The codevelopment of skill at and preference for use of retrieval-based processes for solving addition problems: individual and sex differences from first to sixth grades

Neurobiology of numerical learning

Numerical abilities are complex cognitive skills essential for dealing with requirements of the modern world. Although the brain structures and functions underlying numerical cognition in different species have long been appreciated, genetic and molecular techniques have more recently expanded the knowledge about the mechanisms underlying numerical learning. In this review, we discuss the status of the research related to the neurobiological bases of numerical abilities. We consider how genetic factors have been associated with mathematical capacities and how these link to the current knowledge of brain regions underlying these capacities in human and non-human animals. We further discuss the extent to which significant variations in the levels of specific neurotransmitters may be used as potential markers of individual performance and learning difficulties and take into consideration the therapeutic potential of brain stimulation methods to modulate learning and improve interventional outcomes. The implications of this research for formulating a more comprehensive view of the neural basis of mathematical learning are discussed.

Arithmetic problem size modulates brain activations in females but not in males

Numerous empirical studies have reported that males and females perform equally well in mathematical achievement. However, still to date, very limited is understood about the brain response profiles that are particularly characteristic of males and females when solving mathematical problems. The present study aimed to tackle this issue by manipulating arithmetic problem size to investigate functional significance using functional magnetic resonance imaging (fMRI) in young adults. Participants were instructed to complete two runs of simple calculation tasks with either large or small problem sizes. Behavioural results suggested that the performance did not differ between females and males. Neuroimaging data revealed that sex/gender-related patterns of problem size effect were found in the brain regions that are conventionally associated with arithmetic, including the left middle frontal gyrus (MFG), left intraparietal sulcus (IPS) and insula. Specifically, females demonstrated substantial brain responses of problem size effect in these regions, whereas males showed marginal effects. Moreover, the machine learning method implemented over the brain signal levels within these regions demonstrated that sex/gender is discriminable. These results showed sex/gender effects in the activating patterns varying as a function of the distinct math problem size, even in a simple calculation task. Accordingly, our findings suggested that females and males use two complementary brain resources to achieve equally successful performance levels and highlight the pivotal role of neuroimaging facilities in uncovering neural mechanisms that may not be behaviourally salient.

The association between working memory and mathematical problem solving: A three-level meta-analysis

Although working memory (WM) is an important factor in mathematical problem solving (MPS), it remains unclear how well WM relates to MPS. Thus, we aimed to determine this relationship by using a meta-analysis. We searched electronic databases for studies published between 2000 and 2020 and established operational criteria. We conducted Egger’s regression tests and created funnel plots to test for publication bias. Finally, a three-level meta-analytic model analysis of data from 130 studies involving 43,938 participants and 1,355 effect sizes revealed a moderate relationship between WM and MPS (r = 0.280, 95% CI = [0.263, 0.314]). Moreover, moderator analyses showed that: (1) dressed-up word problems were more strongly tied to WM than to intra-mathematical problems; (2) the central executive function showed the strongest relation with MPS, whereas the phonological loop had the weakest; (3) gender ratio had significant moderating effects; and (4) some of the above-mentioned significant moderating effects were unique after controlling for other factors. Implications for research and practice were also discussed.

Sex Differences in Developmental Pathways to Mathematical Competence

The study tested the hypothesis that there are sex differences in the pathways to mathematical development. Three hundred forty-two adolescents (169 boys) were assessed in various mathematics areas from arithmetic fluency to algebra across 6th to 9th grade, inclusive, and completed a battery of working memory, spatial, and intelligence measures in middle school. Their middle school and 9th grade teachers reported on their in-class attentive behavior. There were no sex differences in overall mathematics performance, but boys had advantages on all spatial measures (ds = .29 to .58) and girls were more attentive in classroom settings (ds = -.28 to -.37). A series of structural equation models indicated that 6th- to 9th-grade mathematical competence was influenced by a combination of general cognitive ability, spatial abilities, and in-class attention. General cognitive ability was important for both sexes but the spatial pathway to mathematical competence was relatively more important for boys and the in-class attention pathway for girls.

The influence of home environmental factors on kindergarten children's addition strategy use

Young children vary widely in their levels of math knowledge, their abilities to solve math problems, and the strategies they use to solve math problems. As much of later math builds on children's early understanding of basic math facts and problem-solving strategies, understanding influences on children's early problem solving is important. Few studies, however, have examined the home environment in relation to children's strategy use during arithmetic problems. We examined how both structural characteristics of children's home environments, such as socioeconomic status (SES), as well as the learning environment, such as engagement in math and literacy activities at home, related to their use of problem-solving strategies for numerical addition problems. Kindergarten children from diverse backgrounds completed a measure of addition problem solving and strategy use, including simple and complex numerical problems. Strategies were coded based on a combination of accuracy and strategy sophistication, with higher scores indicating problems solved correctly with more sophisticated strategies. Parents completed a home activities questionnaire, reporting the frequency with which they and their child had engaged in math and literacy activities at home over the past month. An exploratory factor analysis identified three components of the home activities - a basic activities factor, an advanced math activities factor, and a literacy activities factor. Findings indicated that SES related to children's strategy sophistication, and frequency of engaging in advanced math and literacy activities at home predicted strategy sophistication, however, engaging in activities at home did not moderate the relations between SES and strategy sophistication. This suggests that family engagement in activities at home may promote early arithmetic skills, and that the role of home environmental characteristics should be considered in children's arithmetic strategy use and performance over development.

The Relationship between Working Memory and Arithmetic in Primary School Children: A Meta-Analysis

Working memory (WM) plays a crucial role in the development of arithmetic ability. However, research findings related to which factors influence the relationship between WM and arithmetic skills are inconsistent. The present meta-analysis aimed to examine the links between WM and arithmetic in primary school children and investigate whether this is dependent on WM domains (i.e., verbal, visual, spatial), child age, arithmetic operation type, and arithmetic task type. A total of 11,224 participants with an age range of 6- to 12 years, from 55 independent samples were included in the meta-analysis. Analysis of 46 studies with 187 effect sizes revealed an overall significant and medium correlation between WM and arithmetic. Heterogeneity analyses indicated that verbal WM showed a stronger correlation with arithmetic than visuospatial WM, and that correlations between verbal WM and arithmetic declined with age, whereas correlations between spatial-sequential, and spatial-simultaneous WM and arithmetic remained stable throughout development. Addition and subtraction were more involved in verbal WM than multiplication and division. Moreover, mental and written arithmetic showed comparable correlations with WM in all domains. These findings suggest moderation effects of WM domains, age, and operation types in the WM-arithmetic relationship and highlight the significant role of verbal WM in arithmetic ability in primary school children.

Sex differences in the association of math achievement with visual-spatial and verbal working memory: Does the type of math test matter?

Previous research on sex differences in mathematical achievement shows mixed findings, which have been argued to depend on types of math tests used and the type of solution strategies (i.e., verbal versus visual‐spatial) these tests evoke. The current study evaluated sex differences in (a) performance (development) on two types of math tests in primary schools and (b) the predictive value of verbal and visual‐spatial working memory on math achievement. Children (N = 3175) from grades 2 through five participated. Visual‐spatial and verbal working memory were assessed using online computerized tasks. Math performance was assessed five times during two school years using a speeded arithmetic test (math fluency) and a word problem test (math problem solving). Results from Multilevel Multigroup Latent Growth Modeling, showed that sex differences in level and growth of math performance were mixed and very small. Sex differences in the predictive value of verbal and visual‐spatial working memory for math performance suggested that boys seemed to rely more on verbal strategies than girls. Explanations focus on cognitive and emotional factors and how these may interact to possibly amplify sex differences as children grow older.

Arithmetic learning in children: An fMRI training study

Arithmetic learning is characterized by a change from procedural strategies to fact retrieval. fMRI training studies in adults have revealed that this change coincides with decreased activation in the prefrontal cortex (PFC) and that within the parietal lobe, a shift occurs from the intraparietal sulcus (IPS) to the angular gyrus (AG) during this change. It remains to be determined whether similar changes can be observed in children, particularly because children often recruit the hippocampus (HC) during fact retrieval, an observation that has not consistently been found in adults. In order to experimentally manipulate arithmetic strategy change, 26 typically developing 9- to-10-year-olds completed a six day at-home training of complex multiplication items (e.g. 16 × 4). Before and after training, children were presented with three multiplication conditions during fMRI: (1) complex to-be-trained/trained items, (2) complex untrained items and (3) single-digit items. Behavioral data indicated that training was successful. Similar to adults, children showed greater activity in the IPS and PFC for the untrained condition post-training, indicating that the fronto-parietal network during procedural arithmetic problem solving is already in place in children of this age. We did not observe the expected training-related changes in the HC. In contrast to what has been observed in adults, greater activity in the AG was not observed for the trained items. These results show that the brain processes that accompany the learning of arithmetic facts are different in children as compared to adults.

Predicting learning and achievement using GABA and glutamate concentrations in human development

Previous research has highlighted the role of glutamate and gamma-aminobutyric acid (GABA) in learning and plasticity. What is currently unknown is how this knowledge translates to real-life complex cognitive abilities that emerge slowly and how the link between these neurotransmitters and human learning and plasticity is shaped by development. While some have suggested a generic role of glutamate and GABA in learning and plasticity, others have hypothesized that their involvement shapes sensitive periods during development. Here we used a cross-sectional longitudinal design with 255 individuals (spanning primary school to university) to show that glutamate and GABA in the intraparietal sulcus explain unique variance both in current and future mathematical achievement (approximately 1.5 years). Furthermore, our findings reveal a dynamic and dissociable role of GABA and glutamate in predicting learning, which is reversed during development, and therefore provide novel implications for models of learning and plasticity during childhood and adulthood.

Implications of Change/Stability Patterns in Children's Non-symbolic and Symbolic Magnitude Judgment Abilities Over One Year: A Latent Transition Analysis

Non-symbolic magnitude abilities are often claimed to support the acquisition of symbolic magnitude abilities, which, in turn, are claimed to support emerging math abilities. However, not all studies find links between non-symbolic and symbolic magnitude abilities, or between them and math ability. To investigate possible reasons for these different findings, recent research has analyzed differences in non-symbolic/symbolic magnitude abilities using latent class modeling and has identified four different magnitude ability profiles residing within the general magnitude ability distribution that were differentially related to cognitive and math abilities. These findings may help explain the different patterns of findings observed in previous research. To further investigate this possibility, we (1) attempted to replicate earlier findings, (2) determine whether magnitude ability profiles remained stable or changed over 1 year; and (3) assessed the degree to which stability/change in profiles were related to cognitive and math abilities. We used latent transition analysis to investigate stability/changes in non-symbolic and symbolic magnitude abilities of 109 5- to 6-year olds twice in 1 year. At Time 1 and 2, non-symbolic and symbolic magnitude abilities, number transcoding and single-digit addition abilities were assessed. Visuospatial working memory (VSWM), naming numbers, non-verbal IQ, basic RT was also assessed at Time 1. Analysis showed stability in one profile and changes in the three others over 1 year. VSWM and naming numbers predicted profile membership at Time 1 and 2, and profile membership predicted math abilities at both time points. The findings confirm the existence of four different non-symbolic-symbolic magnitude ability profiles; we suggest the changes over time in them potentially reflect deficit, delay, and normal math developmental pathways.

Prevention: Necessary But Insufficient? A 2-Year Follow-Up of an Effective First-Grade Mathematics Intervention

We present first-grade, second-grade, and third-grade impacts for a first-grade intervention targeting the conceptual and procedural bases that support arithmetic. At-risk students (average age at pretest = 6.5) were randomly assigned to three conditions: a control group (n = 224) and two variants of the intervention (same conceptual instruction but different forms of practice: speeded [n = 211] vs. nonspeeded [n = 204]). Impacts on all first-grade content outcomes were significant and positive, but no follow-up impacts were significant. Many intervention children achieved average mathematics achievement at the end of third grade, and prior math and reading assessment performance predicted which students will require sustained intervention. Finally, projecting impacts 2 years later based on nonexperimental estimates of effects of first-grade math skills overestimates long-term intervention effects.

Different Subcomponents of Executive Functioning Predict Different Growth Parameters in Mathematics: Evidence From a 4-Year Longitudinal Study With Chinese Children

Executive functioning (EF), an umbrella term used to represent cognitive skills engaged in goal-directed behaviors, has been found to be a unique predictor of mathematics performance. However, very few studies have examined how the three core EF subcomponents (inhibition, shifting, and working memory) predict the growth parameters (intercept and slope) in mathematics skills and even fewer studies have been conducted in a non-Western country. Thus, the purpose of this study was to examine how inhibition, shifting, and working memory predict the growth parameters in arithmetic accuracy and fluency in a group of Chinese children (n = 179) followed from Grade 2 (mean age = 97.89 months) to Grade 5 (mean age = 133.43 months). In Grade 2, children were assessed on measures of nonverbal IQ, number sense, speed of processing, inhibition, shifting, and working memory. In addition, in Grades 2–5, they were assessed on arithmetic accuracy and fluency. Results of structural equation modeling showed that nonverbal IQ, speed of processing, and number sense predicted the intercept in arithmetic accuracy, while working memory was the only EF subcomponent to predict the slope (rate of growth) in arithmetic accuracy. In turn, number sense, speed of processing, inhibition, and shifting were significant predictors of the intercept in arithmetic fluency. None of the EF subcomponents predicted the slope in arithmetic fluency. Our findings reinforce those of previous studies in North America and Europe showing that EF contributes to mathematics performance over and above other key predictors of mathematics, and suggest that different EF subcomponents may predict different growth parameters in mathematics.

When is working memory important for arithmetic? The impact of strategy and age

Our ability to perform arithmetic relies heavily on working memory, the manipulation and maintenance of information in mind. Previous research has found that in adults, procedural strategies, particularly counting, rely on working memory to a greater extent than retrieval strategies. During childhood there are changes in the types of strategies employed, as well as an increase in the accuracy and efficiency of strategy execution. As such it seems likely that the role of working memory in arithmetic may also change, however children and adults have never been directly compared. This study used traditional dual-task methodology, with the addition of a control load condition, to investigate the extent to which working memory requirements for different arithmetic strategies change with age between 9–11 years, 12–14 years and young adulthood. We showed that both children and adults employ working memory when solving arithmetic problems, no matter what strategy they choose. This study highlights the importance of considering working memory in understanding the difficulties that some children and adults have with mathematics, as well as the need to include working memory in theoretical models of mathematical cognition.

Cognitive factors affecting children's nonsymbolic and symbolic magnitude judgment abilities: A latent profile analysis

Early math abilities are claimed to be linked to magnitude representation ability. Some claim that nonsymbolic magnitude abilities scaffold the acquisition of symbolic (Arabic number) magnitude abilities and influence math ability. Others claim that symbolic magnitude abilities, and ipso facto math abilities, are independent of nonsymbolic abilities and instead depend on the ability to process number symbols (e.g., 2, 7). Currently, the issue of whether symbolic abilities are or are not related to nonsymbolic abilities, and the cognitive factors associated with nonsymbolic-symbolic relationships, remains unresolved. We suggest that different nonsymbolic-symbolic relationships reside within the general magnitude ability distribution and that different cognitive abilities are likely associated with these different relationships. We further suggest that the different nonsymbolic-symbolic relationships and cognitive abilities in combination differentially predict math abilities. To test these claims, we used latent profile analysis to identify nonsymbolic-symbolic judgment patterns of 124, 5- to 7-year-olds. We also assessed four cognitive factors (visuospatial working memory [VSWM], naming numbers, nonverbal IQ, and basic reaction time [RT]) and two math abilities (number transcoding and single-digit addition abilities). Four nonsymbolic-symbolic ability profiles were identified. Naming numbers, VSWM, and basic RT abilities were differentially associated with the different ability profiles and in combination differentially predicted math abilities. Findings show that different patterns of nonsymbolic-symbolic magnitude abilities can be identified and suggest that an adequate account of math development should specify the inter-relationship between cognitive factors and nonsymbolic-symbolic ability patterns.

Working memory in children's math learning and its disruption in dyscalculia

Working memory (WM) plays an essential role in children's mathematical learning. WM influences both the early foundational phases of number knowledge acquisition and subsequent maturation of problem solving skills. The role of individual WM components in mathematical cognition depends not only on problem complexity but also on individual differences in mathematical abilities. Furthermore, the contributions of individual WM components change dynamically over development with visuospatial processes playing an increasingly important role in learning and enhancing mathematical proficiency. Convergent findings from neuroimaging studies are now providing fundamental insights into the link between WM and mathematical cognition, and the mechanisms by which poor WM contributes to learning disabilities. Evidence to date suggests that visuospatial WM is a specific source of vulnerability in children with mathematical learning disabilities and needs to be considered as a key component in cognitive, neurobiological, and developmental models of typical and atypical mathematical skill acquisition.

The role of physical digit representation and numerical magnitude representation in children's multiplication fact retrieval

Arithmetic facts, in particular multiplication tables, are thought to be stored in long-term memory and to be interference prone. At least two representations underpinning these arithmetic facts have been suggested: a physical representation of the digits and a numerical magnitude representation. We hypothesized that both representations are possible sources of interference that could explain individual differences in multiplication fact performance and/or in strategy use. We investigated the specificity of these interferences on arithmetic fact retrieval and explored the relation between interference and performance on the different arithmetic operations and on general mathematics achievement. Participants were 79 fourth-grade children (M_age=9.6 years) who completed a products comparison and a multiplication production task with verbal strategy reports. Performances on a speeded calculation test including the four operations and on a general mathematics achievement test were also collected. Only the interference coming from physical representations was a significant predictor of the performance across multiplications. However, both the magnitude and physical representations were unique predictors of individual differences in multiplication. The frequency of the retrieval strategy across multiplication problems and across individuals was determined only by the physical representation, which therefore is suggested as being responsible for memory storage issues. Interestingly, this impact of physical representation was not observed when predicting performance on subtraction or on general mathematical achievement. In contrast, the impact of the numerical magnitude representation was more general in that it was observed across all arithmetic operations and in general mathematics achievement.

Memory and cognitive control circuits in mathematical cognition and learning

Numerical cognition relies on interactions within and between multiple functional brain systems, including those subserving quantity processing, working memory, declarative memory, and cognitive control. This chapter describes recent advances in our understanding of memory and control circuits in mathematical cognition and learning. The working memory system involves multiple parietal-frontal circuits which create short-term representations that allow manipulation of discrete quantities over several seconds. In contrast, hippocampal-frontal circuits underlying the declarative memory system play an important role in formation of associative memories and binding of new and old information, leading to the formation of long-term memories that allow generalization beyond individual problem attributes. The flow of information across these systems is regulated by flexible cognitive control systems which facilitate the integration and manipulation of quantity and mnemonic information. The implications of recent research for formulating a more comprehensive systems neuroscience view of the neural basis of mathematical learning and knowledge acquisition in both children and adults are discussed.

Individual differences in children's mathematics achievement: The roles of symbolic numerical magnitude processing and domain-general cognitive functions

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.

Individual differences in algebraic cognition: Relation to the approximate number and semantic memory systems

The relation between performance on measures of algebraic cognition and acuity of the approximate number system (ANS) and memory for addition facts was assessed for 171 ninth graders (92 girls) while controlling for parental education, sex, reading achievement, speed of numeral processing, fluency of symbolic number processing, intelligence, and the central executive component of working memory. The algebraic tasks assessed accuracy in placing x,y pairs in the coordinate plane, speed and accuracy of expression evaluation, and schema memory for algebra equations. ANS acuity was related to accuracy of placements in the coordinate plane and expression evaluation but not to schema memory. Frequency of fact retrieval errors was related to schema memory but not to coordinate plane or expression evaluation accuracy. The results suggest that the ANS may contribute to or be influenced by spatial-numerical and numerical-only quantity judgments in algebraic contexts, whereas difficulties in committing addition facts to long-term memory may presage slow formation of memories for the basic structure of algebra equations. More generally, the results suggest that different brain and cognitive systems are engaged during the learning of different components of algebraic competence while controlling for demographic and domain general abilities.

The role of fingers in the development of counting and arithmetic skills

Interactions between fingers and numbers have been reported in the existing literature on numerical cognition. The aim of the present research was to test whether hand interference movements might have an impact on children performance in counting and basic arithmetic problem solving. In Experiment 1, 5-year-old children had to perform both a one-target and a two-target counting task in three different conditions: with no constraints, while making interfering hand movements or while making interfering foot movements. In Experiment 2, first and fourth graders were required to perform addition problems under the same control and sensori-motor interfering conditions. In both tasks, the hand movements caused more disruption than the foot movements, suggesting that finger-counting plays a functional role in the development of counting and arithmetic.

Profiles of children's arithmetic fact development: a model-based clustering approach

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.

Refining the quantitative pathway of the Pathways to Mathematics model

In the current study, we adopted the Pathways to Mathematics model of LeFevre et al. (2010). In this model, there are three cognitive domains--labeled as the quantitative, linguistic, and working memory pathways--that make unique contributions to children's mathematical development. We attempted to refine the quantitative pathway by combining children's (N=141 in Grades 2 and 3) subitizing, counting, and symbolic magnitude comparison skills using principal components analysis. The quantitative pathway was examined in relation to dependent numerical measures (backward counting, arithmetic fluency, calculation, and number system knowledge) and a dependent reading measure, while simultaneously accounting for linguistic and working memory skills. Analyses controlled for processing speed, parental education, and gender. We hypothesized that the quantitative, linguistic, and working memory pathways would account for unique variance in the numerical outcomes; this was the case for backward counting and arithmetic fluency. However, only the quantitative and linguistic pathways (not working memory) accounted for unique variance in calculation and number system knowledge. Not surprisingly, only the linguistic pathway accounted for unique variance in the reading measure. These findings suggest that the relative contributions of quantitative, linguistic, and working memory skills vary depending on the specific cognitive task.

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

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

State and trait effects on individual differences in children's mathematical development

Substantial longitudinal relations between children's early mathematics achievement and their much later mathematics achievement are firmly established. These findings are seemingly at odds with studies showing that early educational interventions have diminishing effects on children's mathematics achievement across time. We hypothesized that individual differences in children's later mathematical knowledge are more an indicator of stable, underlying characteristics related to mathematics learning throughout development than of direct effects of early mathematical competency on later mathematical competency. We tested this hypothesis in two longitudinal data sets, by simultaneously modeling effects of latent traits (stable characteristics that influence learning across time) and states (e.g., prior knowledge) on children's mathematics achievement over time. Latent trait effects on children's mathematical development were substantially larger than state effects. Approximately 60% of the variance in trait mathematics achievement was accounted for by commonly used control variables, such as working memory, but residual trait effects remained larger than state effects. Implications for research and practice are discussed.

Early predictors of middle school fraction knowledge

Recent findings that earlier fraction knowledge predicts later mathematics achievement raise the question of what predicts later fraction knowledge. Analyses of longitudinal data indicated that whole number magnitude knowledge in first grade predicted knowledge of fraction magnitudes in middle school, controlling for whole number arithmetic proficiency, domain general cognitive abilities, parental income and education, race, and gender. Similarly, knowledge of whole number arithmetic in first grade predicted knowledge of fraction arithmetic in middle school, controlling for whole number magnitude knowledge in first grade and the other control variables. In contrast, neither type of early whole number knowledge uniquely predicted middle school reading achievement. We discuss the implications of these findings for theories of numerical development and for improving mathematics learning.

Fractionating the neural correlates of individual working memory components underlying arithmetic problem solving skills in children

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

Independent contributions of the central executive, intelligence, and in-class attentive behavior to developmental change in the strategies used to solve addition problems

Children's (N=275) use of retrieval, decomposition (e.g., 7=4+3 and thus 6+7=6+4+3), and counting to solve additional problems was longitudinally assessed from first grade to fourth grade, and intelligence, working memory, and in-class attentive behavior was assessed in one or several grades. The goal was to assess the relation between capacity of the central executive component of working memory, controlling for intelligence and in-class attentive behavior, and grade-related changes in children's use of these strategies. The predictor on intercept effects from multilevel models revealed that children with higher central executive capacity correctly retrieved more facts and used the most sophisticated counting procedure more frequently and accurately than their lower capacity peers at the beginning of first grade, but the predictor on slope effects indicated that this advantage disappeared (retrieval) or declined in importance (counting) from first grade to fourth grade. The predictor on slope effects also revealed that from first grade to fourth grade, children with higher capacity adopted the decomposition strategy more quickly than other children. The results remained robust with controls for children's sex, race, school site, speed of encoding Arabic numerals and articulating number words, and mathematics achievement in kindergarten. The results also revealed that intelligence and in-class attentive behavior independently contributed to children's strategy development.