Computational skills, working memory, and conceptual knowledge in older children with mathematics learning disabilities

Knowledge and skill in multiplication were investigated for late elementary-grade students with mathematics learning disabilities (MLD), typically achieving age-matched peers, low-achieving age-matched peers, and ability-matched peers by examining multiple measures of computational skill, working memory, and conceptual knowledge. Poor multiplication fact mastery and calculation fluency and general working memory discriminated children with MLD from typically achieving age-matched peers. Furthermore, children with MLD were slower in executing backup procedures than typically achieving age-matched peers. The performance of children with MLD on multiple measures of multiplication skill and knowledge was most similar to that of ability-matched younger children. MLD may be due to difficulties in computational skills and working memory. Implications for the diagnosis and remediation of MLD are discussed.     

Development of Working Memory from Grade 3 to 5: Differences between Children With and Without Mathematical Learning Difficulties

Based on the finding that children with mathematical learning difficulties (MLD) have deficits in working memory (WM), the question arises as to whether these children differ from typical learners only in the level or also in the developmental trajectories of WM functioning. To this end, the WM of 80 children with MLD and 71 typical learners was assessed longitudinally from third to fifth grade. Typical learners outperformed children with MLD in the phonological, visuospatial and central executive WM functioning in third grade. Latent change analyses indicated that both phonological and central executive WM functioning developed in a parallel pattern in children with MLD and in typical learners. In contrast, visuospatial WM functioning revealed a different development in children with and without MLD since the gap between both groups decreased over time. Overall, despite starting at a lower level, the WM functioning in children with MLD did not develop more slowly.     

Math achievement,numerical processing,and executive functions in girls with Turner syndrome: Do girls with Turner syndrome have math learning disability?

Turner syndrome is a common genetic disorder associated with select deficits in executive functions, working memory and mathematics. In Study 1, we examined growth trajectories of skills in these areas, from grades 1 to 6, among girls with or without Turner syndrome. Rates of growth and performance levels at 6th grade, on an untimed math achievement test, did not suggest that girls with Turner syndrome have math learning difficulty (MLD). However, analyses did reveal lower efficiency on timed executive function tasks, among girls with Turner syndrome, who traded accuracy for speed under mild to moderate working memory demands. In Study 2 we compared numerical processing skills of 6th graders who had either Turner syndrome, MLD, low math achievement, or typical achievement in math. A numerical decomposition task revealed numerical processing deficits for the Turner syndrome and MLD groups, relative to typically achieving students. The relative difficulties in how numerical processing vs. working memory demands affected performance accuracy differed among groups, with the former demands leading to more performance difficulties in the Turner syndrome group. Our findings support the notion that girls with Turner syndrome recruit different strategies than their peers during allegedly basic numerical processing, that numerical processing deficits vs. executive function deficits underlie their difficulties with mathematics, and that math difficulties among these girls may not be apparent on untimed tests. These finding have implications for a possible manifestation of MLD.     

Cognitive mechanisms underlying achievement deficits in children with mathematical learning disability

Using strict and lenient mathematics achievement cutoff scores to define a learning disability, respective groups of children who are math disabled (MLD, n=15) and low achieving (LA, n=44) were identified. These groups and a group of typically achieving (TA, n=46) children were administered a battery of mathematical cognition, working memory, and speed of processing measures (M=6 years). The children with MLD showed deficits across all math cognition tasks, many of which were partially or fully mediated by working memory or speed of processing. Compared with the TA group, the LA children were less fluent in processing numerical information and knew fewer addition facts. Implications for defining MLD and identifying underlying cognitive deficits are discussed.     

Fact retrieval deficits in low achieving children and children with mathematical learning disability

Using 4 years of mathematics achievement scores, groups of typically achieving children (n = 101) and low achieving children with mild (LA-mild fact retrieval; n = 97) and severe (LA-severe fact retrieval; n = 18) fact retrieval deficits and mathematically learning disabled children (MLD; n = 15) were identified. Multilevel models contrasted developing retrieval competence from second to fourth grade with developing competence in executing arithmetic procedures, in fluency of processing quantities represented by Arabic numerals and sets of objects, and in representing quantity on a number line. The retrieval deficits of LA-severe fact retrieval children were at least as debilitating as those of the children with MLD and showed less across-grade improvement. The deficits were characterized by the retrieval of counting string associates while attempting to remember addition facts, suggesting poor inhibition of irrelevant information during the retrieval process. This suggests a very specific form of working memory deficit, one that is not captured by many typically used working memory tasks. Moreover, these deficits were not related to procedural competence or performance on the other mathematical tasks, nor were they related to verbal or nonverbal intelligence, reading ability, or speed of processing, nor would they be identifiable with standard untimed mathematics achievement tests.     

Behind mathematical learning disabilities: What about visual perception and motor skills?

In a sample of 39 children with mathematical learning disabilities (MLD) and 106 typically developing controls belonging to three control groups of three different ages, we found that visual perception, motor skills and visual-motor integration explained a substantial proportion of the variance in either number fact retrieval or procedural calculation. Furthermore, children with MLD performed significantly worse on visual perception, motor skills and visual-motor integration in comparison with age-matched control children. A mild developmental delay in visual perception, visual-motor integration and (fine) motor coordination and a severe delay in motor skills were found in children with MLD. However, not all children with MLD have problems on these domains. They seem to be a heterogeneous group, not only with respect to their mathematical profile but also with respect to their visual perceptual, motor and visual-motor integration skills. Diagnostic implications are discussed.     

Early Cognitive Precursors of Children's Mathematics Learning Disability and Persistent Low Achievement: A 5-Year Longitudinal Study

Mathematical difficulties have been distinguished as mathematics learning disability (MLD) and persistent low achievement (LA). Based on 1,880 Finnish children who were followed from kindergarten (age 6) to fourth grade, this study examined the early risk factors for MLD and LA. Distinct groups of MLD (6.0% of the sample) and LA (25.7%) children were identified on the basis of their mathematics performance between first and fourth grades with latent class growth modeling. Impairment in the same set of cognitive skills, including language, spatial, and counting skills, was found to underlie MLD and LA. The finding highlights the importance of monitoring mathematical development across the early grades and identifying early cognitive precursors of MLD and LA for screening and intervention efforts.     

Differential contribution of specific working memory components to mathematics achievement in 2nd and 3rd graders

The contribution of the three core components of working memory (WM) to the development of mathematical skills in young children is poorly understood. The relation between specific WM components and Numerical Operations, which emphasize computation and fact retrieval, and Mathematical Reasoning, which emphasizes verbal problem solving abilities in 48 2nd and 50 3rd graders was assessed using standardized WM and mathematical achievement measures. For 2nd graders, the central executive and phonological components predicted Mathematical Reasoning skills; whereas the visuo-spatial component predicted both Mathematical Reasoning and Numerical Operations skills in 3rd graders. This pattern suggests that the central executive and phonological loop facilitate performance during early stages of mathematical learning whereas visuo-spatial representations play an increasingly important role during later stages. We propose that these changes reflect a shift from prefrontal to parietal cortical functions during mathematical skill acquisition. Implications for learning and individual differences are discussed.     

State anxiety and working memory in children: A test of processing efficiency theory

This study investigated the effect of individual differences in state anxiety on tasks tapping the central executive, phonological, and visuo‐spatial components of working memory (WM). It was designed to test Eysenck and Calvo’s processing efficiency theory (PET) which suggests that the phonological and executive components of WM may be important in understanding learning outcomes in anxiety. Typically‐developing children aged 9–10 years were split into high and low state anxiety groups. They performed three WM tasks – forward and backward digit span (assumed to measure phonological and central executive components of WM respectively) and a spatial working memory task (measuring the visuo‐spatial component of WM). Measurements of task accuracy were taken as an indicator of performance outcome or effectiveness. Time taken to complete tasks and a subjective rating of mental effort were taken as measurements of performance efficiency. No differences were found between high and low state anxiety groups in task accuracy for any measure. Children in the high state anxiety group, however, took longer to complete the backward digit span task and reported increased mental effort in the forward digit span task, indicating some effect of anxiety on measures of performance efficiency.     

Working Memory and Children’s Mathematical Skills: Implications for mathematical development and mathematics curricula

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 development of working memory in normally achieving and subtypes of learning disabled children

Working memory has been proposed as an important component of reading and arithmetic skills. The development of working memory was studied in normally achieving and subtypes of learning disabled children. The performance of reading disabled (RD), arithmetic disabled (ARITHD), and attentional deficit disordered (ADD) children, age 7-13, was compared to normal achievers (NA) on 2 working memory tasks, 1 involving sentences and the other involving counting. There was a significant growth of working memory as a function of age. In addition, the RD children had significantly lower scores on both tasks. The ARITHD children had significantly lower scores only on the Working Memory--Counting task, and the ADD group had scores similar to the normally achieving children except at the youngest age level in the Working Memory--Sentences task. Thus, a reading disability appears to involve a generalized deficit in working memory. Children with an arithmetic disability do not have a generalized language deficit but have a specific working memory deficit in relation to processing numerical information. As children with ADD did not have deficits in these tasks, working memory may not have significant attentional components. An important component of the development of reading and computational arithmetic skills appears to be the growth of working memory for language and numerical information.     

Visuospatial working memory in adolescents with poor performance in mathematics: variation depending on reading skills

The first purpose of this study was to investigate whether the visuospatial working memory (VSWM) skills of 15–16‐year‐old pupils with difficulties in mathematics differ from those of their normally achieving peers. The goal was to broaden the view of the complex system of VSWM. A set of passive and active VSWM tasks was used. The study’s second purpose was to investigate whether pupils with mathematical difficulties differed in their VSWM skills based on whether they had signs of reading deficits or not. Results indicate that the pupils with poor performance in maths showed poorer performance on certain VSWM tasks. The group with deficits only in maths had less capacity for storing passive visual simultaneous information, while the group with difficulties both in maths and reading had deficits in both storing (passive visual and visuospatial information) and processing, and had less ability to control irrelevant visuospatial information compared to their peers of the same age. The results indicate a general VSWM deficit in pupils with both mathematics and reading problems and a specific VSWM deficit in pupils with only mathematics problems.     

Does strategy knowledge influence working memory in children with mathematical disabilities?

This study investigated the relationship between working memory (WM), declarative strategy knowledge, and math achievement in children with and without mathematical disabilities (MD). Experiment 1 examined the relationship between strategy knowledge, verbal WM, and visual-spatial WM in children with MD as a function of initial, gain, and maintenance conditions. The results showed that after partialing the influence of reading, stable strategy choices rather than specific strategy knowledge was related to verbal and visual-spatial WM span in high demand (maintenance) conditions. Experiment 2 compared children with MD to a group of chronological age-matched children and a group of math ability-matched children on the same conditions as Experiment 1. Age-matched children's verbal and visual-spatial WM performance was superior to that of children with MD, whereas WM performance was statistically comparable between children with MD and younger children matched on math ability. The selection of expert strategies was related to high WM span scores in the initial conditions. After controlling for reading achievement in a regression analysis, verbal and visual-spatial WM, stable verbal strategy choices, and expert strategy choices related to visual-spatial processing all contributed independent variance to math achievement. Overall, these results suggest that WM and math achievement are related to strategy knowledge.     

Additive and multiplicative effects of working memory and test anxiety on mathematics performance in grade 3 students

The aim of this study was to investigate the interplay between test anxiety and working memory (WM) on mathematics performance in younger children. A sample of 624 grade 3 students completed a test battery consisting of a test anxiety scale, WM tasks and the Swedish national examination in mathematics for grade 3. The main effects of test anxiety and WM, and the two-way interaction between test anxiety and WM on mathematics performance, were modelled with structural equation modelling techniques. Additionally, the effects were also tested separately on tasks with high WM demands (mathematical problem-solving) versus low WM demands (basic arithmetic). As expected, WM positively predicted mathematics performance in all three models (overall mathematics performance, problem-solving tasks, and basic arithmetic). Test anxiety had a negative effect on problem-solving on the whole sample level but concerning basic arithmetic only students with lower WM were affected by the negative effects of test anxiety on performance. Thus, students with low WM are more vulnerable to the negative effects of test anxiety in low WM tasks like basic arithmetic. The results are discussed in relation to the early identification of test anxiety.     

A subgroup analysis of working memory in children with reading disabilities: domain-general or domain-specific deficiency?

The purpose of this study was to investigate whether changes in working memory (WM) of children with reading disabilities (RD) were related to a domain-specific or a domain-general system. Based on Daneman and Carpenter's (1980) Sentence Listening Span task, children were subgrouped into a group of high executive processing (high listening span) children without RD, a group of low executive processing (low listening span) children with RD, and a group of children with and without RD matched on executive processing (moderate listening span). Subgroups were compared on phonological, visual-spatial, and semantic WM tasks across initial (no probes or cues), gain (cues that bring performance to an asymptotic level), and maintenance conditions (asymptotic conditions without cues). The results showed that (a) children without RD high in executive processing ability outperformed all other subgroups, (b) the RD subgroup low in executive processing performed poorly relative to all other subgroups across task and memory conditions, (c) children with and without RD matched on executive processing were comparable in WM span and changes in WM for all tasks, and (d) WM performance of children with RD was a strong linear function of the high executive processing group, suggesting that the nature or the specific componential makeup of the tasks are not the main contributors to WM performance. Taken together, the results suggest that a domain-general system may partially contribute to poor WM in children with RD, and that this system may operate independently of their reading deficits.     

A longitudinal analysis of sex differences in math and spatial skills in primary school age children

We report on a longitudinal study designed to assess possible sex differences in math achievement, math ability, and math-related tasks during the primary school age years. Participants included over 200 children from one public school district. Annual assessments included measures of math ability, math calculation achievement scores, rapid naming and decoding tasks, visual perception tests, visual motor tasks, and reading skills. During select years of the study we also administered tests of counting and math facts skills. We examined whether girls or boys were overrepresented among the bottom or top performers on any of these tasks, relative to their peers, and whether growth rates or predictors of math-related skills differed for boys and girls. Our findings support the notion that sex differences in math are minimal or nonexistent on standardized psychometric tests routinely given in assessments of primary school age children. There was no persistent finding suggesting a male or female advantage in math performance overall, during any single year of the study, or in any one area of math or spatial skills. Growth rates for all skills, and early correlates of later math performance, were comparable for boys and girls. The findings fail to support either persistent or emerging sex differences on non-specialized math ability measures during the primary school age years.     

小学数学学习困难儿童的工作记忆广度研究

对数学学习困难儿童的工作记忆广度进行研究,结果发现数学学习困难儿童在言语和视觉空间工作记忆广度上都显著低于正常儿童,但这种不足主要是一种发展的延迟。对有无伴随阅读困难的数学学习困难儿童进行比较,发现二者的言语和视觉空间工作记忆广度都没有表现出显著差异。     

Do different components of working memory underlie different subgroups of reading disabilities?

The purpose of this study was to determine the components of working memory (WM) that underlie less skilled readers' comprehension and word recognition difficulties. Performance of 3 less skilled reading subgroups---children with reading disabilities (RD) in both word recognition and comprehension; children with comprehension deficits only; and children with low verbal IQ, word recognition, and comprehension (poor readers)--was compared to that of skilled readers on WM, short-term memory (STM), processing speed, executive, and phonological processing measures. Ability group comparisons showed that (a) skilled readers outperformed all less skilled readers on measures of WM, updating, and processing speed; (b) children with comprehension deficits only outperformed children with RD on measures of WM, STM, phonological processing, and processing speed; and (c) children with RD outperformed poor readers on WM and phonological processing measures. A hierarchical regression analysis showed that (a) subgroup differences on WM tasks among less skilled readers were moderated by a storage system not specific to phonological skills, and (b) STM and updating contributed significant variance to WM beyond what was contributed by reading group classification. The latter finding suggested that some differences in storage and executive processing emerged between skilled and less skilled readers that were not specific to reading.     

Number magnitude processing and basic cognitive functions in children with mathematical learning disabilities

The study sought out to extend our knowledge regarding the origin of mathematical learning disabilities (MLD) in children by testing different hypotheses in the same samples of children. Different aspects of cognitive functions and number processing were assessed in fifth- and sixth-graders (11–13 years old) with MLD and compared to controls. The MLD group displayed weaknesses with most aspects of number processing (e.g., subitizing, symbolic number comparison, number-line estimation) and two cognitive functions (e.g., visual–spatial working memory). These findings favor the defective approximate number system (ANS) hypothesis, but do not fit well with the access deficit hypothesis. Support is also provided for the defective object-tracking system (OTS) hypothesis, the domain general cognitive deficit hypothesis and to some extent the defective numerosity-coding hypothesis. The study suggests that MLD might be caused by multiple deficits and not a single core deficit.     

Estimating the co-development of executive functions and math achievement throughout the elementary grades using a cross-lagged panel model with fixed effects

Executive functioning (EF) is associated with children’s math skill development, both concurrently and longitudinally. However, it is not known how components of EF might be related to mathematics skills and vice versa over the course of elementary school. The present study addresses this issue by investigating relations between math achievement and two key components of EF -- working memory (WM) and cognitive flexibility (CF) -- from kindergarten to 5th grade, using the large-scale nationally representative dataset (N = 18,174) from the Early Childhood Longitudinal Study-Kindergarten (ECLS-K: 2011). Results from cross-lagged panel models with fixed effects support a transactional theoretical model, demonstrating a long-term reciprocal relationship between WM and math achievement from kindergarten to 5th grade and between CF and math achievement from 2nd grade to 5th grade. However, we found that reciprocal relations decrease as children grow older, suggesting that their math achievement relies less on EF and more on prior math knowledge over time.