Dyscalculia: from brain to education

Patterns of linguistic and numerical performance in aphasia

BACKGROUND

Empirical research on the relationship between linguistic and numerical processing revealed inconsistent results for different levels of cognitive processing (e.g., lexical, semantic) as well as different stimulus materials (e.g., Arabic digits, number words, letters, non-number words). Information of dissociation patterns in aphasic patients was used in order to investigate the dissociability of linguistic and numerical processes. The aim of the present prospective study was a comprehensive, specific, and systematic investigation of relationships between linguistic and numerical processing, considering the impact of asemantic vs. semantic processing and the type of material employed (numbers compared to letters vs. words).

METHODS

A sample of aphasic patients (n = 60) was assessed with a battery of linguistic and numerical tasks directly comparable for their cognitive processing levels (e.g., perceptual, morpho-lexical, semantic).

RESULTS AND CONCLUSIONS

Mean performance differences and frequencies of (complementary) dissociations in individual patients revealed the most prominent numerical advantage for asemantic tasks when comparing the processing of numbers vs. letters, whereas the least numerical advantage was found for semantic tasks when comparing the processing of numbers vs. words. Different patient subgroups showing differential dissociation patterns were further analysed and discussed. A comprehensive model of linguistic and numerical processing should take these findings into account.

Plasticity of left perisylvian white-matter tracts is associated with individual differences in math learning

Plasticity of white matter tracts is thought to be essential for cognitive development and academic skill acquisition in children. However, a dearth of high-quality diffusion tensor imaging (DTI) data measuring longitudinal changes with learning, as well as methodological difficulties in multi-time point tract identification have limited our ability to investigate plasticity of specific white matter tracts. Here, we examine learning-related changes of white matter tracts innervating inferior parietal, prefrontal and temporal regions following an intense 2-month math tutoring program. DTI data were acquired from 18 third grade children, both before and after tutoring. A novel fiber tracking algorithm based on a White Matter Query Language (WMQL) was used to identify three sections of the superior longitudinal fasciculus (SLF) linking frontal and parietal (SLF-FP), parietal and temporal (SLF-PT) and frontal and temporal (SLF-FT) cortices, from which we created child-specific probabilistic maps. The SLF-FP, SLF-FT, and SLF-PT tracts identified with the WMQL method were highly reliable across the two time points and showed close correspondence to tracts previously described in adults. Notably, individual differences in behavioral gains after 2 months of tutoring were specifically correlated with plasticity in the left SLF-FT tract. Our results extend previous findings of individual differences in white matter integrity, and provide important new insights into white matter plasticity related to math learning in childhood. More generally, our quantitative approach will be useful for future studies examining longitudinal changes in white matter integrity associated with cognitive skill development.

Invisible Disabilities: Unique Challenges for Employees and Organizations

Workers with invisible disabilities encounter unique challenges compared to workers with other concealable identities and even workers with visible disabilities. These challenges occur not only in the decisions of whether to disclose the invisible disability in the workplace but also in the detection and acceptance of having a disability to disclose. Disclosure of disabilities in the workplace likely has implications for the individual worker's health, social relationships, and work performance as well as for an employing organization's outcomes. We argue that current legislation and policies might not be sensitive to the unique experiences and disclosure decisions faced by workers with invisible disabilities. We invite researchers and practitioners to consider adjustments to current legislation and workplace practices in order for employing organizations to account for the unique challenges facing workers with invisible disabilities and fully accommodate those workers.

Developmental dyscalculia

Numerical skills are essential in our everyday life, and impairments in the development of number processing and calculation have a negative impact on schooling and professional careers. Approximately 3 to 6 % of children are affected from specific disorders of numerical understanding (developmental dyscalculia (DD)). Impaired development of number processing skills in these children is characterized by problems in various aspects of numeracy as well as alterations of brain activation and brain structure. Moreover, DD is assumed to be a very heterogeneous disorder putting special challenges to define homogeneous diagnostic criteria. Finally, interdisciplinary perspectives from psychology, neuroscience and education can contribute to the design for interventions, and although results are still sparse, they are promising and have shown positive effects on behaviour as well as brain function.

CONCLUSION

In the current review, we are going to give an overview about typical and atypical development of numerical abilities at the behavioural and neuronal level. Furthermore, current status and obstacles in the definition and diagnostics of DD are discussed, and finally, relevant points that should be considered to make an intervention as successful as possible are summarized.

How number line estimation skills relate to neural activations in single digit subtraction problems

The Number Line (NL) task requires judging the relative numerical magnitude of a number and estimating its value spatially on a continuous line. Children's skill on this task has been shown to correlate with and predict future mathematical competence. Neurofunctionally, this task has been shown to rely on brain regions involved in numerical processing. However, there is no direct evidence that performance on the NL task is related to brain areas recruited during arithmetical processing and that these areas are domain-specific to numerical processing. In this study, we test whether 8- to 14-year-old's behavioral performance on the NL task is related to fMRI activation during small and large single-digit subtraction problems. Domain-specific areas for numerical processing were independently localized through a numerosity judgment task. Results show a direct relation between NL estimation performance and the amount of the activation in key areas for arithmetical processing. Better NL estimators showed a larger problem size effect than poorer NL estimators in numerical magnitude (i.e., intraparietal sulcus) and visuospatial areas (i.e., posterior superior parietal lobules), marked by less activation for small problems. In addition, the direction of the activation with problem size within the IPS was associated with differences in accuracies for small subtraction problems. This study is the first to show that performance in the NL task, i.e. estimating the spatial position of a number on an interval, correlates with brain activity observed during single-digit subtraction problem in regions thought to be involved in numerical magnitude and spatial processes.

Of magnitudes and metaphors: explaining cognitive interactions between space, time, and number

Space, time, and number are fundamental to how we act within and reason about the world. These three experiential domains are systematically intertwined in behavior, language, and the brain. Two main theories have attempted to account for cross-domain interactions. A Theory of Magnitude (ATOM) posits a domain-general magnitude system. Conceptual Metaphor Theory (CMT) maintains that cross-domain interactions are manifestations of asymmetric mappings that use representations of space to structure the domains of number and time. These theories are often viewed as competing accounts. We propose instead that ATOM and CMT are complementary, each illuminating different aspects of cross-domain interactions. We argue that simple representations of magnitude cannot, on their own, account for the rich, complex interactions between space, time and number described by CMT. On the other hand, ATOM is better at accounting for low-level and language-independent associations that arise early in ontogeny. We conclude by discussing how magnitudes and metaphors are both needed to understand our neural and cognitive web of space, time and number.

Arithmetic strategy development and its domain-specific and domain-general cognitive correlates: a longitudinal study in children with persistent mathematical learning difficulties

Deficits in arithmetic fact retrieval constitute the hallmark of children with mathematical learning difficulties (MLD). It remains, however, unclear which cognitive deficits underpin these difficulties in arithmetic fact retrieval. Many prior studies defined MLD by considering low achievement criteria and not by additionally taking the persistence of the MLD into account. Therefore, the present longitudinal study contrasted children with persistent MLD (MLD-p; mean age: 9 years 2 months) and typically developing (TD) children (mean age: 9 years 6 months) at three time points, to explore whether differences in arithmetic strategy development were associated with differences in numerical magnitude processing, working memory and phonological processing. Our longitudinal data revealed that children with MLD-p had persistent arithmetic fact retrieval deficits at each time point. Children with MLD-p showed persistent impairments in symbolic, but not in nonsymbolic, magnitude processing at each time point. The two groups differed in phonological processing, but not in working memory. Our data indicate that both domain-specific and domain-general cognitive abilities contribute to individual differences in children's arithmetic strategy development, and that the symbolic processing of numerical magnitudes might be a particular risk factor for children with MLD-p.

Numerical magnitude processing deficits in children with mathematical difficulties are independent of intelligence

Developmental dyscalculia (DD) is thought to arise from difficulties in the ability to process numerical magnitudes. Most research relied on IQ-discrepancy based definitions of DD and only included individuals with normal IQ, yet little is known about the role of intelligence in the association between numerical magnitude processing and mathematical difficulties (MD). The present study examined numerical magnitude processing in matched groups of 7- to 8-year-olds (n=42) who had either discrepant MD (poor math scores, average IQ), nondiscrepant MD (poor math scores, below-average IQ) or no MD. Both groups of children with MD showed similar impairments in numerical magnitudes processing compared to controls, suggesting that the association between numerical magnitude processing deficits and MD is independent of intelligence.

Identification of children with mathematics learning disabilities (MLDs) using latent class growth analysis

The traditional way of identifying children with mathematics learning disabilities (MLDs) using the low-achievement method with one-off assessment suffers from several limitations (e.g., arbitrary cutoff, measurement error, lacking consideration of growth). The present study attempted to identify children with MLD using the latent growth modelling approach, which minimizes the above potential problems. Two hundred and ten Chinese-speaking children were classified into five classes based on their arithmetic performance over 3 years. Their performance on various number-related cognitive measures was also assessed. A potential MLD class was identified, which demonstrated poor achievement over the 3 years and showed smaller improvement over time compared with the average-achieving class. This class had deficits in all number-related cognitive skills, hence supporting the number sense deficit hypothesis. On the other hand, another low-achieving class, which showed little improvement in arithmetic skills over time, was also identified. This class had an average cognitive profile but a low SES. Interventions should be provided to both low-achieving classes according to their needs.

Effect of working memory training on working memory, arithmetic and following instructions

Mathematical ability is dependent on specific mathematical training but also associated with a range of cognitive factors, including working memory (WM) capacity. Previous studies have shown that WM training leads to improvement in non-trained WM tasks, but the results regarding transfer to mathematics are inconclusive. In the present study, 176 children with WM deficits, aged 7-15 years performed 5 weeks of WM training. During the training period, they were assessed five times with a test of complex WM (the Odd One Out), a test of remembering and following instructions and a test of arithmetic. The improvements were compared to the performance of a control group of 304 typically developing children aged 7-15 years who performed the same transfer tasks at the same time intervals, but without training. The training group improved significantly more than the control group on all three transfer tests (all p < 0.0001), after correction for baseline performance, age and sex. The effect size for mathematics was small and the effect sizes for the WM tasks were moderate to large. The transfer increased linearly with the amount of training time and correlated with the amount of improvement on the trained tasks. These results confirm previous findings of training-induced improvements in non-trained WM tasks including the ability to follow instructions, but extend previous findings by showing improvements also for arithmetic. This is encouraging regarding the potential role of cognitive training for education, but it is desirable to find paradigms that would enhance the effect of the training on mathematics. One of the future challenges for studying training effects is combining large sample sizes with high quality and compliance, to detect relevant but smaller effects of cognitive training.

The arithmetic problem size effect in children: an event-related potential study

This study used for the first time event-related potentials (ERPs) to examine the well-known arithmetic problem size effect in children. The electrophysiological correlates of this problem size effect have been well documented in adults, but such information in children is lacking. In the present study, 22 typically developing 12-year-olds were asked to solve single-digit addition problems of small (sum ≤ 10) and large problem size (sum > 10) and to speak the solution into a voice key while ERPs were recorded. Children displayed similar early and late components compared to previous adult studies on the problem size effect. There was no effect of problem size on the early components P1, N1, and P2. The peak amplitude of the N2 component showed more negative potentials on left and right anterior electrodes for large additions compared to small additions, which might reflect differences in attentional and working memory resources between large and small problems. The mean amplitude of the late positivity component which follows the N2, was significantly larger for large than for small additions at right parieto-occipital electrodes, in line with previous adult data. The ERPs of the problem size effect during arithmetic might be a useful neural marker for future studies on fact retrieval impairments in children with mathematical difficulties.

Pre-school children’s bridge to symbolic knowledge: first literature framework for a learning and cognition lab at a South African university

The authors present the theoretical groundwork for a research project on learning and cognitive development of number concepts in the early years of childhood. Giving a background sketch of the genesis of a learning and cognition lab at a university in the metropolitan heartland of South Africa, they present their initial literature framework for inquiries into children’s symbolic learning of number in the pre-school years. They argue that conceptual development of young children is a neglected area in childhood cognition research in South Africa. The study of some of the literature for the first project of the new lab is then introduced with a view of identifying a few of the main components of a conceptual framework for what will become a multiple-year study. The authors propose that this literature can serve as foundation for examining a linguistically diverse group of children’s responses on experimental tasks and in clinical interviews in four or more languages. The designs of these inquiries are imminent. They suggest that the views of leading authors such as Elizabeth Spelke, Susan Carey, and Stanislas Dehaene can shed much light on their understanding of early number concept development of South African children.

Brain hyper-connectivity and operation-specific deficits during arithmetic problem solving in children with developmental dyscalculia

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.

The functional anatomy of single-digit arithmetic in children with developmental dyslexia

Some arithmetic procedures, such as addition of small numbers, rely on fact retrieval mechanisms supported by left hemisphere perisylvian language areas, while others, such as subtraction, rely on procedural-based mechanisms subserved by bilateral parietal cortices. Previous work suggests that developmental dyslexia, a reading disability, is accompanied by subtle deficits in retrieval-based arithmetic, possibly because of compromised left hemisphere function. To test this prediction, we compared brain activity underlying arithmetic problem solving in children with and without dyslexia during addition and subtraction operations using a factorial design. The main effect of arithmetic operation (addition versus subtraction) for both groups combined revealed activity during addition in the left superior temporal gyrus and activity during subtraction in the bilateral intraparietal sulcus, the right supramarginal gyrus and the anterior cingulate, consistent with prior studies. For the main effect of diagnostic group (dyslexics versus controls), we found less activity in dyslexic children in the left supramarginal gyrus. Finally, the interaction analysis revealed that while the control group showed a strong response in the right supramarginal gyrus for subtraction but not for addition, the dyslexic group engaged this region for both operations. This provides physiological evidence in support of the theory that children with dyslexia, because of disruption to left hemisphere language areas, use a less optimal route for retrieval-based arithmetic, engaging right hemisphere parietal regions typically used by good readers for procedural-based arithmetic. Our results highlight the importance of language processing for mathematical processing and illustrate that children with dyslexia have impairments that extend beyond reading.

Are 22q11.2 distal deletions associated with math difficulties?

Approximately 6% of school-aged children have math difficulties (MD). A neurogenetic etiology has been suggested due to the presence of MD in some genetic syndromes such as 22q11.2DS. However, the contribution of 22q11.2DS to the MD phenotype has not yet been investigated. This is the first population-based study measuring the frequency of 22q11.2DS among school children with MD. Children (1,564) were identified in the schools through a screening test for language and math. Of these children, 152 (82 with MD and 70 controls) were selected for intelligence, general neuropsychological, and math cognitive assessments and for 22q11.2 microdeletion screening using MLPA. One child in the MD group had a 22q11.2 deletion spanning the LCR22-4 to LCR22-5 interval. This child was an 11-year-old girl with subtle anomalies, normal intelligence, MD attributable to number sense deficit, and difficulties in social interactions. Only 19 patients have been reported with this deletion. Upon reviewing these reports, we were able to characterize a new syndrome, 22q11.2 DS (LCR22-4 to LCR22-5), characterized by prematurity; pre- and postnatal growth restriction; apparent hypotelorism, short/upslanting palpebral fissures; hypoplastic nasal alae; pointed chin and nose; posteriorly rotated ears; congenital heart defects; skeletal abnormalities; developmental delay, particularly compromising the speech; learning disability (including MD, in one child); intellectual disability; and behavioral problems. These results suggest that 22q11.2 DS (LCR22-4 to LCR22-5) may be one of the genetic causes of MD.

Reading instead of reasoning? Predictors of arithmetic skills in children with cochlear implants

OBJECTIVES

The aim of the present study was to evaluate whether the arithmetic achievement of children with cochlear implants (CI) was lower or comparable to that of their normal hearing peers and to identify predictors of arithmetic achievement in children with CI. In particular we related the arithmetic achievement of children with CI to nonverbal IQ, reading skills and hearing variables.

METHODS

23 children with CI (onset of hearing loss in the first 24 months, cochlear implantation in the first 60 months of life, atleast 3 years of hearing experience with the first CI) and 23 normal hearing peers matched by age, gender, and social background participated in this case control study. All attended grades two to four in primary schools. To assess their arithmetic achievement, all children completed the "Arithmetic Operations" part of the "Heidelberger Rechentest" (HRT), a German arithmetic test. To assess reading skills and nonverbal intelligence as potential predictors of arithmetic achievement, all children completed the "Salzburger Lesetest" (SLS), a German reading screening, and the Culture Fair Intelligence Test (CFIT), a nonverbal intelligence test.

RESULTS

Children with CI did not differ significantly from hearing children in their arithmetic achievement. Correlation and regression analyses revealed that in children with CI, arithmetic achievement was significantly (positively) related to reading skills, but not to nonverbal IQ. Reading skills and nonverbal IQ were not related to each other. In normal hearing children, arithmetic achievement was significantly (positively) related to nonverbal IQ, but not to reading skills. Reading skills and nonverbal IQ were positively correlated. Hearing variables were not related to arithmetic achievement.

CONCLUSIONS

Children with CI do not show lower performance in non-verbal arithmetic tasks, compared to normal hearing peers.

Preterm birth and dyscalculia

OBJECTIVE

To evaluate whether the risk for dyscalculia in preterm children increases the lower the gestational age (GA) and whether small-for-gestational age birth is associated with dyscalculia.

STUDY DESIGN

A total of 922 children ranging from 23 to 41 weeks' GA were studied as part of a prospective geographically defined longitudinal investigation of neonatal at-risk children in South Germany. At 8 years of age, children's cognitive and mathematic abilities were measured with the Kaufman Assessment Battery for Children and with a standardized mathematics test. Dyscalculia diagnoses were evaluated with discrepancy-based residuals of a linear regression predicting children's math scores by IQ and with fixed cut-off scores. We investigated each GA group's ORs for general cognitive impairment, general mathematic impairment, and dyscalculia by using binary logistic regressions.

RESULTS

The risk for general cognitive and mathematic impairment increased with lower GA. In contrast, preterm children were not at increased risk of dyscalculia after statistically adjusting for child sex, family socioeconomic status, and small-for-gestational age birth.

CONCLUSION

The risk of general cognitive and mathematic impairments increases with lower GA but preterm children are not at increased risk of dyscalculia.

Association between individual differences in non-symbolic number acuity and math performance: a meta-analysis

Many recent studies have examined the association between number acuity, which is the ability to rapidly and non-symbolically estimate the quantity of items appearing in a scene, and symbolic math performance. However, various contradictory results have been reported. To comprehensively evaluate the association between number acuity and symbolic math performance, we conduct a meta-analysis to synthesize the results observed in previous studies. First, a meta-analysis of cross-sectional studies (36 samples, N = 4705) revealed a significant positive correlation between these skills (r = 0.20, 95% CI = [0.14, 0.26]); the association remained after considering other potential moderators (e.g., whether general cognitive abilities were controlled). Moreover, a meta-analysis of longitudinal studies revealed 1) that number acuity may prospectively predict later math performance (r = 0.24, 95% CI = [0.11, 0.37]; 6 samples) and 2) that number acuity is retrospectively correlated to early math performance as well (r = 0.17, 95% CI = [0.07, 0.26]; 5 samples). In summary, these pieces of evidence demonstrate a moderate but statistically significant association between number acuity and math performance. Based on the estimated effect sizes, power analyses were conducted, which suggested that many previous studies were underpowered due to small sample sizes. This may account for the disparity between findings in the literature, at least in part. Finally, the theoretical and practical implications of our meta-analytic findings are presented, and future research questions are discussed.

Discovering the structure of mathematical problem solving

The goal of this research is to discover the stages of mathematical problem solving, the factors that influence the duration of these stages, and how these stages are related to the learning of a new mathematical competence. Using a combination of multivariate pattern analysis (MVPA) and hidden Markov models (HMM), we found that participants went through 5 major phases in solving a class of problems: A Define Phase where they identified the problem to be solved, an Encode Phase where they encoded the needed information, a Compute Phase where they performed the necessary arithmetic calculations, a Transform Phase where they performed any mathematical transformations, and a Respond Phase where they entered an answer. The Define Phase is characterized by activity in visual attention and default network regions, the Encode Phase by activity in visual regions, the Compute Phase by activity in regions active in mathematical tasks, the Transform Phase by activity in mathematical and response regions, and the Respond phase by activity in motor regions. The duration of the Compute and Transform Phases were the only ones that varied with condition. Two features distinguished the mastery trials on which participants came to understand a new problem type. First, the duration of late phases of the problem solution increased. Second, there was increased activation in the rostrolateral prefrontal cortex (RLPFC) and angular gyrus (AG), regions associated with metacognition. This indicates the importance of reflection to successful learning.

Children's mapping between non-symbolic and symbolic numerical magnitudes and its association with timed and untimed tests of mathematics achievement

The ability to map between non-symbolic numerical magnitudes and Arabic numerals has been put forward as a key factor in children’s mathematical development. This mapping ability has been mainly examined indirectly by looking at children’s performance on a symbolic magnitude comparison task. The present study investigated mapping in a more direct way by using a task in which children had to choose which of two choice quantities (Arabic digits or dot arrays) matched the target quantity (dot array or Arabic digit), thereby focusing on small quantities ranging from 1 to 9. We aimed to determine the development of mapping over time and its relation to mathematics achievement. Participants were 36 first graders (M = 6 years 8 months) and 46 third graders (M = 8 years 8 months) who all completed mapping tasks, symbolic and non-symbolic magnitude comparison tasks and standardized timed and untimed tests of mathematics achievement. Findings revealed that children are able to map between non-symbolic and symbolic representations and that this mapping ability develops over time. Moreover, we found that children’s mapping ability is related to timed and untimed measures of mathematics achievement, over and above the variance accounted for by their numerical magnitude comparison skills.

Mathematical difficulties as decoupling of expectation and developmental trajectories

Recent years have seen an increase in research articles and reviews exploring mathematical difficulties (MD). Many of these articles have set out to explain the etiology of the problems, the possibility of different subtypes, and potential brain regions that underlie many of the observable behaviors. These articles are very valuable in a research field, which many have noted, falls behind that of reading and language disabilities. Here will provide a perspective on the current understanding of MD from a different angle, by outlining the school curriculum of England and the US and connecting these to the skills needed at different stages of mathematical understanding. We will extend this to explore the cognitive skills which most likely underpin these different stages and whose impairment may thus lead to mathematics difficulties at all stages of mathematics development. To conclude we will briefly explore interventions that are currently available, indicating whether these can be used to aid the different children at different stages of their mathematical development and what their current limitations may be. The principal aim of this review is to establish an explicit connection between the academic discourse, with its research base and concepts, and the developmental trajectory of abstract mathematical skills that is expected (and somewhat dictated) in formal education. This will possibly help to highlight and make sense of the gap between the complexity of the MD range in real life and the state of its academic science.

Preliminary evidence for performance enhancement following parietal lobe stimulation in Developmental Dyscalculia

Nearly 7% of the population exhibit difficulties in dealing with numbers and performing arithmetic, a condition named Developmental Dyscalculia (DD), which significantly affects the educational and professional outcomes of these individuals, as it often persists into adulthood. Research has mainly focused on behavioral rehabilitation, while little is known about performance changes and neuroplasticity induced by the concurrent application of brain-behavioral approaches. It has been shown that numerical proficiency can be enhanced by applying a small-yet constant-current through the brain, a non-invasive technique named transcranial electrical stimulation (tES). Here we combined a numerical learning paradigm with transcranial direct current stimulation (tDCS) in two adults with DD to assess the potential benefits of this methodology to remediate their numerical difficulties. Subjects learned to associate artificial symbols to numerical quantities within the context of a trial and error paradigm, while tDCS was applied to the posterior parietal cortex (PPC). The first subject (DD1) received anodal stimulation to the right PPC and cathodal stimulation to the left PPC, which has been associated with numerical performance's improvements in healthy subjects. The second subject (DD2) received anodal stimulation to the left PPC and cathodal stimulation to the right PPC, which has been shown to impair numerical performance in healthy subjects. We examined two indices of numerical proficiency: (i) automaticity of number processing; and (ii) mapping of numbers onto space. Our results are opposite to previous findings with non-dyscalculic subjects. Only anodal stimulation to the left PPC improved both indices of numerical proficiency. These initial results represent an important step to inform the rehabilitation of developmental learning disabilities, and have relevant applications for basic and applied research in cognitive neuroscience, rehabilitation, and education.

The Contributions of Domain-General and Numerical Factors to Third-Grade Arithmetic Skills and Mathematical Learning Disability

Explanations of the marked individual differences in elementary school mathematical achievement and mathematical learning disability (MLD or dyscalculia) have involved domain-general factors (working memory, reasoning, processing speed, and oral language) and numerical factors that include single-digit processing efficiency and multidigit skills such as number system knowledge and estimation. This study of 3rd graders (N = 258) finds both domain-general and numerical factors contribute independently to explaining variation in 3 significant arithmetic skills: basic calculation fluency, written multidigit computation, and arithmetic word problems. Estimation accuracy and number system knowledge show the strongest associations with every skill, and their contributions are independent of both each other and other factors. Different domain-general factors independently account for variation in each skill. Numeral comparison, a single digit processing skill, uniquely accounts for variation in basic calculation. Subsamples of children with MLD (at or below 10th percentile, n = 29) are compared with low achievement (LA, 11th to 25th percentiles, n = 42) and typical achievement (above 25th percentile, n = 187). Examination of these and subsets with persistent difficulties supports a multiple deficits view of number difficulties: Most children with number difficulties exhibit deficits in both domain-general and numerical factors. The only factor deficit common to all persistent MLD children is in multidigit skills. These findings indicate that many factors matter but multidigit skills matter most in 3rd grade mathematical achievement.

Enumeration of small and large numerosities in adolescents with mathematical learning disorders

The accuracy and speed in an enumeration task were investigated in adolescents with typical and atypically poor development of arithmetic skills. The number naming performances on small and large non-symbolic numerosities of 18 adolescents with mathematical learning disorders (MLD) and 28 typically achieving age-matched (TA) adolescents were compared. A mixed logistic regression model showed that adolescents with MLD were not significantly less accurate on numbers within the subitizing range than control peers. Moreover, no significant differences in reaction times were found between both groups. Nevertheless, we found that within the control group adolescents with higher ability tended to respond faster when taking into account the whole range (1-9) of numerosities. This correlation was much weaker in the MLD group. When looking more closely at the data, however, it became clear that the correlation between accuracy and speed within the control group differed in direction dependent on the range (subitizing or counting) of the numerosities. As such, our findings did not support a limited capacity of subitizing in MLD. However, the data stressed a different correlation between speed and accuracy for both groups of adolescents and a different behavioral pattern depending on the numerosity range as well. Implications for the understanding and approach of MLD are considered.

The association between children's numerical magnitude processing and mental multi-digit subtraction

Children apply various strategies to mentally solve multi-digit subtraction problems and the efficient use of some of them may depend more or less on numerical magnitude processing. For example, the indirect addition strategy (solving 72-67 as "how much do I have to add up to 67 to get 72?"), which is particularly efficient when the two given numbers are close to each other, requires to determine the proximity of these two numbers, a process that may depend on numerical magnitude processing. In the present study, children completed a numerical magnitude comparison task and a number line estimation task, both in a symbolic and nonsymbolic format, to measure their numerical magnitude processing. We administered a multi-digit subtraction task, in which half of the items were specifically designed to elicit indirect addition. Partial correlational analyses, controlling for intellectual ability and motor speed, revealed significant associations between numerical magnitude processing and mental multi-digit subtraction. Additional analyses indicated that numerical magnitude processing was particularly important for those items for which the use of indirect addition is expected to be most efficient. Although this association was observed for both symbolic and nonsymbolic tasks, the strongest associations were found for the symbolic format, and they seemed to be more prominent on numerical magnitude comparison than on number line estimation.

Diagnosing developmental dyscalculia on the basis of reliable single case FMRI methods: promises and limitations

FMRI-studies are mostly based on a group study approach, either analyzing one group or comparing multiple groups, or on approaches that correlate brain activation with clinically relevant criteria or behavioral measures. In this study we investigate the potential of fMRI-techniques focusing on individual differences in brain activation within a test-retest reliability context. We employ a single-case analysis approach, which contrasts dyscalculic children with a control group of typically developing children. In a second step, a support-vector machine analysis and cluster analysis techniques served to investigate similarities in multivariate brain activation patterns. Children were confronted with a non-symbolic number comparison and a non-symbolic exact calculation task during fMRI acquisition. Conventional second level group comparison analysis only showed small differences around the angular gyrus bilaterally and the left parieto-occipital sulcus. Analyses based on single-case statistical procedures revealed that developmental dyscalculia is characterized by individual differences predominantly in visual processing areas. Dyscalculic children seemed to compensate for relative under-activation in the primary visual cortex through an upregulation in higher visual areas. However, overlap in deviant activation was low for the dyscalculic children, indicating that developmental dyscalculia is a disorder characterized by heterogeneous brain activation differences. Using support vector machine analysis and cluster analysis, we tried to group dyscalculic and typically developing children according to brain activation. Fronto-parietal systems seem to qualify for a distinction between the two groups. However, this was only effective when reliable brain activations of both tasks were employed simultaneously. Results suggest that deficits in number representation in the visual-parietal cortex get compensated for through finger related aspects of number representation in fronto-parietal cortex. We conclude that dyscalculic children show large individual differences in brain activation patterns. Nonetheless, the majority of dyscalculic children can be differentiated from controls employing brain activation patterns when appropriate methods are used.

At the intersection of math and reading disabilities: introduction to the special issue

Individuals with comorbid disabilities in mathematics and reading face significant challenges in acquiring the componential skills related to each domain. Persons with these comorbid conditions are significantly understudied and this paucity of work limits how effective practitioners can be at addressing the needs of this population. In the United States, roughly 7% of all children suffer from math disability; of these, an estimated 17% to 66% also has a comorbid reading disability. Underspecification of current conceptualizations of math and reading disabilities, including how to best identify and classify individuals with one or more of these disabilities, hampers our efforts to intervene effectively. To conceptualize how to move forward in these areas, the Eunice Kennedy Shriver National Institute of Child Health and Human Development held a workshop focused on examining the etiology, classification, and remediation of comorbid math and reading disabilities. This special issue, titled At the Intersection of Math and Reading Disabilities, continues that discussion. Contributing authors articulate a path forward to address the needs of these learners and inform the foundational understanding of both conditions in isolation and as they interact.

Neurobiological underpinnings of math and reading learning disabilities

The primary goal of this review is to highlight current research and theories describing the neurobiological basis of math (MD), reading (RD), and comorbid math and reading disability (MD+RD). We first describe the unique brain and cognitive processes involved in acquisition of math and reading skills, emphasizing similarities and differences in each domain. Next we review functional imaging studies of MD and RD in children, integrating relevant theories from experimental psychology and cognitive neuroscience to characterize the functional neuroanatomy of cognitive dysfunction in MD and RD. We then review recent research on the anatomical correlates of MD and RD. Converging evidence from morphometry and tractography studies are presented to highlight distinct patterns of white matter pathways which are disrupted in MD and RD. Finally, we examine how the intersection of MD and RD provides a unique opportunity to clarify the unique and shared brain systems which adversely impact learning and skill acquisition in MD and RD, and point out important areas for future work on comorbid learning disabilities.

The added value of eye-tracking in diagnosing dyscalculia: a case study

The present study compared eye movements and performance of a 9-year-old girl with Developmental Dyscalculia (DD) on a series of number line tasks to those of a group of typically developing (TD) children (n = 10), in order to answer the question whether eye-tracking data from number line estimation tasks can be a useful tool to discriminate between TD children and children with a number processing deficit. Quantitative results indicated that the child with dyscalculia performed worse on all symbolic number line tasks compared to the control group, indicated by a low linear fit (R (2)) and a low accuracy measured by mean percent absolute error. In contrast to the control group, her magnitude representations seemed to be better represented by a logarithmic than a linear fit. Furthermore, qualitative analyses on the data of the child with dyscalculia revealed more unidentifiable fixation patterns in the processing of multi-digit numbers and more dysfunctional estimation strategy use in one third of the estimation trials as opposed to ~10% in the control group. In line with her dyscalculia diagnosis, these results confirm the difficulties with spatially representing and manipulating numerosities on a number line, resulting in inflexible and inadequate estimation or processing strategies. It can be concluded from this case study that eye-tracking data can be used to discern different number processing and estimation strategies in TD children and children with a number processing deficit. Hence, eye-tracking data in combination with number line estimation tasks might be a valuable and promising addition to current diagnostic measures.

The development of numerical magnitude processing and its association with working memory in children with mild intellectual disabilities

The present research examined numerical magnitude processing and its association with working memory in children with mild intellectual disabilities (MID). We investigated the performance of 8-year-old children with MID on a symbolic (Arabic digits) and non-symbolic (dot patterns) magnitude comparison task by means of a chronological-age/ability-level-match design. We also examined whether the predicted problems with numerical magnitude comparison could be explained by working memory by using three working memory tasks. Findings revealed that children with MID performed more poorly than their chronological age-matched peers on both the symbolic and non-symbolic magnitude comparison tasks, suggesting impairments in these children's ability to represent numerical magnitudes. They also performed more poorly on working memory compared to their typically developing age- and ability-matched peers, but when these differences in working memory performance were additionally controlled for, the group differences on the numerical magnitude comparison tasks remained. Both symbolic numerical magnitude processing and central executive functioning predicted addition performance in children with MID.

Numerical matching judgments in children with mathematical learning disabilities

Both deficits in the innate magnitude representation (i.e. representation deficit hypothesis) and deficits in accessing the magnitude representation from symbols (i.e. access deficit hypotheses) have been proposed to explain mathematical learning disabilities (MLD). Evidence for these hypotheses has mainly been accumulated through the use of numerical magnitude comparison tasks. It has been argued that the comparison distance effect might reflect decision processes on activated magnitude representations rather than number processing per se. One way to avoid such decisional processes confounding the numerical distance effect is by using a numerical matching task, in which children have to indicate whether two dot-arrays or a dot-array and a digit are numerically the same or different. Against this background, we used a numerical matching task to examined the representation deficit and access deficit hypotheses in a group children with MLD and controls matched on age, gender and IQ. The results revealed that children with MLD were slower than controls on the mixed notation trials, whereas no difference was found for the non-symbolic trials. This might be in line with the access deficit hypothesis, showing that children with MLD have difficulties in linking a symbol with its quantity representation. However, further investigation is required to exclude the possibility that children with MLD have a deficit in integrating the information from different input notations.

Visuo-spatial working memory is an important source of domain-general vulnerability in the development of arithmetic cognition

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.

Design and evaluation of the computer-based training program Calcularis for enhancing numerical cognition

This article presents the design and a first pilot evaluation of the computer-based training program Calcularis for children with developmental dyscalculia (DD) or difficulties in learning mathematics. The program has been designed according to insights on the typical and atypical development of mathematical abilities. The learning process is supported through multimodal cues, which encode different properties of numbers. To offer optimal learning conditions, a user model completes the program and allows flexible adaptation to a child's individual learning and knowledge profile. Thirty-two children with difficulties in learning mathematics completed the 6–12-weeks computer training. The children played the game for 20 min per day for 5 days a week. The training effects were evaluated using neuropsychological tests. Generally, children benefited significantly from the training regarding number representation and arithmetic operations. Furthermore, children liked to play with the program and reported that the training improved their mathematical abilities.

Infusing Neuroscience into Teacher Professional Development

Bruer (1997) advocated connecting neuroscience and education indirectly through the intermediate discipline of psychology. We argue for a parallel route: the neurobiology of learning, and in particular the core concept of plasticity, have the potential to directly transform teacher preparation and professional development, and ultimately to affect how students think about their own learning. We present a case study of how the core concepts of neuroscience can be brought to in-service teachers - the BrainU workshops. We then discuss how neuroscience can be meaningfully integrated into pre-service teacher preparation, focusing on institutional and cultural barriers.

Development of numerical processing in children with typical and dyscalculic arithmetic skills-a longitudinal study

Numerical processing has been demonstrated to be closely associated with arithmetic skills, however, our knowledge on the development of the relevant cognitive mechanisms is limited. The present longitudinal study investigated the developmental trajectories of numerical processing in 42 children with age-adequate arithmetic development and 41 children with dyscalculia over a 2-year period from beginning of Grade 2, when children were 7; 6 years old, to beginning of Grade 4. A battery of numerical processing tasks (dot enumeration, non-symbolic and symbolic comparison of one- and two-digit numbers, physical comparison, number line estimation) was given five times during the study (beginning and middle of each school year). Efficiency of numerical processing was a very good indicator of development in numerical processing while within-task effects remained largely constant and showed low long-term stability before middle of Grade 3. Children with dyscalculia showed less efficient numerical processing reflected in specifically prolonged response times. Importantly, they showed consistently larger slopes for dot enumeration in the subitizing range, an untypically large compatibility effect when processing two-digit numbers, and they were consistently less accurate in placing numbers on a number line. Thus, we were able to identify parameters that can be used in future research to characterize numerical processing in typical and dyscalculic development. These parameters can also be helpful for identification of children who struggle in their numerical development.

Learning and cognitive disorders: multidiscipline treatment approaches

This article provides a select review of treatments for addressing reading disorder, mathematics disorder, disorder of written expression, auditory processing disorder, and poor working memory. This information will be valuable to practitioners in determining the suitability of certain treatments for these various disorders and problems, which has direct implications for providing comprehensive, multidisciplinary treatment for youth.

[Symptoms diagnosis and treatment of dyscalulia]

Children with dyscalculia show deficits in basic numerical processing which cause difficulties in the acquisition of mathematical skills. This article provides an overview of current research findings regarding the symptoms, cause, and prognosis of dyscalculia, and it summarizes recent developments in the diagnosis, early intervention, and treatment thereof. Diagnosis has improved recently because newly developed tests focus not only on the math curriculum, but also on basic skills found to be impaired in dyscalculia. A controversial debate continues with regard to IQ achievement discrepancy. International studies have demonstrated the effectiveness of specialized interventions. This article summarizes the research findings from intervention studies, describes different treatment approaches, and discusses implications for clinical practice.

Long-term enhancement of brain function and cognition using cognitive training and brain stimulation

Noninvasive brain stimulation has shown considerable promise for enhancing cognitive functions by the long-term manipulation of neuroplasticity. However, the observation of such improvements has been focused at the behavioral level, and enhancements largely restricted to the performance of basic tasks. Here, we investigate whether transcranial random noise stimulation (TRNS) can improve learning and subsequent performance on complex arithmetic tasks. TRNS of the bilateral dorsolateral prefrontal cortex (DLPFC), a key area in arithmetic, was uniquely coupled with near-infrared spectroscopy (NIRS) to measure online hemodynamic responses within the prefrontal cortex. Five consecutive days of TRNS-accompanied cognitive training enhanced the speed of both calculation- and memory-recall-based arithmetic learning. These behavioral improvements were associated with defined hemodynamic responses consistent with more efficient neurovascular coupling within the left DLPFC. Testing 6 months after training revealed long-lasting behavioral and physiological modifications in the stimulated group relative to sham controls for trained and nontrained calculation material. These results demonstrate that, depending on the learning regime, TRNS can induce long-term enhancement of cognitive and brain functions. Such findings have significant implications for basic and translational neuroscience, highlighting TRNS as a viable approach to enhancing learning and high-level cognition by the long-term modulation of neuroplasticity.