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Growing evidence highlights the predictive power of cross-notation magnitude comparison (e.g., 2/5 vs. 0.25) for math outcomes, but the underlying mechanisms remain unknown. Across two studies, we investigated undergraduates’ cross-notation and within-notation comparison skills given equivalent fractions, decimals, and percentages (Study 1, N=220 and Study 2, N=183). We found participants did not perceive equivalent rational numbers equivalently. Cluster analyses revealed that approximately one-quarter of undergraduates exhibited a bias to select percentages as larger in cross-notation comparisons. Compared with the other cluster of undergraduates who showed little-to-no bias, the percentages-are-larger bias cluster performed worse on fraction number line estimation and fraction arithmetic (exact and approximate), as well as reporting lower SAT/ACT scores. Hierarchical linear regression analyses demonstrated that cross-notation comparison accuracy accounted for variance in SAT/ACT beyond within-notation accuracy. Mediation analyses revealed a potential mechanism: stronger cross-notation knowledge equips individuals to evaluate the reasonableness of solutions. Together, these results suggest the importance of an integrated understanding of rational number notations, which may not be fully assessed by within-notation measures alone.

Executive functions (EFs) are linked to positive outcomes across the lifespan. Yet, methodological challenges have prevented precise understanding of the developmental trajectory of their organization. We introduce novel methods to address challenges for both measuring and modeling EFs using an accelerated longitudinal design with a large, diverse sample of students in middle childhood (N = 1,286; ages 8 to 14). We deployed adaptive assessments to equate EF challenge across ages and a data-driven, network analytic approach to reveal the evolving diversity of EFs while simultaneously accounting for their unity. This methodological paradigm shift brings new precision and clarity to EF development, showing number of components stabilizes by age 10, but refinement of component composition continues through at least age 14.

A critical difference between decimal and whole numbers is that among whole numbers the number of digits provides reliable information about the size of the number, e.g., double-digit numbers are larger than single-digit numbers. However, for decimals, fewer digits can sometimes denote a larger number (i.e., 0.8 > 0.27). Accordingly, children and adults perform worse when comparing such Inconsistent decimal pairs relative to Consistent pairs, where the larger number also has more digits (i.e., 0.87 > 0.2). Two explanations have been posited for this effect. The string length congruity account proposes that participants compare each position in the place value system, and they additionally compare the number of digits. The semantic interference account suggests that participants additionally activate the whole number referents of numbers – the numbers unadorned with decimal points (e.g., 8 < 27) – and compare these. The semantic interference account uniquely predicts that for Inconsistent problems with the same actual decimal distance, those with a larger whole number distances should be harder, e.g., 0.9 vs. 0.81 should be harder than 0.3 vs. 0.21 because 9 << 81 whereas 3 < 21. Here we test this prediction in two experiments with college students (Study 1: n = 66 participants, Study 2: n = 85). Across both, we find both a main effect of consistency, reaffirming the string length hypothesis, but also that whole number distance interferes with processing conflicting decimals, supporting the semantic interference proposal. Evidence for both explanations indicates that decimal comparison difficulties arise from multiple competing numerical codes.

Early emerging nonsymbolic proportional skills have been posited as a foundational ability for later fraction learning. A positive relation between nonsymbolic and symbolic proportional reasoning has been reported, as well as successful nonsymbolic training and intervention programs enhancing fraction magnitude skills. However, little is known about the mechanisms underlying this relation. Of particular interest are nonsymbolic representations, which can be in continuous formats that may emphasize proportional relations and in discretized formats that may prompt erroneous whole-number strategies and hamper access to fraction magnitudes. We assessed the proportional comparison skills of 159 middle-school students (mean age = 12.54 years, 43% females, 55% males, 2% other or prefer not to say) across three types of representations: 1) continuous, unsegmented bars, 2) discretized, segmented bars that allowed counting strategies, and 3) symbolic fractions. Using both correlational and cluster approaches, we also examined their relations to symbolic fraction comparison ability. Within each stimulus type, we varied proportional distance, and in the discretize and symbolic stimuli we also manipulated whole-number congruency. Middle-schoolers' performance was modulated by the fraction distance across all formats; however, whole-number information affected discretized and symbolic comparison performance. Further, continuous and discretized nonsymbolic performance was related to fraction comparison ability; however, discretized skills explained variance above and beyond the contributions of continuous skills. Finally, our cluster analyses revealed three nonsymbolic comparison profiles: students who chose the bars with the largest number of segments (whole-number bias), chance-level performers, and high performers. Crucially, students with a whole-number bias profile showed this bias in their fraction skills and failed to show any symbolic distance modulation. Together, these results indicate that the relation between nonsymbolic and symbolic proportional skills may be determined by the (mis)conceptions based on discretized representations, rather than understandings of proportional magnitudes, suggesting that interventions that focus on competence with discretized representations may show dividends for fraction understanding.

Decimal numbers are generally assumed to be a straightforward extension of the base-ten system for whole numbers given their shared place value structure. However, in decimal notation, unlike whole numbers, the same magnitude can be expressed in multiple ways (e.g., 0.8, 0.80, 0.800, etc.). Here, we used a number line task with carefully selected stimuli to investigate how equivalent decimals (e.g., 0.8 and 0.80 on a 0-1 number line) and proportionally equivalent whole numbers (e.g., 80 on a 0-100 number line) are estimated. We find that young adults (N = 88, Mage = 20.22 Years, SD = 1.65, 57 female) have a linear response pattern for both decimals and whole numbers, but that double-digit decimals (e.g., 0.08, 0.82, 0.80) are systematically underestimated relative to proportionally equivalent whole numbers (e.g., 8, 82, 80). Moreover, decimal string length worsens the underestimation, such that single- digit decimals (e.g., 0.8) are perceived as smaller than their equivalent double-digit decimals (e.g., 0.80). Finally, we find that exposing participants to whole number stimuli before decimal stimuli induces magnitude-based underestimation, that is, greater underestimation for larger decimals. Together, these results suggest a small but persistent underestimation bias for decimals less than one, and further that decimal magnitude estimation is fragile and subject to greater underestimation when exposed to whole numbers.

The angular and supramarginal gyri (AG and SMG) together constitute the inferior parietal lobule (IPL) and have been associated with cognitive functions that support reading. How those functions are distributed across the AG and SMG is a matter of debate, the resolution of which is hampered by inconsistencies across stereotactic atlases provided by the major brain image analysis software packages. Schematic results from automated meta-analyses suggest primarily semantic (word meaning) processing in the left AG, with more spatial overlap among phonological (auditory word form), orthographic (visual word form), and semantic processing in the left SMG. To systematically test for correspondence between patterns of neural activation and phonological, orthographic, and semantic representations, we re-analyze a functional magnetic resonance imaging data set of participants reading aloud 465 words. Using representational similarity analysis, we test the hypothesis that within cytoarchitecture-defined subregions of the IPL, phonological representations are primarily associated with the SMG, while semantic representations are primarily associated with the AG. To the extent that orthographic representations can be de-correlated from phonological representations, they will be associated with cortex peripheral to the IPL, such as the intraparietal sulcus. Results largely confirmed these hypotheses, with some nuanced exceptions, which we discuss in terms of neurally inspired computational cognitive models of reading that learn mappings among distributed representations for orthography, phonology, and semantics. De-correlating constituent representations making up complex cognitive processes, such as reading, by careful selection of stimuli, representational formats, and analysis techniques, are promising approaches for bringing additional clarity to brain structure-function relationships.

Fractions have a fundamental importance in the design and construction of mathematical knowledge, being the basis for algebra and other advanced mathematical content (BAILEY et al., 2012; BOOTH; NEWTON, 2012; SIEGLER et al., 2012; TORBEYNS et al., 2015). However, historically this representational form of rational numbers has presented several obstacles in the teaching and learning process, both for students and teachers (PINTO, 2011; SERRAZINA; RODRIGUES, 2018; SIEGLER; THOMPSON; SCHNEIDER, 2011). One of the conceptual bases for the development of fractional thinking is understanding the magnitude of these numbers (SIEGLER et al. 2011, 2013; VAN HOOF et al., 2018), yet how teachers process fractions magnitudes remain unknown. Thus, this study aimed to use the lens of experimental math cognition research to investigate how postgraduate mathematics teachers process the magnitude of fractions. A convergent parallel mixed method was carried out: quantitative data were collected based on research in neuroscience and cognitive psychology, followed by a qualitative task where participants explained their answers on a subset of the comparisons. Teachers’ comparison performance indicated they were not thinking about fractions according to the fractions’ individual components. Instead, they used other strategies to compare symbolic fractions, mostly anchored in the part-whole perspective. The most commonly used strategy was the non-generalizable Gap strategy, where participants choose the fraction with a smaller difference between numerator and denominator rather than fully processing the magnitude of fractions. Thus, we can say that the investigated teachers were not influenced by a componential view, but we cannot affirm a holistic view either, since the most used strategy contains misconceptions and is based on an arithmetic procedure. Interestingly, during the qualitative task, which asked teachers textually to justify the choice of the largest fraction, some users of the Gap strategy recognized that it was not mathematically valid. Therefore, a direct implication of this research is the need for teaching strategies that illustrate failures of the Gap strategy. More broadly, instructional approaches that challenge intuitive thinking, allowing us to understand the magnitude of the fraction (SIEGLER et al., 2011) may prove useful for this population.

By the end of elementary school, students should be able to solve fraction arithmetic and comparison problems. However, less than 30 % of students in the United States have accomplished these educational milestones by the eighth grade. Not only is fraction knowledge as vital to learn more complex math, like algebra, it also has major implications on individuals' health and employment outcomes. Thus, it has become crucial to develop educational methods to enhance students' fraction understanding. In contrast to these struggles with symbolic fractions, young children can work surprisingly well with nonsymbolic representations of proportions, and nonsymbolic ratio processing skills are also positively related to fraction ability in older children and adults. These findings have led to educational interventions which leveraged nonsymbolic skills, primarily via numberlines, to enhance symbolic fraction understanding. This paper aims to offer a systematic review of these emerging intervention and training studies. We identified 19 papers, representing 22 studies, which we grouped in three categories: classroom-based interventions, multiple-session trainings, and single-session training studies. These studies provide an optimistic picture of the malleability of students' fractions skills: Across categories, placing nonsymbolic and symbolic representations of fractions on numberlines successfully enhanced fraction skills of low and typically achieving students, resulting in small-to-large effect sizes. These results suggest that fostering nonsymbolic skills may be important in addressing the persistent bottleneck that fractions pose in mathematical knowledge.

The present study tests two predictions stemming from the hypothesis that a source of difficulty with rational numbers is interference from whole number magnitude knowledge. First, inhibitory control should be an independent predictor of fraction understanding, even after controlling for working memory. Second, if the source of interference is whole number knowledge, then it should hinder fraction understanding. These predictions were tested in a racially and socioeconomically diverse sample of US children (N=765; 337 female) in grades 3 (ages 8-9), 5 (ages 10-11), and 7 (ages 12-13) who completed a battery of computerized tests. The fraction comparison task included problems with both shared components (e.g., 3/5 > 2/5) and distinct components (e.g., 2/3 > 5/9), and problems that were congruent (e.g., 5/6 > 3/4) and incongruent (e.g., 3/4 > 5/7) with whole number knowledge. Inhibitory control predicted fraction comparison performance over and above working memory across component and congruency types. Whole number knowledge did not hinder performance and instead positively predicted performance for fractions with shared components. These results highlight a role for inhibitory control in rational number understanding and suggest that its contribution may be distinct from inhibiting whole number magnitude knowledge.

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 STATEMENTMathematical 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.

Mathematical cognition requires coordinated activity across multiple brain regions, leading to the emergence of resting-state functional connectivity as a method for studying the neural basis of differences in mathematical achievement. Hyper-connectivity of the intraparietal sulcus (IPS), a key locus of mathematical and numerical processing, has been associated with poor mathematical skills in childhood, whereas greater connectivity has been related to better performance in adulthood. No studies to date have considered its role in adolescence. Further, hippocampal connectivity can predict mathematical learning, yet no studies have considered its contributions to contemporaneous measures of math achievement. Here, we used seed-based resting-state fMRI analyses to examine IPS and hippocampal intrinsic functional connectivity relations to math achievement in a group of 31 adolescents (mean age = 16.42 years, range 15-17), whose math performance spanned the 1% to 99% percentile. After controlling for IQ, IPS connectivity was negatively related to math achievement, akin to findings in children. However, the specific temporooccipital regions were more akin to the posterior loci implicated in adults. Hippocampal connectivity with frontal regions was also negatively correlated with concurrent math measures, which contrasts with results from learning studies. Finally, hyper-connectivity was not a global feature of low math performance, as math performance did not modulate connectivity of Heschl's gyrus, a control seed not involved in math cognition. Our results provide preliminary evidence that adolescence is a transitional stage in which patterns found in childhood and adulthood can be observed; most notably, hyper-connectivity continues to be related to low math ability into this period. This article is protected by copyright. All rights reserved.

Executive functions (EFs) are linked to positive outcomes across the lifespan. Yet, methodological challenges have prevented rigorous understanding of the precise ways EFs are organized in childhood and how they develop over time. We introduce novel methods to address these challenges for both measuring and modeling EFs using a large, accelerated longitudinal dataset from a diverse sample of students in middle childhood (approximately ages 8 to 14; N = 1,286). Adaptive assessments allowed us to equate EF challenge across ages and a data-driven, network analytic approach revealed the evolving diversity of EFs while accounting for their unity. Our results suggest EF organization stabilizes around age 10, but continues refining through at least age 14. This approach brings new precision to EFs’ development by removing interpretative ambiguities associated with previous methodologies. By improving EF measurement, the field can move towards improving EF training, to provide a strong foundation for students’ success.

This manuscript describes data collection, cleaning, and quality control procedures used in a large-scale, longitudinal, in-school study of executive function skills (EFs) and academic achievement in middle childhood, Project iLEAD (in-school Longitudinal Executive Function and Academic Achievement Database). Assessments were administered in real-world educational settings in a large sample of third through eighth grade students (ages 7 to 14; N = 1,280) over two years, with eight data collection timepoints in group settings. We assessed students with a novel, mobile EF assessment tool Adaptive Cognitive Evaluation (ACE). This battery included 11 tasks that were each designed to adapt to user performance in a trial-wise manner, allowing the same battery of tasks to be used multiple times within the same individual, and across a wide range of ages. Data quality analyses revealed that the adaptive algorithms were successful in equating challenge levels across ages 7 through 14 for 10 of 11 tasks. Data for each task were found to be approximately normally distributed and split-half reliability was acceptable across both accuracy and reaction time. ACE thus provides a reliable way to assess EFs in middle childhood using the same tasks while maintaining appropriate challenge level without facing ceiling or floor effects.

Individuals on the autism spectrum often have trouble with social and figurative language. As social language is often figurative, it can be challenging to disentangle the cognitive and neural sources of these difficulties. Neural systems for social cognition and language comprehension overlap in areas involved in retrieving linguistic meaning (semantics), such as the anterior temporal lobe (ATL), ventro-medial prefrontal cortex (vmPFC), posterior cingulate cortex (PCC), and posterior middle temporal gyrus (pMTG). Using adjective-noun phrases, we manipulated social/nonsocial and figurative/literal dimensions, which we expected to activate distinct but overlapping regions. We hypothesized that activation differences in the group with autism (AUT) would be greater for more social and figurative stimuli. During fMRI, participants in the AUT group (N = 19) and those in the non-autistic comparison (NAC) group (N = 22) made familiarity judgments to 192 phrases in a balanced 2 × 2 (social/nonsocial x figurative/literal) design. Social phrases activated the PCC in all participants, but only the NAC group activated the vmPFC. Figurative phrases were rated as more literal by the AUT group, with the figurative-literal phrase contrast showing no activation in the AUT group, but activating the PCC and right pMTG in the NAC group. The one significant group-level neural difference was for the social-figurative condition predicted to be most different between groups: greater activation for the AUT group in the right ATL. Differences in the right ATL and pMTG in the AUT group suggest altered engagement of right homologues of the canonical semantic network being recruited for processing combined social and figurative language.

The relationship between executive functions (EF) and academic achievement is well-established, but leveraging this insight to improve educational outcomes remains elusive. Here, we propose a framework for relating the role of specific EF on specific precursor skills that support later academic learning. Starting from the premise that executive functions contribute to general math skills both directly – supporting the execution of problem solving strategies – and indirectly – supporting the acquisition of precursor mathematical content, we hypothesize that the contribution of domain-general EF capacities to precursor skills that support later learning can help explain relations between EF and overall math skills. We test this hypothesis by examining whether the contribution of inhibitory control on general math knowledge can be explained by inhibition's contribution to processing rational number pairs that conflict with individual's prior whole number knowledge. In 97 college students (79 female, age = 20.58 years), we collected three measures of EF: working memory (backwards spatial span), inhibition (color-word Stroop) and cognitive flexibility (task switching), and timed and untimed standardized measures of math achievement. Our target precursor skill was a decimals comparison task where correct responses were inconsistent with prior whole number knowledge (e.g., 0.27 vs. 0.9). Participants performed worse on these trials relative to the consistent decimals pairs (e.g., 0.2 vs. 0.87). Individual differences in the Stroop task predicted performance on inconsistent decimal comparisons, which in turn predicted general math achievement. With respect to relating inhibitory control to math achievement, Stroop performance was an independent predictor of achievement after accounting for age, working memory and cognitive flexibility, but decimal performance mediated this relationship. Finally, we found inconsistent decimals performance mediated the relationship of inhibition with rational number performance, but not other advanced mathematical concepts. These results pinpoint the specific contribution of inhibitory control to rational number understanding, and more broadly are consistent with the hypothesis that acquisition of foundational mathematical content can explain the relationships between executive functions and academic outcomes, making them promising targets for intervention.

Children can successfully compare continuous proportions as early as 4 years of age, yet they struggle to compare discrete proportions at least to 10 years of age, especially when the discrete information is misleading. This study examined whether inhibitory control contributes to individual differences in discrete proportional reasoning and whether reasoning could be enhanced by priming continuous information. A total of 49 second-graders completed two tasks. In the Hearts and Flowers (HF) task, a measure of inhibition, children pressed on either the corresponding or opposite side, depending on the identity of the displayed figure. In the Spinners task, a measure of proportional reasoning, children chose the spinner with the proportionally larger red area across continuous and two discrete formats. In the discrete adjacent format, the continuous stimuli were segmented into sections, which could be compatible with the proportional information or misleading; the discrete mixed format interspersed the colored sections from the discrete adjacent conditions. Finally, two priming groups were formed. Children who saw the continuous format immediately before the discrete adjacent format formed the continuous priming group (n = 26). Children who saw the discrete mixed format immediately before the discrete adjacent format formed the discrete priming group (n = 23). Our results showed that children who performed better on the HF task also had better performance on the discrete counting misleading trials. Furthermore, children in the continuous priming group outperformed children in the discrete priming group, specifically in contexts where discrete information was misleading. These results suggest that children's proportional reasoning may be improved by fostering continuous representations of discrete stimuli and by enhancing inhibitory control skills.

Approximate number sense (ANS), the ability to rapidly and accurately compare quantities presented non-symbolically, has been proposed as a precursor to mathematics skills. Earlier work reported low heritability of approximate number sense, which was interpreted as evidence that approximate number sense acts as a fitness trait. However, viewing ANS as a fitness trait is discordant with findings suggesting that individual differences in approximate number sense acuity correlate with mathematical performance, a trait with moderate genetic effects. Importantly, the shared etiology of approximate number sense, mathematics, and general cognitive ability has remained unexamined. Thus, the etiology of approximate number sense and its overlap with math and general cognitive ability was assessed in the current study with two independent twin samples (N = 451 pairs). Results suggested that ANS acuity had moderate but significant additive genetic influences. ANS also had overlap with generalist genetic mechanisms accounting for variance and covariance in mathematics and general cognitive ability. Furthermore, ANS may have genetic factors unique to covariance with mathematics beyond overlap with general cognitive ability. Evidence across both samples was consistent with the proposal that the etiology of approximate number sense functions similar to that of mathematics and general cognitive skills.

Highlights • We examined neurocognitive mechanisms underlying learning and near transfer in elementary school children. • A five-day tutoring led to marked differentiation of behavioral and brain responses between trained and novel problems. • Children who learned trained problems faster transferred their learnt knowledge to novel problems better. • Faster learners showed greater overlap in local neural representations between trained and novel problems. • More efficient learning led to greater differentiation of functional brain circuits engaged by trained and novel problems., Why some children learn, and transfer their knowledge to novel problems, better than others remains an important unresolved question in the science of learning. Here we developed an innovative tutoring program and data analysis approach to investigate individual differences in neurocognitive mechanisms that support math learning and “near” transfer to novel, but structurally related, problems in elementary school children. Following just five days of training, children performed recently trained math problems more efficiently, with greater use of memory-retrieval-based strategies. Crucially, children who learned faster during training performed better not only on trained problems but also on novel problems, and better discriminated trained and novel problems in a subsequent recognition memory task. Faster learners exhibited increased similarity of neural representations between trained and novel problems, and greater differentiation of functional brain circuits engaged by trained and novel problems. These results suggest that learning and near transfer are characterized by parallel learning-rate dependent local integration and large-scale segregation of functional brain circuits. Our findings demonstrate that speed of learning and near transfer are interrelated and identify the neural mechanisms by which faster learners transfer their knowledge better. Our study provides new insights into the behavioral, mnemonic, and neural mechanisms underlying children’s learning.

Mathematical disabilities (MD) have a negative life-long impact on professional success, employment, and health outcomes. Yet, little is known about the intrinsic functional brain organization that contributes to poor math skills in affected children. It is now increasingly recognized that math cognition requires coordinated interaction within a large-scale fronto-parietal network anchored in the intraparietal sulcus (IPS). Here we characterize intrinsic functional connectivity within this IPS-network in children with MD, relative to a group of typically developing (TD) children who were matched on age, gender, IQ, working memory, and reading abilities. Compared to TD children, children with MD showed hyper-connectivity of the IPS with a bilateral fronto-parietal network. Importantly, aberrant IPS connectivity patterns accurately discriminated children with MD and TD children, highlighting the possibility for using IPS connectivity as a brain-based biomarker of MD. To further investigate regional abnormalities contributing to network-level deficits in children with MD, we performed whole-brain analyses of intrinsic low-frequency fluctuations. Notably, children with MD showed higher low-frequency fluctuations in multiple fronto-parietal areas that overlapped with brain regions that exhibited hyper-connectivity with the IPS. Taken together, our findings suggest that MD in children is characterized by robust network-level aberrations, and is not an isolated dysfunction of the IPS. We hypothesize that intrinsic hyper-connectivity and enhanced low-frequency fluctuations may limit flexible resource allocation, and contribute to aberrant recruitment of task-related brain regions during numerical problem solving in children with MD.

International audience; Autism spectrum disorders (ASD) represent a quintessential example of a clinical population with diverse symptom presentations and marked variation in cognitive abilities. However, the extensive literature lacks rigorous quantitative procedures for characterizing heterogeneity of cognitive abilities in these individuals. Here we employ novel clustering and cross-validation procedures to investigate the stability of heterogeneous patterns of cognitive abilities in reading and math in a relatively large sample (N = 114) of children with ASD and matched controls (N = 96). Our analysis revealed a unique profile of heterogeneity in ASD, consisting of a low-achieving subgroup with poor math skills compared with reading and a high-achieving subgroup who showed superior math skills compared with reading. Verbal and central executive working memory skills further differentiated these subgroups. Findings provide insights into distinct profiles of academic achievement in children with ASD, with implications for educational practice and intervention, and provide a novel framework for quantifying heterogeneity in the disorder.

International audience; Objective. A goal of developmental cognitive neuroscience is to uncover brain mechanisms underlying successful learning. While longitudinal studies capture brain changes following ‘schooling as usual’, short-term training studies can more directly link learning to brain changes. We investigated whether eight weeks of cognitive training recapitulates longitudinal changes in hippocampal engagement and connectivity.Methods. Nineteen children underwent a training program focused on improving arithmetic skills, along with fifteen children in a no-contact control group. Before and after training, or no-contact, both groups performed an arithmetic task during neuroimaging and a strategy assessment.Results. Training increased activity in the anterior hippocampus, and gains in memory-based strategies were associated with decreased lateral fronto-parietal activity and increased hippocampus-parietal connectivity. No changes were observed in the no-contact control group.Conclusions. Our results demonstrate that short-term training can recapitulate long-term neurodevelopmental changes accompanying learning and identifies plasticity of hippocampal responses as a common locus of cognitive skill development in children.

The importance of the hippocampal system for rapid learning and memory is well recognized, but its contributions to a cardinal feature of children's cognitive development-the transition from procedure-based to memory-based problem-solving strategies-are unknown. Here we show that the hippocampal system is pivotal to this strategic transition. Longitudinal functional magnetic resonance imaging (fMRI) in 7-9-year-old children revealed that the transition from use of counting to memory-based retrieval parallels increased hippocampal and decreased prefrontal-parietal engagement during arithmetic problem solving. Longitudinal improvements in retrieval-strategy use were predicted by increased hippocampal-neocortical functional connectivity. Beyond childhood, retrieval-strategy use continued to improve through adolescence into adulthood and was associated with decreased activation but more stable interproblem representations in the hippocampus. Our findings provide insights into the dynamic role of the hippocampus in the maturation of memory-based problem solving and establish a critical link between hippocampal-neocortical reorganization and children's cognitive development.

Baddeley and Hitch's multi-component working memory (WM) model has played an enduring and influential role in our understanding of cognitive abilities. Very little is known, however, about the neural basis of this multi-component WM model and the differential role each component plays in mediating arithmetic problem solving abilities in children. Here, we investigate the neural basis of the central executive (CE), phonological (PL) and visuo-spatial (VS) components of WM during a demanding mental arithmetic task in 7–9 year old children (N = 74). The VS component was the strongest predictor of math ability in children and was associated with increased arithmetic complexity-related responses in left dorsolateral and right ventrolateral prefrontal cortices as well as bilateral intra-parietal sulcus and supramarginal gyrus in posterior parietal cortex. Critically, VS, CE and PL abilities were associated with largely distinct patterns of brain response. Overlap between VS and CE components was observed in left supramarginal gyrus and no overlap was observed between VS and PL components. Our findings point to a central role of visuo-spatial WM during arithmetic problem-solving in young grade-school children and highlight the usefulness of the multi-component Baddeley and Hitch WM model in fractionating the neural correlates of arithmetic problem solving during development.

The study of developmental disorders can provide a unique window into the role of domain-general cognitive abilities and neural systems in typical and atypical development. Mathematical disabilities (MD) are characterized by marked difficulty in mathematical cognition in the presence of preserved intelligence and verbal ability. Although studies of MD have most often focused on the role of core deficits in numerical processing, domain-general cognitive abilities, in particular working memory (WM), have also been implicated. Here we identify specific WM components that are impaired in children with MD and then examine their role in arithmetic problem solving. Compared to typically developing (TD) children, the MD group demonstrated lower arithmetic performance and lower visuo-spatial working memory (VSWM) scores with preserved abilities on the phonological and central executive components of WM. Whole brain analysis revealed that, during arithmetic problem solving, left posterior parietal cortex, bilateral dorsolateral and ventrolateral prefrontal cortex, cingulate gyrus and precuneus, and fusiform gyrus responses were positively correlated with VSWM ability in TD children, but not in the MD group. Additional analyses using a priori posterior parietal cortex regions previously implicated in WM tasks, demonstrated a convergent pattern of results during arithmetic problem solving. These results suggest that MD is characterized by a common locus of arithmetic and VSWM deficits at both the cognitive and functional neuroanatomical levels. Unlike TD children, children with MD do not use VSWM resources appropriately during arithmetic problem solving. This work advances our understanding of VSWM as an important domain-general cognitive process in both typical and atypical mathematical skill development.

Cognitive development is shaped by brain plasticity during childhood, yet little is known about changes in large-scale functional circuits associated with learning in academically relevant cognitive domains such as mathematics. Here, we investigate plasticity of intrinsic brain circuits associated with one-on-one math tutoring and its relation to individual differences in children's learning. We focused on functional circuits associated with the intraparietal sulcus (IPS) and angular gyrus (AG), cytoarchitectonically distinct subdivisions of the human parietal cortex with different roles in numerical cognition. Tutoring improved performance and strengthened IPS connectivity with the lateral prefrontal cortex, ventral temporal-occipital cortex, and hippocampus. Crucially, increased IPS connectivity was associated with individual performance gains, highlighting the behavioral significance of plasticity in IPS circuits. Tutoring-related changes in IPS connectivity were distinct from those of the adjacent AG, which did not predict performance gains. Our findings provide new insights into plasticity of functional brain circuits associated with the development of specialized cognitive skills in children.

The Empathizing-Systemizing (E-S) theory describes a profile of traits that have been linked to autism spectrum disorders, and are thought to encompass a continuum that includes typically developing (TD) individuals. Although systemizing is hypothesized to be related to mathematical abilities, empirical support for this relationship is lacking. We examine the link between empathizing and systemizing tendencies and mathematical achievement in 112 TD children (57 girls) to elucidate how socio-cognitive constructs influence early development of mathematical skills. Assessment of mathematical achievement included standardized tests designed to examine calculation skills and conceptual mathematical reasoning. Empathizing and systemizing were assessed using the Combined Empathy Quotient-Child (EQ-C) and Systemizing Quotient-Child (SQ-C). Contrary to our hypothesis, we found that mathematical achievement was not related to systemizing or the discrepancy between systemizing and empathizing. Surprisingly, children with higher empathy demonstrated lower calculation skills. Further analysis using the Social Responsiveness Scale (SRS) revealed that the relationship between EQ-C and mathematical achievement was mediated by social ability rather than autistic behaviors. Finally, social awareness was found to play a differential role in mediating the relationship between EQ-C and mathematical achievement in girls. These results identify empathy, and social skills more generally, as previously unknown predictors of mathematical achievement.

Although lesion studies over the past several decades have focused on functional dissociations in posterior parietal cortex (PPC) during arithmetic, no consistent view has emerged of its differential involvement in addition, subtraction, multiplication, and division. To circumvent problems with poor anatomical localization, we examined functional overlap and dissociations in cytoarchitectonically defined subdivisions of the intraparietal sulcus (IPS), superior parietal lobule (SPL) and angular gyrus (AG), across these four operations. Compared to a number identification control task, all operations except addition, showed a consistent profile of left posterior IPS activation and deactivation in the right posterior AG. Multiplication and subtraction differed significantly in right, but not left, IPS and AG activity, challenging the view that the left AG differentially subserves retrieval during multiplication. Although addition and multiplication both rely on retrieval, multiplication evoked significantly greater activation in right posterior IPS, as well as the prefrontal cortex, lingual and fusiform gyri, demonstrating that addition and multiplication engage different brain processes. Comparison of PPC responses to the two pairs of inverse operations: division versus multiplication and subtraction versus addition revealed greater activation of left lateral SPL during division, suggesting that processing inverse relations is operation specific. Our findings demonstrate that individual IPS, SPL and AG subdivisions are differentially modulated by the four arithmetic operations and they point to significant functional heterogeneity and individual differences in activation and deactivation within the PPC. Critically, these effects are related to retrieval, calculation and inversion, the three key cognitive processes that are differentially engaged by arithmetic operations. Our findings point to distribute representation of these processes in the human PPC and also help explain why lesion and previous imaging studies have yielded inconsistent findings.

Competency with numbers is essential in today's society; yet, up to 20% of children exhibit moderate to severe mathematical learning disabilities (MLD). Behavioural intervention can be effective, but the neurobiological mechanisms underlying successful intervention are unknown. Here we demonstrate that eight weeks of 1:1 cognitive tutoring not only remediates poor performance in children with MLD, but also induces widespread changes in brain activity. Neuroplasticity manifests as normalization of aberrant functional responses in a distributed network of parietal, prefrontal and ventral temporal–occipital areas that support successful numerical problem solving, and is correlated with performance gains. Remarkably, machine learning algorithms show that brain activity patterns in children with MLD are significantly discriminable from neurotypical peers before, but not after, tutoring, suggesting that behavioural gains are not due to compensatory mechanisms. Our study identifies functional brain mechanisms underlying effective intervention in children with MLD and provides novel metrics for assessing response to intervention., Mathematical learning disabilities (MLD) affect up to 20% of children and are linked to poorer socioeconomic and health outcomes in adulthood. Here, Iuculano et al. show that tutoring induced functional changes in multiple brain systems underlie remediation of poor maths skills in children with MLD.

How the brain develops representations for abstract cognitive problems is a major unaddressed question in neuroscience. Here we tackle this fundamental question using arithmetic problem solving, a cognitive domain important for the development of mathematical reasoning. We first examined whether adults demonstrate common neural representations for addition and subtraction problems, two complementary arithmetic operations that manipulate the same quantities. We then examined how the common neural representations for the two problem types change with development. Whole-brain multivoxel representational similarity (MRS) analysis was conducted to examine common coding of addition and subtraction problems in children and adults. We found that adults exhibited significant levels of MRS between the two problem types, not only in the intra-parietal sulcus (IPS) region of the posterior parietal cortex (PPC), but also in ventral temporal-occipital, anterior temporal and dorsolateral prefrontal cortices. Relative to adults, children showed significantly reduced levels of MRS in these same regions. In contrast, no brain areas showed significantly greater MRS between problem types in children. Our findings provide novel evidence that the emergence of arithmetic problem solving skills from childhood to adulthood is characterized by maturation of common neural representations between distinct numerical operations, and involve distributed brain regions important for representing and manipulating numerical quantity. More broadly, our findings demonstrate that representational analysis provides a powerful approach for uncovering fundamental mechanisms by which children develop proficiencies that are a hallmark of human cognition.

Developmental dyscalculia (DD) is marked by specific deficits in processing numerical and mathematical information despite normal intelligence (IQ) and reading ability. We examined how brain circuits used by young children with DD to solve simple addition and subtraction problems differ from those used by typically developing (TD) children who were matched on age, IQ, reading ability, and working memory. Children with DD were slower and less accurate during problem solving than TD children, and were especially impaired on their ability to solve subtraction problems. Children with DD showed significantly greater activity in multiple parietal, occipito-temporal and prefrontal cortex regions while solving addition and subtraction problems. Despite poorer performance during subtraction, children with DD showed greater activity in multiple intra-parietal sulcus (IPS) and superior parietal lobule subdivisions in the dorsal posterior parietal cortex as well as fusiform gyrus in the ventral occipito-temporal cortex. Critically, effective connectivity analyses revealed hyper-connectivity, rather than reduced connectivity, between the IPS and multiple brain systems including the lateral fronto-parietal and default mode networks in children with DD during both addition and subtraction. These findings suggest the IPS and its functional circuits are a major locus of dysfunction during both addition and subtraction problem solving in DD, and that inappropriate task modulation and hyper-connectivity, rather than under-engagement and under-connectivity, are the neural mechanisms underlying problem solving difficulties in children with DD. We discuss our findings in the broader context of multiple levels of analysis and performance issues inherent in neuroimaging studies of typical and atypical development.

International audience; Background. Autism spectrum disorder (ASD) is a neurodevelopmental disorder characterized by social and communication deficits. While such deficits have been the focus of most research, recent evidence suggests that individuals with ASD may exhibit cognitive strengths in domains such as mathematics. Methods. Cognitive assessments and functional brain imaging were used to investigate mathematical abilities in 18 children with ASD and 18 age-, gender-, and IQ-matched typically developing (TD) children. Multivariate classification and regression analyses were used to investigate whether brain activity patterns during numerical problem solving were significantly different between the groups and predictive of individual mathematical abilities. Results. Children with ASD showed better numerical problem solving abilities and relied on sophisticated decomposition strategies for single-digit addition problems more frequently than TD peers. Although children with ASD engaged similar brain areas as TD children, they showed different multivariate activation patterns related to arithmetic problem complexity in ventral temporal-occipital cortex, posterior parietal cortex, and medial temporal lobe. Furthermore, multivariate activation patterns in ventral temporal-occipital cortical areas typically associated with face processing predicted individual numerical problem solving abilities in children with ASD but not in TD children. Conclusions.Our study suggests that superior mathematical information processing in children with ASD is characterized by a unique pattern of brain organization and that cortical regions typically involved in perceptual expertise may be utilized in novel ways in ASD. Our findings of enhanced cognitive and neural resources for mathematics have critical implications for educational, professional, and social outcomes for individuals with this lifelong disorder.

Now, more than ever, the ability to acquire mathematical skills efficiently is critical for academic and professional success, yet little is known about the behavioral and neural mechanisms that drive some children to acquire these skills faster than others. Here we investigate the behavioral and neural predictors of individual differences in arithmetic skill acquisition in response to 8-wk of one-to-one math tutoring. Twenty-four children in grade 3 (ages 8–9 y), a critical period for acquisition of basic mathematical skills, underwent structural and resting-state functional MRI scans pretutoring. A significant shift in arithmetic problem-solving strategies from counting to fact retrieval was observed with tutoring. Notably, the speed and accuracy of arithmetic problem solving increased with tutoring, with some children improving significantly more than others. Next, we examined whether pretutoring behavioral and brain measures could predict individual differences in arithmetic performance improvements with tutoring. No behavioral measures, including intelligence quotient, working memory, or mathematical abilities, predicted performance improvements. In contrast, pretutoring hippocampal volume predicted performance improvements. Furthermore, pretutoring intrinsic functional connectivity of the hippocampus with dorsolateral and ventrolateral prefrontal cortices and the basal ganglia also predicted performance improvements. Our findings provide evidence that individual differences in morphometry and connectivity of brain regions associated with learning and memory, and not regions typically involved in arithmetic processing, are strong predictors of responsiveness to math tutoring in children. More generally, our study suggests that quantitative measures of brain structure and intrinsic brain organization can provide a more sensitive marker of skill acquisition than behavioral measures.

Unlike natural numbers, negative numbers do not have natural physical referents. How does the brain represent such abstract mathematical concepts? Two competing hypotheses regarding representational systems for negative numbers are a rule-based model, in which symbolic rules are applied to negative numbers to translate them into positive numbers when assessing magnitudes, and an expanded magnitude model, in which negative numbers have a distinct magnitude representation. Using an event-related functional magnetic resonance imaging design, we examined brain responses in 22 adults while they performed magnitude comparisons of negative and positive numbers that were quantitatively near (difference 6). Reaction times (RTs) for negative numbers were slower than positive numbers, and both showed a distance effect whereby near pairs took longer to compare. A network of parietal, frontal, and occipital regions were differentially engaged by negative numbers. Specifically, compared to positive numbers, negative number processing resulted in greater activation bilaterally in intraparietal sulcus (IPS), middle frontal gyrus, and inferior lateral occipital cortex. Representational similarity analysis revealed that neural responses in the IPS were more differentiated among positive numbers than among negative numbers, and greater differentiation among negative numbers was associated with faster RTs. Our findings indicate that despite negative numbers engaging the IPS more strongly, the underlying neural representation are less distinct than that of positive numbers. We discuss our findings in the context of the two theoretical models of negative number processing and demonstrate how multivariate approaches can provide novel insights into abstract number representation.

Recent research into math cognition has identified areas of the brain that are involved in number processing (Dehaene, Piazza, Pinel, & Cohen, 2003) and complex problem solving (Anderson, 2007). Much of this research assumes that participants use a single strategy; yet, behavioral research finds that people use a variety of strategies (LeFevre et al., 1996; Siegler, 1987; Siegler & Lemaire, 1997). In the present study, we examined cortical activation as a function of two different calculation strategies for mentally solving multidigit multiplication problems. The school strategy, equivalent to long multiplication, involves working from right to left. The expert strategy, used by “lightning” mental calculators (Staszewski, 1988), proceeds from left to right. The two strategies require essentially the same calculations, but have different working memory demands (the school strategy incurs greater demands). The school strategy produced significantly greater early activity in areas involved in attentional aspects of number processing (posterior superior parietal lobule, PSPL) and mental representation (posterior parietal cortex, PPC), but not in a numerical magnitude area (horizontal intraparietal sulcus, HIPS) or a semantic memory retrieval area (lateral inferior prefrontal cortex, LIPFC). An ACT-R model of the task successfully predicted BOLD responses in PPC and LIPFC, as well as in PSPL and HIPS.

This article investigates the potential of fMRI to test assumptions about different components in models of complex cognitive tasks. If the components of a model can be associated with specific brain regions, one can make predictions for the temporal course of the BOLD response in these regions. An event-locked procedure is described for dealing with temporal variability and bringing model runs and individual data trials into alignment. Statistical methods for testing the model are described that deal with the scan-to-scan correlations in the errors of measurement of the BOLD signal. This approach is illustrated using a “sacrificial” ACT-R model that involves mapping 6 modules onto 6 brain regions in an experiment from Ravizza, Anderson, and Carter (in press) concerned with equation solving. The model's visual encoding predicted the BOLD response in the fusiform gyrus, its controlled retrieval predicted the BOLD response in the lateral inferior prefrontal cortex, and its subgoal setting predicted the BOLD response in the anterior cingulate cortex. On the other hand, its motor programming failed to predict anticipatory activation in the motor cortex, its representational changes failed to predicted the pattern of activity in the posterior parietal cortex, and its procedural component failed to predict an initial spike in caudate. The results illustrate the power of such data to direct the development of a theory of complex problem solving, both at the level of a specific task model as well as at the level of the cognitive architecture.

We concur with Cohen Kadosh & Walsh (CK&W) that representation of numbers in the parietal cortex is format dependent. In addition, we suggest that all formats do not automatically, and equally, access analog magnitude representation in the intraparietal sulcus (IPS). Understanding how development, learning, and context lead to differential access of analog magnitude representation is a key question for future research.