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We investigated the levels of and changes in mathematics anxiety (MA), symbolic numerical magnitude processing (SNMP) and arithmetic skills, and how those changes are linked to each other. Children's (n = 264) MA, SNMP and arithmetic skills were measured in Grade 1, and again in Grade 2, also including a mathematics performance test. All three constructs correlated significantly within each time point, and the rank-order stability over time was high, particularly in SNMP and arithmetic skills. By means of latent change score modelling, we found overall increases in SNMP and arithmetic skills over time, but not in MA. Most interestingly, changes in arithmetic skills and MA were correlated (i.e. steeper increase in arithmetic skills was linked with less steep increase in MA), as were changes in SNMP and arithmetic skills (i.e. improvement in SNMP was associated with improvement in arithmetic skills). Only the initial level of arithmetic skills and change in it predicted mathematics performance. The only gender difference, in favour of boys, was found in SNMP skills. The differential effects associated with MA (developmentally only linked with arithmetic skills) and gender (predicting only changes in SNMP) call for further longitudinal research on the different domains of mathematical skills.

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Arithmetic is a major building block for children's development of more complex mathematical abilities. Knowing which cognitive factors underlie individual differences in arithmetic is key to gaining further insight into children's mathematical development. The current study investigated the role of executive functions and metacognition (domain-general cognitive factors) as well as symbolic numerical magnitude processing (domain-specific cognitive factor) in arithmetic, enabling the investigation of their unique contribution in addition to each other. Participants were 127 typically developing second graders (7- and 8-year-olds). Our within-participant design took into account different components of executive functions (i.e., inhibition, shifting, and updating), different aspects of metacognitive skills (i.e., task-specific and general metacognition), and different levels of experience in arithmetic, namely addition (where second graders had extensive experience) and multiplication (where second graders were still in the learning phase). This study reveals that both updating and metacognitive monitoring are important unique predictors of arithmetic in addition to each other and to symbolic numerical magnitude processing. Our results point to a strong and unique role of task-specific metacognitive monitoring skills. These individual differences in noticing one's own errors might help one to learn from his or her mistakes.

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Nonsymbolic numerical magnitude processing skills are assumed to be fundamental to mathematical learning. Recent findings suggest that visual-spatial skills account for associations between children's performance in visually presented nonsymbolic numerical magnitude comparison tasks and their performance in visually presented arithmetic tasks. The aim of the current study was to examine whether associations between children's performance in visually presented tasks assessing nonsymbolic numerical magnitude processing skills and their performance in tasks assessing early mathematical skills, which do not involve visual stimulation, may also be mediated by visual-spatial skills. This line of reasoning is based on the assumption that children make use of mental visualization processes when working on tasks assessing early mathematical skills, such as knowledge of the sequence of number words, even when these tasks do not involve visual stimulation. We assessed 4- to 6-year-old children's performance in a nonsymbolic numerical magnitude comparison task, in tasks concerning knowledge of the sequence of number words, and in a developmental test to assess visual-spatial skills. Children's nonsymbolic numerical magnitude processing skills were found to be associated with their number word sequence skills. This association was fully mediated by interindividual differences in visual-spatial skills. The effect size of this mediation effect was small. We assume that the ability to construct mental visualizations constitutes the key factor underlying this mediation effect.

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Objective To explore the mediating role of intelligence in the relationship between in-formation processing speed and numerical magnitude processing of hearing-impaired children. Methods A survey was conducted among 249 hearing-impaired children by using tests of Choice Reaction Time,Numero-sity Comparison,and Raven's Progressive Matrices( simplified version). Results (1) The accurate rate of information processing speed(0. 89±0. 18)of hearing-impaired children was positively correlated with that of numerical magnitude processing(0. 65±0. 15)and intelligence(11. 83±10. 34)(r=0. 165,0. 263,both P<0. 01). The reaction time of information processing speed was negatively correlated with those of numerical mag-nitude processing and intelligence(r=-0. 164,-0. 197,both P<0. 05). And there was significant positive correlation between intelligence and numerical magnitude processing (accurate rate and reaction time ) (r=0. 269,0. 140,both P<0. 05). ( 2) The accurate rate of information processing speed significantly predicted that of numerical magnitude processing in hearing-impaired children( β=0. 17,t=2. 64,both P<0. 01). The reaction time of information processing speed significantly predicted that of numerical magnitude processing in hearing-impaired children(β=-0. 16,t=-2. 61,both P<0. 05). And intelligence played a complete medi-ating role in this relationship and the moderating effect were 36. 71% and 31. 25%. Conclusion Information processing speed has indirect effects on numerical magnitude processing through intelligence in hearing-im-paired children.

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Objective@#To explore the mediating role of intelligence in the relationship between information processing speed and numerical magnitude processing of hearing-impaired children.@*Methods@#A survey was conducted among 249 hearing-impaired children by using tests of Choice Reaction Time, Numerosity Comparison, and Raven's Progressive Matrices(simplified version).@*Results@#(1)The accurate rate of information processing speed(0.89±0.18)of hearing-impaired children was positively correlated with that of numerical magnitude processing(0.65±0.15)and intelligence(11.83±10.34)(r=0.165, 0.263, both P<0.01). The reaction time of information processing speed was negatively correlated with those of numerical magnitude processing and intelligence(r=-0.164, -0.197, both P<0.05). And there was significant positive correlation between intelligence and numerical magnitude processing (accurate rate and reaction time ) (r=0.269, 0.140, both P<0.05). (2)The accurate rate of information processing speed significantly predicted that of numerical magnitude processing in hearing-impaired children(β=0.17, t=2.64, both P<0.01). The reaction time of information processing speed significantly predicted that of numerical magnitude processing in hearing-impaired children(β=-0.16, t=-2.61, both P<0.05). And intelligence played a complete mediating role in this relationship and the moderating effect were 36.71% and 31.25%.@*Conclusion@#Information processing speed has indirect effects on numerical magnitude processing through intelligence in hearing-impaired children.

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Although it has been proposed that inhibition is related to individual differences in mathematical achievement, it is not clear how it is related to specific aspects of mathematical skills, such as arithmetic fact retrieval. The present study therefore investigated the association between inhibition and arithmetic fact retrieval and further examined the unique role of inhibition in individual differences in arithmetic fact retrieval, in addition to numerical magnitude processing. We administered measures of cognitive inhibition (i.e., numerical and non-numerical stroop tasks) and a complementary, more ecologically valid measure of children's inhibition in the classroom (i.e., teacher questionnaire), as well as numerical magnitude processing (i.e., symbolic and non-symbolic numerical magnitude comparison) and arithmetic fact retrieval (i.e., two verification tasks) in 86 typically developing third graders. We used a correlation, a regression and a Bayesian analysis. This study failed to observe a significant association between inhibition and arithmetic fact retrieval. Consequently, our results did not reveal a unique contribution of inhibition to arithmetic fact retrieval in addition to numerical magnitude processing. On the other hand, symbolic numerical magnitude processing turned out to be a very powerful predictor of arithmetic fact retrieval, as indicated by both frequentist and Bayesian approaches.

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This contribution reviewed the available evidence on the domain-specific and domain-general neurocognitive determinants of children's arithmetic development, other than nonsymbolic numerical magnitude processing, which might have been overemphasized as a core factor of individual differences in mathematics and dyscalculia. We focused on symbolic numerical magnitude processing, working memory, and phonological processing, as these determinants have been most researched and their roles in arithmetic can be predicted against the background of brain imaging data. Our review indicates that symbolic numerical magnitude processing is a major determinant of individual differences in arithmetic. Working memory, particularly the central executive, also plays a role in learning arithmetic, but its influence appears to be dependent on the learning stage and experience of children. The available evidence on phonological processing suggests that it plays a more subtle role in children's acquisition of arithmetic facts. Future longitudinal studies should investigate these factors in concert to understand their relative contribution as well as their mediating and moderating roles in children's arithmetic development.

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There are two well-known computation methods for solving multi-digit subtraction items, namely mental and algorithmic computation. It has been contended that mental and algorithmic computation differentially rely on numerical magnitude processing, an assumption that has already been examined in children, but not yet in adults. Therefore, in this study, we examined how numerical magnitude processing was associated with mental and algorithmic computation, and whether this association with numerical magnitude processing was different for mental versus algorithmic computation. We also investigated whether the association between numerical magnitude processing and mental and algorithmic computation differed for measures of symbolic versus nonsymbolic numerical magnitude processing. Results showed that symbolic, and not nonsymbolic, numerical magnitude processing was associated with mental computation, but not with algorithmic computation. Additional analyses showed, however, that the size of this association with symbolic numerical magnitude processing was not significantly different for mental and algorithmic computation. We also tried to further clarify the association between numerical magnitude processing and complex calculation by also including relevant arithmetical subskills, i.e. arithmetic facts, needed for complex calculation that are also known to be dependent on numerical magnitude processing. Results showed that the associations between symbolic numerical magnitude processing and mental and algorithmic computation were fully explained by individual differences in elementary arithmetic fact knowledge.

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Distance effect has been regarded as the best established marker of basic numerical magnitude processes and is related to individual mathematical abilities. A larger behavioral distance effect is suggested to be concomitant with lower mathematical achievement in children. However, the relationship between distance effect and superior mathematical abilities is unclear. One could get superior mathematical abilities by acquiring the skill of abacus-based mental calculation (AMC), which can be used to solve calculation problems with exceptional speed and high accuracy. In the current study, we explore the relationship between distance effect and superior mathematical abilities by examining whether and how the AMC training modifies numerical magnitude processing. Thus, mathematical competencies were tested in 18 abacus-trained children (who accepted the AMC training) and 18 non-trained children. Electroencephalography (EEG) waveforms were recorded when these children executed numerical comparison tasks in both Arabic digit and dot array forms. We found that: (a) the abacus-trained group had superior mathematical abilities than their peers; (b) distance effects were found both in behavioral results and on EEG waveforms; (c) the distance effect size of the average amplitude on the late negative-going component was different between groups in the digit task, with a larger effect size for abacus-trained children; (d) both the behavioral and EEG distance effects were modulated by the notation. These results revealed that the neural substrates of magnitude processing were modified by AMC training, and suggested that the mechanism of the representation of numerical magnitude for children with superior mathematical abilities was different from their peers. In addition, the results provide evidence for a view of non-abstract numerical representation.

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The current longitudinal study tried to capture profiles of individual differences in children's arithmetic fact development. We used a model-based clustering approach to delineate profiles of arithmetic fact development based on empirically derived differences in parameters of arithmetic fact mastery repeatedly assessed at the start of three subsequent school years: third, fourth, and fifth grades. This cluster analysis revealed three profiles in a random sample-slow and variable (n = 8), average (n = 24), and efficient (n = 20)-that were marked by differences in children's development in arithmetic fact mastery from third grade to fifth grade. These profiles did not differ in terms of age, sex, socioeconomic status, or intellectual ability. In addition, we explored whether these profiles varied in cognitive skills that have been associated with individual differences in single-digit arithmetic. The three profiles differed in nonsymbolic and symbolic numerical magnitude processing as well as phonological processing, but not in digit naming or working memory. After also controlling for cluster differences in general mathematics achievement and reading ability, only differences in symbolic numerical magnitude processing remained significant. Taken together, our longitudinal data reveal that symbolic numerical magnitude processing represents an important variable that contributes to individual variability in children's acquisition of arithmetic facts.

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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.

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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.

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Human performance in visual enumeration tasks typically shows two distinct patterns as a function of set size. For small sets, usually up to 4 items, numerosity judgments are extremely rapid, precise and confident, a phenomenon known as subitizing. When this limit is exceeded and serial counting is precluded, exact enumeration gives way to estimation: performance becomes error-prone and more variable. Surprisingly, despite the importance of subitizing and estimation in numerical cognition, only few neuroimaging studies have examined whether the neural activity related to these two phenomena can be dissociated. In the present work, we used multi-channel near-infrared spectroscopy (fNIRS) to measure hemodynamic activity of the bilateral parieto-occipital cortex during a visual enumeration task. Participants had to judge the numerosity of dot arrays and indicate it by means of verbal response. We observed a different hemodynamic pattern in the parietal cortex, both in terms of amplitude modulation and temporal profile, for numerosities below and beyond the subitizing range. Crucially, the neural dissociation between subitizing and estimation was strongest at the level of right IPS. The present findings confirm that fNIRS can be successfully used to detect subtle temporal differences in hemodynamic activity and to produce inferences on the neural mechanisms underlying cognitive functions.

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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.

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The ability to accurately process numerical magnitudes and solve mental arithmetic is of highest importance for schooling and professional career. Although impairments in these domains in disorders such as developmental dyscalculia (DD) are highly detrimental, remediation is still sparse. In recent years, transcranial brain stimulation methods such as transcranial Direct Current Stimulation (tDCS) have been suggested as a treatment for various neurologic and neuropsychiatric disorders. The posterior parietal cortex (PPC) is known to be crucially involved in numerical magnitude processing and mental arithmetic. In this study, we evaluated whether tDCS has a beneficial effect on numerical magnitude processing and mental arithmetic. Due to the unclear lateralization, we stimulated the left, right as well as both hemispheres simultaneously in two experiments. We found that left anodal tDCS significantly enhanced performance in a number comparison and a subtraction task, while bilateral and right anodal tDCS did not induce any improvements compared to sham. Our findings demonstrate that the left PPC is causally involved in numerical magnitude processing and mental arithmetic. Furthermore, we show that these cognitive functions can be enhanced by means of tDCS. These findings encourage to further investigate the beneficial effect of tDCS in the domain of mathematics in healthy and impaired humans.