Continuity and change in children's longitudinal neural responses to numbers

Numerical cognition across the lifespan: A selective review of key developmental stages and neural, cognitive, and affective underpinnings

Numerical cognition constitutes a set of hierarchically related skills and abilities that develop-and may subsequently begin to decline-over developmental time. An innate "number sense" has long been argued to provide a foundation for the development of increasingly complex and applied numerical cognition, such as symbolic numerical reference, arithmetic, and financial literacy. However, evidence for a direct link between basic perceptual mechanisms that allow us to determine numerical magnitude (e.g., "how many" objects are in front of us and whether some of these are of a "greater" or "lesser" quantity), and later symbolic applications for counting and mathematics, has recently been challenged. Understanding how one develops an increasingly precise sense of number and which neurocognitive mechanisms support arithmetic development and achievement is crucial for developing successful mathematics curricula, supporting individual financial literacy and decision-making, and designing appropriate intervention and remediation programs for mathematical learning disabilities as well as mathematics anxiety. The purpose of this review is to provide a broad overview of the cognitive, neural, and affective underpinnings of numerical cognition-spanning the earliest hours of infancy to senior adulthood-and highlight gaps in our knowledge that remain to be addressed.

The calculating brain

The human brain possesses neural networks and mechanisms enabling the representation of numbers, basic arithmetic operations, and mathematical reasoning. Without the ability to represent numerical quantity and perform calculations, our scientifically and technically advanced culture would not exist. However, the origins of numerical abilities are grounded in an intuitive understanding of quantity deeply rooted in biology. Nevertheless, more advanced symbolic arithmetic skills require a cultural background with formal mathematical education. In the past two decades, cognitive neuroscience has seen significant progress in understanding the workings of the calculating brain through various methods and model systems. This review begins by exploring the mental and neuronal representations of nonsymbolic numerical quantity and then progresses to symbolic representations acquired in childhood. During arithmetic operations (addition, subtraction, multiplication, and division), these representations are processed and transformed according to arithmetic rules and principles, leveraging different mental strategies and types of arithmetic knowledge that can be dissociated in the brain. Although it was once believed that number processing and calculation originated from the language faculty, it is now evident that mathematical and linguistic abilities are primarily processed independently in the brain. Understanding how the healthy brain processes numerical information is crucial for gaining insights into debilitating numerical disorders, including acquired conditions like acalculia and learning-related calculation disorders such as developmental dyscalculia.

AI-Enhanced Dyscalculia Screening: A Survey of Methods and Applications for Children

New forms of interaction made possible by developments in special educational technologies can now help students with dyscalculia. Artificial intelligence (AI) has emerged as a promising tool in recent decades, particularly between 2001 and 2010, offering avenues to enhance the quality of education for individuals with dyscalculia. Therefore, the implementation of AI becomes crucial in addressing the needs of students with dyscalculia. Content analysis techniques were used to examine the literature covering the influence of AI on dyscalculia and its potential to assist instructors in promoting education for individuals with dyscalculia. The study sought to create a foundation for a more inclusive dyscalculia education in the future through in-depth studies. AI integration has had a big impact on educational institutions as well as people who struggle with dyscalculia. This paper highlights the importance of AI in improving the educational outcomes of students affected by dyscalculia.

Functional and representational differences between bilateral inferior temporal numeral areas

The processing of numerals as visual objects is supported by an "Inferior Temporal Numeral Area" (ITNA) in the bilateral inferior temporal gyri (ITG). Extant findings suggest some degree of hemispheric asymmetry in how the bilateral ITNAs process numerals. Pollack and Price (2019) reported such a hemispheric asymmetry by which a region in the left ITG was sensitive to digits during a visual search for a digit among letters, and a homologous region in the right ITG that showed greater digit sensitivity in individuals with higher calculation skills. However, the ITG regions were localized with separate analyses without directly contrasting their digit sensitivities and relation to calculation skills. So, the extent of and reasons for these functional asymmetries remain unclear. Here we probe whether the functional and representational properties of the ITNAs are asymmetric by applying both univariate and multivariate region-of-interest analyses to Pollack and Price's (2019) data. Contrary to the implications of the original findings, digit sensitivity did not differ between ITNAs, and digit sensitivity in both left and right ITNAs was associated with calculation skills. Representational similarity analyses revealed that the overall representational geometries of digits in the ITNAs were also correlated, albeit weakly, but the representational contents of the ITNAs were largely inconclusive. Nonetheless, we found a right lateralization in engagement in alphanumeric categorization, and that the right ITNA showed greater discriminability between digits and letters. Greater right lateralization of digit sensitivity and digit discriminability in the left ITNA were also related to higher calculation skills. Our findings thus suggest that the ITNAs may not be functionally identical and should be directly contrasted in future work. Our study also highlights the importance of within-individual comparisons for understanding hemispheric asymmetries, and analyses of individual differences and multivariate features to uncover effects that would otherwise be obscured by averages.

Distinct Contributions of the Cerebellum and Basal Ganglia to Arithmetic Procedures

Humans exhibit complex mathematical skills attributed to the exceptional enlargement of neocortical regions throughout evolution. In the current work, we initiated a novel exploration of the ancient subcortical neural network essential for mathematical cognition. Using a neuropsychological approach, we report that degeneration of two subcortical structures, the cerebellum and basal ganglia, impairs performance in symbolic arithmetic. We identify distinct computational impairments in male and female participants with cerebellar degeneration (CD) or Parkinson's disease (PD). The CD group exhibited a disproportionate cost when the arithmetic sum increased, suggesting that the cerebellum is critical for iterative procedures required for calculations. The PD group showed a disproportionate cost for equations with increasing addends, suggesting that the basal ganglia are critical for chaining multiple operations. In Experiment 2, the two patient groups exhibited intact practice gains for repeated equations at odds with an alternative hypothesis that these impairments were related to memory retrieval. Notably, we discuss how the counting and chaining operations relate to cerebellar and basal ganglia function in other task domains (e.g., motor processes). Overall, we provide a novel perspective on how the cerebellum and basal ganglia contribute to symbolic arithmetic. Our studies demonstrate the constraints on the computational role of two subcortical regions in higher cognition.

Developmental dyscalculia is not associated with atypical brain activation: A univariate fMRI study of arithmetic, magnitude processing, and visuospatial working memory

Functional neuroimaging serves as a tool to better understand the cerebral correlates of atypical behaviors, such as learning difficulties. While significant advances have been made in characterizing the neural correlates of reading difficulties (developmental dyslexia), comparatively little is known about the neurobiological correlates of mathematical learning difficulties, such as developmental dyscalculia (DD). Furthermore, the available neuroimaging studies of DD are characterized by small sample sizes and variable inclusion criteria, which make it problematic to compare across studies. In addition, studies to date have focused on identifying single deficits in neuronal processing among children with DD (e.g., mental arithmetic), rather than probing differences in brain function across different processing domains that are known to be affected in children with DD. Here, we seek to address the limitations of prior investigations. Specifically, we used functional magnetic resonance imaging (fMRI) to probe brain differences between children with and without persistent DD; 68 children (8-10 years old, 30 with DD) participated in an fMRI study designed to investigate group differences in the functional neuroanatomy associated with commonly reported behavioral deficits in children with DD: basic number processing, mental arithmetic and visuo-spatial working memory (VSWM). Behavioral data revealed that children with DD were less accurate than their typically achieving (TA) peers for the basic number processing and arithmetic tasks. No behavioral differences were found for the tasks measuring VSWM. A pre-registered, whole-brain, voxelwise univariate analysis of the fMRI data from the entire sample of children (DD and TA) revealed areas commonly associated with the three tasks (basic number processing, mental arithmetic, and VSWM). However, the examination of differences in brain activation between children with and without DD revealed no consistent group differences in brain activation. In view of these null results, we ran exploratory, Bayesian analyses on the data to quantify the amount of evidence for no group differences. This analysis provides supporting evidence for no group differences across all three tasks. We present the largest fMRI study comparing children with and without persistent DD to date. We found no group differences in brain activation using univariate, frequentist analyses. Moreover, Bayesian analyses revealed evidence for the null hypothesis of no group differences. These findings contradict previous literature and reveal the need to investigate the neural basis of DD using multivariate and network-based approaches to brain imaging.

Novel approaches to functional lateralization: Assessing information in activity patterns across hemispheres and more accurately identifying structural homologues

Functional lateralization is typically measured by comparing activation levels across the right and left hemispheres of the brain. Significant additional information, however, exists within distributed multi-voxel patterns of activity - a format not detectable by traditional activation-based analysis of functional magnetic resonance imaging (fMRI) data. We introduce and test two methods -one anatomical, one functional- that allow hemispheric information asymmetries to be detected. We first introduce and apply a novel tool that draws on brain 'surface fingerprints' to pair every location in one hemisphere with its hemispheric homologue. We use anatomical data to show that this approach is more accurate than the common distance-from-midline method for comparing bilateral regions. Next, we introduce a complementary analysis method that quantifies multivariate laterality in functional data. This new 'multivariate Laterality Index' (mLI) reflects both quantitative and qualitative information-differences across homologous activity patterns. We apply the technique here to functional data collected as participants viewed faces and non-faces. Using the previously generated surface fingerprints to pair-up homologous searchlights in each hemisphere, we use the novel multivariate laterality technique to identify face-information asymmetries across right and left counterparts of the fusiform gyrus, inferior temporal gyrus, superior parietal lobule, and early visual areas. The typical location of the fusiform face area has greater information asymmetry for faces than for shapes. More generally, we argue that the field should consider an information-based approach to lateralization.

The brain lateralization and development of math functions: progress since Sperry, 1974

In 1974, Roger Sperry, based on his seminal studies on the split-brain condition, concluded that math was almost exclusively sustained by the language dominant left hemisphere. The right hemisphere could perform additions up to sums less than 20, the only exception to a complete left hemisphere dominance. Studies on lateralized focal lesions came to a similar conclusion, except for written complex calculation, where spatial abilities are needed to display digits in the right location according to the specific requirements of calculation procedures. Fifty years later, the contribution of new theoretical and instrumental tools lead to a much more complex picture, whereby, while left hemisphere dominance for math in the right-handed is confirmed for most functions, several math related tasks seem to be carried out in the right hemisphere. The developmental trajectory in the lateralization of math functions has also been clarified. This corpus of knowledge is reviewed here. The right hemisphere does not simply offer its support when calculation requires generic space processing, but its role can be very specific. For example, the right parietal lobe seems to store the operation-specific spatial layout required for complex arithmetical procedures and areas like the right insula are necessary in parsing complex numbers containing zero. Evidence is found for a complex orchestration between the two hemispheres even for simple tasks: each hemisphere has its specific role, concurring to the correct result. As for development, data point to right dominance for basic numerical processes. The picture that emerges at school age is a bilateral pattern with a significantly greater involvement of the right-hemisphere, particularly in non-symbolic tasks. The intraparietal sulcus shows a left hemisphere preponderance in response to symbolic stimuli at this age.

Entropy, complexity, and maturity in children’s neural responses to naturalistic video lessons

Temporal characteristics of neural signals are often overlooked in traditional fMRI developmental studies but are critical to studying brain functions in ecologically valid settings. In the present study, we explore the temporal properties of children's neural responses during naturalistic mathematics and grammar tasks. To do so, we introduce a novel measure in developmental fMRI: neural entropy, which indicates temporal complexity of BOLD signals. We show that temporal patterns of neural activity have lower complexity and greater variability in children than in adults in the association cortex but not in the sensory-motor cortex. We also show that neural entropy is associated with both child-adult similarity in functional connectivity and neural synchrony, and that neural entropy increases with the size of functionally connected networks in the association cortex. In addition, neural entropy increases with functional maturity (i.e., child-adult neural synchrony) in content-specific regions. These exploratory findings suggest the hypothesis that neural entropy indexes the increasing breadth and diversity of neural processes available to children for analyzing mathematical information over development.

Neurophysiological signatures of approximate number system acuity in preschoolers

BACKGROUND

A hallmark of the approximate number system (ANS) is ratio dependence. Previous work identified specific event-related potentials (ERPs) that are modulated by numerical ratio throughout the lifespan. In adults, ERP ratio dependence was correlated with the precision of the numerical judgments with individuals who make more precise judgments showing larger ratio-dependent ERP effects. The current study evaluated if this relationship generalizes to preschoolers.

METHOD

ERPs were recorded from 56 4.5 to 5.5-year-olds while they compared the numerosity of two sequentially presented dot arrays. Nonverbal numerical precision, often called ANS acuity, was assessed using a similar behavioral task.

RESULTS

Only children with high ANS acuity exhibited a P2p ratio-dependent effect onsetting ∼250 ms after the presentation of the comparison dot array. Furthermore, P2p amplitude positively correlated with ANS acuity across tasks.

CONCLUSION

Results demonstrate developmental continuity between preschool years and adulthood in the neural basis of the ANS.

Representations within the Intraparietal Sulcus Distinguish Numerical Tasks and Formats

How does our brain understand the number five when it is written as an Arabic numeral, and when presented as five fingers held up? Four facets have been implicated in adult numerical processing: semantic, visual, manual, and phonological/verbal. Here, we ask how the brain represents each, using a combination of tasks and stimuli. We collected fMRI data from adult participants while they completed our novel "four number code" paradigm. In this paradigm, participants viewed one of two stimulus types to tap into the visual and manual number codes, respectively. Concurrently, they completed one of two tasks to tap into the semantic and phonological/verbal number codes, respectively. Classification analyses revealed that neural codes representing distinctions between the number comparison and phonological tasks were generalizable across format (e.g., Arabic numerals to hands) within intraparietal sulcus (IPS), angular gyrus, and precentral gyrus. Neural codes representing distinctions between formats were generalizable across tasks within visual areas such as fusiform gyrus and calcarine sulcus, as well as within IPS. Our results identify the neural facets of numerical processing within a single paradigm and suggest that IPS is sensitive to distinctions between semantic and phonological/verbal, as well as visual and manual, facets of number representations.

Cortical representations of numbers and nonsymbolic quantities expand and segregate in children from 5 to 8 years of age

Number symbols, such as Arabic numerals, are cultural inventions that have transformed human mathematical skills. Although their acquisition is at the core of early elementary education in children, it remains unknown how the neural representations of numerals emerge during that period. It is also unclear whether these relate to an ontogenetically earlier sense of approximate quantity. Here, we used multivariate fMRI adaptation coupled with within- and between-format machine learning to probe the cortical representations of Arabic numerals and approximate nonsymbolic quantity in 89 children either at the beginning (age 5) or four years into formal education (age 8). Although the cortical representations of both numerals and nonsymbolic quantities expanded from age 5 to age 8, these representations also segregated with learning and development. Specifically, a format-independent neural representation of quantity was found in the right parietal cortex, but only for 5-year-olds. These results are consistent with the so-called symbolic estrangement hypothesis, which argues that the relation between symbolic and nonsymbolic quantity weakens with exposure to formal mathematics in children.

Common Neural Functions during Children's Learning from Naturalistic and Controlled Mathematics Paradigms

Two major goals of human neuroscience are to understand how the brain functions in the real world and to measure neural processes under conditions that are ecologically valid. A critical step toward these goals is understanding how brain activity during naturalistic tasks that mimic the real world relates to brain activity in more traditional laboratory tasks. In this study, we used intersubject correlations to locate reliable stimulus-driven cerebral processes among children and adults in a naturalistic video lesson and a laboratory forced-choice task that shared the same arithmetic concept. We show that relative to a control condition with grammatical content, naturalistic and laboratory arithmetic tasks evoked overlapping activation within brain regions previously associated with math semantics. The regions of specific functional overlap between the naturalistic mathematics lesson and laboratory mathematics task included bilateral intraparietal cortex, which confirms that this region processes mathematical content independently of differences in task mode. These findings suggest that regions of the intraparietal cortex process mathematical content when children are learning about mathematics in a naturalistic setting.

Foundational Number Sense Training Gains Are Predicted by Hippocampal-Parietal Circuits

The development of mathematical skills in early childhood relies on number sense, the foundational ability to discriminate among quantities. Number sense in early childhood is predictive of academic and professional success, and deficits in number sense are thought to underlie lifelong impairments in mathematical abilities. Despite its importance, the brain circuit mechanisms that support number sense learning remain poorly understood. Here, we designed a theoretically motivated training program to determine brain circuit mechanisms underlying foundational number sense learning in female and male elementary school-age children (7-10 years). Our 4 week integrative number sense training program gradually strengthened the understanding of the relations between symbolic (Arabic numerals) and nonsymbolic (sets of items) representations of quantity. We found that our number sense training program improved symbolic quantity discrimination ability in children across a wide range of math abilities including children with learning difficulties. Crucially, the strength of pretraining functional connectivity between the hippocampus and intraparietal sulcus, brain regions implicated in associative learning and quantity discrimination, respectively, predicted individual differences in number sense learning across typically developing children and children with learning difficulties. Reverse meta-analysis of interregional coactivations across 14,371 fMRI studies and 89 cognitive functions confirmed a reliable role for hippocampal-intraparietal sulcus circuits in learning. Our study identifies a canonical hippocampal-parietal circuit for learning that plays a foundational role in children's cognitive skill acquisition. Findings provide important insights into neurobiological circuit markers of individual differences in children's learning and delineate a robust target for effective cognitive interventions.SIGNIFICANCE STATEMENT Mathematical skill development relies on number sense, the ability to discriminate among quantities. Here, we develop a theoretically motivated training program and investigate brain circuits that predict number sense learning in children during a period important for acquisition of foundational cognitive skills. Our integrated number sense training program was effective in children across a wide a range of math abilities, including children with learning difficulties. We identify hippocampal-parietal circuits that predict individual differences in learning gains. Our study identifies a brain circuit critical for the acquisition of foundational cognitive skills, which will be useful for developing effective interventions to remediate learning disabilities.

Electrophysiological Signatures of Numerosity Encoding in a Delayed Match-to-Sample Task

The number of elements in a small set of items is appraised in a fast and exact manner, a phenomenon called subitizing. In contrast, humans provide imprecise responses when comparing larger numerosities, with decreasing precision as the number of elements increases. Estimation is thought to rely on a dedicated system for the approximate representation of numerosity. While previous behavioral and neuroimaging studies associate subitizing to a domain-general system related to object tracking and identification, the nature of small numerosity processing is still debated. We investigated the neural processing of numerosity across subitizing and estimation ranges by examining electrophysiological activity during the memory retention period in a delayed numerical match-to-sample task. We also assessed potential differences in the neural signature of numerical magnitude in a fully non-symbolic or cross-format comparison. In line with behavioral performance, we observed modulation of parietal-occipital neural activity as a function of numerosity that differed in two ranges, with distinctive neural signatures of small numerosities showing clear similarities with those observed in visuospatial working memory tasks. We also found differences in neural activity related to numerical information in anticipation of single vs. cross-format comparison, suggesting a top-down modulation of numerical processing. Finally, behavioral results revealed enhanced performance in the mixed-format conditions and a significant correlation between task performance and symbolic mathematical skills. Overall, we provide evidence for distinct mechanisms related to small and large numerosity and differences in numerical encoding based on task demands.

Ancient visual channels have a causal role in arithmetic calculations

Humans exhibit complex arithmetic skills, often attributed to our exceptionally large neocortex. However, the past decade has provided ample evidence that the functional domain of the subcortex extends well beyond basic functions. Using a sensitive behavioral method, for the first time, we explored the contributions of lower-order visual monocular channels to symbolic arithmetic operations, addition and subtraction. The pattern of results from 4 different experiments provides converging evidence for a causal relation between mental arithmetic and primitive subcortical regions. The results have major implications for our understanding of the neuroevolutionary development of general numerical abilities–subcortical regions, which are shared across different species, are essential to complex numerical operations. In a bigger conceptual framework, these findings and others call for a shift from the modal view of the exclusive role of the neocortex in high-level cognition to a view that emphasizes the interplay between subcortical and cortical brain networks.

Developmental brain dynamics of numerical and arithmetic abilities

The development of numerical and arithmetic abilities constitutes a crucial cornerstone in our modern and educated societies. Difficulties to acquire these central skills can lead to severe consequences for an individual's well-being and nation's economy. In the present review, we describe our current broad understanding of the functional and structural brain organization that supports the development of numbers and arithmetic. The existing evidence points towards a complex interaction among multiple domain-specific (e.g., representation of quantities and number symbols) and domain-general (e.g., working memory, visual-spatial abilities) cognitive processes, as well as a dynamic integration of several brain regions into functional networks that support these processes. These networks are mainly, but not exclusively, located in regions of the frontal and parietal cortex, and the functional and structural dynamics of these networks differ as a function of age and performance level. Distinctive brain activation patterns have also been shown for children with dyscalculia, a specific learning disability in the domain of mathematics. Although our knowledge about the developmental brain dynamics of number and arithmetic has greatly improved over the past years, many questions about the interaction and the causal involvement of the abovementioned functional brain networks remain. This review provides a broad and critical overview of the known developmental processes and what is yet to be discovered.

Gray matter volume in left intraparietal sulcus predicts longitudinal gains in subtraction skill in elementary school

Although behavioral studies show large improvements in arithmetic skills in elementary school, we do not know how brain structure supports math gains in typically developing children. While some correlational studies have investigated the concurrent association between math performance and brain structure, such as gray matter volume (GMV), longitudinal studies are needed to infer if there is a causal relation. Although discrepancies in the literature on the relation between GMV and math performance have been attributed to the different demands on quantity vs. retrieval mechanisms, no study has experimentally tested this assumption. We defined regions of interests (ROIs) associated with quantity representations in the bilateral intraparietal sulcus (IPS) and associated with the storage of arithmetic facts in long-term memory in the left middle and superior temporal gyri (MTG/STG), and studied associations between GMV in these ROIs and children's performance on operations having greater demands on quantity vs. retrieval mechanisms, namely subtraction vs. multiplication. The aims of this study were threefold: First, to study concurrent associations between GMV and math performance, second, to investigate the role of GMV at the first time-point (T1) in predicting longitudinal gains in math skill to the second time-point (T2), and third, to study whether changes in GMV over time were associated with gains in math skill. Results showed no concurrent association between GMV in IPS and math performance, but a concurrent association between GMV in left MTG/STG and multiplication skill at T1. This association showed that the higher the GMV in this ROI, the higher the children's multiplication skill. Results also revealed that GMV in left IPS and left MTG/STG predicted longitudinal gains in subtraction skill only for younger children (approximately 10 years old). Whereas higher levels of GMV in left IPS at T1 predicted larger subtraction gains, higher levels of GMV in left MTG/STG predicted smaller gains. GMV in left MTG/STG did not predict longitudinal gains in multiplication skill. No significant association was found between changes in GMV over time and longitudinal gains in math. Our findings support the early importance of brain structure in the IPS for mathematical skills that rely on quantity mechanisms.

Shared Neural Circuits for Visuospatial Working Memory and Arithmetic in Children and Adults

Visuospatial working memory (VSWM) plays an important role in arithmetic problem solving, and the relationship between these two skills is thought to change over development. Even though neuroimaging studies have demonstrated that VSWM and arithmetic both recruit frontoparietal networks, inferences about common neural substrates have largely been made by comparisons across studies. Little work has examined how brain activation for VSWM and arithmetic converge within the same participants and whether there are age-related changes in the overlap of these neural networks. In this study, we examined how brain activity for VSWM and arithmetic overlap in 38 children and 26 adults. Although both children and adults recruited the intraparietal sulcus (IPS) for VSWM and arithmetic, children showed more focal activation within the right IPS, whereas adults recruited the bilateral IPS, superior frontal sulcus/middle frontal gyrus, and right insula. A comparison of the two groups revealed that adults recruited a more left-lateralized network of frontoparietal regions for VSWM and arithmetic compared with children. Together, these findings suggest possible neurocognitive mechanisms underlying the strong relationship between VSWM and arithmetic and provide evidence that the association between VSWM and arithmetic networks changes with age.

Hemispheric asymmetries in processing numerical meaning in arithmetic

Hemispheric asymmetries in arithmetic have been hypothesized based on neuropsychological, developmental, and neuroimaging work. However, it has been challenging to separate asymmetries related to arithmetic specifically, from those associated general cognitive or linguistic processes. Here we attempt to experimentally isolate the processing of numerical meaning in arithmetic problems from language and memory retrieval by employing novel non-symbolic addition problems, where participants estimated the sum of two dot arrays and judged whether a probe dot array was the correct sum of the first two arrays. Furthermore, we experimentally manipulated which hemisphere receive the probe array first using a visual half-field paradigm while recording event-related potentials (ERP). We find that neural sensitivity to numerical meaning in arithmetic arises under left but not right visual field presentation during early and middle portions of the late positive complex (LPC, 400-800 ms). Furthermore, we find that subsequent accuracy for judgements of whether the probe is the correct sum is better under right visual field presentation than left, suggesting a left hemisphere advantage for integrating information for categorization or decision making related to arithmetic. Finally, neural signatures of operational momentum, or differential sensitivity to whether the probe was greater or less than the sum, occurred at a later portion of the LPC (800-1000 ms) and regardless of visual field of presentation, suggesting a temporal and functional dissociation between magnitude and ordinal processing in arithmetic. Together these results provide novel evidence for differences in timing and hemispheric lateralization for several cognitive processes involved in arithmetic thinking.

Neural representations of transitive relations predict current and future math calculation skills in children

A large body of evidence suggests that math learning in children is built upon innate mechanisms for representing numerical quantities in the intraparietal sulcus (IPS). Learning math, however, is about more than processing quantitative information. It is also about understanding relations between quantities and making inferences based on these relations. Consistent with this idea, recent behavioral studies suggest that the ability to process transitive relations (A > B, B > C, therefore A > C) may contribute to math skills in children. Here we used fMRI coupled with a longitudinal design to determine whether the neural processing of transitive relations in children could predict their current and future math skills. At baseline (T₁), children (n = 31) processed transitive relations in an MRI scanner. Math skills were measured at T₁ and again 1.5 years later (T₂). Using a machine learning approach with cross-validation, we found that activity associated with the representation of transitive relations in the IPS predicted math calculation skills at both T₁ and T₂. Our study highlights the potential of neurobiological measures of transitive reasoning for forecasting math skills in children, providing additional evidence for a link between this type of reasoning and math learning.

Functional lateralization of arithmetic processing in the intraparietal sulcus is associated with handedness

Functional lateralization is established for various cognitive functions, but was hardly ever investigated for arithmetic processing. Most neurocognitive models assume a central role of the bilateral intraparietal sulcus (IPS) in arithmetic processing and there is some evidence for more pronounced left-hemispheric activation for symbolic arithmetic. However, evidence was mainly obtained by studies in right-handers. Therefore, we conducted a functional near-infrared spectroscopy (fNIRS) study, in which IPS activation of left-handed adults was compared to right-handed adults in a symbolic approximate calculation task. The results showed that left-handers had a stronger functional right-lateralization in the IPS than right-handers. This finding has important consequences, as the bilateral IPS activation pattern for arithmetic processing seems to be shaped by functional lateralization and thus differs between left- and right-handers. We propose three possible accounts for the observed functional lateralization of arithmetic processing.

The Enculturated Move From Proto-Arithmetic to Arithmetic

The basic human ability to treat quantitative information can be divided into two parts. With proto-arithmetical ability, based on the core cognitive abilities for subitizing and estimation, numerosities can be treated in a limited and/or approximate manner. With arithmetical ability, numerosities are processed (counted, operated on) systematically in a discrete, linear, and unbounded manner. In this paper, I study the theory of enculturation as presented by Menary (2015) as a possible explanation of how we make the move from the proto-arithmetical ability to arithmetic proper. I argue that enculturation based on neural reuse provides a theoretically sound and fruitful framework for explaining this development. However, I show that a comprehensive explanation must be based on valid theoretical distinctions and involve several stages in the development of arithmetical knowledge. I provide an account that meets these challenges and thus leads to a better understanding of the subject of enculturation.

The neural association between arithmetic and basic numerical processing depends on arithmetic problem size and not chronological age

The intraparietal sulcus (IPS) is thought to be an important region for basic number processing (e.g. symbol-quantity associations) and arithmetic (e.g. addition). Evidence for shared circuitry within the IPS is largely based on comparisons across studies, and little research has investigated number processing and arithmetic in the same individuals. It is also unclear how the neural overlap between number processing and arithmetic is influenced by age and arithmetic problem difficulty. This study investigated these unresolved questions by examining basic number processing (symbol-quantity matching) and arithmetic (addition) networks in 26 adults and 42 children. Number processing and arithmetic elicited overlapping activity in the IPS in children and adults, however, the overlap was influenced by arithmetic problem size (i.e. which modulated the need to use procedural strategies). The IPS was recruited for number processing, and for arithmetic problems more likely to be solved using procedural strategies. We also found that the overlap between number processing and small-problem addition in children was comparable to the overlap between number processing and large-problem addition in adults. This finding suggests that the association between number processing and arithmetic in the IPS is related to the cognitive operation being performed rather than age.

Shared genetic aetiology between cognitive performance and brain activations in language and math tasks

Cognitive performance is highly heritable. However, little is known about common genetic influences on cognitive ability and brain activation when engaged in a cognitive task. The Human Connectome Project (HCP) offers a unique opportunity to study this shared genetic etiology with an extended pedigree of 785 individuals. To investigate this common genetic origin, we took advantage of the HCP dataset, which includes both language and mathematics activation tasks. Using the HCP multimodal parcellation, we identified areals in which inter-individual functional MRI (fMRI) activation variance was significantly explained by genetics. Then, we performed bivariate genetic analyses between the neural activations and behavioral scores, corresponding to the fMRI task accuracies, fluid intelligence, working memory and language performance. We observed that several parts of the language network along the superior temporal sulcus, as well as the angular gyrus belonging to the math processing network, are significantly genetically correlated with these indicators of cognitive performance. This shared genetic etiology provides insights into the brain areas where the human-specific genetic repertoire is expressed. Studying the association of polygenic risk scores, using variants associated with human cognitive ability and brain activation, would provide an opportunity to better understand where these variants are influential.

Connectivity of the Human Number Form Area Reveals Development of a Cortical Network for Mathematics

The adult brain contains cortical areas thought to be specialized for the analysis of numbers (the putative number form area, NFA) and letters (the visual word form area, VWFA). Although functional development of the VWFA has been investigated, it is largely unknown when and how the NFA becomes specialized and connected to the rest of the brain. One hypothesis is that NFA and VWFA derive their special functions through differential connectivity, but the development of this differential connectivity has not been shown. Here, we mapped the resting state connectivity of NFA and VWFA to the rest of the brain in a large sample (n = 437) of individuals (age 3.2-21 years). We show that within NFA-math network and within VWFA-reading network the strength of connectivity increases with age. The right NFA is significantly connected to the right intraparietal cortex already at the earliest age tested (age 3), before formal mathematical education has begun. This connection might support or enable an early understanding of magnitude or numerosity In contrast, the functional connectivity from NFA to the left anterior intraparietal cortex and to the right dorsolateral prefrontal cortex is not different from the functional connectivity of VWFA to these regions until around 12-14 years of age. The increase in connectivity to these regions was associated with a gradual increase in mathematical ability in an independent sample. In contrast, VWFA connects significantly to Broca's region around age 6, and this connectivity is correlated with reading ability. These results show how the differential connectivity of the networks for mathematics and reading slowly emerges through years of training and education.

Fronto-parietal numerical networks in relation with early numeracy in young children

Early numeracy provides the foundation of acquiring mathematical skills that is essential for future academic success. This study examined numerical functional networks in relation to counting and number relational skills in preschoolers at 4 and 6 years of age. The counting and number relational skills were assessed using school readiness test (SRT). Resting-state fMRI (rs-fMRI) was acquired in 123 4-year-olds and 146 6-year-olds. Among them, 61 were scanned twice over the course of 2 years. Meta-analysis on existing task-based numeracy fMRI studies identified the left parietal-dominant network for both counting and number relational skills and the right parietal-dominant network only for number relational skills in adults. We showed that the fronto-parietal numerical networks, observed in adults, already exist in 4-year and 6-year-olds. The counting skills were associated with the bilateral fronto-parietal network in 4-year-olds and with the right parietal-dominant network in 6-year-olds. Moreover, the number relational skills were related to the bilateral fronto-parietal and right parietal-dominant networks in 4-year-olds and had a trend of the significant relationship with the right parietal-dominant network in 6-year-olds. Our findings suggested that neural fine-tuning of the fronto-parietal numerical networks may subserve the maturation of numeracy in early childhood.

The neural bases of price estimation: Effects of size and precision of the estimate

People are often confronted with the need of estimating the market price of goods. An important question is how people estimate prices, given the variability of products and prices available. Using event-related fMRI, we investigated how numerical processing modulates the neural bases of retail price estimation by focusing on two numerical dimensions: the size and precision of the estimates. Participants were presented with several product labels and made market price estimates for those products. Measures of product buying frequency and market price variability were also collected. The estimation of higher prices required longer response times, was associated with greater variation in responses across participants, and correlated with increasing medial and lateral prefrontal cortex (PFC) activity. Moreover, price estimates followed Weber's law, a hallmark feature of numerical processing. Increasing accuracy in price estimation, indexed by decreasing Weber fraction, engaged the intraparietal sulcus (IPS), a critical region in numerical processing. Our findings provide evidence for distinguishable neural mechanisms associated with the size and the precision of price estimates.

Specialization of the Right Intraparietal Sulcus for Processing Mathematics During Development

Mathematical ability, especially perception of numbers and performance of arithmetics, is known to rely on the activation of intraparietal sulcus (IPS). However, reasoning ability and working memory, 2 highly associated abilities also activate partly overlapping regions. Most studies aimed at localizing mathematical function have used group averages, where individual variability is averaged out, thus confounding the anatomical specificity when localizing cognitive functions. Here, we analyze the functional anatomy of the intraparietal cortex by using individual analysis of subregions of IPS based on how they are structurally connected to frontal, parietal, and occipital cortex. Analysis of cortical thickness showed that the right anterior IPS, defined by its connections to the frontal lobe, was associated with both visuospatial working memory, and mathematics in 6-year-old children. This region specialized during development to be specifically related to mathematics, but not visuospatial working memory in adolescents and adults. This could be an example of interactive specialization, where interacting with the environment in combination with interactions between cortical regions leads from a more general role of right anterior IPS in spatial processing, to a specialization of this region for mathematics.

Predictive saccades in children and adults: A combined fMRI and eye tracking study

Saccades were assessed in 21 adults (age 24 years, SD = 4) and 15 children (age 11 years, SD = 1), using combined functional magnetic resonance imaging (fMRI) and eye-tracking. Subjects visually tracked a point on a horizontal line in four conditions: time and position predictable task (PRED), position predictable (pPRED), time predictable (tPRED) and visually guided saccades (SAC). Both groups in the PRED but not in pPRED, tPRED and SAC produced predictive saccades with latency below 80 ms. In task versus group comparisons, children's showed less efficient learning compared to adults for predictive saccades (adults = 48%, children = 34%, p = 0.05). In adults brain activation was found in the frontal and occipital regions in the PRED, in the intraparietal sulcus in pPRED and in the frontal eye field, posterior intraparietal sulcus and medial regions in the tPRED task. Group-task interaction was found in the supplementary eye field and visual cortex in the PRED task, and the frontal cortex including the right frontal eye field and left frontal pole, in the pPRED condition. These results indicate that, the basic visuomotor circuitry is present in both adults and children, but fine-tuning of the activation according to the task temporal and spatial demand mature late in child development.

Emergence of the neural network underlying phonological processing from the prereading to the emergent reading stage: A longitudinal study

Numerous studies have shown that phonological skills are critical for successful reading acquisition. However, how the brain network supporting phonological processing evolves and how it supports the initial course of learning to read is largely unknown. Here, for the first time, we characterized the emergence of the phonological network in 28 children over three stages (prereading, beginning reading, and emergent reading) longitudinally. Across these three time points, decreases in neural activation in the left inferior parietal cortex (LIPC) were observed during an audiovisual phonological processing task, suggesting a specialization process in response to reading instruction/experience. Furthermore, using the LIPC as the seed, a functional network consisting of the left inferior frontal, left posterior occipitotemporal, and right angular gyri was identified. The connection strength in this network co-developed with the growth of phonological skills. Moreover, children with above-average gains in phonological processing showed a significant developmental increase in connection strength in this network longitudinally, while children with below-average gains in phonological processing exhibited the opposite trajectory. Finally, the connection strength between the LIPC and the left posterior occipitotemporal cortex at the prereading level significantly predicted reading performance at the emergent reading stage. Our findings highlight the importance of the early emerging phonological network for reading development, providing direct evidence for the Interactive Specialization Theory and neurodevelopmental models of reading.

Applications of Functional Near-Infrared Spectroscopy (fNIRS) in Studying Cognitive Development: The Case of Mathematics and Language

In this review, we aim to highlight the application of functional near-infrared spectroscopy (fNIRS) as a useful neuroimaging technique for the investigation of cognitive development. We focus on brain activation changes during the development of mathematics and language skills in schoolchildren. We discuss how technical limitations of common neuroimaging techniques such as functional magnetic resonance imaging (fMRI) have resulted in our limited understanding of neural changes during development, while fNIRS would be a suitable and child-friendly method to examine cognitive development. Moreover, this technique enables us to go to schools to collect large samples of data from children in ecologically valid settings. Furthermore, we report findings of fNIRS studies in the fields of mathematics and language, followed by a discussion of the outlook of fNIRS in these fields. We suggest fNIRS as an additional technique to track brain activation changes in the field of educational neuroscience.

Mechanisms of interactive specialization and emergence of functional brain circuits supporting cognitive development in children

Cognitive development is thought to depend on the refinement and specialization of functional circuits over time, yet little is known about how this process unfolds over the course of childhood. Here we investigated growth trajectories of functional brain circuits and tested an interactive specialization model of neurocognitive development which posits that the refinement of task-related functional networks is driven by a shared history of co-activation between cortical regions. We tested this model in a longitudinal cohort of 30 children with behavioral and task-related functional brain imaging data at multiple time points spanning childhood and adolescence, focusing on the maturation of parietal circuits associated with numerical problem solving and learning. Hierarchical linear modeling revealed selective strengthening as well as weakening of functional brain circuits. Connectivity between parietal and prefrontal cortex decreased over time, while connectivity within posterior brain regions, including intra-hemispheric and inter-hemispheric parietal connectivity, as well as parietal connectivity with ventral temporal occipital cortex regions implicated in quantity manipulation and numerical symbol recognition, increased over time. Our study provides insights into the longitudinal maturation of functional circuits in the human brain and the mechanisms by which interactive specialization shapes children's cognitive development and learning.

Neural Tuning to Numerosity Relates to Perceptual Tuning in 3-6-Year-Old Children

Neural representations of approximate numerical value, or numerosity, have been observed in the intraparietal sulcus (IPS) in monkeys and humans, including children. Using functional magnetic resonance imaging, we show that children as young as 3-4 years old exhibit neural tuning to cardinal numerosities in the IPS and that their neural responses are accounted for by a model of numerosity coding that has been used to explain neural responses in the adult IPS. We also found that the sensitivity of children's neural tuning to number in the right IPS was comparable to their numerical discrimination sensitivity observed behaviorally, outside of the scanner. Children's neural tuning curves in the right IPS were significantly sharper than in the left IPS, indicating that numerical representations are more precise and mature more rapidly in the right hemisphere than in the left. Further, we show that children's perceptual sensitivity to numerosity can be predicted by the development of their neural sensitivity to numerosity. This research provides novel evidence of developmental continuity in the neural code underlying numerical representation and demonstrates that children's neural sensitivity to numerosity is related to their cognitive development.

SIGNIFICANCE STATEMENT

Here we test for the existence of neural tuning to numerosity in the developing brain in the youngest sample of children tested with fMRI to date. Although previous research shows evidence of numerical distance effects in the intraparietal sulcus of the developing brain, those effects could be explained by patterns of neural activity that do not represent neural tuning to numerosity. These data provide the first robust evidence that from as early as 3-4 years of age there is developmental continuity in how the intraparietal sulcus represents the values of numerosities. Moreover, the study goes beyond previous research by examining the relation between neural tuning and perceptual tuning in children.

Number-space associations without language: Evidence from preverbal human infants and non-human animal species

It is well known that humans describe and think of numbers as being represented in a spatial configuration, known as the 'mental number line'. The orientation of this representation appears to depend on the direction of writing and reading habits present in a given culture (e.g., left-to-right oriented in Western cultures), which makes this factor an ideal candidate to account for the origins of the spatial representation of numbers. However, a growing number of studies have demonstrated that non-verbal subjects (preverbal infants and non-human animals) spontaneously associate numbers and space. In this review, we discuss evidence showing that pre-verbal infants and non-human animals associate small numerical magnitudes with short spatial extents and left-sided space, and large numerical magnitudes with long spatial extents and right-sided space. Together this evidence supports the idea that a more biologically oriented view can account for the origins of the 'mental number line'. In this paper, we discuss this alternative view and elaborate on how culture can shape a core, fundamental, number-space association.

Numerosity representation is encoded in human subcortex

Certain numerical abilities appear to be relatively ubiquitous in the animal kingdom, including the ability to recognize and differentiate relative quantities. This skill is present in human adults and children, as well as in nonhuman primates and, perhaps surprisingly, is also demonstrated by lower species such as mosquitofish and spiders, despite the absence of cortical computation available to primates. This ubiquity of numerical competence suggests that representations that connect to numerical tasks are likely subserved by evolutionarily conserved regions of the nervous system. Here, we test the hypothesis that the evaluation of relative numerical quantities is subserved by lower-order brain structures in humans. Using a monocular/dichoptic paradigm, across four experiments, we show that the discrimination of displays, consisting of both large (5-80) and small (1-4) numbers of dots, is facilitated in the monocular, subcortical portions of the visual system. This is only the case, however, when observers evaluate larger ratios of 3:1 or 4:1, but not smaller ratios, closer to 1:1. This profile of competence matches closely the skill with which newborn infants and other species can discriminate numerical quantity. These findings suggest conservation of ontogenetically and phylogenetically lower-order systems in adults' numerical abilities. The involvement of subcortical structures in representing numerical quantities provokes a reconsideration of current theories of the neural basis of numerical cognition, inasmuch as it bolsters the cross-species continuity of the biological system for numerical abilities.

Primitive Concepts of Number and the Developing Human Brain

Counting is an evolutionarily recent cultural invention of the human species. In order for humans to have conceived of counting in the first place, certain representational and logical abilities must have already been in place. The focus of this review is the origins and nature of those fundamental mechanisms that promoted the emergence of the human number concept. Five claims are presented that support an evolutionary view of numerical development: 1) number is an abstract concept with an innate basis in humans, 2) maturational processes constrain the development of humans' numerical representations between infancy and adulthood, 3) there is evolutionary continuity in the neural processes of numerical cognition in primates, 4) primitive logical abilities support verbal counting development in humans, and 5) primitive neural processes provide the foundation for symbolic numerical development in the human brain. We support these claims by examining current evidence from animal cognition, child development, and human brain function. The data show that at the basis of human numerical concepts are primitive perceptual and logical mechanisms that have evolutionary homologs in other primates and form the basis of numerical development in the human brain. In the final section of the review, we discuss some hypotheses for what makes human numerical reasoning unique by drawing on evidence from human and non-human primate neuroimaging research.

Core mathematical abilities in infants: Number and much more

Adults' ability to process numerical information can be traced back to the first days of life. The cognitive mechanisms underlying numerical representations are functional in preverbal infants, who are able to both track a small number of individuals and to estimate the numerosity of large sets across different modalities. This ability is closely linked to their ability to compute other quantitative dimensions such as spatial extent and temporal duration. In fact, the human mind establishes, early in life, spontaneous links between number, space, and time, which are privileged relative to links with other continuous dimensions (like loudness and brightness). Finally, preverbal infants do not only associate numbers to corresponding spatial extents but also to different spatial positions along a spatial axis. It is argued that these number-space mappings are at the origins of the "mental number line" representation, which is already functional in the first year of life.

Using cognitive training studies to unravel the mechanisms by which the approximate number system supports symbolic math ability

A picture is emerging that preverbal nonsymbolic numerical representations derived from the approximate number system (ANS) play an important role in mathematical development and sustained mathematical thinking. Functional imaging studies are revealing developmental trends in how the brain represents number. We propose that combining behavioral and neuroimaging techniques with cognitive training approaches will help identify the fundamental relationship between the ANS and symbolic mathematics. Understanding this relationship should ultimately benefit educators by providing ways to harness the ANS and hopefully improve math readiness in young children.

Longitudinal development of frontoparietal activity during feedback learning: Contributions of age, performance, working memory and cortical thickness

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We performed a longitudinal study on feedback learning (N = 208, age 8–27 years).

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We tested linear and nonlinear patterns in frontoparietal activity during learning.

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DLPFC and parietal cortex showed a late-adolescent peak in activity.

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SMA showed a linear increase, and ACC a linear decrease in brain activity with age.

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Performance predicted DLPFC and parietal activity, thickness predicted SMA activity.

Feedback learning is a crucial skill for cognitive flexibility that continues to develop into adolescence, and is linked to neural activity within a frontoparietal network. Although it is well conceptualized that activity in the frontoparietal network changes during development, there is surprisingly little consensus about the direction of change. Using a longitudinal design (N = 208, 8–27 years, two measurements in two years), we investigated developmental trajectories in frontoparietal activity during feedback learning. Our first aim was to test for linear and nonlinear developmental trajectories in dorsolateral prefrontal cortex (DLPFC), superior parietal cortex (SPC), supplementary motor area (SMA) and anterior cingulate cortex (ACC). Second, we tested which factors (task performance, working memory, cortical thickness) explained additional variance in time-related changes in activity besides age. Developmental patterns for activity in DLPFC and SPC were best characterized by a quadratic age function leveling off/peaking in late adolescence. There was a linear increase in SMA and a linear decrease with age in ACC activity. In addition to age, task performance explained variance in DLPFC and SPC activity, whereas cortical thickness explained variance in SMA activity. Together, these findings provide a novel perspective of linear and nonlinear developmental changes in the frontoparietal network during feedback learning.

Developmental bias for number words in the intraparietal sulcus

Children and adults show behavioral evidence of psychological overlap between their early, non-symbolic numerical concepts and their later-developing symbolic numerical concepts. An open question is to what extent the common cognitive signatures observed between different numerical notations are coupled with physical overlap in neural processes. We show that from 8 years of age, regions of the intraparietal sulcus (IPS) that exhibit a numerical ratio effect during non-symbolic numerical judgments also show a semantic distance effect for symbolic number words. In both children and adults, the IPS showed a semantic distance effect during magnitude judgments of number words (i.e. larger/smaller number) but not for magnitude judgments of object words (i.e. larger/smaller object size). The results provide novel evidence of conceptual overlap between neural representations of symbolic and non-symbolic numerical values that cannot be explained by a general process, and present the first demonstration of an early-developing dissociation between number words and object words in the human brain.

Functional brain organization for number processing in pre-verbal infants

Humans are born with the ability to mentally represent the approximate numerosity of a set of objects, but little is known about the brain systems that sub-serve this ability early in life and their relation to the brain systems underlying symbolic number and mathematics later in development. Here we investigate processing of numerical magnitudes before the acquisition of a symbolic numerical system or even spoken language, by measuring the brain response to numerosity changes in pre-verbal infants using functional near-infrared spectroscopy (fNIRS). To do this, we presented infants with two types of numerical stimulus blocks: number change blocks that presented dot arrays alternating in numerosity and no change blocks that presented dot arrays all with the same number. Images were carefully constructed to rule out the possibility that responses to number changes could be due to non-numerical stimulus properties that tend to co-vary with number. Interleaved with the two types of numerical blocks were audio-visual animations designed to increase attention. We observed that number change blocks evoked an increase in oxygenated hemoglobin over a focal right parietal region that was greater than that observed during no change blocks and during audio-visual attention blocks. The location of this effect was consistent with intra-parietal activity seen in older children and adults for both symbolic and non-symbolic numerical tasks. A distinct set of bilateral occipital and middle parietal channels responded more to the attention-grabbing animations than to either of the types of numerical stimuli, further dissociating the specific right parietal response to number from a more general bilateral visual or attentional response. These results provide the strongest evidence to date that the right parietal cortex is specialized for numerical processing in infancy, as the response to number is dissociated from visual change processing and general attentional processing.