An open trial assessment of "The Number Race", an adaptive computer game for remediation of dyscalculia

Auditory time perception impairment in children with developmental dyscalculia

Developmental dyscalculia (DD) is a specific learning disability which prevents children from acquiring adequate numerical and arithmetical competences. We investigated whether difficulties in children with DD spread beyond the numerical domain and impact also their ability to perceive time. A group of 37 children/adolescent with and without DD were tested with an auditory categorization task measuring time perception thresholds in the sub-second (0.25-1 s) and supra-second (0.75-3 s) ranges. Results showed that auditory time perception was strongly impaired in children with DD at both time scales. The impairment remained even when age, non-verbal reasoning, and gender were regressed out. Overall, our results show that the difficulties of DD can affect magnitudes other than numerical and contribute to the increasing evidence that frames dyscalculia as a disorder affecting multiple neurocognitive and perceptual systems.

Enhancing Time Reading and Recording Skills in First-Grade Children with Learning Difficulties Using the "Clock Motor Game"

This study aimed to explore the effect of the motor game, "Clock Motor Games", on the improvement of "Reading and Recording of Time" (RRT) in children with Grade 1 mathematical learning difficulties (MLDs). A within-school cluster-randomized intervention study was conducted with 232 children (aged 6-7 years) with limited physical education experience (0.7 ± 0.3 years). The participants were divided into two groups: a control group, which received conventional teaching on time without any additional motor activities, and an experimental group, which incorporated the concept of time with the "Clock Motor Game", for 3 weeks. The Clock-Reading Test was administered before the intervention (T0), immediately after each session (T1), and five weeks after the intervention (T2) in both groups. The results demonstrated that the experimental group exhibited significantly greater improvements in RRT performance compared to the control group (U = 4416.5; p < 0.001; r = 0.3; medium effect). Additionally, the experimental group was more likely to show progress and less likely to experience regression or stagnation compared to the control group (25% vs. 38.4%). The findings suggest that practicing "Clock Motor Games" can positively contribute to the RRT ability in children with Grade 1 MLD.

Visuo-Spatial Working Memory and Mathematical Skills in Children: A Network Analysis Study

Visuo-spatial working memory is one of the main domain-general cognitive mechanisms underlying mathematical abilities and their development in children. However, if visuo-spatial working memory involves different processes and components, then the term 'mathematics' refers to a broad concept that includes multiple domains and skills. The aim of this present study was to investigate the relationship between different visuo-spatial working memory components and several mathematical abilities in a sample of third- to fifth-grade Italian children. To assess the relationships between different visuo-spatial working memory components and different mathematical abilities, we relied on Network Analysis (NA). Results indicate that some but not all visuo-spatial working memory components are associated with some mathematical abilities.

Math computerized games in the classroom: A number line training in primary school children

Several studies have shown that the number line can be a useful tool to support early numeracy development. Here, we conducted a school-based training study to evaluate the effectiveness of the software "The Number Line" ("La Linea Dei Numeri"; Tressoldi and Peroni, 2013) in improving children's mathematical skills. We randomly allocated 10 classes of first, second and third graders (N=183) to one of three experimental groups: one group played with The Number Line; the second group played with Labyrinth, a computerized game designed to train attention skills; the third group had no intervention (business-as-usual). At the end of the first training phase, children in The Number Line group completed another training phase playing with Labyrinth, whereas the other two groups played with The Number Line. After playing with The Number Line, all groups displayed more accuracy when placing numbers in the number line task. However, we observed no evident improvement in other mathematical skills. These results suggest that specific training effects emerge even in the school context, although transfer to other numerical skills may be harder to achieve.

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

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

Evaluation of math anxiety and its remediation through a digital training program in mathematics for first and second graders

INTRODUCTION

Math anxiety severely impacts individuals' learning and future success. However, limited is understood about the profile in East Asian cultures where students genuinely show high-level math anxiety, despite that they outperform their Western counterparts. Here, we investigate the relation between math anxiety and math achievement in children as young as first and second graders in Taiwan. Further, we evaluate whether intensive exposure to digital game-based learning in mathematics could ameliorate math anxiety.

METHODS

The study first evaluated a group of 159 first and second graders' math anxiety and its correlation with math performance. Subsequently, a quasi-experimental design was adopted: 77 of the children continued and participated in multi-component digital game training targeting enumeration, speeded calculation, and working memory. Post-assessment was administered afterward for further evaluation of training-associated effects.

RESULTS

Results confirmed that math anxiety was negatively associated with school math achievement, which assessed numerical knowledge and arithmetic calculation. Furthermore, children's math anxiety was remarkably reduced via digital training in mathematics after 6-week intensive remediation. Crucially, this math anxiety relief was more prominent in those with high-level math anxiety. Although the children who underwent the training showed training-induced math achievement and working memory enhancement, this cognitive improvement appeared to be independent of the math anxiety relief.

CONCLUSION

Our findings demonstrate that students can show highly negative emotions and perceptions toward learning even in high-achieving countries. Auspiciously, the feeling of distress toward learning has the feasibility to be relieved from short-term intensive training. Our study suggests a new approach of early treatments to emotional disturbance that can lead to permanent consequences in individuals.

Testing the role of symbols in preschool numeracy: An experimental computer-based intervention study

Numeracy is of critical importance for scholastic success and modern-day living, but the precise mechanisms that drive its development are poorly understood. Here we used novel experimental training methods to begin to investigate the role of symbols in the development of numeracy in preschool-aged children. We assigned pre-school children in the U.S. and Italy (N = 215; Mean age = 49.15 months) to play one of five versions of a computer-based numerical comparison game for two weeks. The different versions of the game were equated on basic features of gameplay and demands but systematically varied in numerical content. Critically, some versions included non-symbolic numerical comparisons only, while others combined non-symbolic numerical comparison with symbolic aids of various types. Before and after training we assessed four components of early numeracy: counting proficiency, non-symbolic numerical comparison, one-to-one correspondence, and arithmetic set transformation. We found that overall children showed improvement in most of these components after completing these short trainings. However, children trained on numerical comparisons with symbolic aids made larger gains on assessments of one-to-one correspondence and arithmetic transformation compared to children whose training involved non-symbolic numerical comparison only. Further exploratory analyses suggested that, although there were no major differences between children trained with verbal symbols (e.g., verbal counting) and non-verbal visuo-spatial symbols (i.e., abacus counting), the gains in one-to-one correspondence may have been driven by abacus training, while the gains in non-verbal arithmetic transformations may have been driven by verbal training. These results provide initial evidence that the introduction of symbols may contribute to the emergence of numeracy by enhancing the capacity for thinking about exact equality and the numerical effects of set transformations. More broadly, this study provides an empirical basis to motivate further focused study of the processes by which children’s mastery of symbols influences children’s developing mastery of numeracy.

Training basic numerical skills in children with Down syndrome using the computerized game "The Number Race"

Individuals with Down syndrome (DS) present reduced basic numerical skills, which have a negative impact on everyday numeracy and mathematical learning. Here, we evaluated the efficacy of the adaptive (non-commercial) computerized game “The Number Race” in improving basic numerical skills in children with DS. The experimental group (EG; N = 30, M_{age-in-months} 118, range 70–149) completed a training playing with “The Number Race”, whereas children in the control group (CG; N = 31, M_{age-in-months} 138, range 76–207) worked with software aiming at improving their reading skills. The training lasted 10 weeks with two weekly sessions of 20–30 min each. We assessed both groups’ numerical and reading skills before and immediately after the end of the training, as well as at a 3-months follow-up. We found weak evidence for post-training groups differences in terms of overall numeracy score. However, the EG displayed substantial improvements in specific numerical skills and in mental calculation, which were maintained over time, and no improvement in reading. Conversely, the CG showed improvements in their reading skills as well as in number skills but to a lesser extent compared to the EG. Overall, “The Number Race” appears as a suitable tool to improve some aspects of numeracy in DS.

Number Magnitude Processing and Verbal Working Memory in Children with Mild Intellectual Disabilities

This study examined if children (M_age = 14.60) with Mild Intellectual Disabilities (MID) display weaknesses in number processing and verbal working memory. An age-matched and mental age-matched (MA, M_age = 6.17) design extended by a group of 9-10-year-olds, and a group of 11-12-year-olds were used. The MID children's working memory was equal to the MA group but poorer than the other groups. On number tasks, the MID group was faster than the MA group but slower than the other groups. All groups obtained equal Weber fraction scores and distance effects on the number comparison tasks. The MID group performed subitizing and counting faster than the MA group, but slower than the 11-12-year-olds. The results demonstrate that number processing and working memory in children with MID is characterized by a developmental delay, not a deficit. Their main problem is to access the quantitative meaning of Arabic numerals. The development of different types of cognitive abilities is differently affected by educational experience and intellectual ability. The innate number system appears to be unaffected by intellectual capacity or educational experience, while the innate working memory ability is affected by intellectual capacity but not by educational experience. Culturally acquired symbolic number abilities are strongly affected by educational experience.

Learning disabilities: Developmental dyscalculia

Developmental dyscalculia (DD) is a developmental learning disability that manifests as a persistent difficulty in comprehending even the most basic numeric and arithmetic concepts, despite normal intelligence and schooling opportunities. Given the predominant use of numbers in modern society, this condition can pose major challenges in the sufferer's everyday life, both in personal and professional development. Since, to date, we still lack a universally recognized and psychometrically driven definition of DD, its diagnosis has been applied to a wide variety of cognitive profiles. In this chapter, we review the behavioral and neural characterization of DD as well as the different neurocognitive and etiologic accounts of this neurodevelopmental disorder. We underline the multicomponential nature of this heterogeneous disability: different aspects of mathematical competence can be affected by both the suboptimal recruitment of general cognitive functions supporting mathematical cognition (such as attention, memory, and cognitive control) and specific deficits in mastering numeric concepts and operations. Accordingly, both intervention paradigms focused on core numeric abilities and more comprehensive protocols targeting multiple neurocognitive systems have provided evidence for effective positive outcomes.

Short-term numerosity training promotes symbolic arithmetic in children with developmental dyscalculia: The mediating role of visual form perception

Studies have shown that numerosity-based arithmetic training can promote arithmetic learning in typically developing children as well as children with developmental dyscalculia (DD), but the cognitive mechanism underlying this training effect remains unclear. The main aim of the current study was to examine the role of visual form perception in arithmetic improvement through an 8-day numerosity training for DD children. Eighty DD children were selected from four Chinese primary schools. They were randomly divided into the intervention and control groups. The intervention group received training on an apple-collecting game, whereas the control group received an English dictation task. Children's cognitive and arithmetic performances were assessed before and after training. The results showed that the intervention group showed a significant improvement in arithmetic performance, approximate number system (ANS) acuity, and visual form perception, but not in spatial processing and sentence comprehension. The control group showed no significant improvement in any cognitive ability. Mediation analysis further showed that training-related improvement in arithmetic performance was fully mediated by the improvement in visual form perception. The results suggest that short-term numerosity training enhances the arithmetic performance of DD children by improving their visual form perception.

Remediation of Allophonic Perception and Visual Attention Span in Developmental Dyslexia: A Joint Assay

Categorical perception of phonemes and visual attention span are cognitive processes that contribute independently to poor reading skills in developmental dyslexia. We here explored whether training programs specifically targeting one or the other process do improve reading performance in dyslexic children. The dyslexic participants were trained using either the RapDys© program designed to improve phonemic perception or the MAEVA© program targeting visual attention span. Each participant was provided the two programs successively for intensive training. Results show specific effects of RapDys© on phonemic discrimination and pseudo-word reading. MAEVA© specifically improved visual attention span and irregular word reading. Phonemic awareness and regular word reading improved after application of both training programs, suggesting similar positive effects of both methods although effects of concomitant phonic training cannot be ruled out (as there was no control group). The overall findings suggest that both categorical perception and visual attention span remediation contribute to reading.

Testing the Efficacy of Training Basic Numerical Cognition and Transfer Effects to Improvement in Children's Math Ability

The goals of the present study were to test whether (and which) basic numerical abilities can be improved with training and whether training effects transfer to improvement in children's math achievement. The literature is mixed with evidence that does or does not substantiate the efficacy of training basic numerical ability. In the present study, we developed a child-friendly software named "123 Bakery" which includes four training modules; non-symbolic numerosity comparison, non-symbolic numerosity estimation, approximate arithmetic, and symbol-to-numerosity mapping. Fifty-six first graders were randomly assigned to either the training or control group. The training group participated in 6 weeks of training (5 times a week, 30 minutes per day). All participants underwent pre- and post-training assessment of their basic numerical processing ability (including numerosity discrimination acuity, symbolic/non-symbolic magnitude estimation, approximate arithmetic, and symbol-to-numerosity mapping), overall math achievement and intelligence, 6 weeks apart. The acuity for numerosity discrimination (approximate number sense acuity; hereafter ANS acuity) significantly improved after training, but this training effect did not transfer to improvement in symbolic, exact calculation, or any other math ability. We conclude that basic numerical cognition training leads to improvement in ANS acuity, but whether this effect transfers to symbolic math ability remains to be further tested.

Raising Early Achievement in Math With Interactive Apps: A Randomized Control Trial

Improving provision and raising achievement in early math for young children is of national importance. Child-centered apps offer an opportunity to develop strong foundations in learning math as they deliver one-to-one instruction. Reported here is the first pupil-level randomized control trial in the United Kingdom of interactive math apps designed for early years education, with 389 children aged 4–5 years. The original and rigorous research design disentangled the impact of the math apps as a form of quality math instruction from additional exposure to math. It was predicted that using the apps would increase math achievement when implemented by teachers in addition to standard math activities (treatment) or instead of a regular small group-based math activity (time-equivalent treatment) compared with standard math practice only (control). After a 12-week intervention period, results showed significantly greater math learning gains for both forms of app implementation compared with standard math practice. The math apps supported targeted basic facts and concepts and generalized to higher-level math reasoning and problem solving skills. There were no significant differences between the 2 forms of math app implementation, suggesting the math apps can be implemented in a well-balanced curriculum. Features of the interactive apps, which are grounded in instructional psychology and combine aspects of direct instruction with play, may account for the observed learning gains. These novel results suggest that structured, content-rich, interactive apps can provide a vehicle for efficiently delivering high-quality math instruction for all pupils in a classroom context and can effectively raise achievement in early math.

In a pupil-level randomized control trial we evaluated the effectiveness of a new math app intervention to support young children’s early math development. The results showed children using the math apps either as a supplementary intervention or instead of a small group teacher-led math activity made significant learning gains in comparison with children receiving standard practice only. This study suggests high-quality math apps can be used as a form of quality math instruction in a well-rounded curriculum to raise achievement in early math for all children.

Counting and Number Line Trainings in Kindergarten: Effects on Arithmetic Performance and Number Sense

Children’s early numerical capacities form the building blocks for later arithmetic proficiency. Linear number placements and counting skills are indicative of mapping, as an important precursor to arithmetic skills, and have been suggested to be of vital importance to arithmetic development. The current study investigated whether fostering mapping skills is more efficient through a counting or a number line training program. Effects of both programs were compared through a quasi-experimental design, and moderation effects of age and socio-economic status (SES) were investigated. Ninety kindergartners were divided into three conditions: a counting, a number line, and a control condition. Pretests and posttests included an arithmetic (addition) task and a battery of number sense tasks (comparison, number lines, and counting). Results showed significantly greater gains in arithmetic, counting, and symbolic number lines in the counting training group than in the control group. The number line training group did not make significantly greater gains than the control group. Training gains were moderated by age, but not SES. We concluded that counting training improved numerical capacities effectively, whereas no such improvements could be found for the number line training. This suggests that only a counting approach is effective for fostering number sense and early arithmetic skills in kindergarten. Future research should elaborate on the parameters of training programs and the consequences of variation in these parameters.

Effects of non-symbolic arithmetic training on symbolic arithmetic and the approximate number system

The approximate number system (ANS) is an innate cognitive template that allows for the mental representation of approximate magnitude, and has been controversially linked to symbolic number knowledge and math ability. A series of recent studies found that an approximate arithmetic training (AAT) task that draws upon the ANS can improve math skills, which not only supports the existence of this link, but suggests it may be causal. However, no direct transfer effects to any measure of the ANS have yet been reported, calling into question the mechanisms by which math improvements may emerge. The present study investigated the effects of a 7-day AAT and successfully replicated previously reported transfer effects to math. Furthermore, our exploratory analyses provide preliminary evidence that certain ANS-related skills may also be susceptible to training. We conclude that AAT has reproducible effects on math performance, and provide avenues for future studies to further explore underlying mechanisms - specifically, the link between improvements in math and improvements in ANS skills.

Developmental Dyscalculia and Automatic Magnitudes Processing: Investigating Interference Effects between Area and Perimeter

The relationship between numbers and other magnitudes has been extensively investigated in the scientific literature. Here, the objectives were to examine whether two continuous magnitudes, area and perimeter, are automatically processed and whether adults with developmental dyscalculia (DD) are deficient in their ability to automatically process one or both of these magnitudes. Fifty-seven students (30 with DD and 27 with typical development) performed a novel Stroop-like task requiring estimation of one aspect (area or perimeter) while ignoring the other. In order to track possible changes in automaticity due to practice, we measured performance after initial and continuous exposure to stimuli. Similar to previous findings, current results show a significant group × congruency interaction, evident beyond exposure level or magnitude type. That is, the DD group systematically showed larger Stroop effects. However, analysis of each exposure period showed that during initial exposure to stimuli the DD group showed larger Stroop effects in the perimeter and not in the area task. In contrast, during continuous exposure to stimuli no triple interaction was evident. It is concluded that both magnitudes are automatically processed. Nevertheless, individuals with DD are deficient in inhibiting irrelevant magnitude information in general and, specifically, struggle to inhibit salient area information after initial exposure to a perimeter comparison task. Accordingly, the findings support the assumption that DD involves a deficiency in multiple cognitive components, which include domain-specific and domain-general cognitive functions.

Interventions for Primary School Children With Difficulties in Mathematics

Difficulty with arithmetic is a common problem for children and adults, though there has been some work on the topic for a surprisingly long time. This chapter will review some of the research that has been done over the years on interventions with primary school children. Interventions can be of various levels of intensiveness, ranging from whole-class approaches that take account of individual differences through small-group and limited-time individual interventions to extended-time individual interventions. Interventions discussed here include those involving peer tuition and group collaboration; those involving board and computer games; and those that involve assessing children's strengths and weaknesses in different components of mathematics; and targeting remedial activities to the assessed weaknesses. Most of the interventions discussed in this chapter specifically involve mathematics (usually mainly arithmetic), but there is also some discussion of attempts to improve mathematics by training children in domain-general skills, including Piagetian operations, metacognition, and executive functions.

Improving Preschoolers' Arithmetic through Number Magnitude Training: The Impact of Non-Symbolic and Symbolic Training

The numerical cognition literature offers two views to explain numerical and arithmetical development. The unique-representation view considers the approximate number system (ANS) to represent the magnitude of both symbolic and non-symbolic numbers and to be the basis of numerical learning. In contrast, the dual-representation view suggests that symbolic and non-symbolic skills rely on different magnitude representations and that it is the ability to build an exact representation of symbolic numbers that underlies math learning. Support for these hypotheses has come mainly from correlative studies with inconsistent results. In this study, we developed two training programs aiming at enhancing the magnitude processing of either non-symbolic numbers or symbolic numbers and compared their effects on arithmetic skills. Fifty-six preschoolers were randomly assigned to one of three 10-session-training conditions: (1) non-symbolic training (2) symbolic training and (3) control training working on story understanding. Both numerical training conditions were significantly more efficient than the control condition in improving magnitude processing. Moreover, symbolic training led to a significantly larger improvement in arithmetic than did non-symbolic training and the control condition. These results support the dual-representation view.

Sensory-integration system rather than approximate number system underlies numerosity processing: A critical review

It is widely accepted that human and nonhuman species possess a specialized system to process large approximate numerosities. The theory of an evolutionarily ancient approximate number system (ANS) has received converging support from developmental studies, comparative experiments, neuroimaging, and computational modelling, and it is one of the most dominant and influential theories in numerical cognition. The existence of an ANS system is significant, as it is believed to be the building block of numerical development in general. The acuity of the ANS is related to future arithmetic achievements, and intervention strategies therefore aim to improve the ANS. Here we critically review current evidence supporting the existence of an ANS. We show that important shortcomings and confounds exist in the empirical studies on human and non-human animals as well as the logic used to build computational models that support the ANS theory. We conclude that rather than taking the ANS theory for granted, a more comprehensive explanation might be provided by a sensory-integration system that compares or estimates large approximate numerosities by integrating the different sensory cues comprising number stimuli.

Numerical abilities of school-age children with Developmental Coordination Disorder (DCD): A behavioral and eye-tracking study

Developmental Coordination Disorder (DCD) is a disorder of motor coordination which interferes with academic achievement. Difficulties in mathematics have been reported. Performance in the number line task is very sensitive to atypical development of numerical cognition. We used a position-to-number task in which twenty 7-to-10years old children with DCD and 20 age-matched typically developing (TD) children had to estimate the number that corresponded to a hatch mark placed on a 0-100 number line. Eye movements were recorded. Children with DCD were less accurate and slower to respond than their peers. However, they were able to map numbers onto space linearly and used anchoring strategies as control. We suggest that the shift to a linear trend reflects the ability of DCD children to use efficient strategies to solve the task despite a possibly more imprecise underlying numerical acuity.

Dyscalculia and the Calculating Brain

Dyscalculia, like dyslexia, affects some 5% of school-age children but has received much less investigative attention. In two thirds of affected children, dyscalculia is associated with another developmental disorder like dyslexia, attention-deficit disorder, anxiety disorder, visual and spatial disorder, or cultural deprivation. Infants, primates, some birds, and other animals are born with the innate ability, called subitizing, to tell at a glance whether small sets of scattered dots or other items differ by one or more item. This nonverbal approximate number system extends mostly to single digit sets as visual discrimination drops logarithmically to "many" with increasing numerosity (size effect) and crowding (distance effect). Preschoolers need several years and specific teaching to learn verbal names and visual symbols for numbers and school agers to understand their cardinality and ordinality and the invariance of their sequence (arithmetic number line) that enables calculation. This arithmetic linear line differs drastically from the nonlinear approximate number system mental number line that parallels the individual number-tuned neurons in the intraparietal sulcus in monkeys and overlying scalp distribution of discrete functional magnetic resonance imaging activations by number tasks in man. Calculation is a complex skill that activates both visual and spatial and visual and verbal networks. It is less strongly left lateralized than language, with approximate number system activation somewhat more right sided and exact number and arithmetic activation more left sided. Maturation and increasing number skill decrease associated widespread non-numerical brain activations that persist in some individuals with dyscalculia, which has no single, universal neurological cause or underlying mechanism in all affected individuals.

Acquisition of the Cardinal Principle Coincides with Improvement in Approximate Number System Acuity in Preschoolers

Human mathematical abilities comprise both learned, symbolic representations of number and unlearned, non-symbolic evolutionarily primitive cognitive systems for representing quantities. However, the mechanisms by which our symbolic (verbal) number system becomes integrated with the non-symbolic (non-verbal) representations of approximate magnitude (supported by the Approximate Number System, or ANS) are not well understood. To explore this connection, forty-six children participated in a 6-month longitudinal study assessing verbal number knowledge and non-verbal numerical acuity. Cross-sectional analyses revealed a strong relationship between verbal number knowledge and ANS acuity. Longitudinal analyses suggested that increases in ANS acuity were most strongly related to the acquisition of the cardinal principle, but not to other milestones of verbal number acquisition. These findings suggest that experience with culture and language is intimately linked to changes in the properties of a core cognitive system.

Examining pitch and numerical magnitude processing in congenital amusia: A quasi-experimental pilot study

INTRODUCTION

Congenital amusia is a developmental disorder associated with deficits in pitch height discrimination or in integrating pitch sequences into melodies. This quasi-experimental pilot study investigated whether there is an association between pitch and numerical processing deficits in congenital amusia. Since pitch height discrimination is considered a form of magnitude processing, we investigated whether individuals with amusia present an impairment in numerical magnitude processing, which would reflect damage to a generalized magnitude system. Alternatively, we investigated whether the numerical processing deficit would reflect a disconnection between nonsymbolic and symbolic number representations.

METHOD

This study was conducted with 11 adult individuals with congenital amusia and a control comparison group of 6 typically developing individuals. Participants performed nonsymbolic and symbolic magnitude comparisons and number line tasks. Results were available from previous testing using the Montreal Battery of Evaluation of Amusia (MBEA) and a pitch change detection task (PCD).

RESULTS

Compared to the controls, individuals with amusia exhibited no significant differences in their performance on both the number line and the nonsymbolic magnitude tasks. Nevertheless, they showed significantly worse performance on the symbolic magnitude task. Moreover, individuals with congenital amusia, who presented worse performance in the Meter subtest, also presented less precise nonsymbolic numerical representation.

CONCLUSIONS

The relationship between meter and nonsymbolic numerical discrimination could indicate a general ratio processing deficit. The finding of preserved nonsymbolic numerical magnitude discrimination and mental number line representations, with impaired symbolic number processing, in individuals with congenital amusia indicates that (a) pitch height and numerical magnitude processing may not share common neural representations, and (b) in addition to pitch processing, individuals with amusia may present a deficit in accessing nonsymbolic numerical representations from symbolic representations. The symbolic access deficit could reflect a widespread impairment in the establishment of cortico-cortical connections between association areas.

Building Knowledge Structures by Testing Helps Children With Mathematical Learning Difficulty

Mathematical learning difficulty (MLD) is prevalent in the development of mathematical abilities. Previous interventions for children with MLD have focused on number sense or basic mathematical skills. This study investigated whether mathematical performance of fifth grade children with MLD could be improved by developing knowledge structures by testing using a web-based curriculum learning system. A total of 142 children with MLD were recruited; half of the children were in the experimental group (using the system), and the other half were in the control group (not using the system). The children were encouraged to use the web-based learning system at home for at least a 15-min session, at least once a week, for one and a half months. The mean accumulated time of testing on the system for children in the experimental group was 56.2 min. Children in the experimental group had significantly higher scores on their final mathematical examination compared to the control group. The results suggest that web-based curriculum learning through testing that promotes the building of knowledge structures for a mathematical course was helpful for children with MLD.

Cognitive tutoring induces widespread neuroplasticity and remediates brain function in children with mathematical learning disabilities

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.

Foundations of children's numerical and mathematical skills: the roles of symbolic and nonsymbolic representations of numerical magnitude

Numerical and mathematical skills are critical predictors of academic success. The last three decades have seen a substantial growth in our understanding of how the human mind and brain represent and process numbers. In particular, research has shown that we share with animals the ability to represent numerical magnitude (the total number of items in a set) and that preverbal infants can process numerical magnitude. Further research has shown that similar processing signatures characterize numerical magnitude processing across species and developmental time. These findings suggest that an approximate system for nonsymbolic (e.g., dot arrays) numerical magnitude representation serves as the basis for the acquisition of cultural, symbolic (e.g., Arabic numerals) representations of numerical magnitude. This chapter explores this hypothesis by reviewing studies that have examined the relation between individual differences in nonsymbolic numerical magnitude processing and symbolic math abilities (e.g., arithmetic). Furthermore, we examine the extent to which the available literature provides strong evidence for a link between symbolic and nonsymbolic representations of numerical magnitude at the behavioral and neural levels of analysis. We conclude that claims that symbolic number abilities are grounded in the approximate system for the nonsymbolic representation of numerical magnitude are not strongly supported by the available evidence. Alternative models and future research directions are discussed.

Identifying subtypes among children with developmental coordination disorder and mathematical learning disabilities, using model-based clustering

A relationship between motor and mathematical skills has been shown by previous research. However, the question of whether subtypes can be differentiated within developmental coordination disorder (DCD) and/or mathematical learning disability (MLD) remains unresolved. In a sample of children with and without DCD and/or MLD, a data-driven model-based clustering was used to identify subgroups of individuals with relatively homogeneous profiles on measures associated with motor and mathematical skills. One subgroup of children with motor problems was found based on motor variables. Based on mathematical variables, two clinical clusters were found: a subtype with number fact retrieval problems and a subtype with procedural calculation problems. Clustering with motor and mathematical skills revealed two clinical clusters: a cluster with number fact retrieval as well as procedural calculation problems and below average motor and visual-motor integration skills. A second cluster of children had only procedural calculation and visual-motor problems. Our results raise questions about the usefulness of placing children who have below average mathematical skills into a single diagnostic category. Furthermore, we inform ongoing debates about the overlap between DCD and MLD, as below average motor skills were found in both MLD subgroups, although a different motor profile is linked to a different mathematical profile.

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.

Effect of a virtual environment on the development of mathematical skills in children with dyscalculia

In this study, we show the effectiveness of a virtual environment comprising 18 computer games that cover mathematics topics in a playful setting and that can be executed on the Internet with the possibility of player interaction through chat. An arithmetic pre-test contained in the Scholastic Performance Test was administered to 300 children between 7 and 10 years old, including 162 males and 138 females, in the second grade of primary school. Twenty-six children whose scores showed a low level of mathematical knowledge were chosen and randomly divided into the control (CG) and experimental (EG) groups. The EG participated to the virtual environment and the CG participated in reinforcement using traditional teaching methods. Both groups took a post-test in which the Scholastic Performance Test (SPT) was given again. A statistical analysis of the results using the Student's t-test showed a significant learning improvement for the EG and no improvement for the CG (p≤0.05). The virtual environment allows the students to integrate thought, feeling and action, thus motivating the children to learn and contributing to their intellectual development.

Number sense or working memory? The effect of two computer-based trainings on mathematical skills in elementary school

Research on the improvement of elementary school mathematics has shown that computer-based training of number sense (e.g., processing magnitudes or locating numbers on the number line) can lead to substantial achievement gains in arithmetic skills. Recent studies, however, have highlighted that training domain-general cognitive abilities (e.g., working memory [WM]) may also improve mathematical achievement. This study addressed the question of whether a training of domain-specific number sense skills or domain-general WM abilities is more appropriate for improving mathematical abilities in elementary school. Fifty-nine children (M_age = 9 years, 32 girls and 27 boys) received either a computer-based, adaptive training of number sense (n = 20), WM skills (n = 19), or served as a control group (n = 20). The training duration was 20 min per day for 15 days. Before and after training, we measured mathematical ability using a curriculum-based math test, as well as spatial WM. For both training groups, we observed substantial increases in the math posttest compared to the control group (d = .54 for number sense skills training, d = .57 for WM training, respectively). Whereas the number sense group showed significant gains in arithmetical skills, the WM training group exhibited marginally significant gains in word problem solving. However, no training group showed significant posttest gains on the spatial WM task. Results indicate that a short training of either domain-specific or domain-general skills may result in reliable short-term training gains in math performance, although no stable training effects were found in the spatial WM task.

Brief non-symbolic, approximate number practice enhances subsequent exact symbolic arithmetic in children

Recent research reveals a link between individual differences in mathematics achievement and performance on tasks that activate the approximate number system (ANS): a primitive cognitive system shared by diverse animal species and by humans of all ages. Here we used a brief experimental paradigm to test one causal hypothesis suggested by this relationship: activation of the ANS may enhance children's performance of symbolic arithmetic. Over 2 experiments, children who briefly practiced tasks that engaged primitive approximate numerical quantities performed better on subsequent exact, symbolic arithmetic problems than did children given other tasks involving comparison and manipulation of non-numerical magnitudes (brightness and length). The practice effect appeared specific to mathematics, as no differences between groups were observed on a comparable sentence completion task. These results move beyond correlational research and provide evidence that the exercise of non-symbolic numerical processes can enhance children's performance of symbolic mathematics.

Measuring acuity of the approximate number system reliably and validly: the evaluation of an adaptive test procedure

Two studies investigated the reliability and predictive validity of commonly used measures and models of Approximate Number System acuity (ANS). Study 1 investigated reliability by both an empirical approach and a simulation of maximum obtainable reliability under ideal conditions. Results showed that common measures of the Weber fraction (w) are reliable only when using a substantial number of trials, even under ideal conditions. Study 2 compared different purported measures of ANS acuity as for convergent and predictive validity in a within-subjects design and evaluated an adaptive test using the ZEST algorithm. Results showed that the adaptive measure can reduce the number of trials needed to reach acceptable reliability. Only direct tests with non-symbolic numerosity discriminations of stimuli presented simultaneously were related to arithmetic fluency. This correlation remained when controlling for general cognitive ability and perceptual speed. Further, the purported indirect measure of ANS acuity in terms of the Numeric Distance Effect (NDE) was not reliable and showed no sign of predictive validity. The non-symbolic NDE for reaction time was significantly related to direct w estimates in a direction contrary to the expected. Easier stimuli were found to be more reliable, but only harder (7:8 ratio) stimuli contributed to predictive validity.

Training the approximate number system improves math proficiency

Humans and nonhuman animals share an approximate number system (ANS) that permits estimation and rough calculation of quantities without symbols. Recent studies show a correlation between the acuity of the ANS and performance in symbolic math throughout development and into adulthood, which suggests that the ANS may serve as a cognitive foundation for the uniquely human capacity for symbolic math. Such a proposition leads to the untested prediction that training aimed at improving ANS performance will transfer to improvement in symbolic-math ability. In the two experiments reported here, we showed that ANS training on approximate addition and subtraction of arrays of dots selectively improved symbolic addition and subtraction. This finding strongly supports the hypothesis that complex math skills are fundamentally linked to rudimentary preverbal quantitative abilities and provides the first direct evidence that the ANS and symbolic math may be causally related. It also raises the possibility that interventions aimed at the ANS could benefit children and adults who struggle with math.

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.

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.

Comparing apples and pears in studies on magnitude estimations

The present article is concerned with studies on magnitude estimations that strived to uncover the underlying mental representation(s) of magnitudes. We point out a number of methodological differences and shortcomings that make it difficult drawing general conclusions. To solve this problem, we propose a taxonomy by which those studies could be classified, taking into account central methodological aspects of magnitude estimation tasks. Finally, we suggest perspectives for future research on magnitude estimations, which might abandon the hunt for the mathematical model that explains estimations best and turn, instead, to investigate the underlying principles of estimations (e.g., strategies) and ways of their improvement.

[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.

Individual differences in inhibitory control, not non-verbal number acuity, correlate with mathematics achievement

Given the well-documented failings in mathematics education in many Western societies, there has been an increased interest in understanding the cognitive underpinnings of mathematical achievement. Recent research has proposed the existence of an Approximate Number System (ANS) which allows individuals to represent and manipulate non-verbal numerical information. Evidence has shown that performance on a measure of the ANS (a dot comparison task) is related to mathematics achievement, which has led researchers to suggest that the ANS plays a critical role in mathematics learning. Here we show that, rather than being driven by the nature of underlying numerical representations, this relationship may in fact be an artefact of the inhibitory control demands of some trials of the dot comparison task. This suggests that recent work basing mathematics assessments and interventions around dot comparison tasks may be inappropriate.

Are there rapid feedback effects on Approximate Number System acuity?

Humans are believed to be equipped with an Approximate Number System (ANS) that supports non-symbolic representations of numerical magnitude. Correlations between individual measures of the precision of the ANS and mathematical ability have raised the question of whether the precision can be improved by feedback training. A study (DeWind and Brannon, 2012) reported improvement in discrimination precision occurring within 600–700 trials of feedback, suggesting ANS malleability with rapidly improving acuity in response to feedback. We tried to replicate the rapid improvement in a control group design, while controlling for the use of perceptual cues. The results indicate no learning effects, but a minor constant advantage for the feedback group. The measures of motivation suggest that feedback has a positive effect on motivation and that the difference in discrimination is due to the greater motivation of participants with feedback. These results suggest that at least for adults the number sense may not respond to feedback in the short-term.