Non-symbolic Processing Research Articles

Influence de la comparaison symbolique versus non symbolique sur le risque de difficultés d’apprentissage en mathématiques en éducation de la petite enfance

The main goal of this study was to clarify if students at risk of Mathematics Learning Disabilities had deficits in symbolic, non-symbolic processing, or both. Other purpose was to know if the Open Algorithm Based on Numbers (OBN) teaching method offers advantages over Closed Algorithms Based on Ciphers (CBC) method. A total of 307 Early Childhood Education students participated in this study. Based on student’ scores obtained in Test of Early Mathematics Ability (TEMA-3), that assesses formal and informal thinking, three analysis groups were designed: low-performance group, average-performance group, and high-performance group. There were significant differences in symbolic comparison and, consequently, the post hoc analysis showed significant differences between all groups in symbolic comparison except between the High and Average groups calculation. However, there were no differences in non-symbolic comparison between any of the groups. The teaching method and the mathematical competence showed a significant effect only on the symbolic comparison, resulting in a small size of effect for both the method and the Early Math Competence rate. These data support the dual representation hypothesis, in which the non-symbolic magnitudes would activate the Numerical Approximation System (ANS) and the symbolic quantities an exact representation of the numbers. This being a significant source of mathematical learning that could contribute to the improvement of the ANS.

The impact of musical training in symbolic and non-symbolic audiovisual judgements of magnitude.

Quantity estimation can be represented in either an analog or symbolic manner and recent evidence now suggests that analog and symbolic representation of quantities interact. Nonetheless, those two representational forms of quantities may be enhanced by convergent multisensory information. Here, we elucidate those interactions using high-density electroencephalography (EEG) and an audiovisual oddball paradigm. Participants were presented simultaneous audiovisual tokens in which the co-varying pitch of tones was combined with the embedded cardinality of dot patterns. Incongruencies were elicited independently from symbolic and non-symbolic modality within the audio-visual percept, violating the newly acquired rule that “the higher the pitch of the tone, the larger the cardinality of the figure.” The effect of neural plasticity in symbolic and non-symbolic numerical representations of quantities was investigated through a cross-sectional design, comparing musicians to musically naïve controls. Individual’s cortical activity was reconstructed and statistically modeled for a predefined time-window of the evoked response (130–170 ms). To summarize, we show that symbolic and non-symbolic processing of magnitudes is re-organized in cortical space, with professional musicians showing altered activity in motor and temporal areas. Thus, we argue that the symbolic representation of quantities is altered through musical training.

Non-Symbolic Numerosity and Symbolic Numbers are not Processed Intuitively in Children: Evidence From an Event-Related Potential Study

The approximate number system (ANS) theory and the ANS mapping account have been the most prominent theories on non-symbolic numerosity processing and symbolic number processing respectively, over the last 20 years. Recently, there is a growing debate about these theories, mainly based on research in adults. However, whether the ANS theory and ANS mapping account explain the processing of non-symbolic numerosity and symbolic number in childhood has received little attention. In the current ERP study, we first examined whether non-symbolic numerosity processing in 9-to-12-year-old children (N = 34) is intuitive, as proposed by the ANS theory. Second, we examined whether symbolic number processing is rooted in non-symbolic numerosity processing, as proposed the ANS mapping account. ERPs were measured during four same-different match-to-sample tasks with non-symbolic numerosities, symbolic numbers, and combinations of both. We found no evidence for intuitive processing of non-symbolic numerosity. Instead, children processed the visual features of non-symbolic stimuli more automatically than the numerosity itself. Moreover, children do not seem to automatically activate non-symbolic numerosity when processing symbolic numbers. These results challenge the ANS theory and ANS mapping account in 9-to-12-year-old children.

Developmental Changes in ANS Precision Across Grades 1–9: Different Patterns of Accuracy and Reaction Time

The main aim of this study was to analyze the patterns of changes in Approximate Number Sense (ANS) precision from grade 1 (mean age: 7.84 years) to grade 9 (mean age: 15.82 years) in a sample of Russian schoolchildren. To fulfill this aim, the data from a longitudinal study of two cohorts of children were used. The first cohort was assessed at grades 1–5 (elementary school education plus the first year of secondary education), and the second cohort was assessed at grades 5–9 (secondary school education). ANS precision was assessed by accuracy and reaction time (RT) in a non-symbolic comparison test (“blue-yellow dots” test). The patterns of change were estimated via mixed-effect growth models. The results revealed that in the first cohort, the average accuracy increased from grade 1 to grade 5 following a non-linear pattern and that the rate of growth slowed after grade 3 (7–9 years old). The non-linear pattern of changes in the second cohort indicated that accuracy started to increase from grade 7 to grade 9 (13–15 years old), while there were no changes from grade 5 to grade 7. However, the RT in the non-symbolic comparison test decreased evenly from grade 1 to grade 7 (7–13 years old), and the rate of processing non-symbolic information tended to stabilize from grade 7 to grade 9. Moreover, the changes in the rate of processing non-symbolic information were not explained by the changes in general processing speed. The results also demonstrated that accuracy and RT were positively correlated across all grades. These results indicate that accuracy and the rate of non-symbolic processing reflect two different processes, namely, the maturation and development of a non-symbolic representation system.

Frequency-based Dissociation of Symbolic and Nonsymbolic Numerical Processing during Numerical Comparison.

Recent evidence suggests that during numerical calculation, symbolic and nonsymbolic processing are functionally distinct operations. Nevertheless, both roughly recruit the same brain areas (spatially overlapping networks in the parietal cortex) and happen at the same time (roughly 250 msec poststimulus onset). We tested the hypothesis that symbolic and nonsymbolic processing are segregated by means of functionally relevant networks in different frequency ranges: high gamma (above 50 Hz) for symbolic processing and lower beta (12-17 Hz) for nonsymbolic processing. EEG signals were quantified as participants compared either symbolic numbers or nonsymbolic quantities. Larger EEG gamma-band power was observed for more difficult symbolic comparisons (ratio of 0.8 between the two numbers) than for easier comparisons (ratio of 0.2) over frontocentral regions. Similarly, beta-band power was larger for more difficult nonsymbolic comparisons than for easier ones over parietal areas. These results confirm the existence of a functional dissociation in EEG oscillatory dynamics during numerical processing that is compatible with the notion of distinct linguistic processing of symbolic numbers and approximation of nonsymbolic numerical information.

Symbolic Processing Mediates the Relation Between Non-symbolic Processing and Later Arithmetic Performance.

The nature of the relation between non-symbolic and symbolic magnitude processing in the prediction of arithmetic remains a hotly debated subject. This longitudinal study examined whether the influence of non-symbolic magnitude processing on arithmetic is mediated by symbolic processing skills. A sample of 130 children with age-adequate (N = 73) or below-average (N = 57) achievement in early arithmetic was followed from the end of Grade 1 (mean age: 86.9 months) through the beginning of Grade 4. Symbolic comparison of one- and two-digit numbers serially mediated the effect of non-symbolic comparison on later arithmetic. These results support a developmental model in which non-symbolic processing provides a scaffold for single-digit processing, which in turn influences multi-digit processing and arithmetic. In conclusion, both non-symbolic and symbolic processing play an important role in the development of arithmetic during primary school and might be valuable long-term indicators for the early identification of children at risk for low achievement in arithmetic.

The relation between subitizable symbolic and non-symbolic number processing over the kindergarten school year.

A long-standing debate in the field of numerical cognition concerns the degree to which symbolic and non-symbolic processing are related over the course of development. Of particular interest is the possibility that this link depends on the range of quantities in question. Behavioral and neuroimaging research with adults suggests that symbolic and non-symbolic quantities may be processed more similarly within, relative to outside of, the subitizing range. However, it remains unclear whether this unique link exists in young children at the outset of formal education. Further, no study has yet taken numerical size into account when investigating the longitudinal influence of these skills. To address these questions, we investigated the relation between symbolic and non-symbolic processing inside versus outside the subitizing range, both cross-sectionally and longitudinally, in 540 kindergarteners. Cross-sectionally, we found a consistently stronger relation between symbolic and non-symbolic number processing within versus outside the subitizing range at both the beginning and end of kindergarten. We also show evidence for a bidirectional relation over the course of kindergarten between formats within the subitizing range, and a unidirectional relation (symbolic→non-symbolic) for quantities outside of the subitizing range. These findings extend current theories on symbolic and non-symbolic magnitude development by suggesting that non-symbolic processing may in fact play a role in the development of symbolic number abilities, but that this influence may be limited to quantities within the subitizing range.

Spatial-numerical associations: Shared symbolic and non-symbolic numerical representations.

During the last decades, there have been a large number of studies into the number-related abilities of humans. As a result, we know that humans and non-human animals have a system known as the approximate number system that allows them to distinguish between collections based on their number of items, separately from any counting procedures. Dehaene and others have argued for a model on which this system uses representations for numbers that are spatial in nature and are shared by our symbolic and non-symbolic processing of numbers. However, there is a conflicting theoretical perspective in which there are no representations of numbers underlying the approximate number system, but only quantity-related representations. This perspective would then suggest that there are no shared representations between symbolic and non-symbolic processing. We review the evidence on spatial biases resulting from the activation of numerical representations, for both non-symbolic and symbolic tests. These biases may help decide between the theoretical differences; shared representations are expected to lead to similar biases regardless of the format, whereas different representations more naturally explain differences in biases, and thus behaviour. The evidence is not yet decisive, as the behavioural evidence is split: we expect bisection tasks to eventually favour shared representations, whereas studies on the spatial-numerical association of response codes (SNARC) effect currently favour different representations. We discuss how this impasse may be resolved, in particular, by combining these behavioural studies with relevant neuroimaging data. If this approach is carried forward, then it may help decide which of these two theoretical perspectives on number representations is correct.

Contributions of functional Magnetic Resonance Imaging (fMRI) to the study of numerical cognition

Using neuroimaging as a lens through which to understand numerical and mathematical cognition has provided both a different and complementary level of analysis to the broader behavioural literature. In particular, functional magnetic resonance imaging (fMRI) has contributed to our understanding of numerical and mathematical processing by testing and expanding existing psychological theories, creating novel hypotheses, and providing converging evidence with behavioural findings. There now exist several examples where fMRI has provided unique insights into the cognitive underpinnings of basic number processing, arithmetic, and higher-level mathematics. In this review, we discuss how fMRI has contributed to five critical questions in the field including: 1) the relationship between symbolic and nonsymbolic processing; 2) whether arithmetic skills are rooted in an understanding of basic numerical concepts; 3) the role of arithmetic strategies in the development of arithmetic skills; 4) whether basic numerical concepts scaffold higher-level mathematical skills; and 5) the neurobiological origins of developmental dyscalculia. In each of these areas, we review how the fMRI literature has both complemented and pushed the boundaries of our knowledge on these central theoretical issues. Finally, we discuss limitations of current approaches and future directions that will hopefully lead to even greater contributions of neuroimaging to our understanding of numerical cognition.

Numerical Processing Impairment in 22q11.2 (LCR22-4 to LCR22-5) Microdeletion: A Cognitive-Neuropsychological Case Study.

Although progress has been made, the cognitive, biological and, particularly, the genetic underpinnings of math learning difficulties (MD) remain largely unknown. This difficulty stems from the heterogeneity of MD and from the large contribution of environmental factors to its etiology. Understanding endophenotypes, e.g., the role of the Approximate Number System (ANS), may help understanding the nature of MD. MD associated with ANS impairments has been described in some genetic conditions, e.g., 22q11.2 deletion syndrome (22q11.2DS or Velocardiofacial syndrome, VCFS). Recently, a girl with MD was identified in a school population screening. She has a new syndrome resulting from a microdeletion in 22q11.2 (LCR22-4 to LCR22-5), a region adjacent to but not overlapping with region 22q11.2 (LCR22-2 to LCR22-4), typically deleted in VCFS. Here, we describe her cognitive-neuropsychological and numerical-cognitive profiles. The girl was assessed twice, at 8 and 11 years. Her numerical-cognitive performance at both times was compared to demographically similar girls with normal intelligence in a single-case, quasi-experimental study. Neuropsychological assessment was normal, except for relatively minor impairments in executive functions. She presented severe and persistent difficulties in the simplest single-digit calculations. Difficulties in commutative operations improved from the first to the second assessment. Difficulties in subtraction persisted and were severe. No difficulties were observed in Arabic number writing. Difficulties in single-digit calculation co-occurred with basic numerical processing impairments in symbolic and non-symbolic (single-digit comparison, dot sets size comparison and estimation) tasks. Her difficulties suggest ANS impairment. No difficulties were detected in visuospatial/visuoconstructional and in phonological processing tasks. The main contributions of the present study are: (a) this is the first characterization of the neuropsychological phenotype in 22q11.2DS (LCR22-4 to LCR22.5) with normal intelligence; (b) mild forms of specific genetic conditions contribute to persistent MD in otherwise typical persons; (c) heterogeneity of neurogenetic underpinnings of MD is suggested by poor performance in non-symbolic numerical processing, dissociated from visuospatial/visuoconstructional and phonological impairments; (d) similar to what happens in 22q11.2DS (LCR22-2 to LCR22-4), ANS impairments may also characterize 22q11.2DS (LCR22-4 to LCR22-5).

Una aproximación conexionista a los procesos mentales. Entrevista con James L. McClelland

In this interview, James L. McClelland responds to questions regarding connectionist models of cognition, a theory inspired by information processing in the brain. McClelland explains the distinction between symbolic and non-symbolic processing for a better understanding of mental processes. He argues that connectionist models can perform the computations which we know the brain can perform. In addition, he responds to several general questions on the perspectives of computational models of cognition.

Cognitive mechanisms underlying the relation between nonsymbolic and symbolic magnitude processing and their relation to math

Recent studies suggest that the relation between nonsymbolic magnitude processing skills and math competence is mediated by symbolic number processing. However, less is known about whether mapping between nonsymbolic and symbolic magnitude representations also mediates that relation, and whether the mediating role of symbolic number processing is explained by domain-general executive functions. Therefore, the current study examines whether symbolic comparison, mixed-format comparison, and executive function each mediate the relation between nonsymbolic magnitude processing and math. Furthermore, we investigate whether the relation between nonsymbolic and symbolic magnitude comparison is mediated by mapping between the formats and/or domain-general executive functions. Results indicate that symbolic processing mediates the relation between nonsymbolic processing and math, even after controlling for multiple components of executive function, which were also significant mediators. Cross-format comparison (i.e. mapping), on the other hand, did not mediate the relation between nonsymbolic comparison and math, but did mediate the relation between nonsymbolic and symbolic magnitude processing, even after controlling for executive function, which also mediated that relation. Taken together, our results suggest that both domain-specific and domain-general cognitive mechanisms account for the link between nonsymbolic and symbolic magnitude processing and their relation to math.

Common and distinct brain regions in both parietal and frontal cortex support symbolic and nonsymbolic number processing in humans: A functional neuroimaging meta-analysis

In recent years, there has been substantial growth in neuroimaging studies investigating neural correlates of symbolic (e.g. Arabic numerals) and non-symbolic (e.g. dot arrays) number processing. At present it remains contested whether number is represented abstractly, or if number representations in the brain are format-dependent. In order to quantitatively evaluate the available neuroimaging evidence, we used activation likelihood estimation (ALE) to conduct quantitative meta-analyses of the results reported in 57 neuroimaging papers. Consistent with the existence of an abstract representation of number in the brain, conjunction analyses revealed overlapping activation for symbolic and nonsymbolic numbers in frontal and parietal lobes. Consistent with the notion of format-dependent activation, contrast analyses demonstrated anatomically distinct fronto-parietal activation for symbolic and non-symbolic processing. Therefore, symbolic and non-symbolic numbers are subserved by format-dependent and abstract neural systems. Moreover, the present results suggest that regions across the parietal cortex, not just the intraparietal sulcus, are engaged in both symbolic and non-symbolic number processing, challenging the notion that the intraparietal sulcus is the key region for number processing. Additionally, our analyses indicate that regions in the frontal cortex subserve magnitude representations rather than non-numerical cognitive processes associated with number tasks, thereby highlighting the importance of considering both frontal and parietal regions as important for number processing.

Trajectories of Symbolic and Nonsymbolic Magnitude Processing in the First Year of Formal Schooling.

Sensitivity to numerical magnitudes is thought to provide a foundation for higher-level mathematical skills such as calculation. It is still unclear how symbolic (e.g. Arabic digits) and nonsymbolic (e.g. Dots) magnitude systems develop and how the two formats relate to one another. Some theories propose that children learn the meaning of symbolic numbers by scaffolding them onto a pre-existing nonsymbolic system (Approximate Number System). Others suggest that symbolic and nonsymbolic magnitudes have distinct and non-overlapping representations. In the present study, we examine the developmental trajectories of symbolic and nonsymbolic magnitude processing skills and how they relate to each other in the first year of formal schooling when children are becoming more fluent with symbolic numbers. Thirty Grade 1 children completed symbolic and nonsymbolic magnitude processing tasks at three time points in Grade 1. We found that symbolic and nonsymbolic magnitude processing skills had distinct developmental trajectories, where symbolic magnitude processing was characterized by greater gains than nonsymbolic skills over the one-year period in Grade 1. We further found that the development of the two formats only related to one another in the first half of the school year where symbolic magnitude processing skills influenced later nonsymbolic skills. These findings indicate that symbolic and nonsymbolic abilities have different developmental trajectories and that the development of symbolic abilities is not strongly linked to nonsymbolic representations by Grade 1. These findings also suggest that the relationship between symbolic and nonsymbolic processing is not as unidirectional as previously thought.

Individual Differences in Nonsymbolic Ratio Processing Predict Symbolic Math Performance

What basic capacities lay the foundation for advanced numerical cognition? Are there basic nonsymbolic abilities that support the understanding of advanced numerical concepts, such as fractions? To date, most theories have posited that previously identified core numerical systems, such as the approximate number system (ANS), are ill-suited for learning fraction concepts. However, recent research in developmental psychology and neuroscience has revealed a ratio-processing system (RPS) that is sensitive to magnitudes of nonsymbolic ratios and may be ideally suited for supporting fraction concepts. We provide evidence for this hypothesis by showing that individual differences in RPS acuity predict performance on four measures of mathematical competence, including a university entrance exam in algebra. We suggest that the nonsymbolic RPS may support symbolic fraction understanding much as the ANS supports whole-number concepts. Thus, even abstract mathematical concepts, such as fractions, may be grounded not only in higher-order logic and language, but also in basic nonsymbolic processing abilities.

Number sense in kindergarten children: Factor structure and working memory predictors

In the current study, the factor structure of number sense, or the ability to understand, use, and manipulate numbers, was investigated. Previous analyses yielded little consensus concerning number sense factors, other than a distinction between nonsymbolic and symbolic processing. Furthermore, associations between number sense factors and working memory components were investigated to gain insight into working memory involvement in number sense. A total of 441 Dutch kindergartners took part in the study. The factor structure of number sense and associations with working memory were tested using structural equation modelling. Results indicated that there was a distinction between nonsymbolic and symbolic number processing. Nonsymbolic processing was predicted by central executive performance, and symbolic processing was predicted by both central executive and visuospatial sketchpad performance. This implies that symbolic and nonsymbolic processing are distinguishable at this age, and that working memory involvement in symbolic processing is different from that in nonsymbolic processing.

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.

Explaining school mathematics performance from symbolic and nonsymbolic magnitude processing: Similarities and differences between typical and low-achieving children.

Magnitude processing is one of the most central cognitive mechanisms that underlie persistent mathematics difficulties. No consensus has yet been reached about whether these difficulties can be predominantly attributed to deficits in symbolic or nonsymbolic magnitude processing. To investigate this issue, we assessed symbolic and nonsymbolic magnitude representations in children with low or typical achievement in school mathematics. Response latencies and the distance effect were comparable between groups in both symbolic and nonsymbolic tasks. The results indicated that both typical and low achievers were able to access magnitude representation via symbolic and nonsymbolic processing. However, low achievers presented higher error rates than typical achievers, especially in the nonsymbolic task. Furthermore, measures of nonsymbolic magnitude explained individual differences in school mathematics better than measures of symbolic magnitude when considering all of the children together. When examining the groups separately, symbolic magnitude representation explained differences in school mathematics in low achievers but not in typical achievers. These results suggest that symbolic magnitude is more relevant to solving arithmetic problems when mathematics achievement is particularly low. In contrast, individual differences in nonsymbolic processing appear to be related to mathematics achievement in a more general manner.

Subitizing and visual short-term memory in human and non-human species: a common shared system?

Subitizing and visual short-term memory in human and non-human species: a common shared system?

Time and Number Sense Develop in Tandem?

Time and Number Sense Develop in Tandem?