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The purpose of this meta-analysis was to determine the relation between mathematics and working memory (WM) and to identify possible moderators of this relation including domains of WM, types of mathematics skills, and sample type. A meta-analysis of 110 studies with 829 effect sizes found a significant medium correlation of mathematics and WM, r = .35, 95% CI [.32, .37]. Moderation analyses indicated that mathematics showed comparable association with verbal WM, numerical WM, and visuo-spatial WM. Word problem-solving and whole-number calculations showed the strongest relation with WM whereas geometry showed the weakest relation with WM. The relation between WM and mathematics was stronger among individuals with mathematics difficulties that are associated with other disorders or cognitive deficits compared to that among typically developing individuals and individuals with only mathematics difficulties. The implications of these findings with respect to mathematics instruction and WM training are discussed.

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... Following this logic, phonological and visuospatial working memory may have different impacts on QNC levels. It should be noted that the idea of distinguishing the impacts of different working memory task materials on numerical skills is not new (e.g., De Vita et al., 2021;Peng et al., 2016 ); only their relation to different QNC levels have rarely been investigated. ...

... Not only did we identify significant contributions of both working memory types to QNC levels, but we calculated the size of the paths as comparable. These findings are in line with prior research examining the relation between working memory domains and mathematics performance (e.g., Peng et al., 2016 ;Wilson & Swanson, 2001 ). Prior experimental research ( Kroesbergen et al., 2014 ) also revealed no significant differences between the domain-general and domain-specific working memory training groups on children's early numeracy acquisition. ...

... Finally, we did not distinguish numerical working memory and verbal working memory within phonological working memory. There are inconsistent findings in prior research comparing the roles of numerical and verbal working memory for performing early number knowledge (e.g., De Vita et al., 2021;Peng et al., 2016 ). For example, De Vita et al. (2021) focused on preschoolers and suggested that only visuospatial and numerical working memory significantly predicted preschoolers' mathematical knowledge. ...

This meta-analysis systematically investigated the pathways to the Quantity-Number Competencies (QNC) model based on 78 studies involving 21,860 children. The QNC model assumes children acquire early numeracy via three developmental levels: QNC Level I (Basic numerical skills), QNC Level IIa (imprecise quantity to number-word linkage) and QNC Level IIb (precise quantity to number-word linkage), and QNC Level III (relations between numerical quantities). Findings showed (1) all the involved linguistic skills had effects on QNC Levels I and III, whereas only vocabulary yielded support for QNC Level II; (2) phonological and visuospatial working memory made comparable contributions to all QNC levels; (3) most paths within QNC stayed stable with development, with the path coefficient from QNC Level IIa to QNC Level III increased significantly with age; and (4) the path coefficient from QNC Level I to QNC Level IIb increased as the proportion of girls increased in the sample. These findings suggest the phonological awareness might primarily support representations for number words and number facts, whereas vocabulary might facilitate the linkage between number words and quantities. The similar effects of different working memory on QNC levels suggested working memory's general storage and processing abilities underlie early numeracy development.

... Executive functions (EFs)-a set of mental processes that regulate human cognition and behavior (Miyake et al., 2000;Miyake & Friedman, 2012)-and their three arguably most investigated subdimensions (response inhibition, mental set shifting, and updating of working memory; aka inhibition, shifting, and updating) are considered prerequisites for many cognitive skills, such as reading, and correlate with fluid intelligence (Cassidy et al., 2016;Diamond, 2013;Follmer, 2018). In addition to providing empirical evidence of the relation between EFs and these cognitive skills, researchers have repeatedly shown that EFs are linked to math skills, such as basic number knowledge, calculation, spatial skills, and mathematical reasoning, in schoolchildren and adults (see, e.g., Best et al., 2011;Cragg et al., 2017;Frisovan den Bos et al., 2013;Peng et al., 2016;Yeniad et al., 2013) and play a crucial role in the development of math skills (van der Ven, 2011;van der Ven et al., 2012). Therefore, it is critical to examine the preschool years and comprehensively investigate the constructs that have been found to contribute to the development of mathematical skills. ...

... Math skills have been categorized as quantitative knowledge (Gq), fluid intelligence (Gf), and visual processing (Gv; Schneider & McGrew, 2018;Uttal et al., 2013). However, other scholars have argued that EFs and math skills measure correlated but distinct constructs (see, e.g., Best et al., 2011;Cragg et al., 2017;Friedman et al., 2006;Peng et al., 2016;Yeniad et al., 2013). These perspectives differ in whether math skills are conceptualized as broader (e.g., students' grades or performance on teacher-constructed math tests) or narrower (e.g., only intelligence tests with math components). ...

... Typically, EFs and math skills in preschool children are not measured with standardized questionnaires that require the children to read and write (cf. the Behavior Rating Inventory of Executive Function [BRIEF]; Gioia et al., 2000), and researchers have to resort to innovative ways of assessing EFs and math skills. Thus, EFs might vary in their relation to math skills in terms of the measurement properties (e.g., Allan et al., 2014;Cortés Pascual et al., 2019) or study and sample characteristics (David, 2012;Friso-van den Bos et al., 2013;Peng et al., 2016). Investigating the influence of diverse measurement, sample, and study characteristics on the relation between EFs and math skills can provide valuable information to practitioners who want to streamline the assessment of these constructs and researchers who aim to better understand the nature of the relation. ...

Executive functions (EFs) are key skills underlying other cognitive skills that are relevant to learning and everyday life. Although a plethora of evidence suggests a positive relation between the three EF subdimensions inhibition, shifting, and updating, and math skills for schoolchildren and adults, the findings on the magnitude of and possible variations in this relation are inconclusive for preschool children and several narrow math skills (i.e., math intelligence). Therefore, the present meta-analysis aimed to (a) synthesize the relation between EFs and math intelligence (an aggregate of math skills) in preschool children; (b) examine which study, sample, and measurement characteristics moderate this relation; and (c) test the joint effects of EFs on math intelligence. Utilizing data extracted from 47 studies (363 effect sizes, 30,481 participants) from 2000 to 2021, we found that, overall, EFs are significantly related to math intelligence (r = .34, 95% CI [.31, .37]), as are inhibition (r = .30, 95% CI [.25, .35]), shifting (r = .32, 95% CI [.25, .38]), and updating (r = .36, 95% CI [.31, .40]). Key measurement characteristics of EFs, but neither children’s age nor gender, moderated this relation. These findings suggest a positive link between EFs and math intelligence in preschool children and emphasize the importance of measurement characteristics. We further examined the joint relations between EFs and math intelligence via meta-analytic structural equation modeling. Evaluating different models and representations of EFs, we did not find support for the expectation that the three EF subdimensions are differentially related to math intelligence.

... Working memory (WM) is a system for the active maintenance and manipulation of information that is relevant for current task goals (Baddeley, 1992), different from short-term memory that is a passive storage system. WM plays an important role in children's academic performance (Daneman & Merikle, 1996;Friso-van den Bos et al., 2013;Peng et al., 2016Peng et al., , 2018Raghubar et al., 2010;Swanson & Alloway, 2012). This is because many academic tasks involve multiple steps with intermediate information that must be remembered for a short period and simultaneously manipulated to accomplish the task at hand. ...

... Meta-analyses on children with specific learning difficulties also suggests a pattern of domainspecificity of WM in relation to learning. That is, although children with various types of learning difficulties showed WM deficits across (verbal, numerical, and visuospatial) domains (Peng et al., 2016;Swanson & Jerman, 2006), those with specific mathematics difficulties showed more severe numerical WM deficits than their verbal WM deficits (Peng et al., 2016). Children with mathematics difficulties could be differentiated from children with reading difficulties on visuo-spatial WM (Swanson & Jerman, 2006), likely reflecting the salient role of visuo-spatial WM in mathematics learning and mathematics difficulties (Mammarella et al., 2017). ...

... Meta-analyses on children with specific learning difficulties also suggests a pattern of domainspecificity of WM in relation to learning. That is, although children with various types of learning difficulties showed WM deficits across (verbal, numerical, and visuospatial) domains (Peng et al., 2016;Swanson & Jerman, 2006), those with specific mathematics difficulties showed more severe numerical WM deficits than their verbal WM deficits (Peng et al., 2016). Children with mathematics difficulties could be differentiated from children with reading difficulties on visuo-spatial WM (Swanson & Jerman, 2006), likely reflecting the salient role of visuo-spatial WM in mathematics learning and mathematics difficulties (Mammarella et al., 2017). ...

Converging evidence suggests that traditional domain-general working memory (WM) training does not have reliable far-transfer effects, but produces reliable, modest near-transfer effects on structurally similar untrained tasks. Given the critical role of WM in academic development, WM training that incorporates task-specific features may maximize training effects on academic outcomes. In this theory paper, we discuss the training to emphasize the domain-specific function of WM highlighted by recent WM models. That is, WM should be better attuned to the materials being learned through enhancing strategies of linking together WM with the long-term memory knowledge, rather than only the enhancement of a “domain-general” attentional control overall. We provided two example training routes that emphasize explicit instruction and practice on WM-academic tasks (i.e., academic tasks that can be performed using a WM training paradigm) and task-linking strategies (i.e., strategies that can be used in both academic tasks and WM tasks to improve performance efficiency). We also review recent relevant intervention studies that are in line with this approach and report promising effects on academic outcomes. Implications for future studies are also discussed.

... The ability to solve verbal arithmetic problems highly correlates with WM (Peng et al., 2016). The authors conclude that the correlation between WM and mathematics, in general, is of average magnitude (r=0.35). ...

... If we add that, in some higher education systems, a score of less than 7 is a failing grade, all participants (38 women and one man) would fail, regardless of the time elapsed after having had the last contact with arithmetic. The above may be consistent with what is described by Peng et al. (2016), that the numerical calculation ability of elementary education teachers in general training is deficient, which possibly impacts the WM and performance in the evaluation. If we also consider that the students about to graduate obtained the lowest scores, the situation that the pre-service teachers and the children under their care will face is even more worrisome. ...

... Remarkably and in contrast to Peng et al. (2016), the verbal problem solving of the arithmetic exam presents a moderate correlation with the working memory capacity in both domains, being highly significant for the verbal, particularly with the ReadSpan task. A comprehensive reading was required when solving the problems, which on average, handled 27 words. ...

We developed the present study given the impact teachers, in their attitude toward mathematics and their arithmetic ability, can have on children's mathematics learning. For this purpose, we explored the association between working memory, math anxiety, and arithmetic ability levels in 39 pre-service teachers of a private institution in southeastern Mexico. We applied complex span tasks to measure the working memory in verbal and visuospatial domains, the perception of math anxiety with the Mathematical Anxiety Profile scale, and the arithmetic ability with a set of verbal problems extracted from the public guides for the national examination of admission to middle education of the National Evaluation Center for Higher Education in Mexico. As proven in other studies, there is a positive association between working memory capacity and arithmetic ability and a negative one with the level of math anxiety. Its scale of attitudes unfolds future possibilities of improving the first one through adaptive training and verifying the impact on the last two.

... Next, phonological memory is thought to be needed in arithmetic problem solving to store, maintain, and manipulate both phonological and numerical information. For instance, a child may need to temporarily store intermediate numbers when decomposing an arithmetic problem into multiple steps, or a child may need to encode and store the phonological representations of the arithmetic problem to retrieve the arithmetic fact from memory (Geary, 1993;Hecht et al., 2001;Peng et al., 2016). A meta-analysis of 110 studies found that phonological memory was correlated with a wide variety of math tasks including single-and double-digit arithmetic (Peng et al., 2016). ...

... For instance, a child may need to temporarily store intermediate numbers when decomposing an arithmetic problem into multiple steps, or a child may need to encode and store the phonological representations of the arithmetic problem to retrieve the arithmetic fact from memory (Geary, 1993;Hecht et al., 2001;Peng et al., 2016). A meta-analysis of 110 studies found that phonological memory was correlated with a wide variety of math tasks including single-and double-digit arithmetic (Peng et al., 2016). Together, these findings suggest phonological memory is not only related to reading skills (Peng et al., 2018) and impaired in children with dyslexia (Peng & Fuchs, 2014), but also appears to support arithmetic. ...

... Similarly, De Smedt et al. (2010) did not find any correlations between phonological memory and arithmetic, nor did Fuchs et al. (2006) find an association between phonological memory and retrieval or calculation-based arithmetic. A meta-analysis has found evidence for a relationship between phonological memory and arithmetic (Peng et al., 2016), but others have noted that the evidence may be mixed . ...

Phonological processing skills have not only been shown to be important for reading skills, but also for arithmetic skills. Specifically, previous research in typically developing children has suggested that phonological processing skills may be more closely related to arithmetic problems that are solved through fact retrieval (e.g., remembering the solution from memory) than procedural computation (e.g., counting). However, the relationship between phonological processing and arithmetic in children with learning disabilities (LDs) has not been investigated. Yet, understanding these relationships in children with LDs is especially important because it can help elucidate the cognitive underpinnings of math difficulties, explain why reading and math disabilities frequently co‐occur, and provide information on which cognitive skills to target for interventions. In 63 children with LDs, we examined the relationship between different phonological processing skills (phonemic awareness, phonological memory, and rapid serial naming) and arithmetic. We distinguished between arithmetic problems that tend to be solved with fact retrieval versus procedural computation to determine whether phonological processing skills are differentially related to these two arithmetic processes. We found that phonemic awareness, but not phonological memory or rapid serial naming, was related to arithmetic fact retrieval. We also found no association between any phonological processing skills and procedural computation. These results converge with prior research in typically developing children and suggest that phonemic awareness is also related to arithmetic fact retrieval in children with LD. These results raise the possibility that phonemic awareness training might improve both reading and arithmetic fact retrieval skills. This article is protected by copyright. All rights reserved

... arithmetic, mathematical problem solving, geometry, algebra, and mathematical reasoning (Peng et al., 2016(Peng et al., , 2019. In the primary stage, the major mathematics curriculum focuses on three domains across cultures: arithmetic, mathematical problem solving, and mathematical reasoning (Casey et al., 2015;Fuchs et al., 2014;Yang et al., 2021a, b). ...

... While, the Bayesian approach demonstrated anecdotal evidence for the correlation between arithmetic and children's mathematical problem solving, which is consistent with previous findings (Cai et al., 2016). This is probably because that although all of them involved basic number knowledge such as cardinality, ordinality, Arabic digits knowledge, basic axioms, etc. (Peng et al., 2016). Mathematical problem solving involves a more complex process that requires students to understand words, integrate relevant information, build a mental representation of the problem, select an appropriate mathematical strategy, and encode the answer in an acceptable written form (Männamaa et al., 2012;Yang & Meng, 2020). ...

Mathematical abilities are important for children’s academic achievement during the primary education phase. Understanding which cognitive factors underlie individual differences in mathematics is essential to obtaining insights into children’s mathematical development. This study explored the roles of phonological processing skills and visual-spatial skills in arithmetic, mathematical problem solving, and mathematical reasoning among primary school children. Two hundred and fifty-one primary school children (mean age: 8.31 ± 0.89 years old), including 87 first graders, 83 s graders, and 81 third graders participated in this study. Children’s rapid automatized naming was measured using a rapid digit naming task, and phonological awareness was measured with a character rhyming task. Additionally, children’s visual perception was measured with a figure matching task, and mental rotation was measured with a 2D/3D mental rotation task. Children’s mathematical abilities were measured with three mathematics tests: calculation task, mathematical problem solving task, and mathematical reasoning task. Regression analyses and Bayesian hypothesis testing showed that phonological awareness uniquely contributed to children’s mathematical abilities, especially mathematical problem solving. The results suggest that phonological awareness serves as a key precursor of mathematical abilities during the primary education phase.

... However, a meta-analysis by Peng et al. [73] found that verbal WM and visuospatial WM correlated with arithmetic to a similar degree, indicating no moderation effects of WM domains. Given that Peng, Namkung, Barnes and Sun [73] collected the data throughout the life span, whereas the present study focused on a specific age range during primary school, it is possible that despite the similar contribution of each domain of WM to arithmetic across the life span, verbal WM holds a greater impact on arithmetic than visuospatial WM during primary school [74]. ...

... However, a meta-analysis by Peng et al. [73] found that verbal WM and visuospatial WM correlated with arithmetic to a similar degree, indicating no moderation effects of WM domains. Given that Peng, Namkung, Barnes and Sun [73] collected the data throughout the life span, whereas the present study focused on a specific age range during primary school, it is possible that despite the similar contribution of each domain of WM to arithmetic across the life span, verbal WM holds a greater impact on arithmetic than visuospatial WM during primary school [74]. ...

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.

... Updating also appears to have the strongest direct relation with mathematics of all executive functions (Friso-van den Bos et al., 2013). A meta-analysis found that this relation was strongest for whole-number calculations and word problem-solving and weakest for simple geometry (Peng et al., 2016). This may indicate that updating is particularly important for procedural operations in which children are nonfluent (as most studies using whole-number calculations involved younger participants) and for complex math that involves coordinating multiple processes (though this coordination may also overlap with other executive functions). ...

... If executive functions were primarily useful for complex, multistep math content as we had hypothesized, we would expect to see relations between executive functions across all grades (and perhaps more in older grades where math content becomes more difficult). Instead, the consistency of patterns for only third grade may suggest that conceptual and procedural fluency is still being developed in third grade (Peng et al., 2016). With the numeracy findings above, these younger students may be using strategies that are both number heavy and cognitive resource heavy until processes become simplified and automated with expertise, therefore requiring updating and shifting more often. ...

... Current procedures to identify children with potential learning disabilities assume that such children experience cognitive constraints, which impede their ability to perform efficiently on language and achievement measures (e.g., Geary et al., 2017;Kaushanskaya, & Yoo, 2013;Lesaux et al., 2006). Working memory has been identified as one of the most often referred to cognitive processes underlying both RD and MD (WM; David, 2012;Peng, Namkung, Barnes, & Sun, 2016;Peng, Namkung, Fuchs, et al., 2016;Peng et al., 2018;Swanson et al., 2004), which has also been related to achievement difficulties in EL children (e.g., Swanson et al., 2015;Swanson et al., 2006). Working memory (WM) is defined as a limited capacity system of temporary stores, functions related to the preservation of information while simultaneously processing other information, and attention control related to these functions (e.g., Baddeley, 2012). 2 Cowan (2014) defines WM "as the small amount of information that can be held in the mind and used in the execution of cognitive tasks, in contrast with long-term memory, which is the vast amount of information saved in one's life" (p. ...

... Working memory has been shown to contribute significant variance to EL children's reading and math achievement, even when domain-specific processes related to reading (e.g., phonological awareness; Swanson et al., 2015) and math (e.g., estimation, numeracy; are included in the regression modeling. Although the association between WM and reading and/ or math has been established in the literature, the processes of WM that underlie predictions of reading and/or math performance are unclear (see Peng, Namkung, Barnes, & Sun, 2016;Peng, Namkung, Fuchs, et al., 2016;Peng et al., 2018, for review). Some studies have suggested that the storage component of WM (referred to as verbal STM, STM, or the phonological loop) plays a major role in academic performance (e.g., Peng, Namkung, Fuchs, et al., 2016). ...

... Another domain-general cognitive contributor to high mathematics achievement (Myers et al., 2017, for a review) is working memory, or the ability to temporary store and manipulate information (Baddeley, 1992). Working memory is typically measured by examining participants' ability to solve complex, verbal or visualspatial span tasks, which combine concurrent processing and storage, such as the backward digit span (Peng et al., 2016). The association between working memory and mathematics achievement is well established (Peng et al., 2016, for a meta-analysis). ...

... Working memory is typically measured by examining participants' ability to solve complex, verbal or visualspatial span tasks, which combine concurrent processing and storage, such as the backward digit span (Peng et al., 2016). The association between working memory and mathematics achievement is well established (Peng et al., 2016, for a meta-analysis). Higher working memory capacities have also been reported for mathematically high-achieving children (e.g., Berg & McDonald, 2018;Swanson, 2006) and adolescents (e.g., Dark & Benbow, 1991, 1994Leikin et al., 2013). ...

... Indeed, empirical studies corroborated that fluid abilities are a leading predictor of changes in crystallized abilities (Ferrer and McArdle 2004). Consequently, early cognitive abilities such as phonological awareness (Cirino et al. 2018;Vanbinst et al. 2020), rapid automatized naming (Cirino et al. 2018;Korpipää et al. 2017), working memory (Peng et al. 2016(Peng et al. , 2018, or reasoning (Peng et al. 2019) were also comparably associated with reading and mathematics achievement. ...

... Potentially, however, the inclusion of these control variables was not comprehensive enough. For example, we used only a short test to assess students' reasoning abilities and did not include any indicators of working memory (Peng et al. 2016(Peng et al. , 2018 or executive functioning (Bull et al. 2008). Thus, the question remains whether the bidirectional effects found could have been partially spurious results from unacknowledged confounders. ...

Reading and mathematical competencies are important cognitive prerequisites for children’s educational achievement and later success in society. An ongoing debate pertains to potential transfer effects between both domains and whether reading and mathematics influence each other over time. Therefore, the present study on N = 5185 students from the German National Educational Panel Study examined cross-lagged effects between reading and mathematics from Grades 5 to 12. The results revealed, depending on the chosen causal estimand, negligible to small bidirectional effects. Adopting a between-person perspective, students with higher mathematics scores at one point exhibited somewhat higher reading scores at the subsequent measurement. In contrast, when adopting a within-person perspective, both skills predicted longitudinal increases of the other skill in the lower grades but reversed effects in higher grades. Taken together, these findings not only demonstrate that transfer effects between reading and mathematics in secondary education tend to be small but also suggest different patterns of effects depending on the modeling choice.

... Moreover, they are more prone to visuo-spatial processing deficit and eye-hand coordination problems (Stadskleiv et al., 2018), resulting in difficulty to implement early quantification processes such as subitizing (Arp & Fagard, 2005) and counting abilities (Camos et al., 1998). Finally, children with CP often experience working memory impairment (Stadskleiv et al., 2018), leaving them with a long-lasting dependence on finger-based strategies and/or concrete support (Peng et al., 2016). In children with CP a substandard working memory was found to be predictive of mathematics learning disabilities (Jenks et al., 2012;Van Rooijen et al., 2016). ...

Children with cerebral palsy (CP) are at greater risk of mathematical learning disabilities due to associated motor and cognitive limitations. However, there is currently little evidence on how to support the development of arithmetic skills within such a specific profile. The aim of this single-case study was to assess the effectiveness of a neuropsychological rehabilitation of arithmetic skills in NG, a 9-year-old boy with CP who experienced math learning disability and cumulated motor and short-term memory impairments. This issue was explored combining multiple-baseline and changing-criterion designs. The intervention consisted of training NG to solve complex additions applying calculation procedures with a tailor-made computation tool. Based on NG’s strengths, in accordance with evidence-based practice in psychology, the intervention was the result of a co-construction process involving N, his NG’s parents and professionals (therapist and researchers). Results were analyzed by combining graph visual inspections with non-parametric statistics for single-case designs (NAP-scores). Analyses showed a specific improvement in NG’s ability to solve complex additions, which maintained for up to 3 weeks after intervention. The training effect did not generalize to his ability to perform mental additions, and to process the symbolic magnitude.

... Furthermore, performing different kinds of arithmetic tasks (e.g., calculation and word-problem solving) rely on several cognitive subsystems including working memory (DeStefano and LeFevre, 2004;Raghubar et al., 2010;Cragg et al., 2017). Solving a multidigit calculation, for instance, requires the processing of the task at hand, the storage of intermediate steps of the calculation, and the manipulation of this kind of information in mind (DeStefano and LeFevre, 2004;Peng et al., 2016). Further evidence, for the pivotal role of working memory in numeric cognition comes from the neuroimaging literature, which documents overlapping activations during arithmetic and working memory tasks (Owen et al., 2005;Arsalidou and Taylor, 2011;Rottschy et al., 2012;Fedorenko et al., 2013;Matejko and Ansari, 2021). ...

Introduction
The human intraparietal sulcus (IPS) covers large portions of the posterior cortical surface and has been implicated in a variety of cognitive functions. It is, however, unclear how cognitive functions dissociate between the IPS's heterogeneous subdivisions, particularly in perspective to their connectivity profile.
Methods
We applied a neuroinformatics driven system-level decoding on three cytoarchitectural distinct subdivisions (hIP1, hIP2, hIP3) per hemisphere, with the aim to disentangle the cognitive profile of the IPS in conjunction with functionally connected cortical regions.
Results
The system-level decoding revealed nine functional systems based on meta-analytical associations of IPS subdivisions and their cortical coactivations: Two systems–working memory and numeric cognition–which are centered on all IPS subdivisions, and seven systems– attention, language, grasping, recognition memory, rotation, detection of motions/shapes and navigation –with varying degrees of dissociation across subdivisions and hemispheres. By probing the spatial overlap between systems-level co-activations of the IPS and seven canonical intrinsic resting state networks, we observed a trend toward more co-activation between hIP1 and the front parietal network, between hIP2 and hIP3 and the dorsal attention network, and between hIP3 and the visual and somatomotor network.
Discussion
Our results confirm previous findings on the IPS's role in cognition but also point to previously unknown differentiation along the IPS, which present viable starting points for future work. We also present the systems-level decoding as promising approach toward functional decoding of the human connectome.

... This is related to the thinking process of students for sure. The results of research by Peng et al. (2016) found that a student's mathematical ability affected mathematical problem-solving abilities. Students with high mathematical abilities have high ability in problem solving, students with moderate mathematical abilities have good problem-solving abilities, and students who have low mathematical abilities have poor problem-solving abilities. ...

The purpose of the study was to determine the ability of students in solving the national assessment problem model. The study belonged to qualitative approach. The research was conducted at SMPK Putra St. Xavier Kefamenanu. The data conducted from test results of national assessment collection problems and selected-participants interviews. The collected data then identified based on the following stages: data reduction, data presentation, conclusion and verification. The results of data analysis showed that the level of students' ability in completing the national assessment model problems in the percentage is the low-level ability category of 30%, at the medium level by 60% and the high level by 10%. The findings of this study: students with high math ability master all components of MCA in terms of content, cognitive processes, and context; students with moderate math abilities are able to master reading literacy content but are weak in numeracy literacy; students with low math abilities are very weak in content, cognitive processes, and context. In terms of cognitive processes, students with moderate mathematical ability can find and interpret text content but fail to integrate and evaluate text content into mathematical concepts and procedures and the context of the questions that can be worked on are personal and scientific. In cognitive process, students with low mathematical abilities can find information but are unable to interpret it to evaluate the content of the text into a mathematical concept and procedure.

... In both studies, we assessed verbal and visual-spatial WM to better isolate the unique associations between patterning and mathematics knowledge given that WM is related to both constructs (e.g., Peng et al., 2016;Rittle-Johnson et al., 2019). In Study 1, we also measured children's performance on an analogical reasoning task (as recommended by Wijns et al., 2021). ...

This study examined repeating and growing pattern knowledge and their associations with procedural and conceptual arithmetic knowledge in a sample of U.S. children (N = 185; Mage = 79.5 months; 55% female; 88% White) and adults (N = 93; Mage = 19.5 years; 62% female; 66% White) from 2019 to 2020. Three key findings emerged: (1) repeating pattern tasks were easier than growing pattern tasks, (2) repeating pattern knowledge robustly predicted procedural calculation skills over and above growing pattern knowledge and covariates, and (3) growing pattern knowledge modestly predicted procedural and conceptual math outcomes over and above repeating pattern knowledge and covariates. We expand existing theoretical models to incorporate these specific links and discuss implications for supporting math knowledge.

... WM is the function of retaining and processing information and is a necessary mechanism for complex tasks, such as learning, comprehension, and reasoning (Baddeley, 2010). Hence, deficits in WM have been associated with reading disabilities (Swanson et al., 2009), lower mathematical skills (Peng et al., 2016), but also anxiety (Moran, 2016). WM is also of importance for social perception (Phillips et al., 2007), and accordingly, WM deficits have been linked to social problems in children with ADHD (Kofler et al., 2011). ...

The aim of this study was to investigate if training with the memory technique Method of Loci (MoL) is feasible for children and adolescents with ADHD. Twelve children (aged 9-17 years) with ADHD participated. Training with MoL was done using a mobile application, memorizing a sequence of 20-80 pictures, intended to be carried out five times per week for 4 weeks. Feasibility was assessed with pre- and post-intervention ratings, and with interviews after the training. Qualitative data were analyzed with content analysis. Those who trained with MoL performed better on memory test and reported fewer ADHD symptoms after completing the training, as compared to their baseline levels. All of these children would recommend the training to peers but the duration of training varied considerably. The participants and their parents reported that the MoL training was easy and fun to use, although lack of motivation, distractions in every-day life, and lack of routines created challenges. We conclude that training with MoL was considered feasible by most of the participants. Future research should try to make the intervention more acceptable by motivating the participants and limiting potential distractions and involving larger study groups and controls to study the efficacy of the training.

... On the one hand, the content of learning varies across these domains. Specifically, arithmetic deals with numbers, especially the properties of arithmetic operations such as addition and subtraction ( Davenport, 1999 ); geometry involves problem-solving and reasoning related to spatial attributes such as the shape, size, and relative position of a figure ( Peng, Namkung, Barnes, & Sun, 2016 ); and measurement refers to the quantification of an attribute of an object (e.g., length, area; Clements & Stephan, 2004 ). On the other hand, there are some similarities between arithmetic, geometry, and measurement. ...

Based on a sample of 138 children aged approximately 5 years in Hong Kong, this study examined how general cognitive and language skills, including working memory, short-term memory, visual-spatial skills, and receptive vocabulary, were associated with young children's performance in arithmetic, geometry, and measurement. The results showed that these cognitive and language skills yielded unique contributions to different domains of mathematical abilities after controlling for children's sex, age, maternal education, paternal education, and family income. Specifically, working memory, visual-spatial skills, and receptive vocabulary were related to arithmetic performance; receptive vocabulary was associated with measurement performance; and short-term memory was related to geometry performance. These findings' implications for theory and practice are discussed.

... Recent empirical research has largely been focused on associations between mathematical competencies and domain-general abilities. In sum, it has been found that higher competencies are associated with higher information-processing speed (Passolunghi & Lanfranchi, 2012), larger short-term and working memory (Berg & McDonald, 2018;Peng, Namkung, Barnes, & Sun, 2016), higher executive functions (Abreu-Mendoza, Chamorro, Garcia-Barrera, & Matute, 2018;Bull & Lee, 2014), enhanced logical reasoning abilities (Attridge & Inglis, 2013;Morsanyi, Devine, Nobes, & Szucs, 2013), and better visuo-spatial abilities (Benbow & Minor, 1990;Frick, 2018). A review article by Myers, Carey, and Szűcs (2017) showed that these general correlates of mathematical competencies can also be regarded as characteristics of mathematically gifted individuals, even though the authors also noted that the current body of evidence is too limited to draw strong conclusions on this issue. ...

... Number line estimation explained variance proportion of achievement of 5.1% for whole numbers and 19.3% for fractions in our study. By comparison, previous meta-analyses found that nonsymbolic magnitude comparison explains a variance proportion of mathematical achievement of 5.8% (Schneider et al., 2017), symbolic magnitude comparison explains 9.1% (Schneider et al., 2017), working memory explains 12.3% (Peng et al., 2016), spatial ability explains 7.3% (Xie et al., 2020), and fluid intelligence 16.8% . For arithmetic, we did not find a meta-analysis. ...

The number line estimation task is an often-used measure of numerical magnitude understanding. The task also correlates substantially with broader measures of mathematical achievement. This raises the question of whether the task would be a useful component of mathematical achievement tests and instruments to diagnose dyscalculia or mathematical giftedness and whether a stand-alone version of the task can serve as a short screener for mathematical achievement. Previous studies on the relation between number line estimation accuracy and broader mathematical achievement were limited in that they used relatively small nonrepresentative samples and usually did not account for potentially confounding variables. To close this research gap, we report findings from a population-level study with nearly all Luxembourgish ninth-graders (N = 6484). We used multilevel regressions to test how a standardized mathematical achievement test relates to the accuracy in number line estimation on bounded number lines with whole numbers and fractions. We also investigated how these relations were moderated by classroom characteristics, person characteristics, and trial characteristics. Mathematical achievement and number line estimation accuracy were associated even after controlling for potentially confounding variables. Subpopulations of students showed meaningful differences in estimation accuracy, which can serve as benchmarks in future studies. Compared with the number line estimation task with whole numbers, the number line estimation task with fractions was more strongly related to mathematical achievement in students across the entire mathematical achievement spectrum. These results show that the number line estimation task is a valid and useful tool for diagnosing and monitoring mathematical achievement.

... The results indicated that incorporating SRL strategies increased performance in mathematics, learning motivation, and self-efficacy. Moreover, WM is associated with a wide range of cognitive functions, including reasoning and mathematics skills (Barrett et al., 2004;Kane et al., 2004;Peng et al., 2016), and its capacity can be enhanced through various types of WM training tasks (Etherton et al., 2019;Jones et al., 2020). Accordingly, researchers are advised to conduct longitudinal evaluation of SRL and WM training outcomes to verify whether such training continues to improve the AAMS of students of different ages. ...

Studies indicate that learners’ cognitive style (CS), self-regulated learning (SRL), and working memory (WM) are associated with their academic performance. These studies describe the relationship of academic achievement with SRL, CS, or WM individually or pairwise relationships between SRL, CS, and WM rather than the overall relationship between academic achievement and each factor. In this study, a structural equation modelling (SEM) analysis was conducted to explore the overall theoretical relationship. We focused on academic achievements in mathematics and science (AAMS). A total of 191 sixth-grade students (male: 111, female: 80; mean age: 11.08 years, SD = 0.282) from two public elementary schools in Taiwan was selected as valid samples for this study. The findings indicated that CS, WM, and SRL individually had significant influences on AAMS, among which SRL had the largest effect, followed by WM and CS. Furthermore, we discovered that CS was significantly correlated with WM. The results of the analysis of the mediation effect demonstrated that CS both directly affected AAMS and indirectly affected AAMS through SRL. The implication of the findings and recommendations are also discussed.

Poor social skills in autism spectrum disorder (ASD) are associated with reduced independence in daily life. Current interventions for improving the social skills of individuals with ASD fail to represent the complexity of real-life social settings and situations. Virtual reality (VR) may facilitate social skills training in social environments and situations proximal to real life, however, more research is needed for elucidating aspects such as the acceptability, usability, and user experience of VR systems in ASD. Twenty-five participants with ASD attended a neuropsychological evaluation and three sessions of VR social skills training, incorporating 5 social scenarios with three difficulty levels for each. Participants reported high acceptability, system usability, and user experience. Significant correlations were observed between performance in social scenarios, self-reports, and executive functions. Working memory and planning ability were significant predictors of functionality level in ASD and the VR system’s perceived usability respectively. Yet, performance in social scenarios was the best predictor of usability, acceptability, and functionality level in ASD. Planning ability substantially predicted performance in social scenarios, postulating an implication in social skills. Immersive VR social skills training appears effective in individuals with ASD, yet an error-less approach, which is adaptive to the individual’s needs, should be preferred.

Although the role played by finger use in children’s numerical development has been widely investigated, their benefit in arithmetical contexts is still debated today. This scoping review aimed to systematically identify and summarize all studies that have investigated the relation between fingers and arithmetic skills in children. An extensive search on Ovid PsycINFO and Ovid Eric was performed. The reference lists of included articles were also searched for relevant articles. Two reviewers engaged in study selection and data extraction independently, based on the eligibility criteria. Discrepancies were resolved through discussion. Of the 4707 identified studies, 68 met the inclusion criteria and 7 additional papers were added from the reference lists of included studies. A total of 75 studies were included in this review. They came from two main research areas and were conducted with different aims and methods. Studies published in the mathematical education field (n = 29) aimed to determine what finger strategies are used during development and how they support computation skills. Studies published in cognitive psychology and neuroscience (n = 45) specified the cognitive processes and neurobiological mechanisms underlying the fingers/arithmetic relation. Only one study combined issues raised in both research areas. More studies are needed to determine which finger strategy is the most effective, how finger sensorimotor skills mediate the finger strategies/arithmetic relation, and how they should be integrated into educational practice.

Introduction
Development of crucial skills accelerates at the start of formal schooling, although, more knowledge is needed about the relationships between such skills. The current study explored the relationships between visuospatial working memory, letter-sound knowledge, math competence and motor competence, as well as potential effects of gender.
Materials and methods
The sample consisted of 85 (42 girls) 6 to 7 years old first grade children, and was measured with a test battery consisting of tests designed for each skill domain.
Results
Results demonstrated weak to moderate statistically significant correlations between visuospatial working memory, letter-sound knowledge, math competence, with no statistically significant gender differences. Two motor tasks measuring manual dexterity, placing bricks and building bricks, showed a weak statistically significant correlation.
Discussion
We argue that the findings demonstrate the relationships between these skills are low to moderate in first grade. Furthermore, we argue that these skills ought to be trained deliberately. The potential role of visuospatial working memory in procurement of novel skills in early childhood ought to be explored further in future studies.

Background:
Previous studies indicated that working memory (WM) updating and WM capacity play essential roles in mathematical ability. However, it is unclear whether WM capacity mediates the effect of WM updating on mathematics, and whether the cascading effects vary with different mathematical domains.
Aims:
The current study aims to explore the longitudinal mediating role of WM capacity between WM updating and mathematical performance, and how the relations change with the age and domains.
Sample:
A total of 131 Chinese first-graders participated the study.
Methods:
Participants were required to complete tasks on WM updating and WM capacity in Grade 1 and Grade 2, as well as paper-and-pencil tests on mathematics achievement in Grade 3. The role of WM updating and capacity in the development of pupil's mathematical achievement was examined.
Results:
Results revealed that verbal WM updating in Grade 1 predicted basic arithmetic and logical-visuospatial ability in Grade 3 via its cascading effect on verbal WM capacity in Grade 2. Moreover, visuospatial WM updating in Grade 1 predicted visuospatial WM capacity in Grade 2. Visuospatial WM capacity in Grade 1 predicted logical-visuospatial ability in Grade 3 instead of basic arithmetic ability in Grade 3.
Conclusions:
The findings suggested that WM updating exerts effect on pupil's mathematical performance via WM capacity, meanwhile, this effect depends on children's mathematics domain.

Introduction
At present, numerous studies can be found in which influences and relationships between the principal executive functions, reading comprehension, and academic performance associated with reading are reported. However, there is still a lack of convergence regarding the impact of computerized cognitive training on children’s executive development and its transfer in academic reading performance and comprehension of written texts.
Methods
This study analyzes the effect of implementing a cognitive stimulation program on the performance of reading comprehension and academic performance in the subject of Spanish Language and Literature. To this end, a total sample of 196 children from 23 educational centers received the cognitive intervention for 8 weeks, with three weekly sessions of between 15 and 20 min each occurring on non-consecutive days. Pre-test and post-test measurements were collected and analyzed.
Results
The results demonstrate a significant increase in the reading comprehension scores. In addition, a significant impact of the training on the participants’ academic performance in the subject Spanish Language and Literature was found.
Discussion
These results highlight the usefulness of computerized cognitive stimulation programs for reading comprehension enhancement.

This chapter discusses the impact that cognitive deficits have on academic frustration which leads to an increase in distributive and violent behavior in the classroom. Specifically, there are a disproportionate number of learners who are served under the Individuals with Disabilities Act (IDEA) who are referred to law enforcement, restrained, suspended, drop out, expelled, and are bullied. Equipping Minds Cognitive Development Curriculum (EMCDC) is a cognitive training program that has demonstrated improvement in academic skills, executive functioning skills, fluid reasoning skills, and verbal skills in students with neurodevelopmental disorders. By equipping schools to increase cognitive abilities and facilitate academic success, frustration in the classroom will be decreased resulting in a reduction of violent behaviors and provide a safe learning environment.

Developmental dyscalculia is a neurodevelopmental disorder, influencing the learning of mathematics in developing children. In the last two decades, continuous growth of research has helped in the advancement of the state of knowledge of dyscalculia. This upsurge in the number of studies makes it relevant to conduct a systematic review, covering all the empirical evidence, but there is a dearth of review studies synthesizing findings of the studies in the recent past. Therefore, the current study aims to systematically review studies investigating the underlying cognitive causal factors associated with developmental dyscalculia in the last two decades. To investigate the underlying cognitive factors associated with dyscalculia, two prominent approaches have been used: domain-general and domain-specific. While the domain-general approach argues for the deficit in general cognitive abilities, the domain-specific approach argues for the deficit in core numerical abilities. In the present review, the PRISMA method is followed. Articles were searched using two methods: firstly, through database sources of Google Scholar, Web of Science, and ScienceDirect, 1738 abstracts were screened, of which 46 articles met the specific inclusion criteria; and secondly, through recently published systematic reviews and meta-analyses, 29 studies were included. A total of 75 studies, 48 studies from domain-general and 27 studies from domain-specific approaches, have been selected. This review discusses domain-general and domain-specific approaches of developmental dyscalculia, along with specific theories associated with both approaches. Based on the discussed findings, visuospatial working memory and symbolic number processing abilities emerged as the best predictor of math ability in children with dyscalculia.

Central elements of adaptive expertise in arithmetic problem solving are flexibility, using multiple strategies, and adaptivity, selecting the optimal strategy. Research shows that the strategies children actually use do not fully reflect the strategies they know: there is hidden potential. In the current study a sample of 147 third graders from the Netherlands completed a comprehensive assessment of adaptive expertise in the domain of multidigit subtraction, designed to measure, first, the strategies students know and use to solve subtraction problems (potential and practical flexibility). Second, it measured to what extent students know which strategy is optimal and to what extent they use the optimal strategy (potential and practical adaptivity). Findings for flexibility showed that most students consistently used the same strategy across all problems: practical flexibility was low. When prompted, students knew more strategies than they used spontaneously, suggesting hidden potential in flexibility. Findings for adaptivity showed that students hardly ever spontaneously used the task-specific strategy that is efficient for specific problems since it has the fewest and easiest steps. However, almost half of the students could select this strategy from a set of given strategies at least once. Furthermore, an innovative, personalized version of the choice/no-choice method showed that the task-specific strategy was usually not the optimal strategy (fastest strategy leading to a correct answer) for individual students. Finally, students used the strategy with which they performed best more often than the other strategies, but there is hidden potential for the adaptive use of task-specific strategies.

Individuals solve arithmetic problems in different ways and the strategies they choose are indicators of advanced competencies such as adaptivity and flexibility, and predict mathematical achievement. Understanding the factors that encourage or hinder the selection of different strategies is therefore important for helping individuals to succeed in mathematics. Our research contributed to this goal by investigating the skills required for selecting the associativity shortcut-strategy, where problems such as ‘16 + 38 – 35’ are solved by performing the subtraction (38 – 35 = 3) before the addition (3 + 16 = 19). In a well-powered, pre-registered study, adults completed two tasks that involved ‘a + b – c’ problems, and we recorded a) whether and b) when, they identified the shortcut. They also completed tasks that measured domain-specific skills (calculation skill and understanding of the order of operations) and domain-general skills (working memory, inhibition and switching). Of all the measures, inhibition was the most reliable predictor of whether individuals identified the shortcut, and we discuss the roles it may play in selecting efficient arithmetic strategies.

Working memory (WM) training explores whether and how repeated practice on working memory tasks might generalize to a variety of outcome measures. Although this field of research is part of the growing literature in cognitive sciences, it has spawned contentious debates. The controversies are largely driven by inconsistent findings and commercial interests, and as a result, numerous meta-analyses and systematic reviews have focused on the validity of WM training. Similarly, there is an inconsistency in the conclusions drawn by these meta-analyses; while there seems to be an agreement about the generalization to proximal cognitive measures; there is a discrepancy in the interpretation of any translational outcomes (e.g., behavioral, clinical, and academic). In this chapter, we review the collection of meta-analyses with a particular focus on children diagnosed with ADHD and other developmental disabilities, and recommend that the field should focus on improving our understanding of the mechanistic and effectiveness properties of WM training, which might result in the development of valuable alternative and/or supplemental approaches, when traditional interventions might fall short, especially for individuals typically underrepresented and underserved.

Several children in a typical classroom experience persistent learning difficulties that are likely to reflect weak cognitive skills (Holmes et al., 2020). In some cases, these are related to poor working memory. In this chapter, we discuss how limited working memory resources constrain classroom learning, focusing on the impact of poor working memory on children’s abilities to follow both classroom management and learning-activity relevant instructions (e.g., Jaroslawska et al., 2016; 2017). Different ways to help children with poor working memory are discussed. These include an overview of current ideas about memory enhancement by training and brain stimulation (e.g., Byrne et al., 2020), as well as more practical ways for teachers to use action to improve children’s instruction-following.

In this handbook, the world's leading researchers answer fundamental questions about dyslexia and dyscalculia based on authoritative reviews of the scientific literature. It provides an overview from the basic science foundations to best practice in schooling and educational policy, covering research topics ranging from genes, environments, and cognition to prevention, intervention and educational practice. With clear explanations of scientific concepts, research methods, statistical models and technical terms within a cross-cultural perspective, this book will be a go-to reference for researchers, instructors, students, policymakers, educators, teachers, therapists, psychologists, physicians and those affected by learning difficulties.

Background
Many learning methods of mathematical reasoning encourage imitative procedures (algorithmic reasoning, AR) instead of more constructive reasoning processes (creative mathematical reasoning, CMR). Recent research suggest that learning with CMR compared to AR leads to better performance and differential brain activity during a subsequent test. Here, we considered the role of individual differences in cognitive ability in relation to effects of CMR.
Methods
We employed a within-subject intervention (N=72, MAge=18.0) followed by a brain-imaging session (fMRI) one week later. A battery of cognitive tests preceded the intervention. Participants were divided into three cognitive abilitity groups based on their cognitive score (low, intermediate and high).
Results
On mathematical tasks previously practiced with CMR compared to AR we observed better performance, and higher brain activity in key regions for mathematical cognition such as left angular gyrus and left inferior/middle frontal gyrus. The CMR-effects did not interact with cognitive ability, albeit the effects on performance were driven by the intermediate and high cognitive ability groups.
Conclusions
Encouraging pupils to engage in constructive processes when learning mathematical reasoning confers lasting learning effects on brain activation, independent of cognitive ability. However, the lack of a CMR-effect on performance for the low cognitive ability group suggest future studies should focus on individualized learning interventions, allowing more opportunities for effortful struggle with CMR.

A wide literature has studied the predictors of number comprehension and early math learning by considering both domain-general and number-specific prerequisites. However, a consensus has not been reached regarding the specific contribution of these prerequisites. This study aimed to analyze the contribution and interplay of two domain-general functions, working memory (WM) and metacognitive abilities, and number-specific prerequisites in determining number comprehension. The participants, 126 Italian first-graders, were tested on two WM capacity tasks, an early metacognition questionnaire, five number-specific prerequisites tasks (e.g., quantity and/or size comparison; placement of Arabic numeral), and the Number Knowledge Test for whole-number comprehension. We hypothesized that WM capacity would predict number comprehension both directly and indirectly via metacognition and domain-specific prerequisites. This is because both metacognition and domain-specific prerequisites might place an information load on WM to establish schemes for declarative metamemory and metacognitive monitoring and for emerging counting skills, respectively. The results confirmed these hypotheses. WM capacity was positively associated with number comprehension both directly and via increased metacognition and domain-specific prerequisites. These findings offer a model for interpreting the interplay between domain-general and number-specific predictors of whole-number comprehension, but they also underline the multiple ways in which WM capacity affects it.

Objective:
Childhood brain tumor (BT) survivors are at risk for working memory (WM) and processing speed (PS) deficits, which impact other cognitive domains. This study aimed to characterize WM, PS, and untimed mathematics calculation performance in pediatric BT survivors at least 2 years post-diagnosis, identify medical factors associated with deficits in mathematics, and examine whether WM and/or PS predict mathematics performance in this clinical sample.
Methods:
Retrospective data were gathered from 72 BT survivors between 7 and 21 years of age (M=13.64 y, SD=4.01 y) for a clinical neuropsychological evaluation. All participants completed Wechsler measures of WM and PS and a measure of untimed mathematics calculation.
Results:
WM, PS, and the mathematics calculation were significantly lower than the normative mean. Math scores were not correlated with any of the examined medical factors. PS was negatively correlated with the Neurological Predictor Scale and positively correlated with age at diagnosis. Both WM and PS were associated with math outcomes and accounted for 30.4% and 19.2% of the variance, respectively.
Conclusions:
The findings indicate that WM and PS contribute to mathematics performance in pediatric BT survivors. Examining mathematics performance should be a part of clinical neuropsychological evaluations. Interventions to improve mathematics performance in this population should also focus on WM and compensatory strategies for slowed PS.

Background
Previous research suggests that visuospatial working memory (WM) is a unique predictor of mathematics. However, evidence from neuropsychology and cognitive studies suggests dissociations between visual and spatial WM.
Procedure
We examined the differential relationships between visual and spatial WM with mathematics using a new task that 1) utilized the same paradigm across visual and spatial tasks and 2) required executive WM.
Main findings
We found that our new spatial WM task related to mathematics scores while visual WM did not. Spatial WM related to mathematics scores for fourth-graders and not second graders, consistent with previous findings on the relationship between spatial skills and mathematics as mathematics becomes more complex. No relationship was found between spatial WM and reading scores at either grade level.
Conclusions
Our results highlight the dynamic relationship between WM components and mathematics over the elementary school years and suggest that spatial WM is a unique predictor of mathematics starting from middle childhood.

數學詞彙日益受到重視，然而過去研究在一般詞彙和數學詞彙對數學成就之預測路徑未有一致結果。本研究採路徑分析，探討納入非語文智力和語文智力後，前期的一般詞彙知識和數學詞彙知識如何影響後期的數學成就（包含概念理解、程序執行與解題思考）。本研究以737位四、六年級學生為對象，第一年施測一般詞彙知識與數學詞彙知識測驗，第二年施測數學成就與智力測驗。研究結果顯示，在納入非語文智力和語文智力後，第一，四年級學生兩種詞彙知識對數學成就有效果相當的直接預測力，數學詞彙知識在一般詞彙知識與數學成就之間具部分中介效果，代表存有雙重學習路徑。第二，六年級學生數學詞彙知識對數學成就的直接效果大於一般詞彙知識對數學成就的預測力，數學詞彙知識具部分中介效果，代表存有雙重學習路徑。第三，長期而言在國小數學學習上兩種學習路徑並存，前期兩種詞彙知識對後期數學成就的預測力於高年級學生更強，且數學詞彙知識對高年級學生更加關鍵。本路徑模式擴展了過去國小生橫斷面研究結果。本研究結果指出一般詞彙和數學詞彙是建構數學知識的基礎，在數學表現和學習時能促進理解。隨著發展，運用累積的數學詞彙知識作為媒介對學生取得良好數學成就更為重要。
The role of mathematics vocabulary has received considerable attention. Several studies have examined how mathematics vocabulary mediated the relation between general vocabulary and mathematics outcomes, but the findings were inconsistent. This longitudinal study explored how early knowledge of general and mathematics vocabulary related to later mathematics achievement (encompassing conceptual understanding, procedural knowledge, and problem-solving skills) after including non-verbal and verbal intelligence. The participants were 737 students of 4th and 6th grades. The measurements of mathematics vocabulary and general vocabulary were conducted in June 2018, while the intelligence tests and self-edited mathematics achievement tests were administered in May 2019. The results showed that (1) for 4th graders, both general and mathematics vocabulary could predict mathematics achievement. Mathematics vocabulary partially mediated the relation between general vocabulary and mathematics achievement, which indicated dual pathways to mathematics achievement. (2) For 6th graders, the direct effect of mathematics vocabulary on mathematics achievement was larger than that of general vocabulary on mathematics achievement. Mathematics vocabulary partially mediated the relation between general vocabulary and mathematics achievement, which indicated dual pathways to mathematics achievement. (3) In the long run, two developmental pathways to learning mathematics in elementary school coexisted. While the predictive power of early general and mathematical vocabulary for later mathematical achievement increased with grade level, mathematical vocabulary became more critical for senior graders. The longitudinal pattern of results extends findings from cross-sectional studies of elementary students. The findings suggest that general and mathematics vocabulary may provide the foundational building blocks for mathematics knowledge as well as contribute to comprehension during mathematics performance and learning. In sum, the use of cumulative mathematics vocabulary as a medium will become important and will facilitate students' mathematics achievement.

This document presents a review of evidence commissioned by the Education Endowment Foundation to inform the guidance document Improving Mathematics in Key Stages Two and Three (Education Endowment Foundation, 2017). There have been a number of recent narrative and systematic reviews of mathematics education examining how students learn and the implications for teaching (e.g., Anthony & Walshaw, 2009; Conway, 2005; Kilpatrick et al., 2001; Nunes et al., 2010). Although this review builds on these studies, this review has a different purpose and takes a different methodological approach to reviewing and synthesising the literature. The purpose of the review is to synthesise the best available international evidence regarding teaching mathematics to children between the ages of 9 and 14 and to address the question: what is the evidence regarding the effectiveness of different strategies for teaching mathematics? In addition to this broad research question, we were asked to address a set of more detailed topics developed by a group of teachers and related to aspects of pupil learning, pedagogy, the use of resources, the teaching of specific mathematical content, and pupil attitudes and motivation. Using these topics, we derived the 24 research questions that we address in this review. Our aim was to focus primarily on robust, causal evidence of impact, using experimental and quasi-experimental designs. However, there are a very large number of experimental studies relevant to this research question. Hence, rather than identifying and synthesising all these primary studies, we focused instead on working with existing meta-analyses and systematic reviews. This approach has the advantage that we can draw on the findings of a very extensive set of original studies that have already been screened for research quality and undergone some synthesis. Using a systematic literature search strategy, we identified 66 relevant meta-analyses, which synthesise the findings of more than 3000 original studies. However, whilst this corpus of literature is very extensive, there were nevertheless significant gaps. For example, the evidence concerning the teaching of specific mathematical content and topics was limited. In order to address gaps in the meta-analytic literature, we supplemented our main dataset with 22 systematic reviews identified through the same systematic search strategy.

To understand how distraction influences children’s arithmetic performance, we examined effects of irrelevant sounds on children’s performance while they solve arithmetic problems. Third and fifth graders were asked to verify true/false, one-digit addition problems (e.g., 9 + 4 = 12. True? False?) under silence and sound conditions. The sounds began when the problems started to appear on the screen (Experiment 1; N = 76) or slightly after (Experiment 2; N = 92) and continued until participants responded. The results showed that (a) children solved arithmetic problems more quickly in the sound condition than in the silence condition when the sounds started with problem display (phasic arousal effects); (b) children were slower on the arithmetic problem verification task when the sounds was played slightly after the problems started to appear on the screen (distraction effects); (c) phasic arousal effects were found only in third graders, whereas distraction effects were found in both grades, although their magnitudes were smaller in fifth graders; (d) distraction effects increased with increasing latencies in third graders but did not change across the entire latency distribution in fifth graders; and (e) distraction effects on current trials were smaller after sound trials than after silence trials in both age groups (sequential modulations of distraction effects). These findings have important implications for furthering our understanding of effects of irrelevant sounds on arithmetic performance as well as cognitive processes involved in children’s arithmetic.

Some of the worst long-term outcomes of children are associated with the presence of externalizing behavior and low academic achievement. However, the nature of the causal and predictive relationship between these two domains remains contested due to inconsistent findings in previous literature. Leveraging a nationally representative sample (N = 7330) from the Early Childhood Longitudinal Study (ECLS)–Kindergarten Cohort of 2011, we used latent class growth analysis and machine learning cross validation techniques to analyze the association of early math and reading achievement with the development of externalizing behavior trajectories in elementary school. Several theoretically and empirically important covariates were utilized to develop a profile of learners in each trajectory. Results indicated stable teacher ratings of behavior across kindergarten to fifth grade and three primary trajectories, consisting of (1) higher persistent, (2) low persistent, and (3) no problem behavior. Importantly, teacher rated early inattention and approaches to learning behaviors, rather than direct standardized measures of academic achievement, were the strongest malleable predictors to trajectory membership. Student demographics, including being a boy and identifying as Black, contributed to these students being almost twice as likely to belong to the higher problem behavior trajectory as compared to girls and White peers. Educational implications for intervention, as well as the influence of implicit bias and structural racism in the role of teacher ratings, are discussed.

Cognitive training aims to improve skills such as working memory capacity and spatial ability, which have been linked to math skills. In this study, we fit Deep Knowledge Tracing with Transformers (DKTT), Dynamic Key-Value Memory Networks (DKVMN), and Knowledge Tracing Machines (KTM) to a large dataset from a cognitive training system. DKVMN achieved the highest AUC (0.739) of the algorithms. To explore connections between math skills and cognitive skills, the data was split into cognitive and math items. DKVMN’s AUC on the math items was higher (0.745) than on the cognitive (0.706). Notably, the split model AUCs did not differ from skill-level AUCs produced by a model trained on the entire dataset, suggesting that math performance did not improve DKVMN’s cognitive predictions and vice versa.

Children with learning difficulties are commonly assumed to have underlying cognitive deficits by health and educational professionals. However, not all children referred for psycho-educational assessment will be found to have deficits when their abilities are measured by performance on cognitive tasks. The primary aim of this study was to estimate the prevalence of this inconsistent cognitive profile (ICP) in a transdiagnostic sample of children referred by health and education service providers for problems related to attention, learning and memory (N = 715). A second aim was to explore whether elevated mental health problems were associated with ICPs. Findings suggest that approximately half of this sample could be characterised as having an ICP. Cognitive difficulties, whether identified by parent ratings or task performance, were associated with elevated internalising and externalising difficulties. Crucially, a larger discrepancy between a parent’s actual ratings of a child’s cognitive difficulties and the ratings that would be predicted based on the child’s performance on cognitive tasks was associated greater internalising and externalising difficulties for measures of working memory, and greater externalising difficulties for measures of attention. These findings suggest that subjective cognitive difficulties occurring in the absence of any task-based performance deficits may be a functional problem arising from mental health problems.

This is the protocol for a Campbell systematic review. Our primary objective for this systematic review is to examine if preschool and school‐based interventions aimed at improving language, literacy, and/or mathematical skills increase children's and adolescents' executive functions. As a secondary objective, we will examine how the effects of language, literacy, and mathematics interventions on executive functions are moderated by the subject of the intervention, child age or grade, the type of EF measured, and the at‐risk status of participants. We will also explore how the effects are moderated by other study characteristics, and estimate the effects of the included interventions on language, literacy, and mathematical skills.

It is well established that academic performance (AP) depends on a number of factors, such as intellectual capacities, practice, and previous knowledge. We know little about how these factors interact as they are rarely measured simultaneously. Here we present mediated-Factors of Academic Performance (m-FAP) model, which simultaneously assesses direct and indirect, mediated, effects on AP. In a semester-long study with 118 first-year college students, we show that intelligence and working memory only indirectly influenced AP on a familiar, less challenging college course (Introduction to Psychology). Their influence was mediated through previous knowledge and self-regulated learning activities akin to deliberate practice. In a novel and more challenging course (Statistics in Psychology), intellectual capacities influenced performance both directly and indirectly through previous knowledge. The influence of deliberate practice, however, was considerably weaker in the novel course. The amount of time and effort that the students spent on the more difficult course could not offset the advantage of their more intelligent and more knowledgeable peers. The m–FAP model explains previous contradictory results by providing a framework for understanding the extent and limitations of individual factors in AP, which depend not only on each other, but also on the learning context.

This chapter presents the results of collaboration among special education, mathematics education, and psycholinguistic researchers, with a focus on meaningful assessments for students who have difficulties with learning mathematics. Gaining knowledge of the mathematical abilities of students who struggle or who have particular learning disabilities is often limited in traditional assessments. Access to struggling students’ ability to reason and solve problems may be obscured because of inadequate approaches and limitations in assessing the knowledge of these students using traditional test items. Our goal is to address these concerns and other obstacles to assessment and suggest additional and alternate ways of measuring students’ knowledge. It is noteworthy for researchers to attend to what are termed “learning disabilities” in mathematics and the characteristics of goal requirements in students’ individualized education programs (IEPs). IEP requirements are understood in relation to assessment approaches that are used to identify students who struggle to learn mathematics. In particular, it is essential to consider former, existing, and emerging theoretical views of mathematics and special education educators regarding what might be considered the “ability” of students to learn mathematics.KeywordsMathematicsAssessmentStudents with disabilities/difficulties

The developmental literature has revealed a wide variety of factors that correlate with individual differences in mathematical cognition. These factors have been classified along the simple distinction between domain-specific and domain-general factors. Here, I review a series of recent meta-analyses that have summarized the existing evidence for each of the factors separately. These factors all correlate to a similar extent with individual differences in mathematics. These correlations are similar in special populations, such as dyscalculia. The relative contributions of these factors remain after some control for other factors. A nascent body of evidence suggests that these associations between domain-specific/domain-general factors and mathematical performance are bi-directional. Future studies should therefore investigate these factors in concert through carefully designed longitudinal studies.

The study identified the cognitive mechanisms that underly individual differences in mathematics and reading. There is little systematic evidence regarding the connections between cognitive capabilities, mathematics and reading among high-school students with typical development. Moreover, the study aimed to examine whether or not gender is important in predicting cognitive capabilities and performance in mathematics and in reading. Structural equation modelling was used to examine the effects of gender, visuo-spatial working memory (VSWM), numerical working memory (NWM), verbal working memory (VWM), and visual processing (VP) on mathematics and reading performance. A sample of 190 students completed mathematics, reading, VSWM, NWM, VWM, and VP tests. Results showed that mathematical performance was influenced by NWM, VSWM, and VP, while VWM was found to be unrelated to such performance. On the other hand, VWM and VP, but not VSWM, predicted reading performance. Furthermore, while direct correlations between gender, NWM and VP were found, VWM and VSWM were not. The results are important for their theoretical and educational implications.

According to the Pathways to Mathematics model [LeFevre et al. (2010), Child Development, Vol. 81, pp. 1753–1767], children’s cognitive skills in three domains—linguistic, attentional, and quantitative—predict concurrent and future mathematics achievement. We extended this model to include an additional cognitive skill, patterning, as measured by a non-numeric repeating patterning task. Chilean children who attended schools of low or high socioeconomic status (N = 98; 54% girls) completed cognitive measures in kindergarten (Mage = 71 months) and numeracy and mathematics outcomes 1 year later in Grade 1. Patterning and the original three pathways were correlated with the outcomes. Using Bayesian regressions, after including the original pathways and mother’s education, we found that patterning skills predicted additional variability in applied problem solving and arithmetic fluency, but not number ordering, in Grade 1. Similarly, patterning skills were included in the best model for applied problem solving and arithmetic fluency, but not for number ordering, in Grade 1. In accord with the hypotheses of the original Pathways to Mathematics model, patterning varied in its unique and relative contributions to later mathematical performance, depending on the demands of the tasks. We conclude that patterning is a useful addition to the Pathways to Mathematics model, providing further insights into the range of cognitive precursors that are related to children’s mathematical development.

The loneliness of modern people is becoming more and more prominent, and has brought profoundly negative effects on mental health. Social support is an important predictor of loneliness. However, the size of the correlation reported by studies on the relation between social support and loneliness varies greatly. The aim of this meta-analysis is to determine the relation between social support and loneliness. One hundred and seventy-seven articles (N = 113,427) were identified, and robust variance estimation with random effects were used. As expected, higher levels of social support were negatively correlated with loneliness (r = −0.39). The association between social support and loneliness were also moderated by several variables. Specifically, the negative relationship between loneliness and social support among rural populations is stronger than that of urban populations in Chinese samples, the effect of perceived social support (r = −0.45) on loneliness is greater than that of other social supports (r = −0.36), and the friend support (r = −0.48) played a more important role in reducing loneliness than that of two other supports (family support: r = −0.34; significant other support: r = −0.40). The current results support robust links between social support and loneliness, emphasizing the important role of social support in reducing levels of loneliness, this may have some implications for future research and loneliness treatments.

Investigations of the cognitive processes involved in adults’ mental arithmetic have shown the importance of phonological rather than visual codes in performance (e.g. Logie, Gilhooly & Wynn; 1994). However, children’s subjective descriptions of their strategies in arithmetic reveal much more variable strategy use, suggesting that both slave systems of working memory may be involved in arithmetical performance in this age group. This experiment tested children in two age groups (6 to 7 and 8 to 9 years) on a simple mental arithmetic task under three conditions; baseline, phonological disruption and visuospatial disruption, in an attempt to determine the cognitive processes and strategies being used in arithmetic at various points throughout development. The results showed significant and marked lowering of performance in the younger children with concurrent visuospatial disruption, a smaller effect being observed in the older children. Conversely, while the younger children were unaffected by concurrent phonological disruption the older children were. These results support the hypothesis that children use different strategies at different ages; younger children utilise almost exclusively visuospatial strategies in mental arithmetic, whereas older children use a mixture of phonological and visual-spatial strategies.

Working memory training programs have generated great interest, with claims that the training interventions can have profound beneficial effects on children’s academic and intellectual attainment. We describe the criteria by which to evaluate evidence for or against the benefit of working memory training. Despite the promising results of initial research studies, the current review of all of the available evidence of working memory training efficacy is less optimistic. Our conclusion is that working memory training produces limited benefits in terms of specific gains on short-term and working memory tasks that are very similar to the training programs, but no advantage for academic and achievement-based reading and arithmetic outcomes.

This article systematically reviews what is known empirically about the association between executive function and student achievement in both reading and math and critically assesses the evidence for a causal association between the two. Using meta-analytic techniques, the review finds that there is a moderate unconditional association between executive function and achievement that does not differ by executive function construct, age, or measurement type but finds no compelling evidence that a causal association between the two exists.

The study explored the contribution of working memory to mathematical word problem solving in
children. A total of 69 children in grades 2, 3 and 4 were given measures of mathematical problem
solving, reading, arithmetical calculation, fluid IQ and working memory. Multiple regression
analyses showed that three measures associated with the central executive and one measure
associated with the phonological loop contributed unique variance to mathematical problem solving
when the influence of reading, age and IQ were controlled for in the analysis. In addition, the animal
dual-task, verbal fluency and digit span task continued to contribute unique variance when the effects
of arithmetical calculation in addition to reading, fluid IQ, and age were controlled for. These findings
demonstrate that the phonological loop and a number of central executive functions (shifting,
co-ordination of concurrent processing and storage of information, accessing information from
long-term memory) contribute to mathematical problem solving in children.

This study investigated whether individual differences in working memory (WM) moderate effects of 2 variations of intervention designed to improve at-risk 4th graders’ fraction knowledge. We also examined the effects of each intervention condition against a business-as-usual control group and assessed whether children’s measurement interpretation of fractions mediated those effects. At-risk students (n = 243) were randomly assigned to control and 2 intervention conditions. The interventions each lasted 12 weeks, with three 30-min sessions per week. The major focus of both intervention conditions was the measurement interpretation of fractions. Across the 2 conditions, only 5 min of each 30-min session differed. One condition completed activities to build fluency with 4 measurement interpretation topics; in the other, activities were completed to consolidate understanding on the same 4 topics. Results revealed a significant aptitude–treatment interaction, in which students with very weak WM learned better with conceptual activities but children with more adequate (but still low) WM learned better with fluency activities. Both intervention conditions outperformed the control group on all outcomes, and improvement in the measurement interpretation of fractions mediated those effects. (PsycINFO Database Record (c) 2014 APA, all rights reserved)

This study examined the contributions of the different components of the working memory (WM) model to a range of mathematical skills in children, using measures of WM function that did not involve numerical stimuli. A sample of 148 children (78 Year 3, mean age 8 years and 1 month, and 70 Year 5 pupils, mean age 9 years and 10 months) completed WM measures and age‐appropriate mathematics tests designed to assess four mathematical skills defined by the National Curriculum for England. Visuo‐spatial sketchpad and central executive, but not phonological loop, scores predicted unique variance in children's curriculum‐based mathematical attainment but the relative contributions of each component did not vary much across the different skills. Subsequently, the mathematics data were re‐analysed using cluster analysis and new performance‐related mathematics factors were derived. All three components of WM predicted unique variance in these performance‐related skills, but revealed a markedly distinct pattern of associations across the two age groups. In particular, the data indicated a stronger role for the visuo‐spatial sketchpad in the younger children's mathematics performance. We discuss our findings in terms of the importance of WM in the development of early mathematical ability.

The purposes of this study were to investigate the effects of an intervention designed to improve at-risk 4th
graders’ understanding of fractions and to examine the processes by which effects occurred. The intervention
focused more on the measurement interpretation of fractions; the control condition focused more on the
part-whole interpretation of fractions and on procedures. Intervention was also designed to compensate for
at-risk students’ limitations in the domain-general abilities associated with fraction learning. At-risk students
(n � 259) were randomly assigned to intervention and control. Whole-number calculation skill, domaingeneral
abilities (working memory, attentive behavior, processing speed, listening comprehension), and
fraction proficiency were pretested. Intervention occurred for 12 weeks, 3 times per week, 30 min per session,
and then fraction performance was reassessed. On each conceptual and procedural fraction outcome, effects
favored intervention over control (effect sizes � 0.29 to 2.50), and the gap between at-risk and low-risk
students narrowed for the intervention group but not the control group. Improvement in the accuracy of
children’s measurement interpretation of fractions mediated intervention effects. Also, intervention effects
were moderated by domain-general abilities, but not whole-number calculation skill.

In the current study, the factor structure of number sense, or the ability to understand, use, and manipulate numbers, was investigated. Previous analyses yielded little consensus concerning number sense factors, other than a distinction between nonsymbolic and symbolic processing. Furthermore, associations between number sense factors and working memory components were investigated to gain insight into working memory involvement in number sense. A total of 441 Dutch kindergartners took part in the study. The factor structure of number sense and associations with working memory were tested using structural equation modelling (SEM). Results indicated that there was a clear distinction between nonsymbolic and symbolic number processing. Nonsymbolic processing was predicted by central executive performance, and symbolic processing was predicted by both central executive and visuospatial sketchpad performance. This implies that symbolic and nonsymbolic processing are distinguishable at this age, and that working memory involvement in symbolic processing is different from that in nonsymbolic processing.

Explanations of the marked individual differences in elementary school mathematical achievement and mathematical learning disability (MLD or dyscalculia) have involved domain-general factors (working memory, reasoning, processing speed, and oral language) and numerical factors that include single-digit processing efficiency and multidigit skills such as number system knowledge and estimation. This study of 3rd graders (N = 258) finds both domain-general and numerical factors contribute independently to explaining variation in 3 significant arithmetic skills: basic calculation fluency, written multidigit computation, and arithmetic word problems. Estimation accuracy and number system knowledge show the strongest associations with every skill, and their contributions are independent of both each other and other factors. Different domain-general factors independently account for variation in each skill. Numeral comparison, a single digit processing skill, uniquely accounts for variation in basic calculation. Subsamples of children with MLD (at or below 10th percentile, n = 29) are compared with low achievement (LA, 11th to 25th percentiles, n = 42) and typical achievement (above 25th percentile, n = 187). Examination of these and subsets with persistent difficulties supports a multiple deficits view of number difficulties: Most children with number difficulties exhibit deficits in both domain-general and numerical factors. The only factor deficit common to all persistent MLD children is in multidigit skills. These findings indicate that many factors matter but multidigit skills matter most in 3rd grade mathematical achievement.

This study examined whether a minimum level of preschool quality (threshold) is needed in order for a relationship to exist between preschool quality and children's academic, behavioral, and working memory in a sample of children from low-wealth rural communities where quality child care has been found to be lower than more urban communities. Participants included 849 children from two high-poverty, rural regions. Preschool quality was rated using the CLASS observational measure. Child outcomes included direct assessments of early language, mathematics, and working memory, as well as teacher ratings of attention, emotion regulation, problem behaviors, and peer relationships. Analyses included piecewise regression analyses that tested a priori specified cut-points and flexible b-spline analyses that tested for thresholds empirically. Results indicated some evidence for quality thresholds, suggesting that quality was related to children's behavioral outcomes above, but not below, a cut-point. Language, literacy, and working memory did not show evidence of threshold effects. Results are discussed in the context of prior mixed evidence for child care quality thresholds in other samples of predominantly low-income preschoolers in center-based child care in more urban areas.

Although the effects of achievement goals and working memory on academic performance are well established, it is not clear whether they jointly affect academic performance. Children from Primary 4 and 6 (N = 608) were administered (a) measures of working memory and updating from the automated working memory battery and a running span task, (b) performance and mastery goal measures from the inventory of school motivation, and (c) a battery of standardised and curriculum-based mathematical tests. Both mastery and performance goals had direct (positive and negative, respectively) relations with working memory capacity. The negative relation between performance goal and mathematics was stronger for children with lower levels of mastery goal or working memory, than for those with higher levels. These findings suggest that a reduction in the availability of working memory resources may be one reason for a high performance orientation to be associated with poorer academic performance.

Dual-process and dual-system theories in both cognitive and social psychology have been subjected to a number of recently published criticisms. However, they have been attacked as a category, incorrectly assuming there is a generic version that applies to all. We identify and respond to 5 main lines of argument made by such critics. We agree that some of these arguments have force against some of the theories in the literature but believe them to be overstated. We argue that the dual-processing distinction is supported by much recent evidence in cognitive science. Our preferred theoretical approach is one in which rapid autonomous processes (Type 1) are assumed to yield default responses unless intervened on by distinctive higher order reasoning processes (Type 2). What defines the difference is that Type 2 processing supports hypothetical thinking and load heavily on working memory.
© The Author(s) 2013.

Working memory, including central executive functions (inhibition, shifting and updating) are factors thought to play a central role in mathematical skill development. However, results reported with regard to the associations between mathematics and working memory components are inconsistent. The aim of this meta-analysis is twofold: to investigate the strength of this relation, and to establish whether the variation in the association is caused by tests, sample characteristics and study and other methodological characteristics. Results indicate that all working memory components are associated with mathematical performance, with the highest correlation between mathematics and verbal updating. Variation in the strength of the associations can consistently be explained by the type of mathematics measure used: general tests yield stronger correlations than more specific tests. Furthermore, characteristics of working memory measures, age and sample explain variance in correlations in some analyses. Interpretations of the contribution of moderator variables to various models are discussed.

Visuospatial competencies are related to performance in mathematical domains in adulthood, but are not consistently related to mathematics achievement in children. We confirmed the latter for first graders and demonstrated that children who show above average first-to-fifth grade gains in visuospatial memory have an advantage over other children in mathematics. The study involved the assessment of the mathematics and reading achievement of 177 children in kindergarten to fifth grade, inclusive, and their working memory capacity and processing speed in first and fifth grade. Intelligence was assessed in first grade and their second to fourth grade teachers reported on their in-class attentive behavior. Developmental gains in visuospatial memory span (d = 2.4) were larger than gains in the capacity of the central executive (d = 1.6) that in turn were larger than gains in phonological memory span (d = 1.1). First to fifth grade gains in visuospatial memory and in speed of numeral processing predicted end of fifth grade mathematics achievement, as did first grade central executive scores, intelligence, and in-class attentive behavior. The results suggest there are important individual differences in the rate of growth of visuospatial memory during childhood and that these differences become increasingly important for mathematics learning.

This article synthesizes published literature comparing the cognitive functioning of children who have math disabilities (MD) with that of (a) average-achieving children; (b) children who have reading disabilities (RD); and (c) children who have comorbid disabilities (MD+RD). Average achievers outperformed children with MD on measures of verbal problem solving, naming speed, verbal working memory (WM), visual-spatial WM, and long-term memory (LTM). Children with MD outperformed comorbid children on measures of literacy, visual-spatial problem solving, LTM, short-term memory (STM) for words, and verbal WM. Children with MD could be differentiated from children with RD only on naming speed and visual-spatial WM. Differences in cognitive functioning between children with MD and average achievers were related primarily to verbal WM when the effects of all other variables (e.g., age, IQ, and other domain categories) were partialed out.

Two studies were conducted to explore mathematical precocity in young children. Study 1 examined mathematically gifted first and third graders' working memory development. The results showed that mathematically gifted children's working memory growth was similar to that expected of their age peers. Study 2 examined changes in mathematically gifted children's conceptual structures. Mathematically gifted children were roughly a year ahead of their age peers in the rate of development of conceptual structure in the numerical domain. A neo‐Piagetian theory of intellectual development was used to explain these seemingly conflicting findings. The relation between working memory growth and conceptual development was discussed throughout the paper.

Recently, there has been a growing emphasis on basic number processing competencies (such as the ability to judge which of two numbers is larger) and their role in predicting individual differences in school-relevant math achievement. Children's ability to compare both symbolic (e.g. Arabic numerals) and nonsymbolic (e.g. dot arrays) magnitudes has been found to correlate with their math achievement. The available evidence, however, has focused on computerized paradigms, which may not always be suitable for universal, quick application in the classroom. Furthermore, it is currently unclear whether both symbolic and nonsymbolic magnitude comparison are related to children's performance on tests of arithmetic competence and whether either of these factors relate to arithmetic achievement over and above other factors such as working memory and reading ability. In order to address these outstanding issues, we designed a quick (2 minute) paper-and-pencil tool to assess children's ability to compare symbolic and nonsymbolic numerical magnitudes and assessed the degree to which performance on this measure explains individual differences in achievement. Children were required to cross out the larger of two, single-digit numerical magnitudes under time constraints. Results from a group of 160 children from grades 1-3 revealed that both symbolic and nonsymbolic number comparison accuracy were related to individual differences in arithmetic achievement. However, only symbolic number comparison performance accounted for unique variance in arithmetic achievement. The theoretical and practical implications of these findings are discussed which include the use of this measure as a possible tool for identifying students at risk for future difficulties in mathematics.

This chapter is divided into two parts. The first describes the effect of Pat Rabbitt's influence in encouraging the first author to use the increasingly sophisticated methods of ageing research to answer questions about the fundamental characteristics of working memory, together with reflections on why so little of this work reached publication. The second part presents a brief review of the literature on working memory and ageing, followed by an account of more recent work attempting to apply the traditional method of experimental dissociation to research on normal ageing and Alzheimer's disease. The discussion suggests that even such simple methods can throw light on both the processes of ageing and the understanding of working memory.

A longitudinal study of 54 children aged between 4 and 7 years of age investigated whether measures of working memory skills taken shortly after school entry served as useful predictors of children’s attainment levels in National Curriculum assessments at Key Stage 1. Early working memory scores were found to be highly significant predictors of children’s subsequent levels of attainment in literacy, but not in mathematics. Compared with the local education authority baseline assessments also administered at 4 years of age that are designed in large part to predict later attainments, working memory scores accounted for unique variance in children’s spelling and writing scores at 7 years. These findings point to the utility of combining knowledge-based assessments with measures of fluid cognitive ability in order to obtain the best estimates of a child’s chances of future academic success.

Two separate experiments demonstrated that performance on a simple motor task was compromised by performing certain working memory tasks simultaneously. The motor task utilized was a tracking task which was able to differentiate among many of the simultaneously performed working memory tasks. More complex working memory tasks are associated with greater degradation of motor performance. This study also demonstrated that there are large individual differences in which working memory tasks were associated with the greatest decline in performance. This may be partially attributable to individual differences in knowledge and experience that are relevant to particular tasks. ^ For about half of the participants, working memory capacity predicted performance across the different conditions of the tracking task such that those with higher working memory capacity performed better. Working memory capacity did not predict performance for those who performed poorly on the tracking task while performing simple arithmetic, however. Perhaps this condition was able to group participants in a way that afforded some control over individual differences in both mathematical ability and motor skill in tracking.

Cognitive strategies are important tools for children with math difficulties (MD) in learning to solve word problems. The effectiveness of strategy training, however, depends on working memory capacity (WMC). Thus, children with MD but with relatively higher WMC are more likely to benefit from strategy training, whereas children with lower WMC may have their resources overtaxed. Children in Grade 3 (N = 147) were randomly assigned to 1 of 4 conditions: (a) verbal strategies (e. g., underlining question sentence), (b) visual strategies (e.g., correctly placing numbers in diagrams), (c) verbal plus visual strategies, or (d) an untreated control. In line with the predictions, children with MD and higher WMC benefited from verbal or visual strategies relative to those in the control condition on posttest measures of problem solving, calculation, and operation span. In contrast, cognitive strategies decreased problem-solving accuracy in children with low WMC. Thus, improvement in problem solving and related measures, as well as the impairment in learning outcomes, was moderated by WMC.

This study’s hypotheses were that (a) word-problem (WP) solving is a form of text comprehension that involves language comprehension processes, working memory, and reasoning, but (b) WP solving differs from other forms of text comprehension by requiring WP-specific language comprehension as well as general language comprehension. At the start of the 2nd grade, children (n = 206; on average, 7 years, 6 months) were assessed on general language comprehension, working memory, nonlinguistic reasoning, processing speed (a control variable), and foundational skill (arithmetic for WPs; word reading for text comprehension). In spring, they were assessed on WP-specific language comprehension, WPs, and text comprehension. Path analytic mediation analysis indicated that effects of general language comprehension on text comprehension were entirely direct, whereas effects of general language comprehension on WPs were partially mediated by WP-specific language. By contrast, effects of working memory and reasoning operated in parallel ways for both outcomes.

This study investigated contributions of general cognitive abilities and foundational mathematical competencies to numeration understanding (i.e., base-10 structure) versus multidigit calculation skill. Children (n = 394, M = 6.5 years) were assessed on general cognitive abilities and foundational numerical competencies at start of 1st grade; on the same numerical competencies, multidigit calculation skill, and numeration understanding at end of 2nd grade; and on multidigit calculation skill and numeration understanding at end of 3rd grade. Path-analytic mediation analysis revealed that general cognitive predictors exerted more direct and more substantial effects on numeration understanding than on multidigit calculations. Foundational mathematics competencies contributed to both outcomes, but largely via 2nd-grade mathematics achievement, and results suggest a mutually supportive role between numeration understanding and multidigit calculations. (PsycINFO Database Record (c) 2014 APA, all rights reserved)

We reviewed the literature on the role of working memory in the solution of arithmetic problems such as 3 + 4 or 345 + 29. The literature was neither comprehensive nor systematic, but a few conclusions are tenable. First, all three components of the working memory system proposed by Baddeley (i.e., central executive, phonological loop, and visual-spatial sketchpad) play a role in mental arithmetic, albeit under different conditions. Second, mental arithmetic requires central executive resources, even for single-digit problems. Third, further progress in understanding the role of working memory in arithmetic requires that researchers systematically manipulate factors such as presentation conditions (e.g., operand duration, format), problem complexity, task requirements (e.g., verification vs production), and response requirements (e.g., spoken vs written); and that they consider individual differences in solution procedures. Fourth, the encoding-complex model (Campbell, 1994) seems more likely to account for the variability observed in arithmetic solutions than other models of numerical processing. Finally, working memory researchers are urged to use mental arithmetic as a primary task because the results of the present review suggest that solution of problems that involve multiple digits are likely to involve an interaction of all the components of the working memory system.

The present study focused on the role of number skills assessed in kindergarten with regard to their ability to predict mathematical outcomes in grade 1. Number skills included those involving written symbols (symbolic number identification and symbolic number comparison) and counting (procedural and conceptual). Their contributions were contextualized against domain general (working memory, phonological awareness, and behavioral inattention) factors. Both types of kindergarten domain specific skills were strongly correlated with each math outcome in first grade. However, hierarchical regression showed that written symbolic number skills accounted for variance over and above counting predictors. In final models, domain general factors had unique effects (verbal working memory for math fluency, phonological awareness for computation, verbal working memory and phonological awareness for applied problems, and spatial working memory and phonological awareness for story problems). Results highlight the interplay among math precursors and math related domain general factors and their differential roles for different mathematical outcomes.

The ability to connect numbers and magnitudes is an important prerequisite for math learning, here referred to as number–magnitude skills. It has been proposed that working memory plays an important role in constructing these connections. The aim of the current study was to examine if working memory accounts for constructing these connections by testing whether development of number–magnitude skills can be explained by different components of working memory. Number–magnitude skills and working memory skills of 69 children were assessed at age 5, 5.5 and 6. Different results were found for different components of working memory. Whereas no effects were found for the visuospatial sketchpad and phonological loop in development of number–magnitude skills, the central executive predicted variance in intercept and slope of number–magnitude skills. The results of this study provide longitudinal evidence for the involvement of general cognitive skills (i.e. working memory) in the development of number–magnitudes connections. Copyright © 2014 John Wiley & Sons, Ltd.

Children with learning difficulties suffer from working memory (WM) deficits. Yet the specificity of deficits associated with different types of learning difficulties remains unclear. Further research can contribute to our understanding of the nature of WM and the relationship between it and learning difficulties. The current meta-analysis synthesized research on verbal WM and numerical WM among children with reading difficulties (RD), children with mathematics difficulties (MD), and children with reading and mathematics difficulties (RDMD). A total of 29 studies subsuming 110 comparisons were included. Results showed that compared to typically developing children, all learning difficulty groups demonstrated deficits in verbal WM and numerical WM, with RDMD children showing the most severe WM deficits. MD children and RD children showed comparable verbal WM deficits, but MD children showed more severe numerical WM deficits than RD children. Neither severity of learning difficulties nor type of academic screening emerged as a moderator of WM deficit profiles. Although the findings indicate the domain-general nature of WM deficits in RD, MD, and RDMD children, the numerical WM deficits of children with MD and RDMD may reflect the domain-specific nature of WM deficits.

The aim of this study was to examine the development of arithmetic performance and its cognitive precursors in children with CP from 7 till 9 years of age. Previous research has shown that children with CP are generally delayed in arithmetic performance compared to their typically developing peers. In children with CP, the developmental trajectory of the ability to solve addition- and subtraction tasks has, however, rarely been studied, as well as the cognitive factors affecting this trajectory. Sixty children (M = 7.2 years, SD = .23 months at study entry) with CP participated in this study. Standardized tests were administered to assess arithmetic performance, word decoding skills, non-verbal intelligence, and working memory. The results showed that the ability to solve addition- and subtraction tasks increased over a two year period. Word decoding skills were positively related to the initial status of arithmetic performance. In addition, non-verbal intelligence and working memory were associated with the initial status and growth rate of arithmetic performance from 7 till 9 years of age. The current study highlights the importance of non-verbal intelligence and working memory to the development of arithmetic performance of children with CP.

Children’s early mathematics skills develop in a cumulative fashion; foundational skills form a basis for the acquisition of later skills. However, non-mathematical factors such as working memory and language skills have also been linked to mathematical development at a broad level. Unfortunately, little research has been conducted to evaluate the specific relations of these two non-mathematical factors to individual aspects of early mathematics. Thus, the focus of this study was to determine whether working memory and language were related to only individual aspects of early mathematics or related to many components of early mathematics skills. A total of 199 4- to 6-year-old preschool and kindergarten children were assessed on a battery of early mathematics tasks as well as measures of working memory and language. Results indicated that working memory has a specific relation to only a few—but critically important—early mathematics skills and language has a broad relation to nearly all early mathematics skills.

Math anxiety (MA)-working memory (WM) relationships have typically been examined in the context of arithmetic problem solving, and little research has examined the relationship in other math domains (e.g., algebra). Moreover, researchers have tended to examine MA/worry separate from math problem solving activities and have used general WM tasks rather than domain-relevant WM measures. Furthermore, it seems to have been assumed that MA affects all areas of math. It is possible, however, that MA is restricted to particular math domains. To examine these issues, the current research assessed claims about the impact on algebraic problem solving of differences in WM and algebraic worry. A sample of 80 14-year-old female students completed algebraic worry, algebraic WM, algebraic problem solving, nonverbal IQ, and general math ability tasks. Latent profile analysis of worry and WM measures identified four performance profiles (subgroups) that differed in worry level and WM capacity. Consistent with expectations, subgroup membership was associated with algebraic problem solving performance: high WM/low worry>moderate WM/low worry=moderate WM/high worry>low WM/high worry. Findings are discussed in terms of the conceptual relationship between emotion and cognition in mathematics and implications for the MA-WM-performance relationship.

Longitudinal studies of neurodevelopmental disorders that are diagnosed at or before birth and are associated with specific learning difficulties at school-age provide one method for investigating developmental precursors of later-emerging academic disabilities. Spina bifida myelomeningocele (SBM) is a neurodevelopmental disorder associated with particular problems in mathematics, in contrast to well-developed word reading. Children with SBM (n = 30) and typically developing children (n = 35) were used to determine whether cognitive abilities measured at 36 and 60 months of age mediated the effect of group on mathematical and reading achievement outcomes at 8.5 and 9.5 years of age. A series of multiple mediator models showed that: visual–spatial working memory at 36 months and phonological awareness at 60 months partially mediated the effect of group on math calculations, phonological awareness partially mediated the effect of group on small addition and subtraction problems on a test of math fluency, and visual–spatial working memory mediated the effect of group on a test of math problem solving. Groups did not differ on word reading, and phonological awareness was the only mediator for reading fluency and reading comprehension. The findings are discussed with reference to theories of mathematical development and disability and with respect to both common and differing cognitive correlates of math and reading.

Specific language influences have been observed in basic numerical tasks such as magnitude comparison, transcoding, and the number line estimation task. However, so far language influences in more complex calculations have not been reported in children. In this translingual study, 7- to 9-year-old German- and Italian-speaking children were tested on a symbolic addition task. Whereas the order of tens and units in Italian number words follows the order of the Arabic notation, the order is inverted in German number words. For both language groups, addition problems were more difficult when a carry operation was needed, that is, when a manipulation within the place-value structure of the Arabic number system was particularly important. Most important, this carry effect was more pronounced in response latencies for children speaking German, a language with inverted verbal mapping of the place-value structure. In addition, independent of language group, the size of the carry effect was significantly related to verbal working memory. The current study indicates that symbolic arithmetic and the carry effect in particular are modulated by language-specific characteristics. Our results underline the fact that the structure of the language of instruction is an important factor in children's mathematical education and needs to be taken into account even for seemingly nonverbal symbolic Arabic tasks.