed to determine whether the published literature examining particular hypotheses contains evidential value. p-curves demonstrated evidential value for the hypotheses that infants can discriminate between both small and large unimodal and cross-modal numerosities. However, the analyses also revealed that the published data on infants' ability to discriminate between large numerosities is less robust and statistically powered than the data on their ability to discriminate small numerosities. We argue there is a need for adequately powered replication studies to enable stronger inferences in order to use infant data to ground theories concerning the ontogenesis of numerical cognition.

Cues to individuation facilitate 6-month-old infants' visual short-term memory

Infants' ability to perform visual short-term memory (VSTM) tasks develops rapidly between 6 and 8 months. Here we tested the hypothesis that infants' VSTM performance is influenced by their ability to individuate simultaneously presented objects. We used a one-shot change detection task to ask whether 6-month-old infants (N = 47) would detect a change in the color of 1 item in a 2-item array when the stimulus context facilitated individuation of the items. In Experiment 1 the 2 items in the display differed in shape and color and in Experiment 2 the onset and offset times of the 2 items differed. In both experiments, 6-month-old infants detected a change, contrasting with previous results. Thus, young infants' encoding of information about individual items in multiple-item arrays is related to their ability to individuate those items. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

The rewarding effects of number and surface area of food in rats

Visual cues have an important role in food preference for both rats and humans. Here, we aim to isolate the effects of numerosity, density, and surface area on food preference and running speed in rats, by using a forced-choice maze paradigm. In Experiment 1, rats preferred and ran faster for a group of multiple smaller pellets rather than a single large pellet, corroborating previous research (Capaldi, Miller, & Alptekin Journal of Experimental Psychology: Animal Behavior Processes, 15(1), 75-80, 1989). Further experiments tested the prevailing hypothesis that multiple food pieces are more reinforcing because they occupy a larger surface area. Experiment 2 controlled for numerosity by utilizing a continuous food: mashed potatoes flattened to cover a larger surface area or rounded into a ball. The rats preferred and ran faster for the flattened potatoes, suggesting surface area plays a role in quantity estimations. Finally, in Experiment 3, rats displayed no preference or difference in running speed between a group of scattered and clustered pellets when number of pellets were kept constant. Taken together, these results suggest that density has an important role in food perception-that is, the rewarding effect of higher numerosity or larger surface area is removed when the food does not fill out the entire space. Alternative explanations and implications for human diet are discussed.

What is a number? The interplay between number and continuous magnitudes

Leibovich et al. argue that it is impossible to control for all continuous magnitudes in a numerical task. We contend that continuous magnitudes (i.e., perimeter, area, density) can be simultaneously controlled. Furthermore, we argue that shedding light on the interplay between number and continuous magnitudes - rather than considering them independently - will provide a much more fruitful approach to understanding mathematical abilities.

Towards numerical cognition's origin: insights from day-old domestic chicks

Instead of the scepticism on animal numerical understanding that characterized the first half of the twentieth century, in recent decades, a large and increasing body of the literature has shown that adult animals can master a variety of non-symbolic (in the absence of symbols such as mathematical words) numerical tasks. Nonetheless, evidence proving early numerical abilities in non-human animals was sparse. In this paper, I report the ongoing work to investigate numerical cognition in the day-old domestic chick (Gallus gallus). Unlike previous studies on adult animals, chicks can be tested very early in life, which gives us the opportunity to discover the origins of numerical comprehension. Here, I discuss studies revealing that day-old domestic chicks can: (i) discriminate between different numbers of objects; (ii) solve rudimentary arithmetic operations; and (iii) use ordinal information, identifying a target element (e.g. the fourth) in a series of identical elements, on the basis of its serial-numerical position. Some of these abilities are number-specific, while others underlie the interplay between number and continuous extents (continuous-quantity cues that covary with number, such as area and perimeter). These data are discussed in terms of ontogenetic development of mathematical comprehension.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

Children's Non-symbolic and Symbolic Numerical Representations and Their Associations With Mathematical Ability

Most empirical evidence supports the view that non-symbolic and symbolic representations are foundations for advanced mathematical ability. However, the detailed development trajectories of these two types of representations in childhood are not very clear, nor are the different effects of non-symbolic and symbolic representations on the development of mathematical ability. We assessed 253 4- to 8-year-old children's non-symbolic and symbolic numerical representations, mapping skills, and mathematical ability, aiming to investigate the developmental trajectories and associations between these skills. Our results showed non-symbolic numerical representation emerged earlier than the symbolic one. Four-year-olds were capable of non-symbolic comparisons but not symbolic comparisons; five-year-olds performed better at non-symbolic comparisons than symbolic comparisons. This performance difference disappeared at age 6. Children at age 6 or older were able to map between symbolic and non-symbolic quantities. However, as children learn more about the symbolic representation system, their advantage in non-symbolic representation disappeared. Path analyses revealed that a direct effect of children's symbolic numerical skills on their math performance, and an indirect effect of non-symbolic numerical skills on math performance via symbolic skills. These results suggest that symbolic numerical skills are a predominant factor affecting math performance in early childhood. However, the influences of symbolic and non-symbolic numerical skills on mathematical performance both declines with age.

Developmental Dyscalculia and Automatic Magnitudes Processing: Investigating Interference Effects between Area and Perimeter

The relationship between numbers and other magnitudes has been extensively investigated in the scientific literature. Here, the objectives were to examine whether two continuous magnitudes, area and perimeter, are automatically processed and whether adults with developmental dyscalculia (DD) are deficient in their ability to automatically process one or both of these magnitudes. Fifty-seven students (30 with DD and 27 with typical development) performed a novel Stroop-like task requiring estimation of one aspect (area or perimeter) while ignoring the other. In order to track possible changes in automaticity due to practice, we measured performance after initial and continuous exposure to stimuli. Similar to previous findings, current results show a significant group × congruency interaction, evident beyond exposure level or magnitude type. That is, the DD group systematically showed larger Stroop effects. However, analysis of each exposure period showed that during initial exposure to stimuli the DD group showed larger Stroop effects in the perimeter and not in the area task. In contrast, during continuous exposure to stimuli no triple interaction was evident. It is concluded that both magnitudes are automatically processed. Nevertheless, individuals with DD are deficient in inhibiting irrelevant magnitude information in general and, specifically, struggle to inhibit salient area information after initial exposure to a perimeter comparison task. Accordingly, the findings support the assumption that DD involves a deficiency in multiple cognitive components, which include domain-specific and domain-general cognitive functions.

Investigating looking and social looking measures as an index of infant violation of expectation

Accumulated looking time has been widely used to index violation of expectation (VoE) response in young infants. But there is controversy concerning the validity of this measure, with some interpreting infant looking behaviour in terms of perceptual preferences (Cohen & Marks, ; Haith, ). The current study aimed to compare the use of looking time with a recently used measure of social looking (Walden et al., ) in distinguishing between 6-month-old infants' response to novelty/familiarity and a condition in which the object was covertly switched for a different object. Following habituation, infants showed more social looking in response to the object-switch condition than the novel object change, whereas the more commonly used accumulated looking time measure did not distinguish between the two, showing an increase for both. Thus, social looking is a more valid measure of infant VoE than looking time.

A strategy to improve arithmetical performance in four day-old domestic chicks (Gallus gallus)

A large body of literature shows that non-human animals master numerical discriminations, but a limit has been reported in a variety of species in the comparison 3vs.4. Little is known regarding the possibility of using “cognitive strategies” to enable this discrimination. The aims of this study were to investigate: whether domestic chicks discriminated 3vs.4, and if changes in stimuli presentation could improve chicks’ numerical performance. Newly hatched chicks were reared with seven identical objects. On day 4, they underwent 20 consecutive testing trials to assess their capability to discriminate 3vs.4. The objects were presented, one-by-one, to the chicks and hidden behind one of two identical panels. As expected, the chicks did not discriminate (Experiment 1). When objects were presented and hidden in groups comprising one or two objects (2 + 1)vs.(2 + 2), the chicks succeeded (Experiment 2). The grouping strategy did not help in the case of a harder discrimination of (3 + 1)vs.(3 + 2) (Experiment 3), unless chicks were allowed to rest for two hours between testing sessions (Experiment 4). Our results suggest that in some cases, the limits reported for numerical performance in animals do not depend on cognitive limitations but on attentional or motivational factors, which can be overcome employing simple procedural adjustments.

Sex differences in discrimination reversal learning in the guppy

In several mammalian and avian species, females show a higher performance than males in tasks requiring cognitive flexibility such as the discrimination reversal learning. A recent study showed that female guppies are twice as efficient as males in a reversal learning task involving yellow-red discrimination, suggesting a higher cognitive flexibility in female guppies. However, the possibility exists that the superior performance exhibited by females does not reflect a general sex difference in cognitive abilities, but instead, is confined to colour discrimination tasks. To address this issue, we compared male and female guppies in two different discrimination reversal learning tasks and we performed a meta-analysis of these experiments and the previous one involving colour discrimination. In the first experiment of this study, guppies were tested in a task requiring them to learn to select the correct arm of a T-maze in order to rejoin a group of conspecifics. In experiment 2, guppies were observed in a numerical task requiring them to discriminate between 5 and 10 dots in order to obtain a food reward. Although females outperformed males in one condition of the T-maze, we did not find any clear evidence of females' greater reversal learning performance in either experiment. However, the meta-analysis of the three experiments supported the hypothesis of females' greater reversal learning ability. Our data do not completely exclude the idea that female guppies have a generally higher cognitive flexibility than males; however, they suggest that the size of this sex difference might depend on the task.

Mental arithmetic in the bilingual brain: Language matters

How do bilinguals solve arithmetic problems in each of their languages? We investigated this question by exploring the neural substrates of mental arithmetic in bilinguals. Critically, our population was composed of a homogeneous group of adults who were fluent in both of their instruction languages (i.e., German as first instruction language and French as second instruction language). Twenty bilinguals were scanned with fMRI (3T) while performing mental arithmetic. Both simple and complex problems were presented to disentangle memory retrieval occuring in very simple problems from arithmetic computation occuring in more complex problems. In simple additions, the left temporal regions were more activated in German than in French, whereas no brain regions showed additional activity in the reverse constrast. Complex additions revealed the reverse pattern, since the activations of regions for French surpassed the same computations in German and the extra regions were located predominantly in occipital regions. Our results thus highlight that highly proficient bilinguals rely on differential activation patterns to solve simple and complex additions in each of their languages, suggesting different solving procedures. The present study confirms the critical role of language in arithmetic problem solving and provides novel insights into how highly proficient bilinguals solve arithmetic problems.

Quantity discrimination in angelfish (Pterophyllum scalare) is maintained after a 30-s retention interval in the large but not in the small number range

The ability to discriminate between sets that differ in the number of elements can be useful in different contexts and may have survival and fitness consequences. As such, numerical/quantity discrimination has been demonstrated in a diversity of animal species. In the laboratory, this ability has been analyzed, for example, using binary choice tests. Furthermore, when the different number of items first presented to the subjects are subsequently obscured, i.e., are not visible at the moment of making a choice, the task requires memory for the size of the sets. In previous work, angelfish (Pterophyllum scalare) have been found to be able to discriminate shoals differing in the number of shoal members both in the small (less than 4) and the large (4 or more) number range, and they were able to perform well even when a short memory retention interval (2-15 s) was imposed. In the current study, we increased the retention interval to 30 s during which the shoals to choose between were obscured, and investigated whether angelfish could show preference for the larger shoal they saw before this interval. Subjects were faced with a discrimination between numerically small shoals (≤4 fish) and also between numerically large (≥4 fish) shoals of conspecifics. We found angelfish not to be able to remember the location of larger versus smaller shoals in the small number range, but to exhibit significant memory for the larger shoal in the large number range as long as the ratio between these shoals was at least 2:1. These results, together with prior findings, suggest the existence of two separate quantity estimation systems, the object file system for small number of items that does not work with the longer retention interval and the analogue magnitude system for larger number of items that does.

The Faculty of Language Integrates the Two Core Systems of Number

Only humans possess the faculty of language that allows an infinite array of hierarchically structured expressions (Hauser et al., 2002; Berwick and Chomsky, 2015). Similarly, humans have a capacity for infinite natural numbers, while all other species seem to lack such a capacity (Gelman and Gallistel, 1978; Dehaene, 1997). Thus, the origin of this numerical capacity and its relation to language have been of much interdisciplinary interest in developmental and behavioral psychology, cognitive neuroscience, and linguistics (Dehaene, 1997; Hauser et al., 2002; Pica et al., 2004). Hauser et al. (2002) and Chomsky (2008) hypothesize that a recursive generative operation that is central to the computational system of language (called Merge) can give rise to the successor function in a set-theoretic fashion, from which capacities for discretely infinite natural numbers may be derived. However, a careful look at two domains in language, grammatical number and numerals, reveals no trace of the successor function. Following behavioral and neuropsychological evidence that there are two core systems of number cognition innately available, a core system of representation of large, approximate numerical magnitudes and a core system of precise representation of distinct small numbers (Feigenson et al., 2004), I argue that grammatical number reflects the core system of precise representation of distinct small numbers alone. In contrast, numeral systems arise from integrating the pre-existing two core systems of number and the human language faculty. To the extent that my arguments are correct, linguistic representations of number, grammatical number, and numerals do not incorporate anything like the successor function.

An emergentist perspective on the origin of number sense

The finding that human infants and many other animal species are sensitive to numerical quantity has been widely interpreted as evidence for evolved, biologically determined numerical capacities across unrelated species, thereby supporting a 'nativist' stance on the origin of number sense. Here, we tackle this issue within the 'emergentist' perspective provided by artificial neural network models, and we build on computer simulations to discuss two different approaches to think about the innateness of number sense. The first, illustrated by artificial life simulations, shows that numerical abilities can be supported by domain-specific representations emerging from evolutionary pressure. The second assumes that numerical representations need not be genetically pre-determined but can emerge from the interplay between innate architectural constraints and domain-general learning mechanisms, instantiated in deep learning simulations. We show that deep neural networks endowed with basic visuospatial processing exhibit a remarkable performance in numerosity discrimination before any experience-dependent learning, whereas unsupervised sensory experience with visual sets leads to subsequent improvement of number acuity and reduces the influence of continuous visual cues. The emergent neuronal code for numbers in the model includes both numerosity-sensitive (summation coding) and numerosity-selective response profiles, closely mirroring those found in monkey intraparietal neurons. We conclude that a form of innatism based on architectural and learning biases is a fruitful approach to understanding the origin and development of number sense.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

Lookit (Part 1): A New Online Platform for Developmental Research

Many important questions about children’s early abilities and learning mechanisms remain unanswered not because of their inherent scientific difficulty but because of practical challenges: recruiting an adequate number of children, reaching special populations, or scheduling repeated sessions. Additionally, small participant pools create barriers to replication while differing laboratory environments make it difficult to share protocols with precision, limiting the reproducibility of developmental research. Here we introduce a new platform, “Lookit,” that addresses these constraints by allowing families to participate in behavioral studies online via webcam. We show that this platform can be used to test infants (11–18 months), toddlers (24–36 months), and preschoolers (36–60 months) and reliably code looking time, preferential looking, and verbal responses, respectively; empirical results of these studies are presented in Scott, Chu, and Schulz ( 2017 ). In contrast to most laboratory-based studies, participants were roughly representative of the American population with regards to income, race, and parental education. We discuss broad technical and methodological aspects of the platform, its strengths and limitations, recommendations for researchers interested in conducting developmental studies online, and issues that remain before online testing can fulfill its promise.

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.

A dissociation between small and large numbers in young children's ability to "solve for x" in non-symbolic math problems

Solving for an unknown addend in problems like 5+x=17 is challenging for children. Yet, previous work (Kibbe & Feigenson, 2015) found that even before formal math education, young children, aged 4- to 6-years, succeeded when problems were presented using non-symbolic collections of objects rather than symbolic digits. This reveals that the Approximate Number System (ANS) can support pre-algebraic intuitions. Here, we asked whether children also could intuitively "solve for x" when problems contained arrays of four or fewer objects that encouraged representations of individual objects instead of ANS representations. In Experiment 1, we first confirmed that children could solve for an unknown addend with larger quantities, using the ANS. Next, in Experiment 2a, we presented addend-unknown problems containing arrays of four or fewer objects (e.g., 1+x=3). This time, despite the identical task conditions, children were unable to solve for the unknown addend. In Experiment 2b, we replicated this failure with a new sample of children. Finally, in Experiment 3, we confirmed that children's failures in Experiments 2a and b were not due to lack of motivation to compute with small arrays, or to the discriminability of the quantities used: children succeeded at solving for an unknown sum with arrays containing four or fewer objects. Together, these results suggest that children's ability to intuitively solve for an unknown addend may be limited to problems that can be represented using the ANS.

Core mathematical abilities in infants: Number and much more

Adults' ability to process numerical information can be traced back to the first days of life. The cognitive mechanisms underlying numerical representations are functional in preverbal infants, who are able to both track a small number of individuals and to estimate the numerosity of large sets across different modalities. This ability is closely linked to their ability to compute other quantitative dimensions such as spatial extent and temporal duration. In fact, the human mind establishes, early in life, spontaneous links between number, space, and time, which are privileged relative to links with other continuous dimensions (like loudness and brightness). Finally, preverbal infants do not only associate numbers to corresponding spatial extents but also to different spatial positions along a spatial axis. It is argued that these number-space mappings are at the origins of the "mental number line" representation, which is already functional in the first year of life.

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.

Infants use relative numerical group size to infer social dominance

Detecting dominance relationships, within and across species, provides a clear fitness advantage because this ability helps individuals assess their potential risk of injury before engaging in a competition. Previous research has demonstrated that 10- to 13-mo-old infants can represent the dominance relationship between two agents in terms of their physical size (larger agent = more dominant), whereas younger infants fail to do so. It is unclear whether infants younger than 10 mo fail to represent dominance relationships in general, or whether they lack sensitivity to physical size as a cue to dominance. Two studies explored whether infants, like many species across the animal kingdom, use numerical group size to assess dominance relationships and whether this capacity emerges before their sensitivity to physical size. A third study ruled out an alternative explanation for our findings. Across these studies, we report that infants 6-12 mo of age use numerical group size to infer dominance relationships. Specifically, preverbal infants expect an agent from a numerically larger group to win in a right-of-way competition against an agent from a numerically smaller group. In addition, this is, to our knowledge, the first study to demonstrate that infants 6-9 mo of age are capable of understanding social dominance relations. These results demonstrate that infants' understanding of social dominance relations may be based on evolutionarily relevant cues and reveal infants' early sensitivity to an important adaptive function of social groups.

The influence of labels and facts on children's and adults' categorization

Language has been assumed to influence categorization for both adults and children but the precise role and potency of linguistic labels in category formation remains open. Here we explore how linguistic labels help fit objects into categories when relevant perceptual information is either ambiguous or inconsistent with the labels. We also ask how the effects of labels compare to those of other types of information such as facts. We presented 4-year-old children and adults with tasks in which they had to categorize a perceptually ambiguous natural-kind stimulus with one of two equidistant standards (Exp. 1 and 2) or group an ambiguous natural-kind stimulus into a category with a perceptually dissimilar standard (Exp. 3). Participants had access to labels (e.g., "This one is a lorp/pim"), observable facts (e.g., "This one has a long/short beak"), or unobservable facts (e.g., "This one drinks water/milk") that grouped the ambiguous stimulus with one of the standards. Both children and adults followed label- and fact-driven category boundaries for perceptually ambiguous stimuli (Exp. 1 and 2), and continued to do so even when the labels or facts pointed to perceptually incongruent categories (Exp. 3). These findings suggest a strong causal role for both labels and facts in categorization and have implications about theories of how categorization develops in children.

What counts for dogs (Canis lupus familiaris) in a quantity discrimination task?

Numerous studies have reported that animals reliably discriminate quantities of more or less food. However, little attention has been given to the relative salience of numerosity compared to the total amount of food when animals are making their choices. Here we investigated this issue in dogs. Dogs were given choices between two quantities of food items in three different conditions. In the Congruent condition, the total amount of food co-varied with the number of food items; in the Incongruent condition the total amount was pitted against the numerosity; and in the Controlled condition the total amount between the sets was equal. Results show that dogs based their choice on the total amount of edible food rather than on the number of food items, suggesting that, in food choice tasks, amount counts more than number. The presence of the largest individual item in a set did not bias dogs' choices. A control test excluded the possibility that dogs based their choices on olfactory cues alone.