Nonsymbolic Number Processing Research Articles

From spontaneous focusing on numerosity to mathematics achievement: The mediating role of non-symbolic number processing and mapping between symbolic and non-symbolic representations of number

Young children who readily demonstrate a self-initiated orientation, or spontaneous focusing on numerosity (SFON), perform better in mathematics in later years. To further our understanding of the mechanisms behind this relation, the present longitudinal study with 150 Chinese preschoolers examined the potential mediating role of non-symbolic number processing and mapping between symbolic and non-symbolic representations of number. Mediation analysis indicates two independent pathways leading from SFON to math achievement—namely the non-symbolic number processing pathway and the number mapping pathway—providing a more comprehensive model to explain the predictability of SFON on children’s math achievement. Our findings indicate that children with a stronger tendency to focus on the cardinal information of the environment are better at processing set sizes as well as mapping non-symbolic quantity information onto numbers, leading to better math achievement.

An undeniable interplay: Both numerosity and visual features affect estimation of non-symbolic stimuli

Converging lines of evidence suggest that the numerical abilities in Humans are rooted in the approximate number system (ANS): an innate, non-verbal mechanism that enables to estimate the numerosity of a set of items with little effort. Nevertheless, the high correlation between visual features and numerosity in the natural environment always constituted a relevant methodological problem that gathered growing concern throughout the years. This issue led some researchers to cast doubts on the existence of a system able to process numerical information independently from the influence of visual features. In the present study, we sought to shed light on the interplay between numerosity and visual features. To this aim, we implemented a non-symbolic estimation task which included a calibration phase. After performing a pre-calibration block, participants were presented with the calibration image for 20 s, and they were divided in three groups, according to the calibration stimulus they attended to: the three calibration stimuli contained the same number of items (30), but were characterized by a different amount of visual features. Results showed that performance was affected by numerosity and visual features in both phases of the experiment. However, calibration increased the weight of numerosity on performance while decreasing the weight of visual features. These results are hard to be reconciled with theories that attempt to explain human performance in non-symbolic number processing without taking into account both numerical and non-numerical aspects of the stimuli.

Sharpening, focusing, and developing: A study of change in nonsymbolic number comparison skills and math achievement in 1st grade.

Children's ability to discriminate nonsymbolic number (e.g., the number of items in a set) is a commonly studied predictor of later math skills. Number discrimination improves throughout development, but what drives this improvement is unclear. Competing theories suggest that it may be due to a sharpening numerical representation or an improved ability to pay attention to number and filter out non-numerical information. We investigate this issue by studying change in children's performance (N=65) on a nonsymbolic number comparison task, where children decide which of two dot arrays has more dots, from the middle to the end of 1st grade (mean age at time 1=6.85 years old). In this task, visual properties of the dot arrays such as surface area are either congruent (the more numerous array has more surface area) or incongruent. Children rely more on executive functions during incongruent trials, so improvements in each congruency condition provide information about the underlying cognitive mechanisms. We found that accuracy rates increased similarly for both conditions, indicating a sharpening sense of numerical magnitude, not simply improved attention to the numerical task dimension. Symbolic number skills predicted change in congruent trials, but executive function did not predict change in either condition. No factor predicted change in math achievement. Together, these findings suggest that nonsymbolic number processing undergoes development related to existing symbolic number skills, development that appears not to be driving math gains during this period. Children's ability to discriminate nonsymbolic number improves throughout development. Competing theories suggest improvement due to sharpening magnitude representations or changes in attention and inhibition. The current study investigates change in nonsymbolic number comparison performance during first grade and whether symbolic number skills, math skills, or executive function predict change. Children's performance increased across visual control conditions (i.e., congruent or incongruent with number) suggesting an overall sharpening of number processing. Symbolic number skills predicted change in nonsymbolic number comparison performance.

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.

Visual illusions as a tool to hijack numerical perception: Disentangling nonsymbolic number from its continuous visual properties.

The past few years have witnessed a fervent theoretical debate about the exact visual mechanisms supporting nonsymbolic number processing. The idea that quantity information is extracted through a primitive visual segmentation algorithm has been challenged by recent models, which rather tap on low-level features confounded with numerosity (i.e., density, convex hull, or total area). Here we used an original manipulation based on visual illusions to disentangle whether visual numerosity processing operates over discrete units or rather over continuous variables. In particular, we generated a set of stimuli composed by open inducers (e.g., like a pac-man shape) that simulate physical connections with Kanizsa-like illusory contours (ICs). Test sets contained pairs of collinear open inducers items that prompted 0 IC, 2 IC, or 4 IC lines connecting 2 objects. Critically, low-level visual features were fully controlled across connectedness levels. We found a systematic underestimation as we increased the IC connections when participants had to select the larger between 2 sets of objects (Experiment 1) but not in the case of aligned closed inducers preventing illusory lines (Experiments 2A and 2B). We also found a systematic numerosity underestimation when both IC connections and continuous features (e.g., convex hull) were independently manipulated in test stimuli (Experiment 3). Finally, these results were shown to be task independent because the same effects of IC connections were replicated in an estimation task (Experiment 4). Taken together, our findings indicate that numerosity perception relies on basic visual-segmentation mechanisms, pointing out the need of new theoretical frameworks integrating both continuous and discrete perceptual number signals. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

Introducing CUSTOM: A customized, ultraprecise, standardization-oriented, multipurpose algorithm for generating nonsymbolic number stimuli.

When evaluating the properties of a set of elements in a natural environment, an increase in numerosity unavoidably corresponds to an increase in the physical properties of the set: Five apples differ from ten apples not only in numerosity, but also in their visual features, such as volume, density, and surface. Since nonsymbolic number processing is typically investigated through the presentation of arrays of elements, it is mandatory to keep track of the visual features characterizing the stimuli. A plethora of solutions have been proposed to address this complex methodological issue; yet, there is no agreed-upon standard for how to measure and control for visual features. Here we present the "customized ultraprecise standardization-oriented multipurpose" (CUSTOM) algorithm for generating nonsymbolic number stimuli. It is characterized by several core features: The absence of fixed parameters or rules-apart from geometrical constraints-lets the user freely manipulate the visual features of the stimuli; control over the visual features of the stimuli is extremely accurate; no modification is required in order to perform different types of manipulation; and users can re-create any set of stimuli described so far in previous experiments on numerical cognition, for a wide variety of tasks, including comparison, estimation, habituation, and match-to-sample. The CUSTOM algorithm could represent an asset in the field of numerical cognition, as a versatile instrument for effectively generating high-precision visual stimuli within an unbiased theoretical framework.

Network topology of symbolic and nonsymbolic number comparison.

Studies of brain activity during number processing suggest symbolic and nonsymbolic numerical stimuli (e.g., Arabic digits and dot arrays) engage both shared and distinct neural mechanisms. However, the extent to which number format influences large-scale functional network organization is unknown. In this study, using 7 Tesla MRI, we adopted a network neuroscience approach to characterize the whole-brain functional architecture supporting symbolic and nonsymbolic number comparison in 33 adults. Results showed the degree of global modularity was similar for both formats. The symbolic format, however, elicited stronger community membership among auditory regions, whereas for nonsymbolic, stronger membership was observed within and between cingulo-opercular/salience network and basal ganglia communities. The right posterior inferior temporal gyrus, left intraparietal sulcus, and two regions in the right ventromedial occipital cortex demonstrated robust differences between formats in terms of their community membership, supporting prior findings that these areas are differentially engaged based on number format. Furthermore, a unified fronto-parietal/dorsal attention community in the nonsymbolic condition was fractionated into two components in the symbolic condition. Taken together, these results reveal a pattern of overlapping and distinct network architectures for symbolic and nonsymbolic number processing.

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.

Impairment of non-symbolic number processing in children with mathematical learning disability

The functional origin of Mathematical Learning Disability (MLD), an impairment affecting mathematics learning, remains controversial. We aimed to study the number sense deficit hypothesis in children with MLD. We explored the processing of non-symbolic numbers in Quebec French-speaking 8‒9-year-old children with three non-symbolic tasks (number comparison, number matching and dot set selection task). Results showed that children with MLD were as successful as typically developing children in a comparison task, but less successful in a matching task. Their performance in numerical re-production was also similar to that of typically developing children for large (10-99) numerosities, while they were less successful in transcoding small (1-4) and medium (5-9) numerosities. Children with MLD also presented with general cognitive and reading difficulties. Results suggest a deficit in processing small and medium numerosities in children with MLD that could be largely attributed to their poorer cognitive skills.

Disentangling Mathematics from Executive Functions by Investigating Unique Functional Connectivity Patterns Predictive of Mathematics Ability.

What are the underlying neurocognitive mechanisms that give rise to mathematical competence? This study investigated the relationship between tests of mathematical ability completed outside the scanner and resting-state functional connectivity (FC) of cytoarchitectonically defined subdivisions of the parietal cortex in adults. These parietal areas are also involved in executive functions (EFs). Therefore, it remains unclear whether there are unique networks for mathematical processing. We investigate the neural networks for mathematical cognition and three measures of EF using resting-state fMRI data collected from 51 healthy adults. Using 10 ROIs in seed to whole-brain voxel-wise analyses, the results showed that arithmetical ability was correlated with FC between the right anterior intraparietal sulcus (hIP1) and the left supramarginal gyrus and between the right posterior intraparietal sulcus (hIP3) and the left middle frontal gyrus and the right premotor cortex. The connection between the posterior portion of the left angular gyrus and the left inferior frontal gyrus was also correlated with mathematical ability. Covariates of EF eliminated connectivity patterns with nodes in inferior frontal gyrus, angular gyrus, and middle frontal gyrus, suggesting neural overlap. Controlling for EF, we found unique connections correlated with mathematical ability between the right hIP1 and the left supramarginal gyrus and between hIP3 bilaterally to premotor cortex bilaterally. This is partly in line with the "mapping hypothesis" of numerical cognition in which the right intraparietal sulcus subserves nonsymbolic number processing and connects to the left parietal cortex, responsible for calculation procedures. We show that FC within this circuitry is a significant predictor of math ability in adulthood.

Development of a Possible General Magnitude System for Number and Space.

There is strong evidence for a link between numerical and spatial processing. However, whether this association is based on a common general magnitude system is far from conclusive and the impact of development is not yet known. Hence, the present study aimed to investigate the association between discrete non-symbolic number processing (comparison of dot arrays) and continuous spatial processing (comparison of angle sizes) in children between the third and sixth grade (N = 367). Present findings suggest that the processing of comparisons of number of dots or angle are related to each other, but with angle processing developing earlier and being more easily comparable than discrete number representations for children of this age range. Accordingly, results favor the existence of a more complex underlying magnitude system consisting of dissociated but closely interacting representations for continuous and discrete magnitudes.

Frequency of Home Numeracy Activities Is Differentially Related to Basic Number Processing and Calculation Skills in Kindergartners.

Home numeracy has been shown to play an important role in children’s mathematical performance. However, findings are inconsistent as to which home numeracy activities are related to which mathematical skills. The present study disentangled between various mathematical abilities that were previously masked by the use of composite scores of mathematical achievement. Our aim was to shed light on the specific associations between home numeracy and various mathematical abilities. The relationships between kindergartners’ home numeracy activities, their basic number processing and calculation skills were investigated. Participants were 128 kindergartners (Mage = 5.43 years, SD = 0.29, range: 4.88–6.02 years) and their parents. The children completed non-symbolic and symbolic comparison tasks, non-symbolic and symbolic number line estimation tasks, mapping tasks (enumeration and connecting), and two calculation tasks. Their parents completed a home numeracy questionnaire. Results indicated small but significant associations between formal home numeracy activities that involved more explicit teaching efforts (i.e., identifying numerals, counting) and children’s enumeration skills. There was no correlation between formal home numeracy activities and non-symbolic number processing. Informal home numeracy activities that involved more implicit teaching attempts, such as “playing games” and “using numbers in daily life,” were (weakly) correlated with calculation and symbolic number line estimation, respectively. The present findings suggest that disentangling between various basic number processing and calculation skills in children might unravel specific relations with both formal and informal home numeracy activities. This might explain earlier reported contradictory findings on the association between home numeracy and mathematical abilities.

Accumulation of non-numerical evidence during nonsymbolic number processing in the brain: An fMRI study.

Behavioral evidence has shown that when performing a nonsymbolic number comparison task (e.g., deciding which of two dot arrays contains more dots), participants' responses are sensitive to affected by both numerical (e.g., number of items) and non-numerical magnitudes (i.e., area, density, etc.). Thus far it is unclear what brain circuits support this process of accumulating non-numerical variables during nonsymbolic number processing. To investigate this, 21 adult participants were asked to engage in a dot comparison task. To measure the neural correlates of accumulating numerical and non-numerical variables, we manipulated the number of the non-numerical magnitudes that were congruent (correlated with number) or incongruent (anticorrelated with number). In a control task, participants were asked to choose the darker of two gray rectangles (brightness task). The tasks were matched in terms of their difficulty. The results of a whole brain analysis for regions sensitive to the congruity of numerical and non-numerical magnitudes revealed a region in the right inferior frontal gyrus (rIFG). Activation in this region was found to be correlated with the relative congruency of numerical and non-numerical magnitudes. In contrast, this region was not modulated by difficulty of the brightness control task. Accordingly in view of these findings, we suggest that the rIFG supports the accumulation of non-numerical magnitudes that are positively correlated with number. Therefore taken together, this study reveals a brain region whose pattern of activity is influenced by the congruency between numerical and non-numerical variables during nonsymbolic number judgments. Hum Brain Mapp 38:4908-4921, 2017. © 2017 Wiley Periodicals, Inc.

How do individual differences in children's domain specific and domain general abilities relate to brain activity within the intraparietal sulcus during arithmetic? An fMRI study.

Previous research has demonstrated that children recruit the intraparietal sulcus (IPS) during arithmetic, which has largely been attributed to domain-specific processes such as quantity manipulations. However, the IPS has also been found to be important for domain-general abilities, such as visuo-spatial working memory (VSWM). Based on the current literature it is unclear whether individual differences in domain-specific skills, domain-general skills, or a combination of the two, are related to the recruitment of the IPS during arithmetic. This study examines how individual differences in both domain general and domain specific competencies relate to brain activity in the IPS during arithmetic, and whether the relationships are related to how brain activity is measured. In a sample of 44 school-aged children, we found that VSWM was only weakly related to a neural index of arithmetic complexity (neural problem size effect), whereas symbolic number processing skills (symbolic comparison and ordering) were related to overall arithmetic activity (both small and large problems). By simultaneously examining multiple domain-general and domain specific measures, we were also able to determine that symbolic skills were a stronger predictor of brain activity within the IPS than domain general skills such as VSWM and domain specific skills such as non-symbolic number processing. Together, these findings highlight that neural problem size effect may reflect different cognitive processes than brain activity across both small and large arithmetic problems, and that symbolic number processing skills are a critical predictor of variability in IPS activity during arithmetic. Hum Brain Mapp 38:3941-3956, 2017. © 2017 Wiley Periodicals, Inc.

Beyond comparison: The influence of physical size on number estimation is modulated by notation, range and spatial arrangement

Can physical size affect number estimation? Previous studies have shown that physical size influences non-symbolic numerosity in comparison tasks (e.g. which of two dots is larger). The current study investigated the conditions under which physical size can affect numerosity estimation. We employed a line mapping task in order to avoid the context of comparison and the need to provide a verbal label to estimate a quantity. Adult participants were briefly presented with the digits 2–8 or groups of 2–8 dots in 3 different physical sizes and were asked to estimate the position of a presented numerosity on a vertical line from 0 to 10. Physical size affected number estimation only above the subitizing range (i.e., >4) and only for non-symbolic numbers (e.g. dot arrays). Presenting non-symbolic numbers as canonical arrangements (like on a game die) reduced the effect of the physical size in the counting range (5–9). Accordingly, we suggest that the effect of task-irrelevant physical size on performance is modulated by the ability of participants to provide an accurate estimate of number: when the estimated number is easier to perceive (i.e., subitizing range or canonical arrangements), the influence of the physical size is smaller compared to when it is more difficult to give an accurate estimate of number (i.e., counting range, random arrangement). By doing so, we describe the factors that modulate the effect of physical size on number processing and provide another example of the important role continuous properties, such as physical size, play in non-symbolic number processing.

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.

Symbolic and non-symbolic number processing in children with developmental dyslexia

This study examined number processing in 10-year-olds with developmental dyslexia (DD). The phonological deficit and double deficit hypotheses imply that children with DD might have a connection deficit that affects their ability to establish links between number symbols and magnitude representations. The double deficit hypothesis also posits that symbolic number difficulties may emerge due to difficulties with processes underlying rapid automatic naming (RAN). The DD group displayed difficulties with symbolic number processing but not with non-symbolic number processing. However, the underlying processes of this access or connection deficit appeared not to be related to phonological awareness or RAN. The DD group displayed impaired arithmetic fluency and calculation that were accounted for by defective processes underlying RAN. In view of the triple-code model, children with DD have impaired verbal number codes or defective access to verbal number codes but an intact core magnitude representation.

The symbol-grounding problem in numerical cognition: A review of theory, evidence, and outstanding questions.

How do numerical symbols, such as number words, acquire semantic meaning? This question, also referred to as the "symbol-grounding problem," is a central problem in the field of numerical cognition. Present theories suggest that symbols acquire their meaning by being mapped onto an approximate system for the nonsymbolic representation of number (Approximate Number System or ANS). In the present literature review, we first asked to which extent current behavioural and neuroimaging data support this theory, and second, to which extent the ANS, upon which symbolic numbers are assumed to be grounded, is numerical in nature. We conclude that (a) current evidence that has examined the association between the ANS and number symbols does not support the notion that number symbols are grounded in the ANS and (b) given the strong correlation between numerosity and continuous variables in nonsymbolic number processing tasks, it is next to impossible to measure the pure association between symbolic and nonsymbolic numerosity. Instead, it is clear that significant cognitive control resources are required to disambiguate numerical from continuous variables during nonsymbolic number processing. Thus, if there exists any mapping between the ANS and symbolic number, then this process of association must be mediated by cognitive control. Taken together, we suggest that studying the role of both cognitive control and continuous variables in numerosity comparison tasks will provide a more complete picture of the symbol-grounding problem.

Processing of space, time, and number contributes to mathematical abilities above and beyond domain-general cognitive abilities

The current study investigated whether processing of number, space, and time contributes to mathematical abilities beyond previously known domain-general cognitive abilities in a sample of 8- to 10-year-old children (N=133). Multiple regression analyses revealed that executive functions and general intelligence predicted all aspects of mathematics and overall mathematical ability. Working memory capacity did not contribute significantly to our models, whereas spatial ability was a strong predictor of achievement. The study replicates earlier research showing that non-symbolic number processing seems to lose predictive power of mathematical abilities once the symbolic system is acquired. Novel findings include the fact that time discrimination ability was tied to calculation ability. Therefore, a conclusion is that magnitude processing in general contributes to mathematical achievement.

Mathematical difficulties in developmental coordination disorder: Symbolic and nonsymbolic number processing

At school, children with Developmental Coordination Disorder (DCD) struggle with mathematics. However, little attention has been paid to their numerical cognition abilities. The goal of this study was to better understand the cognitive basis for mathematical difficulties in children with DCD.Twenty 7-to-10 years-old children with DCD were compared to twenty age—matched typically developing children using dot and digit comparison tasks to assess symbolic and nonsymbolic number processing and in a task of single digits additions.Results showed that children with DCD had lower performance in nonsymbolic and symbolic number comparison tasks than typically developing children. They were also slower to solve simple addition problems. Moreover, correlational analyses showed that children with DCD who experienced greater impairments in the nonsymbolic task also performed more poorly in the symbolic tasks.These findings suggest that DCD impairs both nonsymbolic and symbolic number processing. A systematic assessment of numerical cognition in children with DCD could provide a more comprehensive picture of their deficits and help in proposing specific remediation.