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This study examined cognitive precursors of hierarchical mathematical development. Six-year-old children (n = 258) were assessed on number skills, cognitive skills, and arithmetic 1 year prior to school entry. Skills in advanced arithmetic and advanced mathematics were assessed in Grades 3 and 6, respectively. Path analyses were computed and provided longitudinal evidence for a hierarchy of mathematics. During development, the reliance on prior skills at lower, most proximal levels becomes increasingly important in order to acquire and succeed at later and higher levels of mathematics. The study extends the Fuchs et al. (2010b) model as the mechanisms underlying 3 different hierarchical levels of mathematics all involve an interplay between domain-specific number abilities (i.e., sequence knowledge, digit comparison) and general cognitive abilities (i.e., logical reasoning, phonological awareness, working memory). The constellation of domain-specific number and domain-general cognitive mechanisms subserving mathematics at different hierarchical levels showed a high degree of similarity. However, the importance of number abilities decreases during development, whereas the importance of general cognitive abilities remains approximately equal across levels of mathematical learning. Early symbolic number skills (i.e., counting sequence knowledge, digit comparison) are subserved by general cognitive mechanisms such as working memory, phonological awareness, and rapid automatic naming processes. (PsycINFO Database Record (c) 2019 APA, all rights reserved)

Highlights • Overlapping neural responses to number discrimination tasks in adults and children. • Ontogenetic maturation of language regions may constrain verbal number processing. • Neural correlates of effortful number discrimination were similar across age groups. • Prefrontal activity attributed to children’s number processing was not replicated. • Results indicative of mature number discrimination abilities around 11 years of age., The triple code model of numerical cognition (TCM) details the neurocognitive mechanisms associated with perceiving and manipulating numerical information in exact symbolic (Arabic digits and number words) and approximate nonsymbolic numerical magnitude (e.g., dot arrays) representation codes. The current study provides a first empirical fMRI-based investigation into neurodevelopmental differences in 30 healthy children’s and 44 healthy adults’ recruitment of neural correlates associated with the Arabic digit, number word, and nonsymbolic magnitude codes. Differences between the two groups were found in cingulate regions commonly associated with domain-general aspects of cognitive control, as opposed to neural correlates of number processing per se. A primary developmental difference was identified in verbal number discrimination, where only adults recruited left-lateralized perisylvian language areas in accordance with the TCM. We therefore call for a revision of the verbal code and a formulation of separate child and adult-specific neurocognitive mechanisms associated with the discrimination of number words. Although further research is necessary, results indicate that numerical discrimination abilities in middle-school-aged children operate close to adult-level maturity. Neurodevelopmental differences may be more apparent in younger children, or on the level of functional network dynamics as opposed to a shift in recruited neural substrates.

This longitudinal study examined whether the cognitive and academic development of children (M-age = 10.52 years) with Borderline Intellectual Functioning (BIF), is characterized by developmental delay or atypical development. Cognitive tasks, arithmetic tasks, and reading tasks were administrated during three succeeding years to the BIF group and a Chronological Age-Matched Comparison (CAMC) group. The BIF children displayed weaknesses in relation to all tasks, and slower developmental rates on four arithmetic tasks and word reading. The results provide evidence in support of the developmental delay model as the BIF children overall displayed similar developmental growth and trends as the CAMC group. Funding Agencies|Swedish Council for Working Life and Social ResearchSwedish Research CouncilSwedish Research Council for Health Working Life & Welfare (Forte) [2010-0078]

This study examined number processing in 10-year-olds with developmental dyslexia (DD). The phonological deficit and double deficit hypotheses imply that children with DD might have a connection deficit that affects their ability to establish links between number symbols and magnitude representations. The double deficit hypothesis also posits that symbolic number difficulties may emerge due to difficulties with processes underlying rapid automatic naming (RAN). The DD group displayed difficulties with symbolic number processing but not with non-symbolic number processing. However, the underlying processes of this access or connection deficit appeared not to be related to phonological awareness or RAN. The DD group displayed impaired arithmetic fluency and calculation that were accounted for by defective processes underlying RAN. In view of the triple-code model, children with DD have impaired verbal number codes or defective access to verbal number codes but an intact core magnitude representation.

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 childrens 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. Funding Agencies|Swedish Research CouncilSwedish Research Council [2015-02157, 721-2011-2872]

Developmental dyscalculia (DD), a pervasive learning difficulty, affects 5-6% of the population. The current study sought to identify the underlying deficits by contrasting third graders (10-year o ...

In contemporary society, it is essential to have adequate mathematical skills. Being numerate has been linked to positive life outcomes and well-being in adults. It is also acknowledged that math anxiety (MA) hampers mathematical skills increasingly with age. Still, the mechanisms by which MA affect performance remain debated. Using structural equation modeling (SEM), we contrast the different ways in which MA has been suggested to interfere with math abilities. Our models indicate that MA may affect math performance through three pathways: (1) indirectly through working memory ability, giving support for the affective drop hypothesis of MAs role in mathematical performance, (2) indirectly through symbolic number processing, corroborating the notion of domain-specific mechanisms pertaining to number, and (3) a direct effect of MA on math performance. Importantly, the pathways vary in terms of their relative strength depending on what type of mathematical problems are being solved. These findings shed light on the mechanisms by which MA may interfere with mathematical performance. Funding Agencies|Marianne and Marcus Wallenberg Foundation [2014-0173]

The mathematical performance of 17 children with developmental dyslexia (DD) was assessed and compared to a control group to examine whether difficulties related to reading and phonological processing affect the development of mathematical skills. The DD group performed worse than the controls on number fact retrieval, multi-step arithmetic problem solving, and multi-digit calculation, whereas their scores on tasks tapping approximate arithmetic and conceptual understanding (i.e., place value, calculation principles) were equal to the controls. In view of the Triple-code model, the findings demonstrate that children with DD have problems with tasks depending on verbal–phonological number codes but have no problems with tasks depending on analogue magnitude representations or visual-Arabic number codes.

In social cognitive theory, self-efficacy is domain-specific. An alternative model, the cross-domain influence model, would predict that self-efficacy beliefs in one domain might influence performance in other domains. Research has also found that children who receive special instruction are not good at estimating their performance. The aim was to test two models of how self-efficacy beliefs influence achievement, and to contrast children receiving special instruction in mathematics with normally-achieving children. The participants were 73 fifth-grade children who receive special instruction and 70 children who do not receive any special instruction. In year four and five, the children's skills in mathematics and reading were assessed by national curriculum tests, and in their fifth year, self-efficacy in mathematics and reading were measured. Structural equation modeling showed that in domains where children do not receive special instruction in mathematics, self-efficacy is a mediating variable between earlier and later achievement in the same domain. Achievement in mathematics was not mediated by self-efficacy in mathematics for children who receive special instruction. For normal achieving children, earlier achievement in the language domain had an influence on later self-efficacy in the mathematics domain, and self-efficacy beliefs in different domains were correlated. Self-efficacy is mostly domain specific, but may play a different role in academic performance depending on whether children receive special instruction. The results of the present study provided some support of the Cross-Domain Influence Model for normal achieving children.

Background Developing sufficient mathematical skills is a prerequisite to function adequately in society today. Given this, an important task is to increase our understanding regarding the cognitive mechanisms underlying young people's acquisition of early number skills and formal mathematical knowledge. Aims The purpose was to examine whether the pathways to mathematics model provides a valid account of the cognitive mechanisms underlying symbolic-number processing and mathematics in adolescents. The pathways model states that the three pathways should provide independent support to symbolic-number skill. Each pathway's unique contribution to formal mathematics varies depending on the complexity and demand of the tasks. Sample The study used a sample of 114 adolescents (71 girls). Their mean age was 14.60 years (SD = 1.00). Methods The adolescents were assessed on tests tapping the three pathways and general cognitive abilities (e.g., working memory). A structural equation path analysis was computed. Results Symbolic-number comparison was predicted by the linguistic pathway, the quantitative pathway, and processing speed. The linguistic pathway, quantitative pathways, and symbolic-number comparison predicted arithmetic fact retrieval. The linguistic pathway, working memory, visual analogies, and symbolic-number comparison predicted percentage calculation. Conclusions There are both similarities and differences in the cognitive mechanisms underlying arithmetic fact retrieval and percentage calculation in adolescents. Adolescents’ symbolic-number processing, arithmetic fact retrieval, and percentage calculation continue to rely on the linguistic pathways, whereas the reliance upon the spatial pathway has ceased. The reliance upon the quantitative pathway varies depending on the task. Funding agencies: Swedish Council for Working Life and Social Research [2010-0078]

A modified pathways to mathematics model was used to examine the cognitive mechanisms underlying arithmetic skills in third graders. A total of 269 children were assessed on tasks tapping the four pathways and arithmetic skills. A path analysis showed that symbolic number processing was directly supported by the linguistic and approximate quantitative pathways. The direct contribution from the four pathways to arithmetic proficiency varied; the linguistic pathway supported single-digit arithmetic and word problem solving, whereas the approximate quantitative pathway supported only multi-digit calculation. The spatial processing and verbal working memory pathways supported only arithmetic word problem solving. The notion of hierarchical levels of arithmetic was supported by the results, and the different levels were supported by different constellations of pathways. However, the strongest support to the hierarchical levels of arithmetic were provided by the proximal arithmetic skills.

Developmental Dyscalculia (DD) has long been thought to be a monolithic learning disorder that can be attributed to a specific neurocognitive dysfunction. However, recent research has increasingly recognized the heterogeneity of DD, where DD can be differentiated into subtypes in which the underlying cognitive deficits and neural dysfunctions may differ. The aim was to further understand the heterogeneity of developmental dyscalculia (DD) from a cognitive psychological perspective. Utilizing four children (8-9 year-old) we administered a comprehensive cognitive test battery that shed light on the cognitive-behavioral profile of each child. The children were compared against norm groups of aged-matched peers. Performance was then contrasted against predominant hypotheses of DD, which would also give insight into candidate neurocognitive correlates. Despite showing similar mathematical deficits, these children showed remarkable interindividual variability regarding cognitive profile and deficits. Two cases were consistent with the approximate number system deficit account and also the general magnitude-processing deficit account. These cases showed indications of having domain-general deficits as well. One case had an access deficit in combination with a general cognitive deficit. One case suffered from general cognitive deficits only. The results showed that DD cannot be attributed to a single explanatory factor. These findings support a multiple deficits account of DD and suggest that some cases have multiple deficits, whereas other cases have a single deficit. We discuss a previously proposed distinction between primary DD and secondary DD, and suggest hypotheses of dysfunctional neurocognitive correlates responsible for the displayed deficits. The results showed that DD cannot be attributed to a single explanatory factor. These findings support a multiple deficits account of DD and suggest that some cases have multiple deficits, whereas other cases have a single deficit. We discuss a previously proposed distinction between primary DD and secondary DD, and suggest hypotheses of dysfunctional neurocognitive correlates responsible for the displayed deficits.

Continuous dimensions, such as time, space, and numerosity, have been suggested to be subserved by common neurocognitive mechanisms. Neuroimaging studies that have investigated either one or two dimensions simultaneously have consistently identified neural correlates in the parietal cortex of the brain. However, the degree of neural overlap across several dimensions has yet to be established, and it remains an open question whether a potential overlap can be conceptualized as a neurocognitive magnitude processing system. The current functional resonance imaging (fMRI) study investigated the potential neurocognitive overlap across three dimensions. A sample of adults (N = 24) performed three different magnitude processing tasks: a temporal discrimination task, a number discrimination task, and a line length discrimination task. A conjunction analysis revealed several overlapping neural substrates across multiple magnitude dimensions, and we argue that these cortical nodes comprise a distributed magnitude processing system. Key components of this predominantly right-lateralized system include the intraparietal sulcus, insula, premotor cortex, inferior frontal gyrus and frontal eye-fields. Together with previous research highlighting IPS, our results suggest that the insula also is a core component of the magnitude processing system. We discuss the functional role of each of these components in the magnitude processing system and suggest that further research of this system may provide insight into the etiology of neurodevelopmental disorders where cognitive deficits in magnitude processing are manifest. Funding Agencies|Swedish Council for Working Life and Social Research [2010-0078]

The current study investigated whether processing of number, space, and time contributes to mathematical abilities beyond previously known domain-general cognitive abilities in a sample of 8- to 10-year-old children (N=133). Multiple regression analyses revealed that executive functions and general intelligence predicted all aspects of mathematics and overall mathematical ability. Working memory capacity did not contribute significantly to our models, whereas spatial ability was a strong predictor of achievement. The study replicates earlier research showing that non-symbolic number processing seems to lose predictive power of mathematical abilities once the symbolic system is acquired. Novel findings include the fact that time discrimination ability was tied to calculation ability. Therefore, a conclusion is that magnitude processing in general contributes to mathematical achievement.

Developmental dyscalculia (DD) is a learning disorder associated with impairments in a preverbal non-symbolic approximate number system (ANS) pertaining to areas in and around the intraparietal sulcus (IPS). The current study sought to enhance our understanding of the developmental trajectory of the ANS and symbolic number processing skills, thereby getting insight into whether a deficit in the ANS precedes or is preceded by impaired symbolic and exact number processing. Recent work has also suggested that humans are endowed with a shared magnitude system (beyond the number domain) in the brain. We therefore investigated whether children with DD demonstrated a general magnitude deficit, stemming from the proposed magnitude system, rather than a specific one limited to numerical quantity. Fourth graders with DD were compared to age-matched controls and a group of ability-matched second graders, on a range of magnitude processing tasks pertaining to space, time, and number. Children with DD displayed difficulties across all magnitude dimensions compared to age-matched peers and showed impaired ANS acuity compared to the younger, ability-matched control group, while exhibiting intact symbolic number processing. We conclude that (1) children with DD suffer from a general magnitude-processing deficit, (2) a shared magnitude system likely exists, and (3) a symbolic number-processing deficit in DD tends to be preceded by an ANS deficit.

This study investigated if developmental dyscalculia (DD) in children with different profiles of mathematical deficits has the same or different cognitive origins. The defective approximate number system hypothesis and the access deficit hypothesis were tested using two different groups of children with DD (11-13 years old): a group with arithmetic fact dyscalculia (AFD) and a group with general dyscalculia (GD). Several different aspects of number magnitude processing were assessed in these two groups and compared with age-matched typically achieving children. The GD group displayed weaknesses with both symbolic and nonsymbolic number processing, whereas the AFD group displayed problems only with symbolic number processing. These findings provide evidence that the origins of DD in children with different profiles of mathematical problems diverge. Children with GD have impairment in the innate approximate number system, whereas children with AFD suffer from an access deficit. These findings have implications for researchers selection procedures when studying dyscalculia, and also for practitioners in the educational setting. Funding Agencies|Swedish Research Council [421-2007-1881]

Previous literature suggests that early number knowledge is important for the development of arithmetic calculation ability. The domain-general ability of verbal working memory also has an impact on arithmetic ability. This longitudinal study tested the impact of early number knowledge and verbal working memory on the arithmetic calculation ability of children in preschool (N=315) and then later in Grade 1 using structural equation modeling. Three models were used to test hypotheses drawn from previous literature. The current study demonstrates that both early number knowledge and the domain-general ability of verbal working memory affect preschool and Grade 1 arithmetic ability. Early number knowledge had a direct impact on the growth of arithmetic ability, whereas verbal working memory had only an indirect effect via number knowledge and preschool arithmetic ability. These results fit well with von Aster and Shalev's developmental model of numerical cognition (Developmental Medicine & Child Neurology, 2007, Vol. 49, pp. 868-873) and highlight the importance of considering arithmetic ability as independent from early number knowledge. Results also emphasize the importance of training early number knowledge before school entry to promote the development of arithmetic ability.