Approximate Number System Task Research Articles

The approximate number system and mathematics achievement: it's complicated. A thorough investigation of different ANS measures and executive functions in mathematics achievement in children

ABSTRACT The ability to represent approximate numerical magnitudes is often referred to as the approximate number system (ANS) and has regularly been proposed as foundational to mathematics achievement. However, some argue that the relation between ANS acuity and mathematics achievement ceases to exist when controlling for domain-general cognitive abilities. The current debate in the literature on whether ANS acuity remains a predictor to mathematics after strict control is applied leads to the need to simultaneously investigate domain-specific and domain-general foundational abilities in different ages. 174 Children took part in two computerised ANS tasks, two executive function tasks, a verbal skills task, two intelligence subscales, and a mathematics achievement task (i.e. global, formal, and informal mathematics achievement). Results demonstrated that, when controlling for intelligence and visuospatial memory, the relation between ANS acuity and mathematics achievement ceased to exist, and that ANS acuity might only play a predictive role in early informal mathematics.

Numerosity Comparison, Estimation and Proportion Estimation Abilities May Predict Numeracy and Cognitive Reflection in Adults.

This study explores whether and how different tasks associated with approximate number system (ANS) ability are related to numeracy and cognitive reflection in adults. We conducted an online experiment using a sample of 300 Japanese adults aged 20–39. Participants were given three ANS tasks (numerosity comparison, numerosity estimation, and proportion estimation) as well as Rasch-based numeracy scale and cognitive reflection test, and we tested the correlation among the measures of these tasks. We explored the hypothesis that the typical measures used to gauge ANS ability, numerosity comparison and numerosity estimation may mediate different cognitive mechanisms in adults. We also introduced a task measuring proportion estimation, added because such estimation requires numerosity perception and the ability to map symbolic numerals. Our findings suggest that there is a weak, but significant correlation among the three ANS-related tasks. Moreover, there is a significant relationship between each of these measures and the numeracy and CRT score, suggesting that the ANS-related ability may be associated with higher cognitive abilities such as numeracy and cognitive reflection. In addition, we found that performances on the numerosity and proportion estimation are more clearly related to CRT score than the numerosity comparison task.

Approximate Number Sense in Students With Severe Hearing Loss: A Modality-Neutral Cognitive Ability.

The Approximate Number System (ANS) allows humans and non-human animals to estimate large quantities without counting. It is most commonly studied in visual contexts (i.e., with displays containing different numbers of dots), although the ANS may operate on all approximate quantities regardless of modality (e.g., estimating the number of a series of auditory tones). Previous research has shown that there is a link between ANS and mathematics abilities, and that this link is resilient to differences in visual experience (Kanjlia et al., 2018). However, little is known about the function of the ANS and its relationship to mathematics abilities in the absence of other types of sensory input. Here, we investigated the acuity of the ANS and its relationship with mathematics abilities in a group of students from the Sichuan Province in China, half of whom were deaf. We found, consistent with previous research, that ANS acuity improves with age. We found that mathematics ability was predicted by Non-verbal IQ and Inhibitory Control, but not visual working memory capacity or Attention Network efficiencies. Even above and beyond these predictors, ANS ability still accounted for unique variance in mathematics ability. Notably, there was no interaction with hearing, which indicates that the role played by the ANS in explaining mathematics competence is not modulated by hearing capacity. Finally, we found that age, Non-verbal IQ and Visual Working Memory capacity were predictive of ANS performance when controlling for other factors. In fact, although students with hearing loss performed slightly worse than students with normal hearing on the ANS task, hearing was no longer significantly predictive of ANS performance once other factors were taken into account. These results indicate that the ANS is able to develop at a consistent pace with other cognitive abilities in the absence of auditory experience, and that its relationship with mathematics ability is not contingent on sensory input from hearing.

Predictors of mathematics in primary school: Magnitude comparison, verbal and spatial working memory measures

We determined the relative importance of the so-called approximate number system (ANS), symbolic number comparison (SNC) and verbal and spatial short-term and working memory (WM) capacity for mathematics achievement in 1,254 Grade 2, 4 and 6 children. The large sample size assured high power and low false report probability and allowed us to determine effect sizes precisely. We used reading decoding as a control outcome measure to test whether findings were specific to mathematics. Bayesian analysis allowed us to provide support for both null and alternative hypotheses. We found very weak zero-order correlations between ANS measures and math achievement. These correlations were not specific to mathematics, became non-significant once intelligence was considered and ANS measures were not selected as predictors of math by regression models. In contrast, overall SNC accuracy and spatial WM measures were reliable and mostly specific predictors of math achievement. Verbal short-term and WM and SNC reaction time were predictors of both reading and math achievement. We conclude that ANS tasks are not suitable as measures of math development in school-age populations. In contrast, all other cognitive functions we studied are promising markers of mathematics development.

Approximate number system task performance: Associations with domain-general and domain-specific cognitive skills in young children

We investigated the associations between young children’s domain-general executive functioning (EF) skills and domain-specific spontaneous focusing on number (SFON) tendencies and their performance on an approximate number system (ANS) task, paying particular attention to variations in associations across different trial types with either congruent or incongruent non-numerical continuous visual cues. We found that children’s EF skills were strongly related to their performance on ANS task trials in which continuous visual cues were incongruent with numerosity. Novel to the current study, we found that children’s SFON tendencies were specifically related to their performance on ANS task trials in which continuous visual cues were congruent with numerosity. Children’s performance on ANS task trials in which children can use both congruent numerical and non-numerical continuous visual cues to approximate large quantities may be related to their unprompted tendency to focus on number in their early environment when there are not salient distractors present. On the other hand, children’s performance on incongruent ANS trials may be less a function of number-specific knowledge but more of children’s domain-general ability to inhibit salient but conflicting or irrelevant stimuli. Importantly, these effects held even when accounting for global math achievement and children’s cardinality knowledge. Overall, results support the consideration of both domain-specific and domain-general cognitive factors in developmental models of children’s early ability to attend to numerosity and provide a possible means for reconciling previous conflicting research findings.

Deficits in Approximate Number System Acuity and Mathematical Abilities in 6.5-Year-Old Children Born Extremely Preterm.

Preterm children are at increased risk for poor academic achievement, especially in math. In the present study, we examined whether preterm children differ from term-born children in their intuitive sense of number that relies on an unlearned, approximate number system (ANS) and whether there is a link between preterm children’s ANS acuity and their math abilities. To this end, 6.5-year-old extremely preterm (i.e., <27 weeks gestation, n = 82) and term-born children (n = 89) completed a non-symbolic number comparison (ANS acuity) task and a standardized math test. We found that extremely preterm children had significantly lower ANS acuity than term-born children and that these differences could not be fully explained by differences in verbal IQ, perceptual reasoning skills, working memory, or attention. Differences in ANS acuity persisted even when demands on visuo-spatial skills and attention were reduced in the ANS task. Finally, we found that ANS acuity and math ability are linked in extremely preterm children, similar to previous results from term-born children. These results suggest that deficits in the ANS may be at least partly responsible for the deficits in math abilities often observed in extremely preterm children.

Is the ANS linked to mathematics performance?

Leibovich et al. argue persuasively that researchers should not assume that approximate number system (ANS) tasks harness an innate sense of number. However, some studies have reported a causal link between ANS tasks and mathematics performance, implicating the ANS in the development of numerical skills. Here we report a p-curve analysis, which indicates that these experimental studies do not contain evidential value.

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.

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.

Children's Approximate Number System in Haptic Modality.

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

Methodological aspects to be considered when measuring the approximate number system (ANS) - a research review.

According to a dominant view, the approximate number system (ANS) is the foundation of symbolic math abilities. Due to the importance of math abilities for education and career, a lot of research focuses on the investigation of the ANS and its relationship with math performance. However, the results are inconsistent. This might be caused by studies differing greatly regarding the operationalization of the ANS (i.e., tasks, dependent variables). Moreover, many methodological aspects vary from one study to the next. In the present review, we discuss commonly used ANS tasks and dependent variables regarding their theoretical foundation and psychometric features. We argue that the inconsistent findings concerning the relationship between ANS acuity and math performance may be partially explained by differences in reliability. Furthermore, this review summarizes methodological aspects of ANS tasks having important impacts on the results, including stimulus range, visual controls, presentation duration of the stimuli and feedback. Based on this review, we give methodological recommendations on how to assess the ANS most reliably and most validly. All important methodological aspects to be considered when designing an ANS task or comparing results of different studies are summarized in two practical checklists.

Early numerical foundations of young children’s mathematical development

This study focused on the relative contributions of the acuity of the approximate number system (ANS) and knowledge of quantitative symbols to young children’s early mathematical learning. At the beginning of preschool, 191 children (Mage=46months) were administered tasks that assessed ANS acuity and explicit knowledge of the cardinal values represented by number words, and their mathematics achievement was assessed at the end of the school year. Children’s executive functions, intelligence, and preliteracy skills and their parents’ educational levels were also assessed and served as covariates. Both the ANS and cardinality tasks were significant predictors of end-of-year mathematics achievement with and without control of the covariates. As simultaneous predictors and with control of the covariates, cardinality remained significantly related to mathematics achievement, but ANS acuity did not. Mediation analyses revealed that the relation between ANS acuity and mathematics achievement was fully mediated by cardinality, suggesting that the ANS may facilitate children’s explicit understanding of cardinal value and in this way may indirectly influence early mathematical learning.

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.

Compromised approximate number system acuity in extremely preterm school‐aged children

The aim of this study was to compare the approximate number system acuity in children born extremely preterm aged 6 years 6 months and typically developing, age-matched peers. This population-based follow-up study included 65 children born before 27 gestational weeks (35 males, 30 females; mean gestational age 25.4 wks [SD 1.1 wk]; mean birthweight 789 g [SD 158 g]) and 47 typically developing children (24 females, 23 males) at the age of 6 years 6 months. A battery of cognitive tests was administered, including a computerized test for measuring approximate number system acuity and tests for general cognition, working memory, processing speed, and visual attention. Approximate number system outcome measures were means of Weber fraction (w) values and response time in milliseconds. The 43 extremely preterm children in whom usable data were obtained performed significantly worse than the typically developing children on the approximate number system task (w=0.30 [SD 0.23] vs. 0.17 [SD 0.13]; p=0.003) and were significantly slower (response time=2934 ms [SD 1102 ms] vs 2376 ms [SD 310 ms]; p=0.002). The differences remained when adjusting for differences in other cognitive functions (p=0.03). Preterm birth has a negative impact on an individual's ability to rapidly approximate and compare numbers of visually presented items. This deficiency is thought to be a consequence of dorsal stream dysfunction. Future studies will investigate whether this deficiency is correlated with lower mathematical proficiency in this group of children.

Visual stimulus parameters seriously compromise the measurement of approximate number system acuity and comparative effects between adults and children.

It has been suggested that a simple non-symbolic magnitude comparison task is sufficient to measure the acuity of a putative Approximate Number System (ANS). A proposed measure of the ANS, the so-called “internal Weber fraction” (w), would provide a clear measure of ANS acuity. However, ANS studies have never presented adequate evidence that visual stimulus parameters did not compromise measurements of w to such extent that w is actually driven by visual instead of numerical processes. We therefore investigated this question by testing non-symbolic magnitude discrimination in seven-year-old children and adults. We manipulated/controlled visual parameters in a more stringent manner than usual. As a consequence of these controls, in some trials numerical cues correlated positively with number while in others they correlated negatively with number. This congruency effect strongly correlated with w, which means that congruency effects were probably driving effects in w. Consequently, in both adults and children congruency had a major impact on the fit of the model underlying the computation of w. Furthermore, children showed larger congruency effects than adults. This suggests that ANS tasks are seriously compromised by the visual stimulus parameters, which cannot be controlled. Hence, they are not pure measures of the ANS and some putative w or ratio effect differences between children and adults in previous ANS studies may be due to the differential influence of the visual stimulus parameters in children and adults. In addition, because the resolution of congruency effects relies on inhibitory (interference suppression) function, some previous ANS findings were probably influenced by the developmental state of inhibitory processes especially when comparing children with developmental dyscalculia and typically developing children.

Quantitative Deficits of Preschool Children at Risk for Mathematical Learning Disability

The study tested the hypothesis that acuity of the potentially inherent approximate number system (ANS) contributes to risk of mathematical learning disability (MLD). Sixty-eight (35 boys) preschoolers at risk for school failure were assessed on a battery of quantitative tasks, and on intelligence, executive control, preliteracy skills, and parental education. Mathematics achievement scores at the end of 1 year of preschool indicated that 34 of these children were at high risk for MLD. Relative to the 34 typically achieving children, the at risk children were less accurate on the ANS task, and a one standard deviation deficit on this task resulted in a 2.4-fold increase in the odds of MLD status. The at risk children also had a poor understanding of ordinal relations, and had slower learning of Arabic numerals, number words, and their cardinal values. Poor performance on these tasks resulted in 3.6- to 4.5-fold increases in the odds of MLD status. The results provide some support for the ANS hypothesis but also suggest these deficits are not the primary source of poor mathematics learning.