The effect of different methods to construct non-symbolic stimuli in numerosity estimation and comparison

Numerosity estimation and comparison tasks are often used to measure the acuity of the approximate number system (ANS), a mechanism which allows extracting numerosity from an array of dots independently from several visual cues (e.g. area extended by the dots). This idea is supported by studies showing that numerosity can be processed while these visual cues are controlled for. Different methods to construct dot arrays while controlling their visual cues have been proposed in the past. In this paper, these methods were contrasted in an estimation and a comparison task. The way of constructing the dot arrays had little impact on estimation. In contrast, in the comparison task, participants' performance was significantly influenced by the method that was used to construct the arrays of dots, indicating better performance when the visual cues of the dot arrays (partly) co-varied with numerosity. The present study therefore shows that estimates of ANS acuity derived from comparison tasks are inconsistent and dependent on how the stimuli are constructed. This makes it difficult to compare studies which utilised different methods to construct the dot arrays in numerosity comparison tasks. In addition, these results question the currently held view of the ANS as capable of robustly extracting numerosity independently from visual cues.     

Differences in the acuity of the Approximate Number System in adults: The effect of mathematical ability

It is largely admitted that processing numerosity relies on an innate Approximate Number System (ANS), and recent research consistently observed a relationship between ANS acuity and mathematical ability in childhood. However, studies assessing this relationship in adults led to contradictory results. In this study, adults with different levels of mathematical expertise performed two tasks on the same pairs of dot collections, based either on numerosity comparison or on cumulative area comparison. Number of dots and cumulative area were congruent in half of the stimuli, and incongruent in the other half. The results showed that adults with higher mathematical ability obtained lower Weber fractions in the numerical condition than participants with lower mathematical ability. Further, adults with lower mathematical ability were more affected by the interference of the continuous dimension in the numerical comparison task, whereas conversely higher-expertise adults showed stronger interference of the numerical dimension in the continuous comparison task. Finally, ANS acuity correlated with arithmetic performance. Taken together, the data suggest that individual differences in ANS acuity subsist in adulthood, and that they are related to mathematical ability.     

Concurrent validity of approximate number sense tasks in adults and children

Reasoning with non-symbolic numerosities is suggested to be rooted in the Approximate Number System (ANS) and evidence pointing to a relationship between the acuity of this system and mathematics is available. In order to use the acuity of this ANS as a screening instrument to detect future math problems, it is important to model ANS acuity over development. However, whether ANS acuity and its development have been described accurately can be questioned. Namely, different tasks were used to examine the developmental trajectory of ANS acuity and studies comparing performances on these different tasks are scarce. In the present study, we examined whether different tasks designed to measure the acuity of the ANS are comparable and lead to related ANS acuity measures (i.e., the concurrent validity of these tasks). We contrasted the change detection task, which is used in infants, with tasks that are more commonly used in older children and adults (i.e., comparison and same-different tasks). Together, our results suggest that ANS acuity measures obtained with different tasks are not related. This poses serious problems for the comparison of ANS acuity measures derived from different tasks and thus for the establishment of the developmental trajectory of ANS acuity.     

The Role of Non-Numerical Stimulus Features in Approximate Number System Training in Preschoolers from Low-Income Homes

Recent findings have suggested that adults’ and children’s approximate number system (ANS) acuity may be malleable through training, but research on ANS acuity has largely been conducted with adults and children who are from middle- to high-income homes. We conducted 2 experiments to test the malleability of ANS acuity in preschool-aged children from low-income homes and to test how non-numerical stimulus features affected performance. In Experiment 1, mixed-effects models indicated that children significantly improved their ratio achieved across training. Children’s change in probability of responding correctly across sessions was qualified by an interaction with surface area features of the arrays such that children improved their probability of answering correctly across sessions on trials in which numerosity conflicted with the total surface area of object sets significantly more than on trials in which total surface area positively correlated with numerosity. In Experiment 2, we found that children who completed ANS acuity training performed better on an ANS acuity task compared with children in a control group, but they only did so on ANS acuity trials in which numerosity conflicted with the total surface area of object sets. These findings suggest that training affects ANS acuity in children from low-income homes by fostering an ability to focus on numerosity in the face of conflicting non-numerical stimulus features.     

学前儿童近似数量系统敏锐度与符号数学能力的关系

选取杭州市122名学前儿童(3~6岁)为被试,以点数比较任务及点数异同任务测量幼儿的近似数量系统敏锐度,以数数测验、基数测验、符号数字知识测验及简单计算来测量幼儿的符号数学能力,以此考察学前儿童近似数量系统敏锐度的发展及与符号数学能力的关系。结果发现:(1)随年龄增长,学前儿童的近似数量加工的敏锐度逐渐提高;(2)点数比较任务与点数异同任务均适合测量学前儿童近似数量系统敏锐度,但儿童完成点数比较任务的正确率要高于点数异同任务的正确率;(3)在抑制控制、短时记忆、工作记忆和言语测验成绩被控制后,根据点数比较任务计算的韦伯系数能显著预测学前儿童的基数和符号数字知识测验分数,总正确率能显著预测学前儿童的数数、基数、符号数字知识测验分数;(4)点数异同任务中只有点数不同试次下的正确率能显著预测学前儿童的符号数字知识测验分数。     

Nonsymbolic numerical magnitude comparison: reliability and validity of different task variants and outcome measures, and their relationship to arithmetic achievement in adults

The numerical ratio effect (NRE) and the Weber fraction (w) are common metrics of the precision of the approximate numbers sense (ANS), a cognitive mechanism suggested to play a role in the development of numerical and arithmetic skills. The task most commonly used to measure the precision of the ANS is the numerical comparison task. Multiple variants of this task have been employed yet it is currently unclear how these affect metrics of ANS acuity, and how these relate to arithmetic achievement. The present study investigates the reliability, validity and relationship to standardized measures of arithmetic fluency of the NRE and w elicited by three variants of the nonsymbolic number comparison task. Results reveal that the strengths of the NRE and w differ between task variants. Moreover, the reliability and validity of the reaction time NRE and the w were generally significant across task variants, although reliability was stronger for w. None of the task variants revealed a correlation between ANS metrics and arithmetic fluency in adults. These results reveal important consistencies across nonsymbolic number comparison tasks, indicating a shared cognitive foundation. However, the relationship between ANS acuity and arithmetic performance remains unclear.     

小学生近似数量系统敏锐度的发展趋势及其与数学能力的关系：抑制控制的中介作用

采用测量法和问卷法考察了172名小学生近似数量系统敏锐度的发展,以及抑制控制在近似数量系统敏锐度与数学能力关系中的中介效应。结果表明：(1)随着年龄增长,小学生的近似数量系统敏锐度逐渐提高;(2)近似数量系统敏锐度(负相关序列中的韦伯系数)和抑制控制均能显著正向预测小学生的数学能力;(3)抑制控制在小学生近似数量系统敏锐度(负相关序列中的韦伯系数)与数学能力的关系中起部分中介作用。     

Comparing performance in discrete and continuous comparison tasks

The approximate number system (ANS) theory suggests that all magnitudes, discrete (i.e., number of items) or continuous (i.e., size, density, etc.), are processed by a shared system and comply with Weber's law. The current study reexamined this notion by comparing performance in discrete (comparing numerosities of dot arrays) and continuous (comparisons of area of squares) tasks. We found that: (a) threshold of discrimination was higher for continuous than for discrete comparisons; (b) while performance in the discrete task complied with Weber's law, performance in the continuous task violated it; and (c) performance in the discrete task was influenced by continuous properties (e.g., dot density, dot cumulative area) of the dot array that were not predictive of numerosities or task relevant. Therefore, we propose that the magnitude processing system (MPS) is actually divided into separate (yet interactive) systems for discrete and continuous magnitude processing. Further subdivisions are discussed. We argue that cooperation between these systems results in a holistic comparison of magnitudes, one that takes into account continuous properties in addition to numerosities. Considering the MPS as two systems opens the door to new and important questions that shed light on both normal and impaired development of the numerical system.     

Preschool acuity of the approximate number system correlates with school math ability

Previous research shows a correlation between individual differences in people's school math abilities and the accuracy with which they rapidly and nonverbally approximate how many items are in a scene. This finding is surprising because the Approximate Number System (ANS) underlying numerical estimation is shared with infants and with non-human animals who never acquire formal mathematics. However, it remains unclear whether the link between individual differences in math ability and the ANS depends on formal mathematics instruction. Earlier studies demonstrating this link tested participants only after they had received many years of mathematics education, or assessed participants' ANS acuity using tasks that required additional symbolic or arithmetic processing similar to that required in standardized math tests. To ask whether the ANS and math ability are linked early in life, we measured the ANS acuity of 200 3- to 5-year-old children using a task that did not also require symbol use or arithmetic calculation. We also measured children's math ability and vocabulary size prior to the onset of formal math instruction. We found that children's ANS acuity correlated with their math ability, even when age and verbal skills were controlled for. These findings provide evidence for a relationship between the primitive sense of number and math ability starting early in life.     

Neural coding partially accounts for the relationship between children’s number-line estimation and number comparison performance

Numerical comparison is a primary measure of the acuity of children’s approximate number system. Approximate number system acuity is associated with key developmental outcomes such as symbolic number skill, standardized test scores, and even employment outcomes (Halberda, Mazzocco, & Feigenson, 2008; Parsons & Bynner, 1997). We examined the relation between children’s performance on the numerical comparison task and the number-line estimation task. It is important to characterize the relation between tasks to develop mathematics interventions that lead to transfer across tasks. We found that number-line performance was significantly predicted by nonsymbolic comparison performance for participants ranging in age from 5 to 8 years. We also evaluated, using a computational model, whether the relation between the 2 tasks could be adequately explained based on known neural correlates of number perception. Data from humans and nonhuman primates characterized neural activity corresponding to the perception of numerosities. Results of behavioral experimentation and computational modeling suggested that though neural coding of numbers predicted a correlation in participants’ performance on the 2 tasks, it could not account for all the variability in the human data. This finding was interpreted as being consistent with accounts of number-line estimation in which number-line estimation does not rely solely on participants’ numerical perception.     

Numerical distance effect size is a poor metric of approximate number system acuity

Individual differences in the ability to compare and evaluate nonsymbolic numerical magnitudes—approximate number system (ANS) acuity—are emerging as an important predictor in many research areas. Unfortunately, recent empirical studies have called into question whether a historically common ANS-acuity metric—the size of the numerical distance effect (NDE size)—is an effective measure of ANS acuity. NDE size has been shown to frequently yield divergent results from other ANS-acuity metrics. Given these concerns and the measure’s past popularity, it behooves us to question whether the use of NDE size as an ANS-acuity metric is theoretically supported. This study seeks to address this gap in the literature by using modeling to test the basic assumption underpinning use of NDE size as an ANS-acuity metric: that larger NDE size indicates poorer ANS acuity. This assumption did not hold up under test. Results demonstrate that the theoretically ideal relationship between NDE size and ANS acuity is not linear, but rather resembles an inverted J-shaped distribution, with the inflection points varying based on precise NDE task methodology. Thus, depending on specific methodology and the distribution of ANS acuity in the tested population, positive, negative, or null correlations between NDE size and ANS acuity could be predicted. Moreover, peak NDE sizes would be found for near-average ANS acuities on common NDE tasks. This indicates that NDE size has limited and inconsistent utility as an ANS-acuity metric. Past results should be interpreted on a case-by-case basis, considering both specifics of the NDE task and expected ANS acuity of the sampled population.     

Measuring the approximate number system

Recent theories in numerical cognition propose the existence of an approximate number system (ANS) that supports the representation and processing of quantity information without symbols. It has been claimed that this system is present in infants, children, and adults, that it supports learning of symbolic mathematics, and that correctly harnessing the system during tuition will lead to educational benefits. Various experimental tasks have been used to investigate individuals' ANSs, and it has been assumed that these tasks measure the same system. We tested the relationship across six measures of the ANS. Surprisingly, despite typical performance on each task, adult participants' performances across the tasks were not correlated, and estimates of the acuity of individuals' ANSs from different tasks were unrelated. These results highlight methodological issues with tasks typically used to measure the ANS and call into question claims that individuals use a single system to complete all these tasks.     

Visual Field asymmetries for verbal and dot localization tasks in monolingual and bilingual subjects

Visual Field asymmetries for verbal and dot localization tasks were examined in monolingual and bilingual subjects. Consistent right-visual-field advantages were found for verbal material in all groups, although bilingual subjects showed a reduced laterality for their second language in comparison with their native language, Monolingual subjects displayed left-visual-field advantages on the dot localization task, but no consistent asymmetries were shown by the bilingual subjects. The overall pattern of results is consistent with left-hemisphere involvement for the processing of verbal material, but the heterogeneity of performance on the dot localization task suggests that processing of such a task may be influenced by subjects' linguistic backgrounds.     

Measuring the approximate number system

Recent theories in numerical cognition propose the existence of an approximate number system (ANS) that supports the representation and processing of quantity information without symbols. It has been claimed that this system is present in infants, children, and adults, that it supports learning of symbolic mathematics, and that correctly harnessing the system during tuition will lead to educational benefits. Various experimental tasks have been used to investigate individuals' ANSs, and it has been assumed that these tasks measure the same system. We tested the relationship across six measures of the ANS. Surprisingly, despite typical performance on each task, adult participants' performances across the tasks were not correlated, and estimates of the acuity of individuals' ANSs from different tasks were unrelated. These results highlight methodological issues with tasks typically used to measure the ANS and call into question claims that individuals use a single system to complete all these tasks.     

Cultural differences in visual attention: Implications for distraction processing

We investigated differences between participants of East Asian and Western descent in attention to and implicit memory for irrelevant words which participants were instructed to ignore while completing a target task (a Stroop Task in Experiment 1 and a 1‐back task on pictures in Experiment 2 ). Implicit memory was measured using two conceptual priming tasks (category generation in Experiment 1 and general knowledge in Experiment 2 ). Participants of East Asian descent showed reliable implicit memory for previous distractors relative to those of Western descent with no evidence of differences on target task performance. We also found differences in a Corsi Block spatial memory task in both studies, with superior performance by the East Asian group. Our findings suggest that cultural differences in attention extend to task‐irrelevant background information, and demonstrate for the first time that such information can boost performance when it becomes relevant on a subsequent task.     

Acuity of the approximate number system and preschoolers’ quantitative development

The study assessed the relations among acuity of the inherent approximate number system (ANS), performance on measures of symbolic quantitative knowledge, and mathematics achievement for a sample of 138 (64 boys) preschoolers. The Weber fraction (a measure of ANS acuity) and associated task accuracy were significantly correlated with mathematics achievement following one year of preschool, and predicted performance on measures of children's explicit knowledge of Arabic numerals, number words, and cardinal value, controlling for age, sex, parental education, intelligence, executive control, and preliteracy knowledge. The relation between ANS acuity, as measured by the Weber fraction and task accuracy, and mathematics achievement was fully mediated by children's performance on the symbolic quantitative tasks, with knowledge of cardinal value emerging as a particularly important mediator. The overall pattern suggests that ANS acuity facilitates the early learning of symbolic quantitative knowledge and indirectly influences mathematics achievement through this knowledge.     

The Acuity for Numerosity (but Not Continuous Magnitude) Discrimination Correlates with Quantitative Problem Solving but Not Routinized Arithmetic

Approximate number sense (ANS) acuity refers to the ability to non-symbolically recognize, estimate and operate upon large numerosities. ANS acuity has been reported to be correlated with math achievement in children and adolescents. However, reports of this relationship in adults have been inconsistent. The present study aimed to resolve the inconsistency in the relationship between adults’ ANS acuity and math achievement by contrasting between different kinds of mathematical problem solving. We hypothesized that the correlation between ANS acuity mathematical performance would be stronger when deep quantitative processing is required during problem solving. In Experiment 1, ANS acuity was correlated with Mathematical Reasoning but not Directed Calculation performance. In Experiment 2, ANS acuity was correlated with Two-digit Subtraction (but not Addition) performance only when Regrouping (i.e., borrowing) was required. The results from two experiments demonstrated that ANS acuity was correlated with mathematical performance only when problem solving involved effortful, quantitative processing that goes beyond automatized, routinized arithmetic. In addition, ANS acuity was distinguishable from Area acuity regarding its unique relationship with math achievement, which was unconfounded by the influence of demographic variables and fluid intelligence. Overall, the present results help resolve the inconsistency in previous reports of the correlation between ANS acuity and math achievement in adults.     

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.     

Children’s representation of abstract relations in relational/array match-to-sample tasks

Five experiments compared preschool children’s performance to that of adults and of non-human animals on match to sample tasks involving 2-item or 16-item arrays that varied according to their composition of same or different items (Array Match-to-Sample, AMTS). They establish that, like non-human animals in most studies, 3- and 4-year-olds fail 2-item AMTS (the classic relational match to sample task introduced into the literature by Premack, 1983), and that robust success is not observed until age 6. They also establish that 3-year-olds, like non-human animal species, succeed only when they are able to encode stimuli in terms of entropy, a property of an array (namely its internal variability), rather than relations among the individuals in the array (same vs. different), whereas adults solve both 2-item and 16-item AMTS on the basis of the relations same and different. As in the case of non-human animals, the acuity of 3- and 4-year-olds’ representation of entropy is insufficient to solve the 2-item same-different AMTS task. At age 4, behavior begins to contrast with that of non-human species. On 16-item AMTS, a subgroup of 4-year-olds induce a categorical rule matching all-same arrays to all-same arrays, while matching other arrays (mixed arrays of same and different items) to all-different arrays. These children tend to justify their choices using the words “same” and “different.” By age 4 a number of our participants succeed at 2-item AMTS, also justifying their choices by explicit verbal appeals using words for same and different. Taken together these results suggest that the recruitment of the relational representations corresponding to the meaning of these words contributes to the better performance over the preschool years at solving array match-to-sample tasks.     

心算形式对不同近似数量系统敏锐度儿童心算表现的影响

选取112名二年级小学生,以点阵比较任务测量近似数量系统敏锐度,同时以工作记忆测验成绩为协变量,探究了不同心算形式（视算、读算）对不同近似数量系统敏锐度儿童心算表现的潜在影响。结果显示：（1）心算形式显著影响心算的正确率,读算形式下儿童的心算表现最好;（2）控制工作记忆影响后,心算形式与近似数量系统敏锐分组均对心算正确率影响显著。总体来讲,读算可能是提高小学儿童简单心算表现的有效形式,并能提高低近似数量系统敏锐度儿童的心算表现。