Non-symbolic Number 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.

Single-neuron representation of nonsymbolic and symbolic number zero in the human medial temporal lobe

The number zero holds a special status among numbers, indispensable for developing a comprehensive number theory.1,2,3,4 Despite its importance in mathematics, the neuronal foundation of zero in the human brain is unknown. We conducted single-neuron recordings in neurosurgical patients5,6,7 while they made judgments involving nonsymbolic number representations (dot numerosity), including the empty set, and symbolic numbers (Arabic numerals), including numeral zero. Neurons showed responsiveness to either the empty set or numeral zero, but not both. Neuronal activity to zero in both nonsymbolic and symbolic formats exhibited a numerical distance effect, indicating that zero representations are integrated together with countable numerosities and positive integers at the low end of the number line.8,9 A boundary in neuronal coding existed between the nonsymbolic empty set and small numerosities, correlating with the relative difficulty in discriminating numerosity zero behaviorally. Conversely, no such boundary was found for symbolic zero activity, suggesting that symbolic representations integrate zero with other numerals along the number line, reconciling its outlier role. The status of zero as a special nonsymbolic numerical quantity is reflected in the activity of neurons in the human brain, which seems to serve as a scaffold for more advanced representations of zero as a symbolic number.

Weaker number sense accounts for impaired numerosity perception in dyscalculia: Behavioral and computational evidence.

Impaired numerosity perception in developmental dyscalculia (low "number acuity") has been interpreted as evidence of reduced representational precision in the neurocognitive system supporting non-symbolic number sense. However, recent studies suggest that poor numerosity judgments might stem from stronger interference from non-numerical visual information, in line with alternative accounts that highlight impairments in executive functions and visuospatial abilities in the etiology of dyscalculia. To resolve this debate, we used a psychophysical method designed to disentangle the contribution of numerical and non-numerical features to explicit numerosity judgments in a dot comparison task and we assessed the relative saliency of numerosity in a spontaneous categorization task. Children with dyscalculia were compared to control children with average mathematical skills matched for age, IQ, and visuospatial memory. In the comparison task, the lower accuracy of dyscalculics compared to controls was linked to weaker encoding of numerosity, but not to the strength of non-numerical biases. Similarly, in the spontaneous categorization task, children with dyscalculia showed a weaker number-based categorization compared to the control group, with no evidence of a stronger influence of non-numerical information on category choice. Simulations with a neurocomputational model of numerosity perception showed that the reduction of representational resources affected the progressive refinement of number acuity, with little effect on non-numerical bias in numerosity judgments. Together, these results suggest that impaired numerosity perception in dyscalculia cannot be explained by increased interference from non-numerical visual cues, thereby supporting the hypothesis of a core number sense deficit. RESEARCH HIGHLIGHTS: A strongly debated issue is whether impaired numerosity perception in dyscalculia stems from a deficit in number sense or from poor executive and visuospatial functions. Dyscalculic children show reduced precision in visual numerosity judgments and weaker number-based spontaneous categorization, but no increasing reliance on continuous visual properties. Simulations with deep neural networks demonstrate that reduced neural/computational resources affect the developmental trajectory of number acuity and account for impaired numerosity judgments. Our findings show that weaker number acuity in developmental dyscalculia is not necessarily related to increased interference from non-numerical visual cues.

Mathematics anxiety and number processing: The link between executive functions, cardinality, and ordinality.

One important factor that hampers children's learning of mathematics is math anxiety (MA). Still, the mechanisms by which MA affects performance remain debated. The current study investigated the relationship between MA, basic number processing abilities (i.e., cardinality and ordinality processing), and executive functions in school children enrolled in grades 4-7 (N = 127). Children were divided into a high math anxiety group (N = 29) and a low math anxiety group (N = 31) based on the lowest quartile and the highest quartile. Using a series of analyses of variances, we find that highly math-anxious students do not perform worse on cardinality processing tasks (i.e., digit comparison and non-symbolic number sense), but that they perform worse on numerical and non-numerical ordinality processing tasks. We demonstrate that children with high MA show poorer performance on a specific aspect of executive functions-shifting ability. Our models indicate that shifting ability is tied to performance on both the numerical and non-numerical ordinality processing tasks. A central factor seems to be the involvement of executive processes during ordinality judgements, and executive functions may constitute the driving force behind these delays in numerical competence in math-anxious children.

Intuitive mapping between nonsymbolic quantity and observed action across development

Adults’ concurrent processing of numerical and action information yields bidirectional interference effects consistent with a cognitive link between these two systems of representation. This link is in place early in life: infants create expectations of congruency across numerical and action-related stimuli (i.e., a small [large] hand aperture associated with a smaller [larger] numerosity). Although these studies point to a developmental continuity of this mapping, little is known about the later development and thus how experience shapes such relationships. We explored how number–action intuitions develop across early and later childhood using the same methodology as in adults. We asked 3-, 6-, and 8-year-old children, as well as adults, to relate the magnitude of an observed action (a static hand shape, open vs. closed, in Experiment 1; a dynamic hand movement, opening vs. closing, in Experiment 2) to either a small or large nonsymbolic quantity (numerosity in Experiment 1 and numerosity and/or object size in Experiment 2). From 6 years of age, children started performing in a systematic congruent way in some conditions, but only 8-year-olds (added in Experiment 2) and adults performed reliably above chance in this task. We provide initial evidence that early intuitions guiding infants’ mapping between magnitude across nonsymbolic number and observed action are used in an explicit way only from late childhood, with a mapping between action and size possibly being the most intuitive. An initial coarse mapping between number and action is likely modulated with extensive experience with grasping and related actions directed to both arrays and individual objects.

Electrophysiological Comparison of Cumulative Area and Non-Symbolic Number Judgments.

Despite the importance of representing different magnitudes (i.e., number and cumulative area) for action planning and formal mathematics, there is much debate about the nature of these representations, particularly the extent to which magnitudes interact in the mind and brain. Early interaction views suggest that there are shared perceptual processes that form overlapping magnitude representations. However, late interaction views hold that representations of different magnitudes remain distinct, interacting only when preparing a motor response. The present study sheds light on this debate by examining the temporal onset of ratio and congruity effects as participants made ordinal judgments about number and cumulative area. Event-related potentials (ERPs) were recorded to identify whether the onset of such effects aligned with early versus late views. Ratio effects for both magnitudes were observed starting in the P100. Moreover, a congruity effect emerged within the P100. That interactions were observed early in processing, at the same time that initial ratio effects occurred, suggests that number and cumulative area processes interacted when magnitude representations were being formed, prior to preparing a decision response. Our findings are consistent with an early interaction view of magnitude processing, in which number and cumulative area may rely on shared perceptual mechanisms.

The importance of domain-specific number abilities and domain-general cognitive abilities for early arithmetic achievement and development.

Children's numerical and arithmetic skills differ greatly already at an early age. Although research focusing on accounting for these large individual differences clearly demonstrates that mathematical performance draws upon several cognitive abilities, our knowledge concerning key abilities underlying mathematical skill development is still limited. First, to identify key cognitive abilities contributing to children's development of early arithmetic skills. Second, to examine the extent to which early arithmetic performance and early arithmetic development rely on different or similar constellations of domain-specific number abilities and domain-general cognitive abilities. In all, 134 Swedish children (Mage = 6 years and 4 months, SD = 3 months, 74 boys) participated in this study. Verbal and non-verbal logical reasoning, non-symbolic number comparison, counting knowledge, spatial processing, verbal working memory and arithmetic were assessed. Twelve months later, arithmetic skills were reassessed. A latent change score model was computed to determine whether any of the abilities accounted for variations in arithmetic development. Arithmetic performance was supported by counting knowledge, verbal and non-verbal logical reasoning and spatial processing. Arithmetic skill development was only supported by spatial processing. Results show that young children's early arithmetic performance and arithmetic development are supported by different cognitive processes. The findings regarding performance supported Fuchs et al.'s model (Dev Psychol, 46, 2010b, 1731) but the developmental findings did not. The developmental findings align partially to Geary et al.'s (J Educ Psychol, 109, 2017, 680) hypothesis stating that young children's early arithmetic development is more dependent on general cognitive abilities than number abilities.

Absolute or relative size: What do we perceive when we look at a glass that is half full?

Given that both children and adults struggle with fractions in mathematics education, we investigated the processing of nonsymbolic fractions in a continuous form of part-of-the-whole. Continuous features of nonsymbolic numbers (e.g., the size of dots in an array) were found to influence numerosity judgment, but it should be noted that the (continuous) size of a part can be processed relative to a whole or as an absolute size. This study tested which of these size types (i.e., absolute and relative) influences comparison of parts. In two Stroop-like comparison tasks, we measured the interference of each size type on the processing of the other. In Experiment 1, stimuli were three-dimensional-like partially filled glasses of water. In both tasks, congruent trials (in which the larger absolute size was also the larger part-of-the-whole) were processed more efficiently than incongruent trials (in which the larger absolute size was the smaller part-of-the-whole). In Experiment 2, where stimuli were two-dimensional rectangles, this result was replicated under improved experimental control. We conclude that both absolute size and relative size of a part are automatically processed. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Early neural markers for individual difference in mathematical achievement determined from rational number processing

The neural markers for individual differences in mathematical achievement have been studied extensively using magnetic resonance imaging; however, high temporal resolution electrophysiological evidence for individual differences in mathematical achievement require further elucidation. This study evaluated the event-related potential (ERP) when 48 college students with high or low mathematical achievement (HA vs. LA) matched non-symbolic and symbolic rational numbers. Behavioral results indicated that HA students had better performance in the discretized non-symbolic matching, although the two groups showed similar performances in the continuous matching. ERP data revealed that even before non-symbolic stimulus presentation, HA students had greater Bereitschaftspotential (BP) amplitudes over posterior central electrodes. After the presentation of non-symbolic numbers, HA students had larger N1 amplitudes at 160 ms post-stimulus, over left-lateralized parieto-occipital electrodes. After the presentation of symbolic numbers, HA students displayed more profound P1 amplitudes at 100 ms post-stimulus, over left parietal electrodes. Furthermore, larger BP and N1 amplitudes were associated with the shorter reaction times, and larger P1 amplitudes corresponded to lower error rates. The BP effect could indicate preparation processing, and early left-lateralized N1 and P1 effects could reflect the non-symbolic and symbolic number processing along the dorsal neural pathways. These results suggest that the left-lateralized P1 and N1 components elicited by matching non-symbolic and symbolic rational numbers can be considered as neurocognitive markers for individual differences in mathematical achievement.

The SNARC Effect for Nonsymbolic Numbers is Not Observed When Stimuli Spatially Orient Attention

The SNARC Effect for Nonsymbolic Numbers is Not Observed When Stimuli Spatially Orient Attention

Groupitizing reflects conceptual developments in math cognition and inequities in math achievement from childhood through adolescence.

Understanding the cognitive processes central to mathematical development is crucial to addressing systemic inequities in math achievement. We investigate the "Groupitizing" ability in 1209 third to eighth graders (mean age at first timepoint=10.48, 586 girls, 39.16% Asian, 28.88% Hispanic/Latino, 18.51% White), a process that captures the ability to use grouping cues to access the exact value of a set. Groupitizing improves each year from late childhood to early adolescence (d=3.29), is a central predictor of math achievement (beta weight=.30), is linked to conceptual processes in mathematics (minimum d=0.69), and helps explain the dynamic between the ongoing development of non-symbolic number concepts, systemic educational inequities in school associated with SES, and mathematics achievement (minimum beta weight=.11) in ways that explicit symbolic measures may miss.

Severe Developmental Dyscalculia Is Characterized by Core Deficits in Both Symbolic and Nonsymbolic Number Sense

A long-standing debate concerns whether developmental dyscalculia is characterized by core deficits in processing nonsymbolic or symbolic numerical information as well as the role of domain-general difficulties. Heterogeneity in recruitment and diagnostic criteria make it difficult to disentangle this issue. Here, we selected children (n = 58) with severely compromised mathematical skills (2 SD below average) but average domain-general skills from a large sample referred for clinical assessment of learning disabilities. From the same sample, we selected a control group of children (n = 42) matched for IQ, age, and visuospatial memory but with average mathematical skills. Children with dyscalculia showed deficits in both symbolic and nonsymbolic number sense assessed with simple computerized tasks. Performance in the digit-comparison task and the numerosity match-to-sample task reliably separated children with developmental dyscalculia from controls in cross-validated logistic regression (area under the curve = .84). These results support a number-sense-deficit theory and highlight basic numerical abilities that could be targeted for early identification of at-risk children as well as for intervention.

Neurodevelopmental differences in task-evoked number network connectivity: Comparing symbolic and nonsymbolic number discrimination in children and adults

Numerical cognition can take place in multiple representational formats, such as Arabic digits (e.g., 1), verbal number words (e.g., “two”), and nonsymbolic (e.g., •••) numerical magnitude. Basic numerical discrimination abilities are key factors underlying the development of arithmetic abilities, acting as an important developmental precursor of adult-level numeracy. While prior research has begun to detail the neural correlates associated with basic numerical discrimination skills in different representational formats, the interactions between functional neural circuits are less understood. A growing body of evidence suggests that the functional networks recruited by number discrimination tasks differ between children and adults, which may provide valuable insights into the development of numerical cognition. To this end, we posed two questions: how do the interactions between functional circuits associated with number processing differ in children and adults? Are differences in functional network connectivity modulated by numerical representational codes? A theoretically motivated 22 ROI analysis indicated significant functional connectivity differences between children and adults across all three codes. Adults demonstrated sparser and more consistent connectivity patterns across codes, indicative of developmental domain-specialization for number processing. Although neural activity in children and adults is similar, the functional connectivity supporting number processing appears subject to substantial developmental maturation effects.

Unpacking the home numeracy environment: Examining dimensions of number activities in early childhood

A growing body of research has examined parents’ practices to support their young children's number learning at home, that is, the home numeracy environment. Many of these studies focus on formal and informal domains of numeracy activities, which are inconsistently defined and related to children's math learning. In this study, we explore dimensions of the home numeracy environment and examine their relations with children's math skills among a sample of 4-year-old children and their parents over the course of 1 year. Parents reported on the frequency of 21 numeracy activities when children were 4 and 5. Exploratory and confirmatory factor analyses revealed a 2-factors solution: number-related play activities and use of educational materials with numbers. Frequency of play with numbers was positively related to children's ability to solve applied math problems at age 5, controlling for prior number skills, child age, and socioeconomic status. In contrast, neither measure of the home numeracy environment predicted symbolic number knowledge or non-symbolic number sense when controlling for covariates. These findings underscore the need to differentiate between factors of the home numeracy environment and to develop clear theoretical definitions of these factors.

The association between non-symbolic number comparison and mathematical abilities depends on fluency.

Numerous studies have explored the correlation between non-symbolic number comparison and mathematical abilities in children, but the results have been inconsistent. The underlying mental processing featuring fluency may affect the correlation. The current study tested the fluency hypothesis that non-symbolic number comparison is associated with mathematical fluency in the development of mathematical ability. Non-symbolic number comparison, arithmetic computation, mathematical reasoning, non-symbolic number estimation, symbolic number comparison, and a series of basic cognitive processing tasks, including mental rotation, non-verbal matrix reasoning, and choice reaction time, were administered to 1072 first- to fourth-grade children. The results show that non-symbolic number comparison (measured via numerosity comparison) was the only independent predictor of arithmetic computation in higher grades, even after controlled for age, gender, basic cognitive processing, non-symbolic number estimation (measured via numerosity estimation), and symbolic number comparison (measured via digit comparison). However, it did not correlate with mathematical reasoning in any grade. These findings support the fluency hypothesis for developmental correlation between non-symbolic number comparison and mathematical abilities. That is, non-symbolic number comparison correlates with mathematical ability featuring fluency.

The role of parent-led and child-led home numeracy activities in early mathematical skills

Existing studies have shown mixed evidence for the role of the home numeracy environment (HNE) in supporting children’s early numeracy skills. To address some of the limitations of the existing literature, the present study used a multi-method approach to assess the parent-led HNE. Parents of children aged 3–5 years completed a questionnaire to assess the frequency of engagement in home numeracy activities and parent and child number talk was coded from play-based observations to assess the quality of engagement. This study also assessed the role of child-led home numeracy activities, including child number talk and Spontaneous Focusing on Numerosity (SFON) on early numeracy skills. Children (n = 164) were assessed on six early number skills (counting, cardinal knowledge, ordering skills, digit naming, arithmetic and symbolic to non-symbolic number mapping). Parent-led activities (questionnaire-assessed HNE activities and parent number talk) were not significantly associated with the composite of these six numeracy skills. There was also no significant association between parent-reported frequency of engagement in HNE activities and parent number talk. Child-led skills (SFON and child number talk) were not significantly associated with the composite numeracy score. Children’s and parents’ use of cardinal number talk was associated with children’s performance on the cardinality task, although associations were small (rs =.22 to.31). This study adds further moderate evidence that parent-led home numeracy activities may not be associated with overall early numeracy skills, and we consider next steps for researchers seeking to understand the role of the HNE.

Predictors of middle school students’ growth in symbolic number comparison performance

The ability to efficiently compare number symbols, such as digits, is associated with mathematics competence across the lifespan. Performance on symbolic number comparison tasks differ across age groups; young students who are developing fluency with digits improve on symbolic number comparison, and performance is better in adults than children. However, whether this improvement continues for older students who are fluent with number symbols, and what cognitive factors relate to this improvement, is unknown. This study used a longitudinal sample of U.S. middle school students (n = 394) to examine whether symbolic number comparison performance changes over middle school (i.e., students aged 11-14), whether there are individual differences in students’ rate of change, and potential predictors of that change. Students completed measures of single-digit symbolic number comparison, nonsymbolic number comparison, executive function (EF), and mathematics competence in Grade 5 (M = 11.02 years; SD = 0.32), and double-digit symbolic number comparison in Grades 6-8. Results showed that, on average, students’ symbolic number comparison performance improved from Grades 6-8. Grade 5 Symbolic number comparison performance predicted Grade 8 symbolic number comparison and rate of change over Grades 6-8. Grade 5 nonsymbolic number comparison, EF, and mathematics competence predicted Grade 8 symbolic number comparison performance. Results suggest that numerical magnitude processing, executive functions, and mathematics competence are related to symbolic number processing well into middle school, and that students continue to refine their ability to process number symbols into adolescence.

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.

Working memory and arithmetic impairments in children with FMR1 premutation and gray zone alleles.

ABSTRACT.Expansive mutations in familial mental retardation 1 (FMR1) gene have been associated with different phenotypes. Full mutations are associated with intellectual disability and autism spectrum disorder; premutations are associated with math learning difficulties and working memory impairments. In gray zone, neuropsychological development has not yet been described.Objectives:This study aimed to describe the frequency of FMR1 premutation and gray zone alleles in a school population sample representing a broad spectrum of variation in math achievement and detail school achievement and cognitive performance in the children identified with FMR1 premutation or gray zone alleles.Methods:We described a two-phase study. In the first phase, 2,195 school-age children were screened for math achievement. In the second phase, 378 children with normal intelligence were neuropsychologically assessed and genotyped for FMR1. Of these, 121 children (61 girls) performed below percentile 25 in mathematics (MD group) and 257 children (146 girls) performed above percentile 25 (control group).Results:Four pupils presented expanded alleles, one premutation and three gray zone alleles. The girl with the premutation and one boy with a gray zone allele presented impairments in working memory and arithmetic performance below percentile 6, compatible with the diagnosis of developmental dyscalculia. These children’s difficulties were not associated with inaccuracy of nonsymbolic number representations or literacy impairments. Dyscalculia in these children seems to be associated mainly with working memory impairments.Conclusions: FMR1 expansions in the gray zone may contribute to dyscalculia in otherwise healthy and normally intelligent children.

Predicting children's math skills from task-based and resting-state functional brain connectivity.

A critical goal of cognitive neuroscience is to predict behavior from neural structure and function, thereby providing crucial insights into who might benefit from clinical and/or educational interventions. Across development, the strength of functional connectivity among a distributed set of brain regions is associated with children's math skills. Therefore, in the present study we use connectome-based predictive modeling to investigate whether functional connectivity during numerical processing and at rest "predicts" children's math skills (N = 31, Mage = 9.21years, 14 Female). Overall, we found that functional connectivity during symbolic number comparison and rest, but not during nonsymbolic number comparison, predicts children's math skills. Each task revealed a largely distinct set of predictive connections distributed across canonical brain networks and major brain lobes. Most of these predictive connections were negatively correlated with children's math skills so that weaker connectivity predicted better math skills. Notably, these predictive connections were largely nonoverlapping across task states, suggesting children's math abilities may depend on state-dependent patterns of network segregation and/or regional specialization. Furthermore, the current predictive modeling approach moves beyond brain-behavior correlations and toward building models of brain connectivity that may eventually aid in predicting future math skills.