Cognitive components of a mathematical processing network in 9-year-old children

Stress, Time Pressure, Strategy Selection and Math Anxiety in Mathematics: A Review of the Literature

We review how stress induction, time pressure manipulations and math anxiety can interfere with or modulate selection of problem-solving strategies (henceforth “strategy selection”) in arithmetical tasks. Nineteen relevant articles were identified, which contain references to strategy selection and time limit (or time manipulations), with some also discussing emotional aspects in mathematical outcomes. Few of these take cognitive processes such as working memory or executive functions into consideration. We conclude that due to the sparsity of available literature our questions can only be partially answered and currently there is not much evidence of clear associations. We identify major gaps in knowledge and raise a series of open questions to guide further research.

Visuospatial and verbal memory in mental arithmetic

Working memory allows complex information to be remembered and manipulated over short periods of time. Correlations between working memory and mathematics achievement have been shown across the lifespan. However, only a few studies have examined the potentially distinct contributions of domain-specific visuospatial and verbal working memory resources in mental arithmetic computation. Here we aimed to fill this gap in a series of six experiments pairing addition and subtraction tasks with verbal and visuospatial working memory and interference tasks. In general, we found higher levels of interference between mental arithmetic and visuospatial working memory tasks than between mental arithmetic and verbal working memory tasks. Additionally, we found that interference that matched the working memory domain of the task (e.g., verbal task with verbal interference) lowered working memory performance more than mismatched interference (verbal task with visuospatial interference). Findings suggest that mental arithmetic relies on domain-specific working memory resources.

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.

How do individual differences in children's domain specific and domain general abilities relate to brain activity within the intraparietal sulcus during arithmetic? An fMRI study

Previous research has demonstrated that children recruit the intraparietal sulcus (IPS) during arithmetic, which has largely been attributed to domain-specific processes such as quantity manipulations. However, the IPS has also been found to be important for domain-general abilities, such as visuo-spatial working memory (VSWM). Based on the current literature it is unclear whether individual differences in domain-specific skills, domain-general skills, or a combination of the two, are related to the recruitment of the IPS during arithmetic. This study examines how individual differences in both domain general and domain specific competencies relate to brain activity in the IPS during arithmetic, and whether the relationships are related to how brain activity is measured. In a sample of 44 school-aged children, we found that VSWM was only weakly related to a neural index of arithmetic complexity (neural problem size effect), whereas symbolic number processing skills (symbolic comparison and ordering) were related to overall arithmetic activity (both small and large problems). By simultaneously examining multiple domain-general and domain specific measures, we were also able to determine that symbolic skills were a stronger predictor of brain activity within the IPS than domain general skills such as VSWM and domain specific skills such as non-symbolic number processing. Together, these findings highlight that neural problem size effect may reflect different cognitive processes than brain activity across both small and large arithmetic problems, and that symbolic number processing skills are a critical predictor of variability in IPS activity during arithmetic. Hum Brain Mapp 38:3941-3956, 2017. © 2017 Wiley Periodicals, Inc.

The relationship between non-symbolic multiplication and division in childhood

Children without formal education in addition and subtraction are able to perform multi-step operations over an approximate number of objects. Further, their performance improves when solving approximate (but not exact) addition and subtraction problems that allow for inversion as a shortcut (e.g., a + b - b = a). The current study examines children's ability to perform multi-step operations, and the potential for an inversion benefit, for the operations of approximate, non-symbolic multiplication and division. Children were trained to compute a multiplication and division scaling factor (*2 or /2, *4 or /4), and were then tested on problems that combined two of these factors in a way that either allowed for an inversion shortcut (e.g., 8*4/4) or did not (e.g., 8*4/2). Children's performance was significantly better than chance for all scaling factors during training, and they successfully computed the outcomes of the multi-step testing problems. They did not exhibit a performance benefit for problems with the a*b/b structure, suggesting that they did not draw upon inversion reasoning as a logical shortcut to help them solve the multi-step test problems.

Fadeout in an early mathematics intervention: Constraining content or preexisting differences?

A robust finding across research on early childhood educational interventions is that the treatment effect diminishes over time, with children not receiving the intervention eventually catching up to children who did. One popular explanation for fadeout of early mathematics interventions is that elementary school teachers may not teach the kind of advanced content that children are prepared for after receiving the intervention, so lower-achieving children in the control groups of early mathematics interventions catch up to the higher-achieving children in the treatment groups. An alternative explanation is that persistent individual differences in children's long-term mathematical development result more from relatively stable preexisting differences in their skills and environments than from the direct effects of previous knowledge on later knowledge. We tested these 2 hypotheses using data from an effective preschool mathematics intervention previously known to show a diminishing treatment effect over time. We compared the intervention group to a matched subset of the control group with a similar mean and variance of scores at the end of treatment. We then tested the relative contributions of factors that similarly constrain learning in children from treatment and control groups with the same level of posttreatment achievement and preexisting differences between these 2 groups to the fadeout of the treatment effect over time. We found approximately 72% of the fadeout effect to be attributable to preexisting differences between children in treatment and control groups with the same level of achievement at posttest. These differences were fully statistically attenuated by children's prior academic achievement. (PsycINFO Database Record

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.

Subtypes and comorbidity in mathematical learning disabilities: Multidimensional study of verbal and visual memory processes is key to understanding

A large body of research suggests that mathematical learning disability (MLD) is related to working memory impairment. Here, I organize part of this literature through a meta-analysis of 36 studies with 665 MLD and 1049 control participants. I demonstrate that one subtype of MLD is associated with reading problems and weak verbal short-term and working memory. Another subtype of MLD does not have associated reading problems and is linked to weak visuospatial short-term and working memory. In order to better understand MLD we need to precisely define potentially modality-specific memory subprocesses and supporting executive functions, relevant for mathematical learning. This can be achieved by taking a multidimensional parametric approach systematically probing an extended network of cognitive functions. Rather than creating arbitrary subgroups and/or focus on a single factor, highly powered studies need to position individuals in a multidimensional parametric space. This will allow us to understand the multidimensional structure of cognitive functions and their relationship to mathematical performance.

Basic and advanced numerical performances relate to mathematical expertise but are fully mediated by visuospatial skills

Recent studies have highlighted the potential role of basic numerical processing in the acquisition of numerical and mathematical competences. However, it is debated whether high-level numerical skills and mathematics depends specifically on basic numerical representations. In this study mathematicians and nonmathematicians performed a basic number line task, which required mapping positive and negative numbers on a physical horizontal line, and has been shown to correlate with more advanced numerical abilities and mathematical achievement. We found that mathematicians were more accurate compared with nonmathematicians when mapping positive, but not negative numbers, which are considered numerical primitives and cultural artifacts, respectively. Moreover, performance on positive number mapping could predict whether one is a mathematician or not, and was mediated by more advanced mathematical skills. This finding might suggest a link between basic and advanced mathematical skills. However, when we included visuospatial skills, as measured by block design subtest, the mediation analysis revealed that the relation between the performance in the number line task and the group membership was explained by non-numerical visuospatial skills. These results demonstrate that relation between basic, even specific, numerical skills and advanced mathematical achievement can be artifactual and explained by visuospatial processing.

Cognitive and numerosity predictors of mathematical skills in middle school

There is a strong research base on the underlying concomitants of early developing math skills. Fewer studies have focused on later developing skills. Here, we focused on direct and indirect contributions of cognitive measures (e.g., language, spatial skills, working memory) and numerosity measures, as well as arithmetic proficiency, on key outcomes of fraction performance, proportional reasoning, and broad mathematics achievement at sixth grade (N=162) via path analysis. We expected a hierarchy of skill development, with predominantly indirect effects of cognitive factors via number and arithmetic. Results controlling for age showed that the combination of cognitive, number, and arithmetic variables cumulatively accounted for 38% to 44% of the variance in fractions, proportional reasoning, and broad mathematics. There was consistency across outcomes, with more proximal skills providing direct effects and with the effects of cognitive skills being mediated by number and by more proximal skills. Results support a hierarchical progression from domain-general cognitive processes through numerosity and arithmetic skills to proportional reasoning to broad mathematics achievement.

Thinking about quantity: the intertwined development of spatial and numerical cognition

There are many continuous quantitative dimensions in the physical world. Philosophical, psychological, and neural work has focused mostly on space and number. However, there are other important continuous dimensions (e.g., time and mass). Moreover, space can be broken down into more specific dimensions (e.g., length, area, and density) and number can be conceptualized discretely or continuously (i.e., natural vs real numbers). Variation on these quantitative dimensions is typically correlated, e.g., larger objects often weigh more than smaller ones. Number is a distinctive continuous dimension because the natural numbers (i.e., positive integers) are used to quantify collections of discrete objects. This aspect of number is emphasized by teaching of the count word sequence and arithmetic during the early school years. We review research on spatial and numerical estimation, and argue that a generalized magnitude system is the starting point for development in both domains. Development occurs along several lines: (1) changes in capacity, durability, and precision, (2) differentiation of the generalized magnitude system into separable dimensions, (3) formation of a discrete number system, i.e., the positive integers, (4) mapping the positive integers onto the continuous number line, and (5) acquiring abstract knowledge of the relations between pairs of systems. We discuss implications of this approach for teaching various topics in mathematics, including scaling, measurement, proportional reasoning, and fractions.

Early language and executive skills predict variations in number and arithmetic skills in children at family-risk of dyslexia and typically developing controls

Two important foundations for learning are language and executive skills. Data from a longitudinal study tracking the development of 93 children at family-risk of dyslexia and 76 controls was used to investigate the influence of these skills on the development of arithmetic. A two-group longitudinal path model assessed the relationships between language and executive skills at 3-4 years, verbal number skills (counting and number knowledge) and phonological processing skills at 4-5 years, and written arithmetic in primary school. The same cognitive processes accounted for variability in arithmetic skills in both groups. Early language and executive skills predicted variations in preschool verbal number skills, which in turn, predicted arithmetic skills in school. In contrast, phonological awareness was not a predictor of later arithmetic skills. These results suggest that verbal and executive processes provide the foundation for verbal number skills, which in turn influence the development of formal arithmetic skills. Problems in early language development may explain the comorbidity between reading and mathematics disorder.

A general number-to-space mapping deficit in developmental dyscalculia

Previous research on developmental dyscalculia (DD) suggested that deficits in the number line estimation task are related to a failure to represent number magnitude linearly. This conclusion was derived from the observation of logarithmically shaped estimation patterns. However, recent research questioned this idea of an isomorphic relationship between estimation patterns and number magnitude representation. In the present study, we evaluated an alternative hypothesis: impairments in the number line estimation task are due to a general deficit in mapping numbers onto space. Adults with DD and a matched control group had to learn linear and non-linear layouts of the number line via feedback. Afterwards, we assessed their performance how well they learnt the new number-space mappings. We found irrespective of the layouts worse performance of adults with DD. Additionally, in case of the linear layout, we observed that their performance did not differ from controls near reference points, but that differences between groups increased as the distance to reference point increased. We conclude that worse performance of adults with DD in the number line task might be due a deficit in mapping numbers onto space which can be partly overcome relying on reference points.

Early numerical abilities and cognitive skills in kindergarten children

In this study, a unitary path analysis model was developed to investigate the relationship between cognitive variables (derived from published studies) and early numerical abilities in children attending the last year of kindergarten. We tested 100 children starting their last year of kindergarten on the following cognitive abilities: intelligence, phonological abilities, counting, verbal and visuospatial short-term memory and working memory, processing speed, and early numerical abilities. The same children were tested again on early numerical abilities at the end of the same year. The children's early numerical abilities at the beginning of the final year of kindergarten were found to be directly related to their verbal intelligence, phonological abilities, processing speed, and working memory and to be indirectly related to their nonverbal intelligence. Early numerical abilities at the end of the same year are directly related not only to early numerical abilities assessed at the beginning of the year but also to working memory and phonological abilities as well as have an indirect relationship with verbal and nonverbal intelligence. Overall, our results showed that both general and specific abilities are related to early mathematic learning in kindergarten-age children.

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.

Working memory and early numeracy training in preschool children

Many factors influence children's performance in mathematical achievement, including both domain-specific and domain-general factors. This study aimed to verify and compare the effects of two types of training on early numerical skills. One type of training focused on the enhancement of working memory, a domain-general precursor, while the other focused on the enhancement of early numeracy, a domain-specific precursor. The participants were 48 five-year-old preschool children. Both the working memory and early numeracy training programs were implemented for 5 weeks. The results showed that the early numeracy intervention specifically improved early numeracy abilities in preschool children, whereas working memory intervention improved not only working memory abilities but also early numeracy abilities. These findings stress the importance of performing activities designed to train working memory abilities, in addition to activities aimed to enhance more specific skills, in the early prevention of learning difficulties during preschool years.

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.