Two core systems of numerical representation in infants

Perceptual subitizing performance in 3- and 4-year-olds: The impact of visual features of sets

Perceptual subitizing is a pivotal skill in children's mathematical development. It is defined as the rapid identification of small numerosities. Previous studies pointed to the contribution of visual features of sets to perceptual subitizing performance in adults. Insights into the contribution of visual features to subitizing performance in the critical 3- to 4-year age range are scant. This study aimed to address this gap by investigating the impact of visual features on perceptual subitizing performance (accuracy and response time) in 3- and 4-year-olds. Participants (119 3- and 4-year-olds) were offered a subitizing task that incorporated pictures of sets of three to five objects. The pictures systematically varied across four visual features: (a) pictorial context (distractors present vs. absent), (b) set homogeneity (homogeneous vs. heterogeneous objects), (c) set arrangement (linearly vs. randomly arranged objects), and (d) set differentiation (distinct vs. overlapping objects). Pictures with distractors, heterogeneous objects, randomly arranged objects, or overlapping objects were associated with lower subitizing accuracy and longer response times compared with pictures without distractors, homogeneous objects, linearly arranged objects, or distinct objects, respectively. Pictures with randomly arranged or overlapping objects along with distractors were associated with even lower subitizing accuracy. Pictures featuring a simple visual design-without distractors and with homogeneous, linearly arranged, and distinct sets-yielded the best subitizing performance in terms of accuracy and response time. Our findings might be explained by the cognitive processes underlying 3- and 4-year-olds' subitizing performance. The findings offer building blocks for future research in the domain and preschool educational practice.

"Catastrophic" set size limits on infants' capacity to represent objects: A systematic review and Bayesian meta-analysis

Decades of research has revealed that humans can concurrently represent small quantities of three-dimensional objects as those objects move through space or into occlusion. For infants (but not older children or adults), this ability apparently comes with a significant limitation: when the number of occluded objects exceeds three, infants experience what has been characterized as a "catastrophic" set size limit, failing to represent even the approximate quantity of the hidden array. Infants' apparent catastrophic representational failures suggest a significant information processing limitation in the first years of life, and the evidence has been used as support for prominent theories of the development of object and numerical cognition. However, the evidence for catastrophic failure consists of individual small-n experiments that use null hypothesis significance testing to obtain null results (i.e., p > 0.05). Whether catastrophic representational failures are robust or reliable across studies, methods, and labs is not known. Here we report a systematic review and Bayesian meta-analysis to examine the strength of the evidence in favor of catastrophic representational failures in infancy. Our analysis of 22 experiments across 12 reports, with a combined total of n = 367 infants aged 10-20 months, revealed strong support for the evidence for catastrophic set size limits. A complementary analysis found moderate support for infants' success when representing fewer than four objects. We discuss the implications of our findings for theories of object and numerical cognitive development. RESEARCH HIGHLIGHTS: Previous work has suggested that infants are unable to concurrently represent four or more objects-a "catastrophic" set size limit. We reviewed this work and conducted a Bayesian meta-analysis to examine the robustness of this limit across individual small-n experiments. We found strong support for the evidence for catastrophic set size limits, and moderate support for infants' success when representing fewer than four objects.

Quantity discrimination in newly hatched zebrafish suggests hardwired numerical abilities

An intriguing hypothesis to explain the ubiquity of numerical abilities is that all vertebrates are born with hardwired neuronal networks for processing numbers. To date, only studies on human foetuses have clearly supported this hypothesis. Zebrafish hatch 48-72 h after fertilisation with an embryonic nervous system, providing a unique opportunity for investigating this hypothesis. Here, we demonstrated that zebrafish larvae exposed to vertical bars at birth acquired an attraction for bar stimuli and we developed a numerical discrimination task based on this preference. When tested with a series of discriminations of increasing difficulty (1vs.4, 1vs.3, 1vs.2, and 2vs.4 bars), zebrafish larvae reliably selected the greater numerosity. The preference was significant when stimuli were matched for surface area, luminance, density, and convex hull, thereby suggesting a true capacity to process numerical information. Converging results from two phylogenetically distant species suggests that numerical abilities might be a hallmark feature of vertebrates' brains.

The role of spatial information in an approximate cross-modal number matching task

The approximate number system (ANS) is thought to be an innate cognitive system that allows humans to perceive numbers (>4) in a fuzzy manner. One assumption of the ANS is that numerosity is represented amodally due to a mechanism, which filters out nonnumerical information from stimulus material. However, some studies show that nonnumerical information (e.g., spatial parameters) influence the numerosity percept as well. Here, we investigated whether there is a cross-modal transfer of spatial information between the haptic and visual modality in an approximate cross-modal number matching task. We presented different arrays of dowels (haptic stimuli) to 50 undergraduates and asked them to compare haptically perceived numerosity to two visually presented dot arrays. Participants chose which visually presented array matched the numerosity of the haptic stimulus. The distractor varied in number and displayed a random pattern, whereas the matching (target) dot array was either spatially identical or spatially randomized (to the haptic stimulus). We hypothesized that if a "numerosity" percept is based solely on number, neither spatially identical nor spatial congruence between the haptic and the visual target arrays would affect the accuracy in the task. However, results show significant processing advantages for targets with spatially identical patterns and, furthermore, that spatial congruency between haptic source and visual target facilitates performance. Our results show that spatial information was extracted from the haptic stimuli and influenced participants' responses, which challenges the assumption that numerosity is represented in a truly abstract manner by filtering out any other stimulus features.

Get in touch with numbers - an approximate number comparison task in the haptic modality

The Approximate Number System (ANS) is conceptualized as an innate cognitive system that allows humans to perceive numbers of objects or events (>4) in a fuzzy, imprecise manner. The representation of numbers is assumed to be abstract and not bound to a particular sense. In the present study, we test the assumption of a shared cross-sensory system. We investigated approximate number processing in the haptic modality and compared performance to that of the visual modality. We used a dot comparison task (DCT), in which participants compare two dot arrays and decide which one contains more dots. In the haptic DCT, 67 participants had to compare two simultaneously presented dot arrays with the palms of their hands; in the visual DCT, participants inspected and compared dot arrays on a screen. Tested ratios ranged from 2.0 (larger/smaller number) to 1.1. As expected, in both the haptic and the visual DCT responses similarly depended on the ratio of the numbers of dots in the two arrays. However, on an individual level, we found evidence against medium or stronger positive correlations between “ANS acuity” in the visual and haptic DCTs. A regression model furthermore revealed that besides number, spacing-related features of dot patterns (e.g., the pattern’s convex hull) contribute to the percept of numerosity in both modalities. Our results contradict the strong theory of the ANS solely processing number and being independent of a modality. According to our regression and response prediction model, our results rather point towards a modality-specific integration of number and number-related features.

Parent-based training of basic number skills in children with Down syndrome using an adaptive computer game

BACKGROUND

Numeracy is an area of difficulty for children with Down syndrome (DS). It has been demonstrated that The Number Race, a non-commercial adaptive computer game designed to foster basic mathematical abilities, represents a promising instrument to enhance these skills in children with DS when delivered by an expert in a clinical setting.

AIMS

In the present study, we assessed the efficacy of The Number Race when administered at home by properly instructed and remotely supervised parents.

METHODS AND PROCEDURES

Basic numerical skills were assessed before and after training, as well as at three-months follow-up. Performance of children with DS who worked at home with the parent (PG) was compared with that of children who received the training by an expert (EG). For both groups, the training lasted ten weeks, with two weekly sessions of 20-30 min.

OUTCOMES AND RESULTS

Results show that both groups improved across various measures of numerical proficiency, including the overall score of the numeracy assessment battery, while only the EG showed an improvement in a measure of mental calculation. The improvements were maintained three months after the end of the training.

CONCLUSIONS AND IMPLICATIONS

These findings confirm the efficacy of The Number Race and extend it to an home-based setting, whereby parents administer the training with external supervision.

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.

Symbol grounding of number words in the subitization range

In three experiments, we explored whether number words are grounded in a nonsymbolic representation of numerosity. We used a sentence-picture verification task, where participants are required to check whether the concept given in a sentence corresponds to the subsequently presented object. We concurrently manipulated numerical congruency by orthogonally varying the number word attached to the concept and the quantity of objects. The number words and numerosities varied from one to four in Experiment 1 and from six to nine in Experiment 2. In Experiment 3, we employed number words six and eight with the constraint that, in the incongruent condition, a constant number-to-numerosity ratio of 2:1 was used. In Experiment 1, we found that participants were faster and more efficient when concept-object matches were accompanied by numerical congruency relative to incongruency. On the other hand, no such difference was observed in Experiments 2 and 3 for numbers falling outside of the subitization range. The results are consistent with the hypothesis that number words from one to four are grounded in a nonsymbolic representation of the size of small sets.

Absence perception and the philosophy of zero

Zero provides a challenge for philosophers of mathematics with realist inclinations. On the one hand it is a bona fide cardinal number, yet on the other it is linked to ideas of nothingness and non-being. This paper provides an analysis of the epistemology and metaphysics of zero. We develop several constraints and then argue that a satisfactory account of zero can be obtained by integrating (1) an account of numbers as properties of collections, (2) work on the philosophy of absences, and (3) recent work in numerical cognition and ontogenetic studies.

The Open Algorithm Based on Numbers (ABN) Method: An Effective Instructional Approach to Domain-Specific Precursors of Arithmetic Development

This article presents the results of a comparative study regarding the impact and contribution of two instructional approaches to formal and informal mathematical reasoning with two groups of Spanish students, aged four and five. Data indicated that for both age groups, children under the ABN method [Open Algorithm Based on Numbers (ABN)] (n = 147) achieved better results than the group under the CBC approach (Closed Algorithms Based on Ciphers) (n = 82), which is the widespread approach in Spanish schools to teach formal and informal mathematical reasoning. Furthermore, the comparative analyses showed that the effect is higher in the group of students who received more instruction on skills considered domain-specific predictors of later arithmetic performance. Statistically significant differences were found in 9 of the 10 dimensions evaluated by TEMA-3 (p < 0.01), as well as on estimation tasks in the number-line for the 5-year-old-group. However, the 4-year-old group only presented significant results in calculation and concepts tasks about informal mathematical reasoning. We discuss that these differences arise by differential exposure to specific number-sense tasks, since the groups proved to be equivalent in terms of receptive vocabulary, processing speed, and working memory. The educational consequences of these results were also analyzed.

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'.

Infants' intermodal numerical knowledge

Two-system theory as the dominant approach in the field of infant numerical representation is characterized by three features: precise representation of small sets of objects, approximate representation of large magnitudes and failure to compare small and large sets. Comparison of single- and multimodal numerical abilities suggests that infants' performance in multimodal conditions is consistent with these three features. Nevertheless, the influence of multimodal stimulation on infants' numerical representation is characterized by preventing the formation of perceptual overlaps across different sensory modalities which can lead to an understanding of numerical values of small sets and also by creating a conceptual overlap about numbers that increases infants' accuracy for discriminating quantities when numerical information is presented bimodally and synchronously. Such multisensory benefits provide numerical capabilities beyond what is depicted by the two-system view.

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.

Short-Term Memory Effects on Crossing the Boundary: Discrimination between Large and Small Quantities in Angelfish (Pterophyllum scalare)

Rudimentary quantification abilities are found in numerous animal species and in human infants all demonstrating the ability to discriminate between quantities differing in numerical size. An open question is whether individuals rely on different underlying systems to discriminate between large (analogue magnitude system (AMS) for number of items exceeding 3) and small quantities (object-file system (OFS) for number of items below 4), or they use only one system (AMS) for the entire number range. The two-system hypothesis has been supported by finding reduced ability to discriminate between quantities that cross the large-small boundary in several species. Recently, the role of cognitive representation, i.e., memory, in quantity discrimination has also been recognized. Here, we investigated whether angelfish can discriminate quantities across the boundary under two memory conditions. In a binary choice test, single angelfish were allowed to see groups (shoals) of conspecifics of different numerical size on the two sides of their test tank. In Experiment 1, their choice was recorded after a 2-sec retention interval during which shoal size information was unavailable. Angelfish were able to discriminate the larger shoal across the boundary when the shoals differed by a 2:1 or higher ratio, but not when the ratio was lower. In Experiment 2, however, with a 15-sec retention interval, angelfish could only detect a four-fold difference in ratio but failed to detect a three- or a two-fold difference across the boundary. These results suggest that angelfish can remember smaller differences for a short (2 sec) but not for a longer (15 sec) period. Together with previous findings, the current results support the idea that angelfish use two distinct systems for representing quantity, but they may recruit the AMS even for the small number range under some circumstances, e.g., when higher memory demand is imposed by a greater retention interval.

A Place for Zero in the Brain

It has long been thought that the primary cognitive and neural systems responsible for processing numerosities are not predisposed to encode empty sets (i.e., numerosity zero). A new study challenges this view by demonstrating that zero is translated into an abstract quantity along the numerical continuum by the primate parietofrontal magnitude system.

Cognitive and brain systems underlying early mathematical development

We review current debate regarding the core competencies that support early mathematics learning, focusing on the contributions of the inherent system for representing approximate magnitudes, and domain-general systems that facilitate learning across academic domains. The latter include the executive control system that enables explicit processing of quantitative symbols, such as Arabic numerals, and the logical problem-solving abilities (intelligence) that facilitate learning the relations among numerals. The neural systems that underlie these abilities, as related to mathematical learning, are also discussed, albeit briefly. We place the contributions of inherent quantitative abilities and domain-general mechanisms in an evolutionary context and provide some discussion as to how they interact during the learning of evolutionarily novel mathematics.

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.

Signal clarity: an account of the variability in infant quantity discrimination tasks

Infants have shown variable success in quantity comparison tasks, with infants of a given age sometimes successfully discriminating numerical differences at a 2:3 ratio but requiring 1:2 and even 1:4 ratios of change at other times. The current explanations for these variable results include the two-systems proposal - a theoretical framework that suggests that there are multiple systems at play and that these systems do not communicate early in infancy, leading to failure in certain numerical comparisons. An alternative proposal is that infants may be attending to continuous extent dimensions in these tasks rather than number per se. However, neither of these two main proposals is independently capable of accounting for the previously published data. Recently the Signal Clarity Hypothesis was proposed to account for and predict the variability (Cantrell & Smith, 2013). According to this hypothesis, infants' variable success may be understood from a framework of statistical learning taken together with the signal-to-noise ratio generated by control procedures in habituation tasks. Here we test specific predictions made by the Signal Clarity Hypothesis. Across four experiments assessing 9-month old discriminations of small and large sets (2 vs. 4 and 3 vs. 4), we demonstrate that infant success can be predicted by this novel approach and, further, that infants may discriminate smaller ratios of change than previously believed (3:4 numerical change and 2:3 cumulative area change).

Refining the quantitative pathway of the Pathways to Mathematics model

In the current study, we adopted the Pathways to Mathematics model of LeFevre et al. (2010). In this model, there are three cognitive domains--labeled as the quantitative, linguistic, and working memory pathways--that make unique contributions to children's mathematical development. We attempted to refine the quantitative pathway by combining children's (N=141 in Grades 2 and 3) subitizing, counting, and symbolic magnitude comparison skills using principal components analysis. The quantitative pathway was examined in relation to dependent numerical measures (backward counting, arithmetic fluency, calculation, and number system knowledge) and a dependent reading measure, while simultaneously accounting for linguistic and working memory skills. Analyses controlled for processing speed, parental education, and gender. We hypothesized that the quantitative, linguistic, and working memory pathways would account for unique variance in the numerical outcomes; this was the case for backward counting and arithmetic fluency. However, only the quantitative and linguistic pathways (not working memory) accounted for unique variance in calculation and number system knowledge. Not surprisingly, only the linguistic pathway accounted for unique variance in the reading measure. These findings suggest that the relative contributions of quantitative, linguistic, and working memory skills vary depending on the specific cognitive task.