Early identification and interventions for students with mathematics difficulties

This article highlights key findings from the small body of research on mathematics difficulties (MD) relevant to early identification and early intervention. The research demonstrates that (a) for many children, mathematics difficulties are not stable over time; (b) the presence of reading difficulties seems related to slower progress in many aspects of mathematics; (c) almost all students with MD demonstrate problems with accurate and automatic retrieval of basic arithmetic combinations, such as 6 + 3. The following measures appear to be valid and reliable indicators of potential MD in kindergartners: (a) magnitude comparison (i.e., knowing which digit in a pair is larger), (b) sophistication of counting strategies, (c) fluent identification of numbers, and (d) working memory (as evidenced by reverse digit span). These are discussed in terms of the components of number sense. Implications for early intervention strategies are explored.     

Mathematical thinking in second-grade children with different forms of LD

Based on their performance on a standardized achievement test, second-grade children (N = 49) were classified as having mathematics difficulties with normal reading achievement (MD only), both mathematics and reading difficulties (MD/RD), reading difficulties with normal mathematics achievement (RD only) and normal mathematics and reading achievement (NA). Each child was given a series of tasks so that we might assess their thinking across four areas of mathematics: number facts, story problems, place value, and written calculation. Children with MD/RD performed significantly worse than NA children in most areas of mathematical thinking, whereas children with MD only performed worse than NA children only on complex story problems. The MD-only group outperformed the MD/RD group on story problems and written calculation. No significant differences were found between the RD-only and NA groups on any of the tasks. The results suggested that among children with mathematics difficulties, the MD/RD subgroup is distinct from the MD-only subgroup, with the former being characterized by pervasive deficiencies in mathematical thinking and the latter by more specific deficits in problem solving.     

Toward a Two‐Factor Theory of One Type of Mathematics Disabilities

A two‐factor theory is proposed in an attempt to explain the difficulty that children with math disabilities have in mastering the basic number facts. The theory is based on the premise that weak cognitive representations lead to poorer retrieval of information from long‐term memory. Two groups of children with disabilities are discussed: those with math disabilities alone (MD) and those with co‐morbid math and reading disabilities (MD/RD). It is proposed that weak phonological processing abilities underlie the learning difficulties of MD/RD children, and that weak number sense is a causal factor in the math‐fact learning difficulties of MD only and some MD/RD children.     

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.     

The role of numerical and non-numerical ordering abilities in mathematics and reading in middle childhood

This study investigated whether different types of ordering skills were related to mathematics achievement in children (n = 100) in middle childhood, after the effects of age, socio-economic status, IQ, and processing speed were taken into account. The relations between ordering skills and magnitude processing were also investigated, as well as the possibility that some of the shared variance between math and reading is explained by ordering abilities. The ordering tasks included the ordering of familiar numerical and non-numerical sequences, a parental report of children’s everyday ordering skills, and an order working memory task. Three magnitude processing tasks (symbolic and non-symbolic comparison and number line estimation), were also administered, as well as measures of inhibition and spatial working memory. From this set of measures, number ordering, order working memory and number line estimation emerged as the most important predictors of mathematics skills. We found that number ordering mediated the effect of both symbolic and non-symbolic comparison skills on mathematics, further confirming that this task captures some essential skills related to mathematics. Additionally, order working memory mediated the effect of both number comparison and reading skills on math. Finally, whereas non-symbolic comparison and number line estimation are considered important indicators of magnitude processing skills, there was no relationship between these abilities, but there was a correlation between each of these abilities and reading skills, with number line estimation also mediating the effect of reading skills on math. These novel findings could contribute to a better understanding of the basic processes underlying math ability, and why math and reading are strongly related in typical populations and in children with learning difficulties.     

数学学优生与学困生两种抑制能力的比较研究

以小学五年级学生为被试,选取数学学优生30名,数学学困生32名,通过设计不同条件的负启动实验,对两组被试的分心抑制能力和记忆提取抑制能力进行了比较。结果表明：（1）数学学困生在特性抑制和记忆提取抑制上显著差于学优生;（2）与学困生相比,数学学优生对存在于知觉水平的干扰物反应更快。     

The role of sentence recall in reading and language skills of children with learning difficulties

The present study explores the relationship between sentence recall and reading and language skills in a group of 7–11-year-old children with learning difficulties. While recent studies have found that performance on sentence recall tasks plays a role in learning, it is possible that this contribution is a reflection of shared resources with working memory. In order to investigate whether sentence recall was uniquely associated with reading and language skills, differences associated with IQ and working memory capacity were statistically controlled. A sample of 72 children was tested on measures of verbal complex memory, verbal short-term memory, sentence recall, expressive vocabulary, verbal and performance IQ, reading and language skills. Both sentence recall and verbal complex memory shared unique links with reading skill, and sentence recall was uniquely associated with language skills. This finding indicates that resources in long-term memory also play an important diagnostic role in reading and language abilities. The implications for educational practice are discussed.     

Phonological Awareness and Rapid Naming Skills of Children with Reading Disabilities and Children with Reading Disabilities Who Are At Risk for Mathematics Difficulties

Limited research has examined the skills of children with a reading disability (RD) and children with RD and a mathematics disability (MD). Even less research has examined the phonological awareness (PA) and rapid automatized naming (RAN) skills in these two groups of children and how these skills relate to reading and math achievement. Additionally, various classification criteria are frequently implemented to classify children with MD. The purpose of this study, therefore, was to examine the PA and RAN skills in children who met different criteria for RD only and children with RD who are at risk for mathematics difficulties (MDR). Participants were 114 second‐ or third‐grade students with RD from public elementary schools in three large metropolitan areas. Students were classified as at risk for mathematics difficulties utilizing a 25th‐percentile cutoff and a 15th‐percentile cutoff as assessed by the KeyMath‐Revised Test ( Connolly, 1988 ). A series of PA and RAN measures were administered along with a range of reading and mathematics measures. Hierarchical regression analyses indicated that children with RD only evidenced a different pattern of results compared to children with RD + MDR. Additionally, using a more stringent criterion to classify children at risk for mathematics difficulties resulted in a differential pattern of results when compared to a less stringent classification criterion.     

Poor performance on serial visual tasks in persons with reading disabilities: impaired working memory?

The present study investigates the performance of persons with reading disabilities (PRD) on a variety of sequential visual-comparison tasks that have different working-memory requirements. In addition, mediating relationships between the sequential comparison process and attention and memory skills were looked for. Our findings suggest that PRD perform worse than normally achieving readers (NAR) when the task requires more than a minimal amount of working memory, unrelated to presentation rate. We also demonstrate high correlations between performance on the task with the most working-memory demands and reading-related skills, suggesting that poor working-memory abilities may be one of the underlying mechanisms of dyslexia. The mediating model analysis indicates that order judgment tasks are mediating to verbal working memory, suggesting that visual sequence memory precedes auditory sequence memory. We further suggest that visual tasks involving sequential comparisons could probe for poor working memory in PRD.     

Comprehension difficulties after year 4: Actioning appropriately

Abstract

Many children identified with average intelligence exhibit ongoing reading comprehension difficulties after Year 4. They typically have language delays that are possibly due to their disengagement with reading and they do not process information efficiently in working memory. As a consequence they do not adequately learn from their encounters with print and they develop a reluctance to use appropriate self‐monitoring strategies. Many intervention programs have a limited focus by concentrating on word level processes but these may be largely inappropriate for older children with reading comprehension difficulties. What may be more suitable is the use of scaffolding techniques that promote cognitive and motivational engagement in a rich personalised language environment. The article investigates the problems experienced by such children and suggests some appropriate text processing and reorganisation strategies that can be applied to teaching practice.     

Reading,syntactic, and working memory skills of bilingual Portuguese-English Canadian children

The purpose of the study was to examine the nature of language, memory, and reading skills of bilingual students and to determine the relationship between reading problems in English and reading problems in Portuguese. The study assessed the reading, language, and memory skills of 37 bilingual Portuguese-Canadian children, aged 9–12 years. English was their main instructional language and Portuguese was the language spoken at home. All children attended a Heritage Language Program at school where they were taught to read and write Portuguese. The children were administered word and pseudoword reading, language, and working memory tasks in English and Portuguese. The majority of the children (67%) showed at least average proficiency in both languages. The children who had low reading scores in English also had significantly lower scores on the Portuguese tasks. There was a significant relationship between the acquisition of word and pseudoword reading, working memory, and syntactic awareness skills in the two languages. The Portuguese-Canadian children who were normally achieving readers did not differ from a comparison group of monolingual English speaking normally achieving readers except that the bilingual children had significantly lower scores on the English syntactic awareness task. The bilingual reading disabled children had similar scores to the monolingual reading disabled children on word reading and working memory but lower scores on the syntactic awareness task. However, the bilingual reading disabled children had significantlyhigher scores than the monolingual English speaking reading disabled children on the English pseudoword reading test and the English spelling task, perhaps reflecting a positive transfer from the more regular grapheme phoneme conversion rules of Portuguese. In this case, bilingualism does not appear to have negative consequences for the development of reading skills. In both English and Portuguese, reading difficulties appear to be strongly related to deficits in phonological processing.     

Localizing the development of ability differences in organized memory

Although it is well know that learning disabled children tend to perform more poorly than normally achieving children on most memory tasks, the basis of this poor performance remains unclear. Recently, evidence has been accumulating which suggests that disabled children have difficulty with the basic processes of storage and retrieval. The purpose of the present research was to investigate this hypothesis using a recently developed model of memory that provides a factoring procedure for measuring storage and retrieval processes. In particular, we were interested in localizing the source of the development of ability differences in organized recall in early (grade 2) and later (grade 6) elementary school. All of the 600 children who participated had measured IQs in the 97–107 range; however, the disabled students were at least 1 year behind in either reading or arithmetic but not both. All of the children learned a 16-item list that was either unrelated or categorized using either a free or cued recall procedure. The results indicated that while ability differences were present at storage and retrieval, differences tended to be larger at retrieval than at storage, on categorized rather than on uncategorized lists, and in the cued rather than in the free recall conditions. These results illustrate that the ability to execute purposive components of retrieval develops more slowly in disabled than in nondisabled children. Consequently, it is these processes that should be the primary target in remedial memory programs.     

Differences in the Mathematics‐Vocabulary Knowledge of Fifth‐Grade Students With and Without Learning Difficulties

The purpose of this pilot study was to explore the impact of mathematics and reading learning difficulties on the mathematics‐vocabulary understanding of fifth‐grade students. Students (n = 114) completed three measures: mathematics computation, general vocabulary, and mathematics vocabulary. Based on performance on the mathematics computation and general vocabulary measures, students were categorized with no learning difficulty (i.e., typical), mathematics difficulty without reading difficulty (MD‐only), reading difficulty without mathematics difficulty (RD‐only), and combined mathematics and reading difficulties (MDRD). On the mathematics‐vocabulary measure, students with MD‐only or RD‐only scored significantly lower than typical students, and students with MDRD demonstrated significantly lower performance than students with MD‐only or RD‐only.     

Do children with mathematics learning disability in Hong Kong perceive word problems differently?

The current study aimed at identifying the difficulties experienced by children with mathematics learning disability (MLD) in the problem representation phase of arithmetic word problem solving using a novel problem types identification task. An MLD group (n = 66) and a typically achieving control group (n = 139) were recruited for an assessment on problem type identification as well as some domain-general and mathematics-related cognitive abilities. Results from ANCOVA showed that the MLD group scored significantly lower than the typically achieving control group on this assessment, after controlling for the effect of cognitive correlates, reading achievement and arithmetic performance. Furthermore, this assessment significantly predicted MLD membership even after taking children's arithmetic competency into account. The current study confirmed the difficulties in problem representation of arithmetic word problems experienced by students with MLD and provided evidence for the need to introduce schema instructions in mathematics classes.     

Orthographic awareness,phonemic awareness,syntactic processing,and working memory skill in hyperlexic children

Children classified as hyperlexic learn to readwords spontaneously before age five, areimpaired in both reading and listeningcomprehension, and exhibit word recognitionskills above their linguistic and cognitiveabilities. Despite their strong wordrecognition skills, previous studies have shownthat the phonemic awareness skills ofhyperlexic children are low and notcommensurate with their word reading skill, inpart because of their limited comprehension of phonemic awareness tasks. Heretofore, a verylimited number of studies have investigateddirectly the orthographic processing, syntacticprocessing, and working memory skills ofchildren with hyperlexia. In the presentstudy, measures of orthographic processing,syntactic processing, and working memory skillwere administered to three hyperlexic childrenand three normally achieving readers; inaddition, measures of phonemic awareness,academic achievement, and cognitive abilitywere also administered. Results showed thatthe hyperlexic children performed aboveexpectations on the orthographic processingmeasures based on their cognitive andlinguistic abilities. The children withhyperlexia did not exhibit orthographic skillsthat were superior to the normally achievingreaders, although ceiling effects on theorthographic tasks may not have allowed them todemonstrate this skill. The three children withhyperlexia achieved lower scores than thenormally achieving readers on the syntacticprocessing measures and had great difficulty onthe phonemic awareness measures. Only one ofthe three hyperlexic children performed at alevel consistent with that of normallyachieving readers on the working memorymeasures. Findings suggest the hyperlexicchildren had levels of orthographic processingsimilar to that of normally achieving readers,read words using strategies similar to those ofnormal readers, and had phonemic awarenessskill that appears to be adequate for wordanalysis but could not be demonstrated ontraditional phonemic awareness measures.     

Effects of Cognitive Strategy Interventions and Cognitive Moderators on Word Problem Solving in Children at Risk for Problem Solving Difficulties

This study investigated the role of strategy instruction and cognitive abilities on word problem solving accuracy in children with math difficulties (MD). Elementary school children (N = 120) with and without MD were randomly assigned to 1 of 4 conditions: general‐heuristic (e.g., underline question sentence), visual‐schematic presentation (diagrams), general‐heuristic + visual‐schematic, and an untreated control. When compared to the control condition that included children with MD, an advantage at posttest was found for children with MD for the visual‐schematic‐alone condition on measures of problem solving and calculation accuracy, whereas all strategy conditions facilitated posttest performance in correctly identifying problem solving components. The results also suggested that strategy conditions drew upon different cognitive resources. The General‐heuristic condition drew primarily upon the executive component of working memory (WM), Visual‐schematic condition drew upon the visual component of WM and the combined strategies condition drew upon number processing skills.     

我国数学学习困难儿童的教育与干预研究回顾与展望

近年来,我国对数学学习困难儿童的表现、诊断、认知特点等有了较多的研究,这为数学学习困难儿童的干预策略提供了基础。通过对我国数学学习困难儿童的教育干预研究进行综述,发现主要有知觉加工干预策略、工作记忆干预策略、问题表征干预策略和元记忆干预策略,并在借鉴国外数学学习困难研究的基础上,对我国未来数学学习困难的研究提供若干建议。     

A Four Year Follow‐up Study of Low Socioeconomic Status,Latin American Children with Reading Difficulties

This paper presents the results of a follow‐up study of reading difficulties among Spanish‐speaking Latin American children of low socioeconomic background (SES) over four school years. The research started with a group of 93 children with reading difficulties (RD) by 2nd and 3rd grade teachers in four schools and a control group of 63 children without reading difficulties (ND), belonging to the same schools. Both groups were equivalent in age, grade, gender and SES. All children were tested on reading decoding and comprehension, reading and writing pseudo‐words, verbal abilities, visual perception, phonological processing, and the WISC‐R. Neither SES nor IQ accounted significantly for the reading difficulties in the RD group. The most predictive variables of the final reading level in the RD group were phonological processing, verbal abilities, and the initial level of decoding. After four years, 17% of the initial RD children had reached an average level of reading, but 11% of this group remained with severe reading difficulties and may be considered as dyslexic children. The RD children who became average readers were not significantly different from the control group in verbal abilities, nor in phonological processing. Nevertheless, they were significantly different in terms of IQ scores. The neuropsychological characteristics of the children with severe reading difficulties appear similar to those found in children with dyslexia in more developed countries.     

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.     

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.