Nonsymbolic Numerical Magnitude Processing Research Articles

Different impacts of long-term abacus training on symbolic and non-symbolic numerical magnitude processing in children

Abacus-based mental calculation (AMC) has been shown to be effective in promoting math ability in children. Given that AMC relies on a visuospatial strategy to perform rapid and precise arithmetic, previous studies mostly focused on the promotion of AMC training on arithmetic ability and mathematical visual-spatial ability, as well as its transfer of advanced cognitive ability. However, little attention has been given to its impact on basic numerical comparison ability. Here, we aim to examine whether and how long-term AMC training impacts symbolic and non-symbolic numerical comparisons. The distance effect (DE) was utilized as a marker, indicating that the comparison between two numbers becomes faster as their numerical distance enlarges. In the current study, forty-one children matched for age and sex were recruited at primary school entry and randomly assigned to the AMC group and the control group. After three years of training, the event-related potential (ERP) recording technique was used to explore the temporal dynamics of number comparison, of which tasks were given in symbolic (Arabic number) or non-symbolic (dot array) format. In the symbolic task, the children in the AMC group showed a smaller DE than those in the control group. Two ERP components, N1 and P2p, located in parietal areas (PO7, PO8) were selected as neural markers of numerical processing. Both groups showed DE in the P2p component in both tasks, but only the children in the AMC group showed DE in the N1 component in the non-symbolic task. In addition, the DE size calculated from reaction times and ERP amplitudes was correlated with higher cognitive capacities, such as coding ability. Taken together, the present results provide evidence that long-term AMC training may be beneficial for numerical processing in children, which may be associated with neurocognitive indices of parietal brain regions.

Nonsymbolic and Symbolic Numerical Magnitude Processing in the Brazilian Children with Mathematics Difficulties.

ABSTRACTIt is still debated if the main deficit in mathematical difficulties (MD) is nonsymbolic or symbolic numerical magnitude processing.Objectives:In the present study, our main goal was to investigate nonsymbolic and symbolic numerical magnitude processing in MD and the relationship between these abilities and arithmetic.Methods:The Brazilian school-age children with MD completed a nonsymbolic and a symbolic numerical magnitude comparison task and an arithmetic task. We compared their performance with a group of children with typical achievement (TA) and investigated the association between numerical magnitude processing and arithmetic with a series of regression analyses.Results:Results indicated that children with MD had low performance in the nonsymbolic numerical magnitude comparison task. Performance in both nonsymbolic and symbolic numerical magnitude comparison tasks predicted arithmetic abilities in children with TA, but not in children with MD.Conclusions:These results indicate that children with MD have difficulties in nonsymbolic numerical magnitude processing, and do not engage basic numerical magnitude representations to solve arithmetic.

Identifying Children with Persistent Developmental Dyscalculia from a 2‐min Test of Symbolic and Nonsymbolic Numerical Magnitude Processing

ABSTRACTDevelopmental dyscalculia (DD) is a mathematical learning disability that occurs in around 5%–7% of the population. At present, there are only a handful of screening tools to identify children that might be at risk of developing DD. The present study evaluated the classification accuracy of one such tool: The Numeracy Screener, a 2‐min test of symbolic (Arabic numerals) and nonsymbolic (dot arrays) discrimination ability. A sample of 222 children who demonstrated persistent deficits (n = 55), inconsistent deficits (n = 51), or typical performance (n = 116) on standardized tests of math achievement over multiple observations was tested. The Numeracy Screener correctly classified children in all three groups. Notably, the symbolic condition has greater sensitivity in discriminating children with persistent DD from the other two groups. Screening tools that assess early numeracy skills may be promising for identifying children at risk for developing severe mathematical difficulties.

The Contribution of General Cognitive Abilities and Specific Numerical Abilities to Mathematics Achievement in Students Who are Deaf or Hard-of-Hearing

General cognitive and specific numerical abilities that underlay mathematical performance have been heavily investigated among hearing students; however, the inquiry has rarely been applied to students who are deaf or hard-of-hearing (d/Dhh). We examined whether general cognitive abilities (i.e. nonverbal IQ, processing speed, and spatial ability) and specific numerical abilities (i.e. symbolic and non-symbolic numerical magnitude processing) are related to mathematics achievement in 198 d/Dhh students in Grades 3 to 9. The results of our regression models indicated that, the three general cognitive abilities independently explained the variance in mathematics achievement when entered in one step; and spatial ability and processing speed had an independent contribution to mathematics achievement in the presence of the specific numerical abilities. The specific numerical abilities independently explained the variance in mathematics achievement when entered in one step; however, none of them had an independent contribution to mathematics achievement in the presence of the general cognitive abilities. These findings suggested that mathematics achievement in d/Dhh students depended more on general cognitive abilities, such as spatial ability and processing speed, than on specific numerical abilities.

How Are Symbols and Nonsymbolic Numerical Magnitudes Related? Exploring Bidirectional Relationships in Early Numeracy

ABSTRACTWhat is the nature of the relationship between different lower‐level numerical skills and their role in developing arithmetic skills? We consider the hypothesis of a reciprocal relationship between the development of symbolic (e.g., Arabic numerals) and nonsymbolic (e.g., arrays of objects) numerical magnitude processing. Evidence for bidirectional relationships between symbolic and nonsymbolic numerical magnitude skill development is examined. Overall, present evidence is more indicative of an influence of symbolic numerical magnitude skills on the development of nonsymbolic numerical magnitude skills than vice versa. Looking forward, methodological issues pertinent to measuring the direction of such relationships are discussed. Also discussed is the important role that training studies need to play to further understand the complex relationships between basic number skills, and in turn their relationship with arithmetic. It is important that assumptions about relationships between lower and higher‐level cognitive skills are tested empirically and that seemingly counterintuitive relationships are given serious consideration.

The influence of visual–spatial skills on the association between processing of nonsymbolic numerical magnitude and number word sequence skills

Nonsymbolic numerical magnitude processing skills are assumed to be fundamental to mathematical learning. Recent findings suggest that visual–spatial skills account for associations between children’s performance in visually presented nonsymbolic numerical magnitude comparison tasks and their performance in visually presented arithmetic tasks. The aim of the current study was to examine whether associations between children’s performance in visually presented tasks assessing nonsymbolic numerical magnitude processing skills and their performance in tasks assessing early mathematical skills, which do not involve visual stimulation, may also be mediated by visual–spatial skills. This line of reasoning is based on the assumption that children make use of mental visualization processes when working on tasks assessing early mathematical skills, such as knowledge of the sequence of number words, even when these tasks do not involve visual stimulation. We assessed 4- to 6-year-old children’s performance in a nonsymbolic numerical magnitude comparison task, in tasks concerning knowledge of the sequence of number words, and in a developmental test to assess visual–spatial skills. Children’s nonsymbolic numerical magnitude processing skills were found to be associated with their number word sequence skills. This association was fully mediated by interindividual differences in visual–spatial skills. The effect size of this mediation effect was small. We assume that the ability to construct mental visualizations constitutes the key factor underlying this mediation effect.

Differences in arithmetic performance between Chinese and German adults are accompanied by differences in processing of non-symbolic numerical magnitude.

Human beings are assumed to possess an approximate number system (ANS) dedicated to extracting and representing approximate numerical magnitude information. The ANS is assumed to be fundamental to arithmetic learning and has been shown to be associated with arithmetic performance. It is, however, still a matter of debate whether better arithmetic skills are reflected in the ANS. To address this issue, Chinese and German adults were compared regarding their performance in simple arithmetic tasks and in a non-symbolic numerical magnitude comparison task. Chinese participants showed a better performance in solving simple arithmetic tasks and faster reaction times in the non-symbolic numerical magnitude comparison task without making more errors than their German peers. These differences in performance could not be ascribed to differences in general cognitive abilities. Better arithmetic skills were thus found to be accompanied by a higher speed of retrieving non-symbolic numerical magnitude knowledge but not by a higher precision of non-symbolic numerical magnitude representations. The group difference in the speed of retrieving non-symbolic numerical magnitude knowledge was fully mediated by the performance in arithmetic tasks, suggesting that arithmetic skills shape non-symbolic numerical magnitude processing skills.

The association between symbolic and nonsymbolic numerical magnitude processing and mental versus algorithmic subtraction in adults

There are two well-known computation methods for solving multi-digit subtraction items, namely mental and algorithmic computation. It has been contended that mental and algorithmic computation differentially rely on numerical magnitude processing, an assumption that has already been examined in children, but not yet in adults. Therefore, in this study, we examined how numerical magnitude processing was associated with mental and algorithmic computation, and whether this association with numerical magnitude processing was different for mental versus algorithmic computation. We also investigated whether the association between numerical magnitude processing and mental and algorithmic computation differed for measures of symbolic versus nonsymbolic numerical magnitude processing. Results showed that symbolic, and not nonsymbolic, numerical magnitude processing was associated with mental computation, but not with algorithmic computation. Additional analyses showed, however, that the size of this association with symbolic numerical magnitude processing was not significantly different for mental and algorithmic computation. We also tried to further clarify the association between numerical magnitude processing and complex calculation by also including relevant arithmetical subskills, i.e. arithmetic facts, needed for complex calculation that are also known to be dependent on numerical magnitude processing. Results showed that the associations between symbolic numerical magnitude processing and mental and algorithmic computation were fully explained by individual differences in elementary arithmetic fact knowledge.

Associations of non-symbolic and symbolic numerical magnitude processing with mathematical competence: a meta-analysis.

Many studies have investigated the association between numerical magnitude processing skills, as assessed by the numerical magnitude comparison task, and broader mathematical competence, e.g. counting, arithmetic, or algebra. Most correlations were positive but varied considerably in their strengths. It remains unclear whether and to what extent the strength of these associations differs systematically between non-symbolic and symbolic magnitude comparison tasks and whether age, magnitude comparison measures or mathematical competence measures are additional moderators. We investigated these questions by means of a meta-analysis. The literature search yielded 45 articles reporting 284 effect sizes found with 17,201 participants. Effect sizes were combined by means of a two-level random-effects regression model. The effect size was significantly higher for the symbolic (r=.302, 95% CI [.243, .361]) than for the non-symbolic (r=.241, 95% CI [.198, .284]) magnitude comparison task and decreased very slightly with age. The correlation was higher for solution rates and Weber fractions than for alternative measures of comparison proficiency. It was higher for mathematical competencies that rely more heavily on the processing of magnitudes (i.e. mental arithmetic and early mathematical abilities) than for others. The results support the view that magnitude processing is reliably associated with mathematical competence over the lifespan in a wide range of tasks, measures and mathematical subdomains. The association is stronger for symbolic than for non-symbolic numerical magnitude processing. So symbolic magnitude processing might be a more eligible candidate to be targeted by diagnostic screening instruments and interventions for school-aged children and for adults.

Individual differences in children's mathematics achievement: The roles of symbolic numerical magnitude processing and domain-general cognitive functions.

This contribution reviewed the available evidence on the domain-specific and domain-general neurocognitive determinants of children's arithmetic development, other than nonsymbolic numerical magnitude processing, which might have been overemphasized as a core factor of individual differences in mathematics and dyscalculia. We focused on symbolic numerical magnitude processing, working memory, and phonological processing, as these determinants have been most researched and their roles in arithmetic can be predicted against the background of brain imaging data. Our review indicates that symbolic numerical magnitude processing is a major determinant of individual differences in arithmetic. Working memory, particularly the central executive, also plays a role in learning arithmetic, but its influence appears to be dependent on the learning stage and experience of children. The available evidence on phonological processing suggests that it plays a more subtle role in children's acquisition of arithmetic facts. Future longitudinal studies should investigate these factors in concert to understand their relative contribution as well as their mediating and moderating roles in children's arithmetic development.

Foundations of children's numerical and mathematical skills: the roles of symbolic and nonsymbolic representations of numerical magnitude.

Numerical and mathematical skills are critical predictors of academic success. The last three decades have seen a substantial growth in our understanding of how the human mind and brain represent and process numbers. In particular, research has shown that we share with animals the ability to represent numerical magnitude (the total number of items in a set) and that preverbal infants can process numerical magnitude. Further research has shown that similar processing signatures characterize numerical magnitude processing across species and developmental time. These findings suggest that an approximate system for nonsymbolic (e.g., dot arrays) numerical magnitude representation serves as the basis for the acquisition of cultural, symbolic (e.g., Arabic numerals) representations of numerical magnitude. This chapter explores this hypothesis by reviewing studies that have examined the relation between individual differences in nonsymbolic numerical magnitude processing and symbolic math abilities (e.g., arithmetic). Furthermore, we examine the extent to which the available literature provides strong evidence for a link between symbolic and nonsymbolic representations of numerical magnitude at the behavioral and neural levels of analysis. We conclude that claims that symbolic number abilities are grounded in the approximate system for the nonsymbolic representation of numerical magnitude are not strongly supported by the available evidence. Alternative models and future research directions are discussed.

A Two-Minute Paper-and-Pencil Test of Symbolic and Nonsymbolic Numerical Magnitude Processing Explains Variability in Primary School Children's Arithmetic Competence

Recently, there has been a growing emphasis on basic number processing competencies (such as the ability to judge which of two numbers is larger) and their role in predicting individual differences in school-relevant math achievement. Children’s ability to compare both symbolic (e.g. Arabic numerals) and nonsymbolic (e.g. dot arrays) magnitudes has been found to correlate with their math achievement. The available evidence, however, has focused on computerized paradigms, which may not always be suitable for universal, quick application in the classroom. Furthermore, it is currently unclear whether both symbolic and nonsymbolic magnitude comparison are related to children’s performance on tests of arithmetic competence and whether either of these factors relate to arithmetic achievement over and above other factors such as working memory and reading ability. In order to address these outstanding issues, we designed a quick (2 minute) paper-and-pencil tool to assess children’s ability to compare symbolic and nonsymbolic numerical magnitudes and assessed the degree to which performance on this measure explains individual differences in achievement. Children were required to cross out the larger of two, single-digit numerical magnitudes under time constraints. Results from a group of 160 children from grades 1–3 revealed that both symbolic and nonsymbolic number comparison accuracy were related to individual differences in arithmetic achievement. However, only symbolic number comparison performance accounted for unique variance in arithmetic achievement. The theoretical and practical implications of these findings are discussed which include the use of this measure as a possible tool for identifying students at risk for future difficulties in mathematics.

How do symbolic and non-symbolic numerical magnitude processing skills relate to individual differences in children's mathematical skills? A review of evidence from brain and behavior

Many studies tested the association between numerical magnitude processing and mathematics achievement, but results differ depending on the number format used. For symbolic numbers (digits), data are consistent and robust across studies and populations: weak performance correlates with low math achievement and dyscalculia. For non-symbolic formats (dots), many conflicting findings have been reported. These inconsistencies might be explained by methodological issues. Alternatively, it might be that the processes measured by non-symbolic tasks are not critical for school-relevant mathematics. A few neuroimaging studies revealed that brain activation during number comparison correlates with children's mathematics achievement level, but the consistency of such relationships for symbolic and non-symbolic processing is unclear. These neurocognitive data provided ground for educational interventions, which seem to have positive effects on children's numerical development in (a)typical populations.

An electrophysiological investigation of non-symbolic magnitude processing: Numerical distance effects in children with and without mathematical learning disabilities

IntroductionThe aim of the present study was to probe electrophysiological effects of non-symbolic numerical processing in 20 children with mathematical learning disabilities (mean age = 99.2 months) compared to a group of 20 typically developing matched controls (mean age = 98.4 months). MethodsEEG data were obtained while children were tested with a standard non-symbolic numerical comparison paradigm that allowed us to investigate the effects of numerical distance manipulations for different set sizes, i.e., the classical subitizing, counting and estimation ranges. Effects of numerical distance manipulations on event-related potential (ERP) amplitudes as well as activation patterns of underlying current sources were analyzed. ResultsIn typically developing children, the amplitudes of a late parietal positive-going ERP component showed systematic numerical distance effects that did not depend on set size. For the group of children with mathematical learning disabilities, ERP distance effects were found only for stimuli within the subitizing range. Current source density analysis of distance-related group effects suggested that areas in right inferior parietal regions are involved in the generation of the parietal ERP amplitude differences. ConclusionOur results suggest that right inferior parietal regions are recruited differentially by controls compared to children with mathematical learning disabilities in response to non-symbolic numerical magnitude processing tasks, but only for stimuli with set sizes that exceed the subitizing range.

Numerical Magnitude Representations and Individual Differences in Children's Arithmetic Strategy Use

Against the background of neuroimaging studies on how the brain processes numbers, there is now converging evidence that numerical magnitude representations are crucial for successful mathematics achievement. One major drawback of this research is that it mainly investigated mathematics performance as measured through general standardized achievement tests. We extended this research by investigating the association between numerical magnitude representations and children's strategy use during single‐digit arithmetic. Our findings reveal that children's symbolic but not nonsymbolic numerical magnitude processing skills are associated with individual differences in arithmetic. Children with better access to magnitude representations from symbolic digits retrieve more facts from their memory and are faster in executing fact retrieval as well as procedural strategies. These associations remain even when intellectual ability, digit naming, and general mathematics achievement are additionally controlled for. All this indicates that particularly the access to numerical meaning from Arabic symbols is key for children's arithmetic strategy development, which suggests that educators and remedial teachers should focus on connecting Arabic symbols to the quantities they represent.

Differences between literates and illiterates on symbolic but not nonsymbolic numerical magnitude processing

The study of numerical magnitude processing provides a unique opportunity to examine interactions between phylogenetically ancient systems of semantic representations and those that are the product of enculturation. While nonsymbolic representations of numerical magnitude are processed similarly by humans and nonhuman animals, symbolic representations of numerical magnitude (e.g., Hindu-Arabic numerals) are culturally invented symbols that are uniquely human. Here, we report a comparison of symbolic and nonsymbolic numerical magnitude processing in two groups of participants who differ substantially in their level of literacy. In this study, level of literacy is used as an index of level of school-based numeracy skill. The data from these groups demonstrate that while the processing of nonsymbolic numerical magnitude (numerical distance effect) is unaffected by an individual's level of literacy, the processing of Hindu-Arabic numerals differs between literate and illiterate individuals who live in a literature culture and have limited symbolic recognition skills. These findings reveal that nonsymbolic numerical magnitude processing is unaffected by enculturation, while the processing of numerical symbols is modulated by literacy.