Developmental change in the acuity of the "Number Sense": The Approximate Number System in 3-, 4-, 5-, and 6-year-olds and adults

Behavioral, neuropsychological, and brain imaging research points to a dedicated system for processing number that is shared across development and across species. This foundational Approximate Number System (ANS) operates over multiple modalities, forming representations of the number of objects, sounds, or events in a scene. This system is imprecise and hence differs from exact counting. Evidence suggests that the resolution of the ANS, as specified by a Weber fraction, increases with age such that adults can discriminate numerosities that infants cannot. However, the Weber fraction has yet to be determined for participants of any age between 9 months and adulthood, leaving its developmental trajectory unclear. Here we identify the Weber fraction of the ANS in 3-, 4-, 5-, and 6-year-old children and in adults. We show that the resolution of this system continues to increase throughout childhood, with adultlike levels of acuity attained surprisingly late in development.     

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.     

Adolescents’ inhibitory control: keep it cool or lose control

Inhibitory control (i.e., the ability to resist automatisms, temptations, distractions, or interference and to adapt to conflicting situations) is a determinant of cognitive and socio‐emotional development. In light of the discrepancies of previous findings on the development of inhibitory control in affectively charged contexts, two important issues need to be addressed. We need to determine (a) whether cool inhibitory control (in affectively neutral contexts) and hot inhibitory control (in affectively charged contexts) follow the same developmental pattern and (b) the degree of specificity of these two types of inhibitory control at different ages. Thus, in the present study, we investigated the developmental patterns of cool and hot inhibitory control and the degree of specificity of these abilities in children, adolescents and adults. Typically developing children, adolescents, and adults performed two Stroop‐like tasks: an affectively neutral one (Cool Stroop task) and an affectively charged one (Hot Stroop task). In the Cool Stroop task, the participants were asked to identify the ink color of the words independent of color that the words named; in the Hot Stroop task, the participants were asked to identify the emotional expression of a face independent of the emotion named by a simultaneously displayed written word. We found that cool inhibitory control abilities develop linearly with age, whereas hot inhibitory control abilities follow a quadratic developmental pattern, with adolescents displaying worse hot inhibitory control abilities than children and adults. In addition, cool and hot inhibitory control abilities were correlated in children but not in adolescents and adults. The present study suggests (a) that cool and hot inhibitory control abilities develop differently from childhood to adulthood – i.e., that cool inhibition follows a linear developmental pattern and hot inhibition follows an adolescent‐specific pattern – and (b) that they become progressively more domain‐specific with age.     

Probing the nature of deficits in the ‘Approximate Number System’ in children with persistent Developmental Dyscalculia

In the present study we examined whether children with Developmental Dyscalculia (DD) exhibit a deficit in the so‐called ‘Approximate Number System’ (ANS). To do so, we examined a group of elementary school children who demonstrated persistent low math achievement over 4 years and compared them to typically developing (TD), aged‐matched controls. The integrity of the ANS was measured using the Panamath ( www.panamath.org ) non‐symbolic numerical discrimination test. Children with DD demonstrated imprecise ANS acuity indexed by larger Weber fraction (w) compared to TD controls. Given recent findings showing that non‐symbolic numerical discrimination is affected by visual parameters, we went further and investigated whether children performed differently on trials on which number of dots and their overall area were either congruent or incongruent with each other. This analysis revealed that differences in w were only found between DD and TD children on the incongruent trials. In addition, visuo‐spatial working memory strongly predicts individual differences in ANS acuity (w) during the incongruent trials. Thus the purported ANS deficit in DD can be explained by a difficulty in extracting number from an array of dots when area is anti‐correlated with number. These data highlight the role of visuo‐spatial working memory during the extraction process, and demonstrate that close attention needs to be paid to perceptual processes invoked by tasks thought to represent measures of the ANS.     

Late‐ but not early‐onset blindness impairs the development of audio‐haptic multisensory integration

Integrating different senses to reduce sensory uncertainty and increase perceptual precision can have an important compensatory function for individuals with visual impairment and blindness. However, how visual impairment and blindness impact the development of optimal multisensory integration in the remaining senses is currently unknown. Here we first examined how audio‐haptic integration develops and changes across the life span in 92 sighted (blindfolded) individuals between 7 and 70 years of age. We used a child‐friendly task in which participants had to discriminate different object sizes by touching them and/or listening to them. We assessed whether audio‐haptic performance resulted in a reduction of perceptual uncertainty compared to auditory‐only and haptic‐only performance as predicted by maximum‐likelihood estimation model. We then compared how this ability develops in 28 children and adults with different levels of visual experience, focussing on low‐vision individuals and blind individuals that lost their sight at different ages during development. Our results show that in sighted individuals, adult‐like audio‐haptic integration develops around 13–15 years of age, and remains stable until late adulthood. While early‐blind individuals, even at the youngest ages, integrate audio‐haptic information in an optimal fashion, late‐blind individuals do not. Optimal integration in low‐vision individuals follows a similar developmental trajectory as that of sighted individuals. These findings demonstrate that visual experience is not necessary for optimal audio‐haptic integration to emerge, but that consistency of sensory information across development is key for the functional outcome of optimal multisensory integration.     

Bidirectional,Longitudinal Associations Between Math Ability and Approximate Number System Precision in Childhood

Research suggests that individual differences in math abilities correlate with approximate representations of quantity that are supported by a primitive Approximate Number System (ANS). However, relatively little research has addressed the direction of this association in early childhood. Here we examined the development of the ANS and math ability longitudinally in 3- to 5-year-old children. Children were observed at three time points roughly six months apart; they completed a nonsymbolic numerical comparison task that measured ANS precision and a standardized math assessment. A series of cross-lagged panel models was then estimated to explore the associations between ANS precision and math ability over time. Bidirectional associations between ANS precision and math ability emerged: Early ANS precision was related to children’s later math skills, and early math ability also significantly predicted children’s later ANS precision. Evidence for mutual enhancement over time between the ANS and symbolic math ability adds to our growing understanding of the ANS and how the ANS and math knowledge interact.     

Development of approximate number sense across the elementary school years: A cross‐cultural longitudinal study

In recent years, there has been growing interest among researchers in exploring approximate number sense (ANS)—the ability to estimate and discriminate quantities without the use of symbols. Despite the growing number of studies on ANS, there have been no cross‐cultural longitudinal studies to estimate both the development of ANS and the cross‐cultural differences in ANS growth trajectories. In this study, we aimed to estimate the developmental trajectories of ANS from the beginning of formal education to the end of elementary school in two countries, Russia and Kyrgyzstan, which have similar organization of their educational systems but differences in socioeconomic status (SES) and in the results of large‐scale educational assessments. To assess the developmental trajectories of ANS, we used a four‐wave longitudinal study with 416 participants from two countries and applied the mixed effect growth approach and the latent class growth approach. Our analysis revealed that the rate of growth in ANS accuracy was higher for the Russian sample than for the Kyrgyz sample and that this difference remained significant even after controlling for fluid intelligence. We identified two latent classes of growth trajectories: the first class had a significant growth in ANS, whereas the second class had no growth. Comparing the distribution of latent classes within the two countries revealed that there was a significantly larger proportion of schoolchildren from the second class in Kyrgyzstan than in Russia.     

Understanding the mapping between numerical approximation and number words: evidence from Williams syndrome and typical development

All numerate humans have access to two systems of number representation: an exact system that is argued to be based on language and that supports formal mathematics, and an Approximate Number System (ANS) that is present at birth and appears independent of language. Here we examine the interaction between these two systems by comparing the profiles of people with Williams syndrome (WS) with those of typically developing children between ages 4 and 9 years. WS is a rare genetic deficit marked by fluent and well‐structured language together with severe spatial deficits, deficits in formal math, and abnormalities of the parietal cortex, which is thought to subserve the ANS. One of our tasks, requiring approximate number comparison but no number words, revealed that the ANS precision of adolescents with WS was in the range of typically developing 2‐ to 4‐year‐olds. Their precision improved with age but never reached the level of typically developing 6‐ or 9‐year‐olds. The second task, requiring verbal number estimation using number words, revealed that the estimates produced by adolescents with WS were comparable to those of typically developing 6‐ and 9‐year‐olds, i.e. were more advanced than their ANS precision. These results suggest that ANS precision is somewhat separable from the mapping between approximate numerosities and number words, as the former can be severely damaged in a genetic disorder without commensurate impairment in the latter.     

Is there really a link between exact‐number knowledge and approximate number system acuity in young children?

Although everyone perceives approximate numerosities, some people make more accurate estimates than others. The accuracy of this estimation is called approximate number system (ANS) acuity. Recently, several studies have reported that individual differences in young children's ANS acuity are correlated with their knowledge of exact numbers such as the word ‘six’ (Mussolin et al., 2012, Trends Neurosci. Educ., 1, 21; Shusterman et al., 2011, Connecting early number word knowledge and approximate number system acuity; Wagner & Johnson, 2011, Cognition, 119, 10; see also Abreu‐Mendoza et al., 2013, Front. Psychol., 4, 1). This study argues that this correlation should not be trusted. It seems to be an artefact of the procedure used to assess ANS acuity in children. The correlation arises because (1) some experimental designs inadvertently allow children to answer correctly based on the size (rather than the number) of dots in the display and/or (2) young children with little exact‐number knowledge may not understand the phrase ‘more dots’ to mean numerically more. When the task is modified to make sure that children respond on the basis of numerosity, the correlation between ANS acuity and exact‐number knowledge in normally developing children disappears.     

Infants' enumeration of actions: numerical discrimination and its signature limits

Are abstract representations of number – representations that are independent of the particular type of entities that are enumerated – a product of human language or culture, or do they trace back to human infancy? To address this question, four experiments investigated whether human infants discriminate between sequences of actions (jumps of a puppet) on the basis of numerosity. At 6 months, infants successfully discriminated four‐ versus eight‐jump sequences, when the continuous variables of sequence duration, jump duration, jump rate, jump interval and duration, and extent of motion were controlled, and rhythm was eliminated. In contrast, infants failed to discriminate two‐ versus four‐jump sequences, suggesting that infants fail to form cardinal number representations of small numbers of actions. Infants also failed to discriminate between sequences of four versus six jumps at 6 months, and succeeded at 9 months, suggesting that infants’ number representations are imprecise and increase in precision with age. All of these findings agree with those of studies using visual–spatial arrays and auditory sequences, providing evidence that a single, abstract system of number representation is present and functional in infancy.     

近似数量系统敏锐度与数学能力的关系

近年来,来自认知发展、比较认知、跨文化认知和神经生物学的研究证据都表明近似数量系统的存在,并且相较于一般认知能力,它更可能是决定个体数学能力差异最为重要的因素。本文综述了有关近似数量系统敏锐度与数学能力相互关系的横断研究、纵向研究、训练研究及认知神经科学的研究成果,分析了影响二者关系的因素,包括个体年龄、数学能力高低、抑制控制等,并总结了多种理论对二者间显著正相关关系的解释。未来研究需要在确定更具信效度的测量范式的基础上探讨近似数量系统与数学能力各维度的关系,以及这种相互关系背后的原因,并将研究结论运用于数学教学及计算障碍个体的干预。     

Indexing space and time in film understanding

We investigated the extent to which understanders monitor shifts in time and space during film comprehension. Participants viewed a feature‐length film and identified those points in the film in which they perceived a change in situation. We performed an a priori analysis of the films to identify the shifts in time, the movement of characters, and region. The relationship between the theoretical analysis of the films and the participants judgements of situational change was assessed. The results provide support for the Event Indexing Model and suggest that situation models for filmed events are indexed along multiple dimensions of situational continuity. Furthermore, the pattern of results was similar for narrative film as they are for narrative text. This finding suggest that there are general mechanisms for event understanding that operate independently of medium or mode of experience. Copyright © 2001 John Wiley & Sons, Ltd.     

Small and large number processing in infants and toddlers with Williams syndrome

Previous studies have suggested that typically developing 6‐month‐old infants are able to discriminate between small and large numerosities. However, discrimination between small numerosities in young infants is only possible when variables continuous with number (e.g. area or circumference) are confounded. In contrast, large number discrimination is successful even when variables continuous with number are systematically controlled for. These findings suggest the existence of different systems underlying small and large number processing in infancy. How do these develop in atypical syndromes? Williams syndrome (WS) is a rare neurocognitive developmental disorder in which numerical cognition has been found to be impaired in older children and adults. Do impairments of number processing have their origins in infancy? Here this question is investigated by testing the small and large number discrimination abilities of infants and toddlers with WS. While infants with WS were able to discriminate between 2 and 3 elements when total area was confounded with numerosity, the same infants did not discriminate between 8 and 16 elements, when number was not confounded with continuous variables. These findings suggest that a system for tracking the features of small numbers of object (object‐file representation) may be functional in WS, while large number discrimination is impaired from an early age onwards. Finally, we argue that individual differences in large number processing in infancy are more likely than small number processing to be predictive of later development of numerical cognition.     

The approximate number system and its relation to early math achievement: Evidence from the preschool years

Humans rely on two main systems of quantification; one is nonsymbolic and involves approximate number representations (known as the approximate number system or ANS), and the other is symbolic and allows for exact calculations of number. Despite the pervasiveness of the ANS across development, recent studies with adolescents and school-aged children point to individual differences in the precision of these representations that, importantly, have been shown to relate to symbolic math competence even after controlling for general aspects of intelligence. Such findings suggest that the ANS, which humans share with nonhuman animals, interfaces specifically with a uniquely human system of formal mathematics. Other findings, however, point to a less straightforward picture, leaving open questions about the nature and ontogenetic origins of the relation between these two systems. Testing children across the preschool period, we found that ANS precision correlated with early math achievement but, critically, that this relation was nonlinear. More specifically, the correlation between ANS precision and math competence was stronger for children with lower math scores than for children with higher math scores. Taken together, our findings suggest that early-developing connections between the ANS and mathematics may be fundamentally discontinuous. Possible mechanisms underlying such nonlinearity are discussed.     

Previous reward decreases errors of commission on later ‘No‐Go’ trials in children 4 to 12 years of age: evidence for a context monitoring account

Inhibitory control is widely hypothesized to be the cornerstone of executive function in childhood and the central deficit in a number of developmental disorders, including attention‐deficit/hyperactivity disorder (ADHD). However, recent evidence from adults indicates that performance on response inhibition tasks may primarily reflect non‐inhibitory attentional control (context monitoring) processes. Yet it may be that inhibition plays a more central role in childhood – a time when the architecture of cognitive processes might be more transparent due to wide variability in skill level. Here we directly test inhibitory and context monitoring explanations of task performance on a Go/No‐Go task in a large group of children 4–12 years of age. We conclude that traditional inhibitory conceptualizations of task performance on the Go/No‐Go task cannot account for our findings, calling into question evidence supporting a central role for inhibitory control in cognitive development or developmental psychopathology.     

ANS acuity and mathematics ability in preschoolers from low‐income homes: contributions of inhibitory control

Recent findings by Libertus, Feigenson, and Halberda (2011) suggest that there is an association between the acuity of young children's approximate number system (ANS) and their mathematics ability before exposure to instruction in formal schooling. The present study examined the generalizability and validity of these findings in a sample of preschoolers from low‐income homes. Children attending Head Start (N = 103) completed measures to assess ANS acuity, mathematics ability, receptive vocabulary, and inhibitory control. Results showed only a weak association between ANS acuity and mathematics ability that was reduced to non‐significance when controlling for a direct measure of receptive vocabulary. Results also revealed that inhibitory control plays an important role in the relation between ANS acuity and mathematics ability. Specifically, ANS acuity accounted for significant variance in mathematics ability over and above receptive vocabulary, but only for ANS acuity trials in which surface area conflicted with numerosity. Moreover, this association became non‐significant when controlling for inhibitory control. These results suggest that early mathematical experiences prior to formal schooling may influence the strength of the association between ANS acuity and mathematics ability and that inhibitory control may drive that association in young children.     

Children's learning of number words in an indigenous farming‐foraging group

We show that children in the Tsimane', a farming‐foraging group in the Bolivian rain‐forest, learn number words along a similar developmental trajectory to children from industrialized countries. Tsimane' children successively acquire the first three or four number words before fully learning how counting works. However, their learning is substantially delayed relative to children from the United States, Russia, and Japan. The presence of a similar developmental trajectory likely indicates that the incremental stages of numerical knowledge – but not their timing — reflect a fundamental property of number concept acquisition which is relatively independent of language, culture, age, and early education.     

Development of children's ability to judge relative numerosity

Three- to seven-year-old chilren were trained through reinforcement to select either the more or less numerous of two rows of squares. One group was shown rows in which number covaried with row length, one group saw rows in which number covaried with row density, and a third group saw rows in which number did not covary with other dimensions. The children's operational stage was assessed by a number conservation test. All children successfully judged relative numerosity when number covaried with length or with density, but only concrete operational children were successful when number did not covary with other dimensions. Preoperational children are thus not able to base judgments of relative numerosity solely on number.     

Number discrimination in 10‐month‐old infants

Two experiments investigated developmental changes in large number discrimination with visual‐spatial arrays. Previous studies found that 6‐month‐old infants were able to discriminate arrays that differ by a ratio of 1:2 but not 2:3. We found that by 10 months, infants were able to reliably discriminate 8 from 12 elements (2:3) but not 8 from 10 elements (4:5). Thus, number discrimination improves in precision during the first year, and these findings converge with studies using auditory stimuli.     

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).