Approximate Number System Precision Research Articles

Directionality in the interrelations between approximate number, verbal number, and mathematics in preschool-aged children.

A fundamental question in numerical development concerns the directional relation between an early-emerging non-verbal approximate number system (ANS) and culturally acquired verbal number and mathematics knowledge. Using path models on longitudinal data collected in preschool children (Mage =3.86 years; N=216; 99 males; 80.8% White; 10.8% Multiracial, 3.8% Latino; 1.9% Black; collected 2013-2017) over 1 year, this study showed that earlier verbal number knowledge was associated with later ANS precision (average β=.32), even after controlling for baseline differences in numerical, general cognitive, and language abilities. In contrast, earlier ANS precision was not associated with later verbal number knowledge (β=-.07) or mathematics abilities (average β=.10). These results suggest that learning about verbal numbers is associated with a sharpening of pre-existing non-verbal numerical abilities.

Emergence of the Link Between the Approximate Number System and Symbolic Math Ability.

Experimentally manipulating Approximate Number System (ANS) precision has been found to influence children's subsequent symbolic math performance. Here in three experiments (N=160; 81 girls; 3-5year old) we replicated this effect and examined its duration and developmental trajectory. We found that modulation of 5-year-olds' ANS precision continued to affect their symbolic math performance after a 30-min delay. Furthermore, our cross-sectional investigation revealed that children 4.5years and older experienced a significant transfer effect of ANS manipulation on math performance, whereas younger children showed no such transfer, despite experiencing significant changes in ANS precision. These findings support the existence of a causal link between nonverbal numerical approximation and symbolic math performance that first emerges during the preschool years.

The relationship between children’s approximate number certainty and symbolic mathematics

Why do some children excel in mathematics while others struggle? A large body of work has shown positive correlations between children’s Approximate Number System (ANS) and school-taught symbolic mathematical skills, but the mechanism explaining this link remains unknown. One potential mediator of this relationship might be children’s numerical metacognition: children’s ability to evaluate how sure or unsure they are in understanding and manipulating numbers. While previous work has shown that children’s math abilities are uniquely predicted by symbolic numerical metacognition, we focus on the extent to which children’s non-symbolic/ANS numerical metacognition, in particular sensitivity to certainty, might be predictive of math ability, and might mediate the relationship between the ANS and symbolic math. A total of 72 children aged 4–6 years completed measures of ANS precision, ANS metacognition sensitivity, and the Test of Early Mathematical Ability (TEMA-3). Our results replicate many established findings in the literature, including the correlation between ANS precision and the TEMA-3, particularly on the Informal subtype questions. However, we did not find that ANS metacognition sensitivity was related to TEMA-3 performance, nor that it mediated the relationship between the ANS and the TEMA-3. These findings suggest either that metacognitive calibration may play a larger role than metacognitive sensitivity, or that metacognitive differences in the non-symbolic number perception do not robustly contribute to symbolic math performance.

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

ABSTRACTResearch 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.

Children's intuitive sense of number develops independently of their perception of area, density, length, and time.

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

Scale and the evolutionarily based approximate number system: an exploratory study

ABSTRACTCrosscutting concepts such as scale, proportion, and quantity are recognised by U.S. science standards as a potential vehicle for students to integrate their scientific and mathematical knowledge; yet, U.S. students and adults trail their international peers in scale and measurement estimation. Culturally based knowledge of scale such as measurement units may be built on evolutionarily-based systems of number such as the approximate number system (ANS), which processes approximate representations of numerical magnitude. ANS is related to mathematical achievement in pre-school and early elementary students, but there is little research on ANS among older students or in science-related areas such as scale. Here, we investigate the relationship between ANS precision in public school U.S. seventh graders and their accuracy estimating the length of standard units of measurement in SI and U.S. customary units. We also explored the relationship between ANS and science and mathematics achievement. Accuracy estimating the metre was positively and significantly related to ANS precision. Mathematics achievement, science achievement, and accuracy estimating other units were not significantly related to ANS. We thus suggest that ANS precision may be related to mathematics understanding beyond arithmetic, beyond the early school years, and to the crosscutting concepts of scale, proportion, and quantity.

Approximate number sense correlates with math performance in gifted adolescents

Nonhuman animals, human infants, and human adults all share an Approximate Number System (ANS) that allows them to imprecisely represent number without counting. Among humans, people differ in the precision of their ANS representations, and these individual differences have been shown to correlate with symbolic mathematics performance in both children and adults. For example, children with specific math impairment (dyscalculia) have notably poor ANS precision. However, it remains unknown whether ANS precision contributes to individual differences only in populations of people with lower or average mathematical abilities, or whether this link also is present in people who excel in math. Here we tested non-symbolic numerical approximation in 13- to 16-year old gifted children enrolled in a program for talented adolescents (the Center for Talented Youth). We found that in this high achieving population, ANS precision significantly correlated with performance on the symbolic math portion of two common standardized tests (SAT and ACT) that typically are administered to much older students. This relationship was robust even when controlling for age, verbal performance, and reaction times in the approximate number task. These results suggest that the Approximate Number System is linked to symbolic math performance even at the top levels of math performance.

Individual differences in nonverbal number skills predict math anxiety

Math anxiety (MA) involves negative affect and tension when solving mathematical problems, with potentially life-long consequences. MA has been hypothesized to be a consequence of negative learning experiences and cognitive predispositions. Recent research indicates genetic and neurophysiological links, suggesting that MA stems from a basic level deficiency in symbolic numerical processing. However, the contribution of evolutionary ancient purely nonverbal processes is not fully understood. Here we show that the roots of MA may go beyond symbolic numbers. We demonstrate that MA is correlated with precision of the Approximate Number System (ANS). Individuals high in MA have poorer ANS functioning than those low in MA. This correlation remains significant when controlling for other forms of anxiety and for cognitive variables. We show that MA mediates the documented correlation between ANS precision and math performance, both with ANS and with math performance as independent variable in the mediation model. In light of our results, we discuss the possibility that MA has deep roots, stemming from a non-verbal number processing deficiency. The findings provide new evidence advancing the theoretical understanding of the developmental etiology of MA.

Better together: Multiple lines of evidence for a link between approximate and exact number representations: A reply to Merkley, Matejko, and Ansari

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

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

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

Cognitive abilities that mediate SES’s effect on elementary mathematics learning: The Uruguayan tablet-based intervention

In unequal societies the effectiveness of formal education depends on the socioeconomic status (SES) of students. Studies have shown that poverty affects the development of the brain in ways that might compromise future learning, thus increasing the differences between groups with different SES. Interest is growing in the development of tools that might change this state of affairs. This article presents a tablet-based study aimed at determining the cognitive abilities related to primary school children’s math learning. The study followed the students’ changes during a short intervention, the purpose of which was to improve students’ performance of some of the core components of mathematical cognition; in particular, of the approximate number system (ANS), a system that supports one’s ability to estimate quantities and to compare time intervals. The article presents the study’s characteristics and shows how the variables that were evaluated—ANS precision, time discrimination accuracy, digit span, and mathematical achievement—depend on SES. We employ multiple regressions to show that the variance in mathematics performance attributed to SES can be explained by differences in underlying cognitive factors. The study also indicates that those students of low-SES schools who participated in more tablet activities increased their performance more than students who did fewer activities. Although the intervention’s initial objective was to influence mathematical development and the study is not a randomized double-blind study, we argue that training the ANS can have positive effects in mathematics learning, and that this might benefit children living in low-SES contexts more than those in the general population, perhaps because of the former’s initially low levels of performance in school mathematics.

Changing the precision of preschoolers’ approximate number system representations changes their symbolic math performance

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

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.

Measuring the approximate number system in children: Exploring the relationships among different tasks

Research has demonstrated that children and adults have an Approximate Number System (ANS) which allows individuals to represent and manipulate the representations of the approximate number of items within a set. It has been suggested that individual differences in the precision of the ANS are related to individual differences in mathematics achievement. One difficulty with understanding the role of the ANS, however, is a lack of consistency across studies in tasks used to measure ANS performance. Researchers have used symbolic or nonsymbolic comparison and addition tasks with varying types and sizes of stimuli. Recent studies with adult participants have shown that performance on different ANS tasks is unrelated. Across two studies we demonstrate that, in contrast to adults, children's performance across different ANS tasks, such as symbolic and nonsymbolic comparison or approximate addition, is related. These findings suggest that there are differences across development in the extent to which performance on nonsymbolic and symbolic tasks reflects ANS precision.

Intuitive sense of number correlates with math scores on college-entrance examination

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

Preschoolers' Precision of the Approximate Number System Predicts Later School Mathematics Performance

The Approximate Number System (ANS) is a primitive mental system of nonverbal representations that supports an intuitive sense of number in human adults, children, infants, and other animal species. The numerical approximations produced by the ANS are characteristically imprecise and, in humans, this precision gradually improves from infancy to adulthood. Throughout development, wide ranging individual differences in ANS precision are evident within age groups. These individual differences have been linked to formal mathematics outcomes, based on concurrent, retrospective, or short-term longitudinal correlations observed during the school age years. However, it remains unknown whether this approximate number sense actually serves as a foundation for these school mathematics abilities. Here we show that ANS precision measured at preschool, prior to formal instruction in mathematics, selectively predicts performance on school mathematics at 6 years of age. In contrast, ANS precision does not predict non-numerical cognitive abilities. To our knowledge, these results provide the first evidence for early ANS precision, measured before the onset of formal education, predicting later mathematical abilities.

Impaired Acuity of the Approximate Number System Underlies Mathematical Learning Disability (Dyscalculia)

Many children have significant mathematical learning disabilities (MLD, or dyscalculia) despite adequate schooling. The current study hypothesizes that MLD partly results from a deficiency in the Approximate Number System (ANS) that supports nonverbal numerical representations across species and throughout development. In this study of 71 ninth graders, it is shown that students with MLD have significantly poorer ANS precision than students in all other mathematics achievement groups (low, typically, and high achieving), as measured by psychophysical assessments of ANS acuity (w) and of the mappings between ANS representations and number words (cv). This relation persists even when controlling for domain-general abilities. Furthermore, this ANS precision does not differentiate low-achieving from typically achieving students, suggesting an ANS deficit that is specific to MLD.