Effects of problem size, operation, and working-memory span on simple-arithmetic strategies: differences between children and adults?

Structural and transcriptional signatures of arithmetic abilities in children

Arithmetic ability is critical for daily life, academic achievement, career development, and future economic success. Individual differences in arithmetic skills among children and adolescents are related to variations in brain structures. Most existing studies have used hypothesis-driven region of interest analysis. To identify distributed brain regions related to arithmetic ability, we used data-driven cross-validated predictive models to analyze cross-sectional behavioral and structural MRI data in children and adolescents. The gray matter volume (GMV) of widespread brain regions reliably predicted arithmetic abilities measured by the Comprehensive Mathematical Abilities Test. Furthermore, we applied neuroimaging-transcriptome association analysis to explore transcriptional signatures associated with structural patterns of arithmetic ability. Structural patterns of arithmetic ability primarily correlated with transcriptional profiles enriched for genes involved in transmembrane transport and synaptic signaling. Our findings enhance our understanding of the neural and genetic mechanisms underlying children's arithmetic ability and offer a practical predictor for arithmetic skills during development.

Evidence for Automatic, Stimulus Driven, Arithmetic Processing of Single-digit Multiplication Problems

Certain stimuli can automatically trigger different behaviors in a stimulus-driven manner. To investigate whether mathematical equations automatically trigger the tendency to engage in arithmetic processing, we asked whether the presentation of multiplication equations in an irrelevant dimension can trigger the automatic task of arithmetic processing and if so, which processes are involved. To that end, we employed a color-naming task in which participants had to name the color of different stimuli, such as: mathematical equations (e.g., 4 × 6 = 24), neutral-symbols (e.g., ####), neutral-words (e.g., building), and same-number strings (e.g., 11111), which appeared as one of four different colors. We found that mathematical equations and regular words in the irrelevant dimension triggered more task conflict (i.e., color naming's reaction time was longer) as compared to same-number strings. In addition, we found evidence for the automatic activation of different numerical processes; such that large-size equations (7 × 9 = 63) triggered more conflict as compared with small-size (2 × 3 = 6) equations and same-parity incorrect equations (3 × 2 = 8) triggered more conflict as compared to different-parity incorrect equations (4 × 2 = 9). We found no evidence indicating a distinction between the correct and incorrect equations. We discussed the relevance of the findings to the automaticity of arithmetic abilities and other domains in numerical cognition.

The neural correlates of retrieval and procedural strategies in mental arithmetic: A functional neuroimaging meta-analysis

Mental arithmetic is a complex skill of great importance for later academic and life success. Many neuroimaging studies and several meta-analyses have aimed to identify the neural correlates of mental arithmetic. Previous meta-analyses of arithmetic grouped all problem types into a single meta-analytic map, despite evidence suggesting that different types of arithmetic problems are solved using different strategies. We used activation likelihood estimation (ALE) to conduct quantitative meta-analyses of mental arithmetic neuroimaging (n = 31) studies, and subsequently grouped contrasts from the 31 studies into problems that are typically solved using retrieval strategies (retrieval problems) (n = 18) and problems that are typically solved using procedural strategies (procedural problems) (n = 19). Foci were compiled to generate probabilistic maps of activation for mental arithmetic (i.e., all problem types), retrieval problems, and procedural problems. Conjunction and contrast analyses were conducted to examine overlapping and distinct activation for retrieval and procedural problems. The conjunction analysis revealed overlapping activation for retrieval and procedural problems in the bilateral inferior parietal lobules, regions typically associated with magnitude processing. Contrast analyses revealed specific activation in the left angular gyrus for retrieval problems and specific activation in the inferior frontal gyrus and cingulate gyrus for procedural problems. These findings indicate that the neural bases of arithmetic systematically differs according to problem type, providing new insights into the dynamic and task-dependent neural underpinnings of the calculating brain.

On doing multi-act arithmetic: A multitrait-multimethod approach of performance dimensions in integrated multitasking

Here we present a systematic plan to the experimental study of test-retest reliability in the multitasking domain, adopting the multitrait-multimethod (MTMM) approach to evaluate the psychometric properties of performance in Düker-type speeded multiple-act mental arithmetic. These form of tasks capacitate the experimental analysis of integrated multi-step processing by combining multiple mental operations in flexible ways in the service of the overarching goal of completing the task. A particular focus was on scoring methodology, particularly measures of response speed variability. To this end, we present data of two experiments with regard to (a) test-retest reliability, (b) between-measures correlational structure, (c) and stability (test-retest practice effects). Finally, we compared participants with high versus low performance variability to assess ability-related differences in measurement precision (typically used as proxy to "simulate" patient populations), which is especially relevant in the applied fields of clinical neuropsychology. The participants performed two classic integrated multi-act arithmetic tasks, combining addition and verification (Exp. 1) and addition and comparison (Exp. 2). The results revealed excellent test-retest reliability for the standard and the variability measures. The analysis of between-measures correlational structure revealed the typical pattern of convergent and discriminant relationships, and also, that absolute response speed variability was highly correlated with average speed (r > 0.85), indicating that these measures mainly deliver redundant information. In contrast, speed-adjusted (relativized) variability revealed discriminant validity being correlated to a much lesser degree with average speed, indicating that this measure delivers additional information not already provided by the speed measure. Furthermore, speed-adjusted variability was virtually unaffected by test-retest practice, which makes this measure interesting in situations with repeated testing.

When multiplying is meaningful in memory: Electrophysiological signature of the problem size effect in children

Children are less fluent at verifying the answers to larger single-digit arithmetic problems compared with smaller ones. This problem size effect may reflect the structure of memory for arithmetic facts. In the current study, typically developing third to fifth graders judged the correctness of single-digit multiplication problems, presented as a sequence of three digits, that were either small (e.g., 4 3 12 vs. 4 3 16) or large (e.g., 8 7 56 vs. 8 7 64). We measured the N400, an index of access to semantic memory, along with accuracy and response time. The N400 was modulated by problem size only for correct solutions, with larger amplitude for large problems than for small problems. This suggests that only solutions that exist in memory (i.e., correct solutions) reflect a modulation of semantic access likely based on the relative frequency of encountering small versus large problems. The absence of an N400 problem size effect for incorrect solutions suggests that the behavioral problem size effects were not due to differences in initial access to memory but instead were due to a later stage of cognitive processing that was reflected in a post-N400 main effect of problem size. A second post-N400 main effect of correctness at occipital electrodes resembles the beginning of an adult-like brain response observed in prior studies. In sum, event-related brain potentials revealed different cognitive processes for correct and incorrect solutions. These results allude to a gradual transition to an adult-like brain response, from verifying multiplication problems using semantic memory to doing so using more automatic categorization.

Functional neuroanatomy of arithmetic in monolingual and bilingual adults and children

Prior studies on the brain bases of arithmetic have not focused on (or even described) their participants' language backgrounds. Yet, unlike monolinguals, early bilinguals have the capacity to solve arithmetic problems in both of their two languages. This raises the question whether this ability, or any other experience that comes with being bilingual, affects brain activity for arithmetic in bilinguals relative to monolinguals. Here, we used functional magnetic resonance imaging to compare brain activity in 44 English monolinguals and 44 Spanish-English early bilinguals, during the solving of arithmetic problems in English. We used a factorial design to test for a main effect of bilingual Language Experience. Based on the known modulating roles of arithmetic operation and age, we used two arithmetic tasks (addition and subtraction) and studied two age groups (adults and children). When collapsing across operations and age, we found broad bilateral activation for arithmetic in both the monolingual group and the bilingual group. However, an analysis of variance revealed that there was no effect of Language Experience, nor an interaction of Language Experience with Operation or Age Group. Bayesian analyses within regions of interest chosen for their role in arithmetic further supported the finding of no effect of Language Experience on brain activity underlying arithmetic. We conclude that early bilingualism does not influence the functional neuroanatomy of simple arithmetic.

Math Performance and Academic Anxiety Forms, from Sociodemographic to Cognitive Aspects: a Meta-analysis on 906,311 Participants

The relationship between anxiety and mathematics has often been investigated in the literature. Different forms of anxiety have been evaluated, with math anxiety (MA) and test anxiety (TA) consistently being associated with various aspects of mathematics. In this meta-analysis, we have evaluated the impact of these forms of anxiety, distinguishing between different types of mathematical tasks. In investigating this relationship, we have also included potential moderators, such as age, gender, working memory, type of task, and type of material. One hundred seventy-seven studies met the inclusion criteria, providing an overall sample of 906,311 participants. Results showed that both MA and TA had a significant impact on mathematics. Sociodemographic factors had modest moderating effects. Working memory (WM) also mediated the relationship between MA and TA with mathematics; however, this indirect effect was weak. Theoretical and educational implications, as well as future directions for research in this field, are discussed.

Emerging neurodevelopmental perspectives on mathematical learning

Strong foundational skills in mathematical problem solving, acquired in early childhood, are critical not only for success in the science, technology, engineering, and mathematical (STEM) fields but also for quantitative reasoning in everyday life. The acquisition of mathematical skills relies on protracted interactive specialization of functional brain networks across development. Using a systems neuroscience approach, this review synthesizes emerging perspectives on neurodevelopmental pathways of mathematical learning, highlighting the functional brain architecture that supports these processes and sources of heterogeneity in mathematical skill acquisition. We identify the core neural building blocks of numerical cognition, anchored in the posterior parietal and ventral temporal-occipital cortices, and describe how memory and cognitive control systems, anchored in the medial temporal lobe and prefrontal cortex, help scaffold mathematical skill development. We highlight how interactive specialization of functional circuits influences mathematical learning across different stages of development. Functional and structural brain integrity and plasticity associated with math learning can be examined using an individual differences approach to better understand sources of heterogeneity in learning, including cognitive, affective, motivational, and sociocultural factors. Our review emphasizes the dynamic role of neurodevelopmental processes in mathematical learning and cognitive development more generally.

Aerobic fitness relates to superior exact and approximate arithmetic processing in college-aged adults

BACKGROUND

Aerobic fitness relates to superior math achievement, but the underlying reasons remain unclear. This study tested how more efficient processing (efficiency hypothesis) or enhanced allocation of cognitive resources (resources hypothesis) underly fitness-related differences in arithmetic cognition in a sample of 138 college-aged adults.

METHOD

Participants completed an arithmetic task while pupillary measures were recorded prior to an aerobic fitness test.

RESULTS

Higher aerobic fitness was associated with shorter reaction time for all problems and greater pupillary reactivity for problems requiring approximate and exact arithmetic.

CONCLUSIONS

Superior aerobic fitness relates to greater cognitive resources available to execute exact and approximate arithmetic faster. Fitness-related differences in math achievement may be driven by the cognitive resources underlying arithmetic strategy. These differences may extend beyond educational achievement and affect the motivation to engage in health behaviors based on quantitative information. Thus, improving cardiovascular fitness has the potential to also ameliorate health numeracy.

Arithmetic processing in children with dyscalculia: an event-related potential study

Introduction

Dyscalculia is a specific learning disorder affecting the ability to learn certain math processes, such as arithmetic data recovery. The group of children with dyscalculia is very heterogeneous, in part due to variability in their working memory (WM) deficits. To assess the brain response to arithmetic data recovery, we applied an arithmetic verification task during an event-related potential (ERP) recording. Two effects have been reported: the N400 effect (higher negative amplitude for incongruent than for congruent condition), associated with arithmetic incongruency and caused by the arithmetic priming effect, and the LPC effect (higher positive amplitude for the incongruent compared to the congruent condition), associated with a reevaluation process and modulated by the plausibility of the presented condition. This study aimed to (a) compare arithmetic processing between children with dyscalculia and children with good academic performance (GAP) using ERPs during an addition verification task and (b) explore, among children with dyscalculia, the relationship between WM and ERP effects.

Materials and Methods

EEGs of 22 children with dyscalculia (DYS group) and 22 children with GAP (GAP group) were recorded during the performance of an addition verification task. ERPs synchronized with the probe stimulus were computed separately for the congruent and incongruent probes, and included only epochs with correct answers. Mixed 2-way ANOVAs for response times and correct answers were conducted. Comparisons between groups and correlation analyses using ERP amplitude data were carried out through multivariate nonparametric permutation tests.

Results

The GAP group obtained more correct answers than the DYS group. An arithmetic N400 effect was observed in the GAP group but not in the DYS group. Both groups displayed an LPC effect. The larger the LPC amplitude was, the higher the WM index. Two subgroups were found within the DYS group: one with an average WM index and the other with a lower than average WM index. These subgroups displayed different ERPs patterns.

Discussion

The results indicated that the group of children with dyscalculia was very heterogeneous and therefore failed to show a robust LPC effect. Some of these children had WM deficits. When WM deficits were considered together with dyscalculia, an atypical ERP pattern that reflected their processing difficulties emerged. Their lack of the arithmetic N400 effect suggested that the processing in this step was not useful enough to produce an answer; thus, it was necessary to reevaluate the arithmetic-calculation process (LPC) in order to deliver a correct answer.

Conclusion

Given that dyscalculia is a very heterogeneous deficit, studies examining dyscalculia should consider exploring deficits in WM because the whole group of children with dyscalculia seems to contain at least two subpopulations that differ in their calculation process.

Dissociating Arithmetic Operations in the Parietal Cortex Using 1 Hz Repetitive Transcranial Magnetic Stimulation: The Importance of Strategy Use

The triple-code model (TCM) of number processing suggests the involvement of distinct parietal cortex areas in arithmetic operations: the bilateral horizontal segment of the intraparietal sulcus (hIPS) for arithmetic operations that require the manipulation of numerical quantities (e.g., subtraction) and the left angular gyrus (AG) for arithmetic operations that require the retrieval of answers from long-term memory (e.g., multiplication). Although neuropsychological, neuroimaging, and brain stimulation studies suggest the dissociation of these operations into distinct parietal cortex areas, the role of strategy (online calculation vs. retrieval) is not yet fully established. In the present study, we further explored the causal involvement of the left AG for multiplication and left hIPS for subtraction using a neuronavigated repetitive transcranial magnetic stimulation (rTMS) paradigm. Stimulation sites were determined based on an fMRI experiment using the same tasks. To account for the effect of strategy, participants were asked whether they used retrieval or calculation for each individual problem. We predicted that the stimulation of the left AG would selectively disrupt the retrieval of the solution to multiplication problems. On the other hand, stimulation of the left hIPS should selectively disrupt subtraction. Our results revealed that left AG stimulation was detrimental to the retrieval and online calculation of solutions for multiplication problems, as well as, the retrieval (but not online calculation) of the solutions to subtraction problems. In contrast, left hIPS stimulation had no detrimental effect on both operations regardless of strategy.

The neural association between arithmetic and basic numerical processing depends on arithmetic problem size and not chronological age

The intraparietal sulcus (IPS) is thought to be an important region for basic number processing (e.g. symbol-quantity associations) and arithmetic (e.g. addition). Evidence for shared circuitry within the IPS is largely based on comparisons across studies, and little research has investigated number processing and arithmetic in the same individuals. It is also unclear how the neural overlap between number processing and arithmetic is influenced by age and arithmetic problem difficulty. This study investigated these unresolved questions by examining basic number processing (symbol-quantity matching) and arithmetic (addition) networks in 26 adults and 42 children. Number processing and arithmetic elicited overlapping activity in the IPS in children and adults, however, the overlap was influenced by arithmetic problem size (i.e. which modulated the need to use procedural strategies). The IPS was recruited for number processing, and for arithmetic problems more likely to be solved using procedural strategies. We also found that the overlap between number processing and small-problem addition in children was comparable to the overlap between number processing and large-problem addition in adults. This finding suggests that the association between number processing and arithmetic in the IPS is related to the cognitive operation being performed rather than age.

Understanding Strategy Change: Contextual, Individual, and Metacognitive Factors

Learning, development, and response to instruction often involve changes in the strategies that learners use to solve problems. In this chapter, our focus is on mathematical problem solving in both children and adults. We offer a selective review of research on three classes of factors that may influence processes of strategy change in mathematical problem solving: contextual factors, individual factors, and metacognitive factors. Contextual factors involve information that learners encounter in the learning context, such as feedback about prior strategies and examples of alternative strategies. Individual factors involve the abilities, dispositions, and knowledge that learners bring to the learning context. Metacognitive factors involve knowledge about strategies and factors that affect the application of strategies-including perceptions of problem difficulty, confidence in the strategies one already knows, and judgments about the qualities of alternative strategies. These factors operate both independently and in combination to influence learners' behavior. Therefore, we argue that scientific progress in understanding strategy change will require comprehensive conceptual models that specify how different factors come together to explain behavior. We discuss several such models, including vulnerability-trigger models, cumulative risk models, and dynamic systems models. Research guided by such models will contribute to greater progress in understanding processes of strategy use and strategy change.

Effects of Working Memory, Strategy Use, and Single-Step Mental Addition on Multi-Step Mental Addition in Chinese Elementary Students

The aim of this paper was to examine the roles of working memory, single-step mental addition skills, and strategy use in multi-step mental addition in two independent samples of Chinese elementary students through different approaches to manipulate two dimensions of task characteristics (the primary task). In Study 1, we manipulated strategy types through the dimension of schema automaticity (whether intermediate sums were 10s) and the dimension of working memory load (WML, two steps versus four steps). A hierarchical linear model (HLM) analysis was conducted at case level, strategy level, and individual level. In Study 2, we manipulated task characteristics through schema automaticity (one-time versus two-time regrouping) and the WML (partial versus complete decomposition). A three-level HLM analysis was applied. The general findings of Study 1 and Study 2 suggested that shorter response time on single-step mental addition corresponded to shorter response time on multi-step mental addition. The use of strategies (from easier to more difficult strategies) negatively predicted response time on multi-step mental addition. Easier strategy was associated with shorter response time on multi-step mental addition. Better phonological loop was associated with shorter response time on multi-step mental addition. The findings in both studies highlighted the important role of phonological loop in mental addition in Chinese children, suggesting that the involvement of a specific subcomponent of working memory in mental arithmetic might be subject to linguistic, instructional, and contextual factors.

Is a fact retrieval deficit the main characteristic of children with mathematical learning disabilities?

Although a fact retrieval deficit is widely considered to be the hallmark of children with mathematical learning disabilities (MLD), recent studies suggest that even adults use procedural strategies to solve small additions, except for ties that are unanimously considered to be solved by retrieval. Our study, based on how MLD children process ties and non-ties compared to typically developing (TD) children, sheds new light on their retrieval and procedural difficulties. Our results show that, by the end of the second grade, MLD children do not differ in their ability to solve the tie problems that are certainly solved by retrieval, but they do struggle with both small and large non-ties. These findings emphasize the extend of the difficulties that MLD children exhibit in procedural strategies relatively to retrieval ones.

Children's Strategy Choices on Complex Subtraction Problems: Individual Differences and Developmental Changes

We examined how children's strategy choices in solving complex subtraction problems are related to grade and to variations in problem complexity. In two studies, third- and fifth-grade children (N≈160 each study) solved multi-digit subtraction problems (e.g., 34–18) and described their solution strategies. In the first experiment, strategy selection was investigated by means of a free-choice paradigm, whereas in the second study a discrete-choice approach was implemented. In both experiments, analyses of strategy repertoire indicated that third-grade children were more likely to report less-efficient strategies (i.e., counting) and relied more on the right-to-left solution algorithm compared to fifth-grade children who more often used efficient memory-based retrieval and conceptually-based left-to-right (i.e., decomposition) strategies. Nevertheless, all strategies were reported or selected by both older and younger children and strategy use varied with problem complexity and presentation format for both age groups. These results supported the overlapping waves model of strategy development and provide detailed information about patterns of strategy choice on complex subtraction problems.

Non-adaptive strategy selection in adults with high mathematical anxiety

Participants with mathematical anxiety (MA) tend to show particular difficulty in mathematical operations with high working memory (WM) demands compared to operations with lower WM demands. Accordingly, we examined strategy selection to test the cognitive mechanism underlying the observed weakness of high MA participants in mathematical operations with high WM demands. We compared two groups of college students with high or low MA, in the solution of simple non-carry addition problems (e.g., 54 + 63) and complex carryover addition problems (e.g., 59 + 63). The results indicated that high MA participants showed particular difficulty in the harder carry condition. Testing the strategy selection mechanism among high MA participants, we found in the carry condition 1) they used the common strategy less often compared to low MA participants and 2) employed unusual strategies more often compared to low MA participants. Therefore, high MA participants were less efficient in their strategy selection, which may be due to weaker spatial representations, numerical difficulties, or less experience solving complex problems. These primitive representations are not adaptive, and can negatively impact performance in math tasks with high WM demands.

Where arithmetic and phonology meet: The meta-analytic convergence of arithmetic and phonological processing in the brain

Arithmetic facts can be solved using different strategies. Research suggests that some arithmetic problems, particularly those solved by fact retrieval, are related to phonological processing ability and elicit activity in left-lateralized brain regions that support phonological processing. However, it is unclear whether common brain regions support both retrieval-based arithmetic and phonological processing, and if these regions differ across children and adults. This study used activation likelihood estimation to investigate functional neural overlap between arithmetic and phonological processing, separately for children and adults. The meta-analyses in children showed six clusters of overlapping activation concentrated in bilateral frontal regions and in the left fusiform gyrus. The meta-analyses in adults yielded two clusters of concordant activity, one in the left inferior frontal gyrus and one in the left inferior parietal lobule. A qualitative comparison across the two age groups suggests that children show more bilateral and diffuse activation than adults, which may reflect attentional processes that support more effortful processing in children. The present meta-analyses contribute novel insights into the relationship between retrieval-based arithmetic and phonological processing in the brain across children and adults, and brain regions that may support processing of more complex symbolic representations, such as arithmetic facts and words.

Impact of ageing on problem size and proactive interference in arithmetic facts solving

Arithmetic facts (AFs) are required when solving problems such as “3 × 4” and refer to calculations for which the correct answer is retrieved from memory. Currently, two important effects that modulate the performance in AFs have been highlighted: the problem size effect and the proactive interference effect. The aim of this study is to investigate possible age-related changes of the problem size effect and the proactive interference effect in AF solving. To this end, the performance of young and older adults was compared in a multiplication production task. Furthermore, an independent measure of proactive interference was assessed to further define the architecture underlying this effect in multiplication solving. The results indicate that both young and older adults were sensitive to the effects of interference and of the problem size. That is, both interference and problem size affected performance negatively: the time needed to solve a multiplication problem increases as the level of interference and the size of the problem increase. Regarding the effect of ageing, the problem size effect remains constant with age, indicating a preserved AF network in older adults. Interestingly, sensitivity to proactive interference in multiplication solving was less pronounced in older than in younger adults suggesting that part of the proactive interference has been overcome with age.

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.

Strategy over operation: neural activation in subtraction and multiplication during fact retrieval and procedural strategy use in children

Arithmetic development is characterized by strategy shifts between procedural strategy use and fact retrieval. This study is the first to explicitly investigate children's neural activation associated with the use of these different strategies. Participants were 26 typically developing 4th graders (9- to 10-year-olds), who, in a behavioral session, were asked to verbally report on a trial-by-trial basis how they had solved 100 subtraction and multiplication items. These items were subsequently presented during functional magnetic resonance imaging. An event-related design allowed us to analyze the brain responses during retrieval and procedural trials, based on the children's verbal reports. During procedural strategy use, and more specifically for the decomposition of operands strategy, activation increases were observed in the inferior and superior parietal lobes (intraparietal sulci), inferior to superior frontal gyri, bilateral areas in the occipital lobe, and insular cortex. For retrieval, in comparison to procedural strategy use, we observed increased activity in the bilateral angular and supramarginal gyri, left middle to inferior temporal gyrus, right superior temporal gyrus, and superior medial frontal gyrus. No neural differences were found between the two operations under study. These results are the first in children to provide direct evidence for alternate neural activation when different arithmetic strategies are used and further unravel that previously found effects of operation on brain activity reflect differences in arithmetic strategy use. Hum Brain Mapp 38:4657-4670, 2017. © 2017 Wiley Periodicals, Inc.

Functional neuroanatomy of arithmetic and word reading and its relationship to age

Arithmetic and written language are uniquely human skills acquired during early schooling and used daily. While prior studies have independently characterized the neural bases for arithmetic and reading, here we examine both skills in a single study to capture their shared and unique cognitive mechanisms, as well as the role of age/experience in modulating their neural representations. We used functional MRI in 7- to 29-year-olds who performed single-digit subtraction, single-digit addition, and single-word reading. Using a factorial design, we examined the main effects of Task (subtraction, addition, reading) and Age (as a continuous variable), and their interactions. A main effect of Task revealed preferential activation for subtraction in bilateral intraparietal sulci and supramarginal gyri, right insula, inferior frontal gyrus, and cingulate. The right middle temporal gyrus and left superior temporal gyrus were preferentially active for both addition and reading, and left fusiform gyrus was preferentially active for reading. A main effect of Age revealed increased activity in older participants in right angular gyrus, superior temporal sulcus, and putamen, and less activity in left supplementary motor area, suggesting a left frontal to right temporo-parietal shift of activity with increasing age/experience across all tasks. Interactions for Task by Age were found in right hippocampus and left middle frontal gyrus, with older age invoking greater activity for addition and at the same time less activity for subtraction and reading. Together, in a study conducted in the same participants using similar task and acquisition parameters, the results reveal the neural substrates of these educationally relevant cognitive skills in typical participants in the context of age/experience.

Early but not late blindness leads to enhanced arithmetic and working memory abilities

Behavioural and neurophysiological evidence suggest that vision plays an important role in the emergence and development of arithmetic abilities. However, how visual deprivation impacts on the development of arithmetic processing remains poorly understood. We compared the performances of early (EB), late blind (LB) and sighted control (SC) individuals during various arithmetic tasks involving addition, subtraction and multiplication of various complexities. We also assessed working memory (WM) performances to determine if they relate to a blind person's arithmetic capacities. Results showed that EB participants performed better than LB and SC in arithmetic tasks, especially in conditions in which verbal routines and WM abilities are needed. Moreover, EB participants also showed higher WM abilities. Together, our findings demonstrate that the absence of developmental vision does not prevent the development of refined arithmetic skills and can even trigger the refinement of these abilities in specific tasks.

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.

The Contribution of Numerical Magnitude Comparison and Phonological Processing to Individual Differences in Fourth Graders' Multiplication Fact Ability

Although numerical magnitude processing has been related to individual differences in arithmetic, its role in children’s multiplication performance remains largely unknown. On the other hand, studies have indicated that phonological awareness is an important correlate of individual differences in children’s multiplication performance, but the involvement of phonological memory, another important phonological processing skill, has not been studied in much detail. Furthermore, knowledge about the relative contribution of above mentioned processes to the specific arithmetic operation of multiplication in children is lacking. The present study therefore investigated for the first time the unique contributions of numerical magnitude comparison and phonological processing in explaining individual differences in 63 fourth graders’ multiplication fact ability (mean age = 9.6 years, SD = .67). The results showed that children’s multiplication fact competency correlated significantly with symbolic and nonsymbolic magnitude comparison as well as with phonological short-term memory. A hierarchical regression analysis revealed that, after controlling for intellectual ability and general reaction time, both symbolic and nonsymbolic magnitude comparison and phonological short-term memory accounted for unique variance in multiplication fact performance. The ability to compare symbolic magnitudes was found to contribute the most, indicating that the access to numerical magnitudes by means of Arabic digits is a key factor in explaining individual differences in children’s multiplication fact ability.

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.

ERP measures of math anxiety: how math anxiety affects working memory and mental calculation tasks?

There have been several attempts to account for the impact of Mathematical Anxiety (MA) on brain activity with variable results. The present study examines the effects of MA on ERP amplitude during performance of simple arithmetic calculations and working memory tasks. Data were obtained from 32 university students as they solved four types of arithmetic problems (one- and two-digit addition and multiplication) and a working memory task comprised of three levels of difficulty (1, 2, and 3-back task). Compared to the Low-MA group, High-MA individuals demonstrated reduced ERP amplitude at frontocentral (between 180-320 ms) and centroparietal locations (between 380-420 ms). These effects were independent of task difficulty/complexity, individual performance, and general state/trait anxiety levels. Results support the hypothesis that higher levels of self-reported MA are associated with lower cortical activation during the early stages of the processing of numeric stimuli in the context of cognitive tasks.

Efficiency of Arithmetic Procedures Modulates the Problem-Size Effect in Subtraction

The aim of this study was to examine whether differences in strategy selection and/or strategy efficiency can explain the modulation of the problem-size effect by arithmetic skill. More specifically, we wondered whether arithmetic skill increases the use of retrieval strategy in large problems, and/or enhances the efficiency of either retrieval or procedural strategies. The performance of highly-skilled (HS) and less highly-skilled (LS) individuals on a subtraction verification task was analyzed according to problem size and to the strategy reported on a trial-by-trial basis after each problem. The problem size effect was larger for LS individuals than for their HS peers, both in response time and in hit rate. Nevertheless, groups did not differ regarding the strategy reported for each subtraction size. As expected, problems in which retrieval strategy was reported were solved more quickly and more accurately than problems solved by procedural strategies. Responses using retrieval strategy were equally fast in the two groups, but HS individuals performed better than LS when using procedural strategies. The results therefore suggest that the differences in behavioral measures between groups might specifically be due to differences in the efficiency of procedural strategies.

The arithmetic problem size effect in children: an event-related potential study

This study used for the first time event-related potentials (ERPs) to examine the well-known arithmetic problem size effect in children. The electrophysiological correlates of this problem size effect have been well documented in adults, but such information in children is lacking. In the present study, 22 typically developing 12-year-olds were asked to solve single-digit addition problems of small (sum ≤ 10) and large problem size (sum > 10) and to speak the solution into a voice key while ERPs were recorded. Children displayed similar early and late components compared to previous adult studies on the problem size effect. There was no effect of problem size on the early components P1, N1, and P2. The peak amplitude of the N2 component showed more negative potentials on left and right anterior electrodes for large additions compared to small additions, which might reflect differences in attentional and working memory resources between large and small problems. The mean amplitude of the late positivity component which follows the N2, was significantly larger for large than for small additions at right parieto-occipital electrodes, in line with previous adult data. The ERPs of the problem size effect during arithmetic might be a useful neural marker for future studies on fact retrieval impairments in children with mathematical difficulties.

The influence of cardiorespiratory fitness on strategic, behavioral, and electrophysiological indices of arithmetic cognition in preadolescent children

The current study investigated the influence of cardiorespiratory fitness on arithmetic cognition in forty 9-10 year old children. Measures included a standardized mathematics achievement test to assess conceptual and computational knowledge, self-reported strategy selection, and an experimental arithmetic verification task (including small and large addition problems), which afforded the measurement of event-related brain potentials (ERPs). No differences in math achievement were observed as a function of fitness level, but all children performed better on math concepts relative to math computation. Higher fit children reported using retrieval more often to solve large arithmetic problems, relative to lower fit children. During the arithmetic verification task, higher fit children exhibited superior performance for large problems, as evidenced by greater d' scores, while all children exhibited decreased accuracy and longer reaction time for large relative to small problems, and incorrect relative to correct solutions. On the electrophysiological level, modulations of early (P1, N170) and late ERP components (P3, N400) were observed as a function of problem size and solution correctness. Higher fit children exhibited selective modulations for N170, P3, and N400 amplitude relative to lower fit children, suggesting that fitness influences symbolic encoding, attentional resource allocation and semantic processing during arithmetic tasks. The current study contributes to the fitness-cognition literature by demonstrating that the benefits of cardiorespiratory fitness extend to arithmetic cognition, which has important implications for the educational environment and the context of learning.

Left fronto-parietal white matter correlates with individual differences in children's ability to solve additions and multiplications: a tractography study

Functional neuroimaging data have pointed to the activation of a fronto-parietal network during calculation tasks, the activity of which is modulated by arithmetic operation and arithmetical competence. As the cortical brain regions of this network are distant, it is crucial to investigate the white matter connections between them and to examine how these connections are related to different arithmetic operations and individual differences in arithmetical competence. By using diffusion tensor imaging (DTI) tractography in eighteen 12-year-olds, we tested whether white matter pathways connecting these distant regions were related to children's arithmetical competence and how this association was modulated by operation. For each child, we delineated the three subcomponents of the arcuate fasciculus, a bundle of pathways linking frontal and temporo-parietal regions that are commonly active during calculation tasks. Fractional anisotropy in the left anterior portion of the arcuate fasciculus was positively correlated with addition and multiplication, but not with subtraction and division, suggesting a specific role of this left anterior segment in the solution of those problems that are expected to be solved with fact retrieval. The observed correlation was not explained by age, intelligence and working memory. Follow-up control analyses using different types of reading measures revealed that the observed correlation only disappeared when measures that draw heavily on phonological processing, such as non-word reading, were controlled for, suggesting that the association between the left arcuate fasciculus-anterior and addition/multiplication reflects the involvement of phonological processing. These results are the first to demonstrate that individual differences in fronto-parietal white matter are associated with arithmetical competence in typically developing children of a very narrow age range and indicate that this association is modulated by arithmetic operation.

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.

Eating breakfast enhances the efficiency of neural networks engaged during mental arithmetic in school-aged children

To determine the influence of a morning meal on complex mental functions in children (8-11 y), time-frequency analyses were applied to electroencephalographic (EEG) activity recorded while children solved simple addition problems after an overnight fast and again after having either eaten or skipped breakfast. Power of low frequency EEG activity [2 Hertz (Hz) bands in the 2-12 Hz range] was determined from recordings over frontal and parietal brain regions associated with mathematical thinking during mental calculation of correctly answered problems. Analyses were adjusted for background variables known to influence or reflect the development of mathematical skills, i.e., age and measures of math competence and math fluency. Relative to fed children, those who continued to fast showed greater power increases in upper theta (6-8 Hz) and both alpha bands (8-10 Hz; 10-12 Hz) across sites. Increased theta suggests greater demands on working memory. Increased alpha may facilitate task-essential activity by suppressing non-task-essential activity. Fasting children also had greater delta (2-4 Hz) and greater lower-theta (4-6 Hz) power in left frontal recordings-indicating a region-specific emphasis on both working memory for mental calculation (theta) and activation of processes that suppress interfering activity (delta). Fed children also showed a significant increase in correct responses while children who continued to fast did not. Taken together the findings suggest that neural network activity involved in processing numerical information is functionally enhanced and performance is improved in children who have eaten breakfast, whereas greater mental effort is required for this mathematical thinking in children who skip breakfast.

Different components of working memory have different relationships with different mathematical skills

A comprehensive working memory battery and tests of mathematical skills were administered to 90 children-41 in Year 1 (5-6 years of age) and 49 in Year 3 (7-8 years of age). Working memory could explain statistically significant variance in number writing, magnitude judgment, and single-digit arithmetic, but the different components of working memory had different relationships with the different skills. Visual-spatial sketchpad (VSSP) functioning predicted unique variance in magnitude judgments and number writing. Central executive functioning explained unique variance in the addition accuracy of Year 1 children. The unique variance explained in Year 3 multiplication explained by phonological loop functioning just missed conventional levels of significance (p=.06). The results are consistent with the VSSP having a role in the development of number writing and magnitude judgments but a lesser role in early arithmetic.

The effect of problem structure on problem-solving: an fMRI study of word versus number problems

It has long been thought that word problems are more difficult to solve than number/equation problems. However, recent findings have begun to bring this broadly believed idea into question. The current study examined the processing differences between these two types of problems. The behavioral results presented here failed to show an overwhelming advantage for number problems. In fact, there were more errors for the number problems than the word problems. The neuroimaging results reported demonstrate that there is significant overlap in the processing of what, on the surface, appears to be completely different problems that elicit different problem-solving strategies. Word and number problems rely on a general network responsible for problem-solving that includes the superior posterior parietal cortex, the horizontal segment of the intraparietal sulcus which is hypothesized to be involved in problem representation and calculation as well as the regions that have been linked to executive aspects of working memory such as the pre-SMA and basal ganglia. While overlap was observed, significant differences were also found primarily in language processing regions such as Broca's and Wernicke's areas for the word problems and the horizontal segment of the intraparietal sulcus for the number problems.

Coherent motion sensitivity predicts individual differences in subtraction

Recent findings suggest deficits in coherent motion sensitivity, an index of visual dorsal stream functioning, in children with poor mathematical skills or dyscalculia, a specific learning disability in mathematics. We extended these data using a longitudinal design to unravel whether visual dorsal stream functioning is able to predict individual differences in subsequent specific mathematical skills, i.e., single-digit subtraction and multiplication. We measured children's sensitivity to coherent motion in kindergarten (mean age: 5 years 8 months) and evaluated their subtraction and multiplication skills in third grade (mean age 8 years 3 months). Findings revealed an association between subtraction but not multiplication performance and coherent motion sensitivity. This association remained significant even when intellectual ability and reading ability were additionally controlled for. Subtractions are typically solved by means of quantity-based procedural strategies, which reliably recruit the intraparietal sulcus. Against the background of a neural overlap between the intraparietal sulcus and visual dorsal stream functioning, we hypothesize that low-level visuospatial mechanisms might set constraints on the development of quantity representations, which are used during calculation, particularly in subtraction.

Effects of problem size and arithmetic operation on brain activation during calculation in children with varying levels of arithmetical fluency

Most studies on mathematics learning in the field of educational neuroscience have focused on the neural correlates of very elementary numerical processing skills in children. Little is known about more complex mathematical skills that are formally taught in school, such as arithmetic. Using functional magnetic resonance imaging, the present study investigated how brain activation during single-digit addition and subtraction is modulated by problem size and arithmetic operation in 28 children aged 10-12 years with different levels of arithmetical fluency. Commensurate with adult data, large problems and subtractions activated a fronto-parietal network, including the intraparietal sulci, the latter of which indicates the influence of quantity-based processes during procedural strategy execution. Different from adults, the present findings revealed that particularly the left hippocampus was active during the solution of those problems that are expected to be solved by means of fact retrieval (i.e. small problems and addition), suggesting a specific role of the hippocampus in the early stages of learning arithmetic facts. Children with low levels of arithmetical fluency showed higher activation in the right intraparietal sulcus during the solution of problems with a relatively small problem size, indicating that they continued to rely to a greater extent on quantity-based strategies on those problems that the children with relatively higher arithmetical fluency already retrieved from memory. This might represent a neural correlate of fact retrieval impairments in children with mathematical difficulties.

Comparing 5/7 and 2/9: Adults can do it by accessing the magnitude of the whole fractions

This study investigated adults' ability to compare the magnitude of fractions without common components (e.g., 5/7 and 3/8), and the representation accessed in that process. We hypothesized that the absence of common components would enhance access to the magnitude of the fractions (i.e., a holistic representation) rather than a direct comparison of the numerators or the denominators. This hypothesis was tested in four between-subject conditions. Two types of experimental pairs were used that differed in the congruity of the magnitude of the denominator and the magnitude of the fraction. Each type of experimental pair was presented either alone or with filler pairs that introduced variability into the congruity of the components. In all four conditions, accuracy was above chance and the effect of the distance between the fractions on response times was significant, indicating an access to the magnitude of the fractions. Nevertheless, the variability of the congruity of the components had also a significant effect on performance, suggesting that the relative magnitude of the components was also processed. In conclusion, the representation of the fraction magnitude is hybrid, rather than purely holistic, in a magnitude-comparison task on fractions without common components.

Intrinsic motivation inventory: an adapted measure for schizophrenia research

This article describes the psychometric validation of a scale designed to measure intrinsic motivation (IM) in schizophrenia. Recent studies have highlighted the relationship between motivation and functional outcome in schizophrenia and identified IM as an important mediating factor between neurocognition and psychosocial outcome. It therefore becomes imperative to have validated measures of IM for empirical use. To that end, we validated a self-report IM scale that gauges the central motivational structures identified by Self-determinism Theory as pertinent to cognitive task engagement, skill acquisition, treatment compliance, and remediation outcome. Participants were schizophrenia outpatients involved in a cognitive remediation study (n = 58), a convenience subsample of clinically stable schizophrenia outpatients (n = 15), and a group of healthy normals (n = 22). The Intrinsic Motivation Inventory for Schizophrenia Research (IMI-SR) is a concise instrument, possessing good internal consistency (alpha = .92) and test-retest reliability (intraclass correlation = .77). Data were analyzed to abridge the original 54 items into a final 21-item questionnaire comprised of 3 domains relevant to motivation for treatments (interest/enjoyment, perceived choice, value/usefulness). The scale was highly associated with germane constructs of motivation for health-related behaviors, including perceived competency for attempting challenging tasks and autonomous treatment engagement. Importantly, the scale was able to distinguish improvers and nonimprovers on a cognitive task and actual learning exercises, delineate high vs low treatment attendance, and demonstrate sensitivity to motivational changes due to intervention variation. The IMI-SR is a viable instrument to measure IM in schizophrenia as part of a cognitive remediation protocol or psychosocial rehabilitation program.