Number sense or working memory? The effect of two computer-based trainings on mathematical skills in elementary school

Research on the improvement of elementary school mathematics has shown that computer-based training of number sense (e.g., processing magnitudes or locating numbers on the number line) can lead to substantial achievement gains in arithmetic skills. Recent studies, however, have highlighted that training domain-general cognitive abilities (e.g., working memory [WM]) may also improve mathematical achievement. This study addressed the question of whether a training of domain-specific number sense skills or domain-general WM abilities is more appropriate for improving mathematical abilities in elementary school. Fifty-nine children (M_age = 9 years, 32 girls and 27 boys) received either a computer-based, adaptive training of number sense (n = 20), WM skills (n = 19), or served as a control group (n = 20). The training duration was 20 min per day for 15 days. Before and after training, we measured mathematical ability using a curriculum-based math test, as well as spatial WM. For both training groups, we observed substantial increases in the math posttest compared to the control group (d = .54 for number sense skills training, d = .57 for WM training, respectively). Whereas the number sense group showed significant gains in arithmetical skills, the WM training group exhibited marginally significant gains in word problem solving. However, no training group showed significant posttest gains on the spatial WM task. Results indicate that a short training of either domain-specific or domain-general skills may result in reliable short-term training gains in math performance, although no stable training effects were found in the spatial WM task.     

Working memory and early numeracy training in preschool children

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

Basic number processing in children with specific learning disorders: Comorbidity of reading and mathematics disorders

As well as being the hallmark of mathematics disorders, deficits in number processing have also been reported for individuals with reading disorders. The aim of the present study was to investigate separately the components of numerical processing affected in reading and mathematical disorders within the framework of the Triple Code Model. Children with reading disorders (RD), mathematics disorders (MD), comorbid deficits (RD + MD), and typically developing children (TD) were tested on verbal, visual-verbal, and nonverbal number tasks. As expected, children with MD were impaired across a broad range of numerical tasks. In contrast, children with RD were impaired in (visual-)verbal number tasks but showed age-appropriate performance in nonverbal number skills, suggesting their impairments were domain specific and related to their reading difficulties. The comorbid group showed an additive profile of the impairments of the two single-deficit groups. Performance in speeded verbal number tasks was related to rapid automatized naming, a measure of visual-verbal access in the RD but not in the MD group. The results indicate that deficits in number skills are due to different underlying cognitive deficits in children with RD compared to children with MD: a phonological deficit in RD and a deficit in processing numerosities in MD.     

Mathematics ability and related skills in preschoolers born very preterm

Children born very preterm (VPT) are at risk for academic, behavioral, and/or emotional problems. Mathematics is a particular weakness and better understanding of the relationship between preterm birth and early mathematics ability is needed, particularly as early as possible to aid in early intervention. Preschoolers born VPT (n = 58) and those born full term (FT; n = 29) were administered a large battery of measures within 6 months of beginning kindergarten. A multiple-mediation model was utilized to characterize the difference in skills underlying mathematics ability between groups. Children born VPT performed significantly worse than FT-born children on a measure of mathematics ability as well as full-scale IQ, verbal skills, visual–motor integration, phonological awareness, phonological working memory, motor skills, and executive functioning. Mathematics was significantly correlated with verbal skills, visual–motor integration, phonological processing, and motor skills across both groups. When entered into the mediation model, verbal skills, visual–motor integration, and phonological awareness were significant mediators of the group differences. This analysis provides insights into the pre-academic skills that are weak in preschoolers born VPT and their relationship to mathematics. It is important to identify children who will have difficulties as early as possible, particularly for VPT children who are at higher risk for academic difficulties. Therefore, this model may be used in evaluating VPT children for emerging difficulties as well as an indicator that if other weaknesses are found, an assessment of mathematics should be conducted.     

Mathematical problem solving and working memory in children with learning disabilities: both executive and phonological processes are important

The purpose of this investigation was to explore the relationship between working memory (WM) and mathematical problem solving in children with learning disabilities (LD). Children with LD (age 11.5 years) were compared to chronologically age-matched (CA-M) and younger comprehension/computation achievement-matched children (age 8.9 years) on measures of verbal and visual-spatial WM, phonological processing, components of problem solving, and word-problem solving accuracy. The results showed that (1) children with LD were inferior on measures of word solution accuracy, components of problem solving, phonological processing, domain-general WM, and verbal WM when compared to children who were CA-M, (2) children with LD were comparable to younger children on all processing measures, except measures of domain-general WM, visual-spatial WM, phonemic deletion, and identifying problem goals, (3) measures of verbal and visual-spatial WM contributed significant variance to solution accuracy independent of phonological processing, and (4) the influence of WM on solution accuracy was mediated by long-term memory (LTM) processes related to the knowledge of algorithms. The results support the notion that information activated from LTM, rather than phonological processing, mediates the relationship between executive processing and solution accuracy in children with LD.     

Phonological storage and executive function deficits in children with mathematics difficulties

Children with mathematics difficulties suffer from working memory deficits. This study investigated the deficit profile of phonological storage and executive functions in working memory among children with mathematics difficulties. Based on multiple instruments and two assessment points, 68 children were screened out of 805 fifth graders. Of these 68 children, 18 were classified as children with only mathematics difficulties (MD), 20 were classified as children with mathematics and reading difficulties (MDRD), and 30 were typically developing (TD) peers matched on age and general ability. Measures for phonological storage, dual-task performance, inhibition, and updating of verbal and numerical materials were administered individually. Results showed that compared with the TD group, children with MD exhibited storage and inhibition deficits specific to numerical information and dual-task deficits of both verbal and numerical information, whereas children with MDRD showed extensive deficits on phonological storage and executive functions on both verbal and numerical tasks. Moreover, executive function deficits were not confined to phonological storage deficits. Implications of the findings for the working memory deficit profile and working memory training among children with mathematics difficulties were discussed.     

The complexities of complex span: explaining individual differences in working memory in children and adults

Two studies are presented that investigated the constraints underlying working memory performance in children and adults. In each case, independent measures of processing efficiency and storage capacity are assessed to determine their relative importance in predicting performance on complex span tasks,which measure working memory capacity. Results show that complex span performance was independently constrained by individual differences in domain-general processing efficiency and domain-specific storage capacity. Residual variance, which may reflect the ability to coordinate storage and processing, also predicted academic achievement. These results challenge the view that complex span taps a limited-capacity resource pool shared between processing and storage operations. Rather, they are consistent with a multiple-component model in which separate resource pools support the processing and storage functions of working memory.     

First-grade predictors of mathematical learning disability: A latent class trajectory analysis

Kindergarten to third grade mathematics achievement scores from a prospective study of mathematical development (n = 306) were subjected to latent growth trajectory analyses. The four corresponding classes included children with mathematical learning disability (MLD, 6% of sample), and low (LA, 50%), typically (TA, 39%) and high (HA, 5%) achieving children. The groups were administered a battery of intelligence (IQ), working memory, and mathematical-cognition measures in first grade. The children with MLD had general deficits in working memory and IQ and potentially more specific deficits on measures of number sense. The LA children did not have working memory or IQ deficits but showed moderate deficits on these number sense measures and for addition fact retrieval. The distinguishing features of the HA children were a strong visuospatial working memory, a strong number sense, and frequent use of memory-based processes to solve addition problems. Implications for the early identification of children at risk for poor mathematics achievement are considered.     

Working memory deficit in children with mathematical difficulties: a general or specific deficit?

This study examined whether children with mathematical difficulties (MDs) or comorbid mathematical and reading difficulties have a working memory deficit and whether the hypothesized working memory deficit includes the whole working memory system or only specific components. In the study, 31 10-year-olds with MDs and 37 10-year-olds with both mathematical and reading difficulties were compared with 47 age-matched and 50 younger controls (9-year-olds) on a number of working memory tasks. Compared with the age-matched controls, both groups of children with MDs performed worse on tasks tapping the central executive (e.g., visual matrix span) and the phonological loop (e.g., word span). More important, the MD group performed worse than the younger controls on the counting span task, whereas the group with comorbid mathematical and reading difficulties performed worse on the counting span task and the visual matrix span task. These findings provide support for the assumption that children with MDs have a working memory deficit. More specifically, children with MDs have a central executive deficit connected to concurrent processing and storage of numerical and visual information.     

Working memory and access to numerical information in children with disability in mathematics

The relationship among working memory, mathematic ability, and the cognitive impairment of children with difficulties in mathematics was examined. A group of children with difficulties in mathematics (MD) was compared with a group of children with a normal level of achievement matched for vocabulary, age, and gender (N = 49). The children were required to perform a variety of working memory and short-term memory tasks that had been administered 1 year previously. Moreover, the children were asked to perform tasks designed to provide information about speed of articulation. The results suggest a general working memory deficit in children with MD, specifically in the central executive component of Baddeley's model and primarily in the ability to inhibit irrelevant information. However, the MD children were not impaired in speech rate and counting speed tasks, which mainly involve the role of the articulatory loop.     

The relationship between phonological processing and arithmetic in children with learning disabilities

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.

Research Highlights

• Relationships between phonological processing and various arithmetic skills were investigated in children with learning disabilities (LDs) for the first time.
• We found phonemic awareness was related to arithmetic involving fact retrieval, but not to arithmetic involving procedural computation in LDs.
• The results suggest that phonemic awareness is not only important to skilled reading, but also to some aspects of arithmetic.
• These results raise the question of whether intervention in phonemic awareness might improve arithmetic fact retrieval skills.

     

Increased arithmetic complexity is associated with domain-general but not domain-specific magnitude processing in children: A simultaneous fNIRS-EEG study

The investigation of the neural underpinnings of increased arithmetic complexity in children is essential for developing educational and therapeutic approaches and might provide novel measures to assess the effects of interventions. Although a few studies in adults and children have revealed the activation of bilateral brain regions during more complex calculations, little is known about children. We investigated 24 children undergoing one-digit and two-digit multiplication tasks while simultaneously recording functional near-infrared spectroscopy (fNIRS) and electroencephalography (EEG) data. FNIRS data indicated that one-digit multiplication was associated with brain activity in the left superior parietal lobule (SPL) and intraparietal sulcus (IPS) extending to the left motor area, and two-digit multiplication was associated with activity in bilateral SPL, IPS, middle frontal gyrus (MFG), left inferior parietal lobule (IPL), and motor areas. Oscillatory EEG data indicated theta increase and alpha decrease in parieto-occipital sites for both one-digit and two-digit multiplication. The contrast of two-digit versus one-digit multiplication yielded greater activity in right MFG and greater theta increase in frontocentral sites. Activation in frontal areas and theta band data jointly indicate additional domain-general cognitive control and working memory demands for heightened arithmetic complexity in children. The similarity in parietal activation between conditions suggests that children rely on domain-specific magnitude processing not only for two-digit but—in contrast to adults—also for one-digit multiplication problem solving. We conclude that in children, increased arithmetic complexity tested in an ecologically valid setting is associated with domain-general processes but not with alteration of domain-specific magnitude processing.     

Intellectual growth in children as a function of domain specific and domain general working memory subgroups

This study examined whether children's growth on measures of fluid (Raven Colored Progressive Matrices) and crystallized (reading and math achievement) intelligence was attributable to domain-specific or domain-general functions of working memory (WM). A sample of 290 elementary school children was tested on measures of intelligence across three testing waves. Two methods, a hierarchical factor model predicting latent growth and a double dissociation design comparing children divided into high and low performers, tested whether general and/or specific components accounted for performance on intelligence measures. A general domain model for the total sample provided the best fit for latent growth on all measures. Further, the lack of a significant WM subgroup × domain interaction suggested that a general WM system underlies the individual differences on measures of fluid and crystallized intelligence. Overall, the findings support a domain-general view of WM capacity on intelligence measures in children, but also suggest that domain specific storage systems may come into play on isolated measures.     

Representation and retrieval of arithmetical facts: Developmental difficulties

One form of developmental difficulty with arithmetic affects the storage or retrieval of arithmetical facts, such as tables, which are required to implement arithmetical computations (Temple, 1991, 1994). Such difficulties may arise because of impairment in a specialized system for the storage of arithmetical facts or as a result of causally linked impairment in another cognitive domain. This study explored issues concerning the representation and retrieval of arithmetical facts in children with number fact disorders (NF) and in normal children, in particular the status of hypothesized linked impairments: short-term memory (STM) spans, counting skills, speed of speech, and speed of number fact and lexical retrieval. There was no evidence that NF children had weak STM spans on any span measure or that STM spans related to arithmetical fact skills. There was also no evidence that NF children had weak counting abilities or free counting speeds. The NF children were slower in speeded counting, which also correlated with number fact skill. The significance or not of this is discussed. The NF children were also slower than controls in speed of speech and on some measures of speed of access. However, the absence of correlation with number fact skill, the absence of generality across tasks, and the possibility that delayed speeds in fact retrieval reflect the use of alternative strategies, together suggest that the increased speeds are not causally linked to number fact skill. The results are consistent with modular accounts, in which there is a specialized system for the storage and retrieval of arithmetical facts.     

The generality of working memory capacity: a latent-variable approach to verbal and visuospatial memory span and reasoning

A latent-variable study examined whether verbal and visuospatial working memory (WM) capacity measures reflect a primarily domain-general construct by testing 236 participants in 3 span tests each of verbal WM. visuospatial WM, verbal short-term memory (STM), and visuospatial STM. as well as in tests of verbal and spatial reasoning and general fluid intelligence (Gf). Confirmatory' factor analyses and structural equation models indicated that the WM tasks largely reflected a domain-general factor, whereas STM tasks, based on the same stimuli as the WM tasks, were much more domain specific. The WM construct was a strong predictor of Gf and a weaker predictor of domain-specific reasoning, and the reverse was true for the STM construct. The findings support a domain-general view of WM capacity, in which executive-attention processes drive the broad predictive utility of WM span measures, and domain-specific storage and rehearsal processes relate more strongly to domain-specific aspects of complex cognition.     

How does processing affect storage in working memory tasks? Evidence for both domain-general and domain-specific effects

Two studies that examine whether the forgetting caused by the processing demands of working memory tasks is domain-general or domain-specific are presented. In each, separate groups of adult participants were asked to carry out either verbal or nonverbal operations on exactly the same processing materials while maintaining verbal storage items. The imposition of verbal processing tended to produce greater forgetting even though verbal processing operations took no longer to complete than did nonverbal processing operations. However, nonverbal processing did cause forgetting relative to baseline control conditions, and evidence from the timing of individuals' processing responses suggests that individuals in both processing groups slowed their responses in order to "refresh" the memoranda. Taken together the data suggest that processing has a domain-general effect on working memory performance by impeding refreshment of memoranda but can also cause effects that appear domain-specific and that result from either blocking of rehearsal or interference.     

The role of planning in different mathematical skills

Despite the fact that planning has been found to be a significant predictor of reading (particularly of reading comprehension), much less is known about its contribution to mathematics. The purpose of this study was to examine the role of two levels of planning (operation planning and action planning) in three mathematical skills (calculation fluency, math problem-solving, and math reasoning). Eighty Grade 2 children from Shanghai, China were assessed on measures of nonverbal cognitive ability (nonverbal matrices), working memory (digit span backwards and N-back), operation planning (matching numbers, planned codes, and planned search), action planning (crack the code), and mathematics (calculation fluency, math problem-solving, and math reasoning). The results of regression analyses showed that both levels of planning accounted for unique variance in mathematics over and above the effects of nonverbal cognitive ability and working memory. The effects of action planning were particularly strong in math problem-solving. These findings suggest that measures of planning could be used along with measures of working memory to detect children at-risk for mathematics disabilities and that intervention programmes targeting planning could be developed to boost children's mathematics performance.     

Roles of working memory capacity and long-term working memory skill in complex task performance

In the present research, we examined the relative roles of domain-general and domain-specific individual difference characteristics in complex cognitive task performance. Specifically, we examined the impact both of working memory (WM) capacity and of acquired skills used to encode presented information in an accessible form in long-term working memory (LTWM) on performance in a complex aviation task environment. Measures of WM capacity and LTWM skill served as performance predictors. A criterion measure of task performance was related to the predictor measures. The results indicated that an increase in LTWM skill decreases the role of WM capacity as the determinant of complex task performance, although both measures are important performance predictors. We discuss how the two distinct WM constructs coexist and interact to support complex task performance.     

Adaptive training leads to sustained enhancement of poor working memory in children

Working memory plays a crucial role in supporting learning, with poor progress in reading and mathematics characterizing children with low memory skills. This study investigated whether these problems can be overcome by a training program designed to boost working memory. Children with low working memory skills were assessed on measures of working memory, IQ and academic attainment before and after training on either adaptive or non‐adaptive versions of the program. Adaptive training that taxed working memory to its limits was associated with substantial and sustained gains in working memory, with age‐appropriate levels achieved by the majority of children. Mathematical ability also improved significantly 6 months following adaptive training. These findings indicate that common impairments in working memory and associated learning difficulties may be overcome with this behavioral treatment.     

Cognitive Analysis of Children's Mathematics Difficulties

The aim of the study is to investigate the informal and formal mathematical knowledge of children suffering from "mathematics difficulty" (MD). The research involves comparisons among three groups: fourth-grade children performing poorly in mathematics but normal in intelligence; fourth-grade peers matched for intelligence but experiencing no apparent difficulties in mathematics; and a randomly selected group of third graders. These children were individually presented with a large number of tasks designed to measure key mathematical concepts and skills. The findings suggest that: (1) MD children are not seriously deficient in key informal mathematical concepts and skills; (2) MD children seem to have elementary concepts of base ten notation but experience difficulty in related enumeration skills, particularly when large numbers are involved; (3) MD children's calculational errors often result from common error strategies; (4) MD children display severe difficulty in recalling common addition facts; and (5) in the area of problem solving, MD children are capable of "insightful" solutions and can solve simple forms of word problems, but experience difficulty with complex word problems. MD children are in many respects similar to normal, younger peers; an hypothesis of "essential cognitive normality" is advanced. The only and dramatic exception occurs in the area of number facts. While clinical experience corroborates this finding, its explanation is not evident.