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.     

Explicitly Teaching for Transfer: Effects on the Mathematical Problem‐Solving Performance of Students with Mathematics Disabilities

The purpose of this study was to explore methods to enhance mathematical problem solving for students with mathematics disabilities (MD). A small‐group problem‐solving tutoring treatment incorporated explicit instruction on problem‐solution rules and on transfer. The transfer component was designed to increase awareness of the connections between novel and familiar problems by broadening the categories by which students group problems requiring the same solution methods and by prompting students to search novel problems for these broad categories. To create a stringent test of efficacy, we incorporated a computer‐assisted practice condition, which provided students with direct practice on real‐world problem‐solving tasks. We randomly assigned 40 students to problem‐solving tutoring, computer‐assisted practice, problem‐solving tutoring plus computer‐assisted practice, or control, and pre‐ and posttested students on three problem‐solving tasks. On story problems and transfer story problems, tutoring (with or without computer‐assisted practice) effected reliably stronger growth compared to control; effects on real‐world problem solving, although moderate to large, were not statistically significant. Computer‐assisted practice added little value beyond tutoring but, alone, yielded moderate effects on two measures.     

Social support in inclusive middle schools: Perceptions of youth with learning disabilities

This cross‐sectional study examined the perceptions of social support reported by middle‐school students with multiple learning disabilities (LD) (reading + math) in inclusive settings. Comparison groups included youths with a learning disability in reading only (RD), mathematics only (MD), and no LD (i.e., normally achieving) (NA). One hundred twenty middle‐school students, 15 boys and 15 girls in each group, were included in the current study. Participants were surveyed using an established measure of perceived social support. Effect size differences and MANOVA were used in the data analyses. Learning disability type explained 21% ( p < .001) of the variance in perceptions of parent, classmate, and friend support. Students with multiple LD (RD + MD) reported the lowest perceived social support on these dependent variables. Follow‐up analyses revealed that eighth‐grade boys reported the lowest perceived parent support and that boys in general reported lower perceived friend support than girls. Sixth‐grade students with multiple LD reported the lowest perceived friend support, and sixth‐grade males reported the lowest perceived teacher support. Implications for practicing school psychologists and recommendations for future research are discussed. © 2006 Wiley Periodicals, Inc. Psychol Schs 43: 197–209, 2006.     

Arithmetic education and learning disabilities in Italy

In the first part of this article, we briefly present the context of mathematics education in Italy and its specific programs, teaching and assessment methods, use of psychological concepts, and policies in favor of pupils with difficulties. We illustrate the importance and the role of mathematics teachers' associations. In the second part, we focus more directly on the issue of arithmetic learning and learning disabilities in Italy. Despite the absence of a specific strong tradition in the field, new perspectives regarding mathematics education are evident in Italy. The preparation of new assessment tools has proceeded together with an effort toward the identification of subtypes of arithmetic learning disabilities. A basic distinction has been made between difficulties in number processing/calculation and problem-solving tasks. Within the area of calculation, a finer distinction between numerical knowledge, speed, and accuracy seems critical, but a more articulated differentiation of subcomponents relying on modern cognitive-neuropsychological models can be used also.     

A preliminary discourse analysis of constructivist-oriented mathematics instruction for a student with learning disabilities

Reform efforts in mathematics education arose, in part, in response to constructivist works on conceptual learning. However, little research has examined how students with learning disabilities (LD) respond to constructivist-oriented instruction in mathematics, particularly in moment-to-moment interactions. To understand the nature of constructivist-oriented mathematics instruction involving students with LD, the authors conducted a case study to analyze teacher–student interactions during constructivist-oriented small group instruction involving a student with LD. The student demonstrated, to a certain degree, the ability to reason mathematically when provided with appropriate opportunities and prompting. However, given the limited intervention time, his reasoning and problem solving did not seem to go beyond the semiconcrete level of operation, which may have inhibited his solving of complex word problems with large numbers. Findings indicate that more efforts are needed to support students, those with LD in particular, in their transitions from concrete or semiconcrete to abstract conceptual understanding and problem solving.     

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.     

The relative difficulty of signed arithmetic story problems for primary level deaf and hard-of-hearing students

This study determines the relative difficulty and associated strategy use of arithmetic (addition and subtraction) story problems when presented in American Sign Language to primary level (K-3) deaf and hard-of-hearing students. Results showed that deaf and hard-of-hearing students may consider and respond to arithmetic story problems differently than their hearing peers, with the critical dimension in problem difficulty being based on the operation typically used to solve the problem, not the story within the problem. The types of strategies used by the students supported the order of problem difficulty. The visual-spatial nature of the problem presentation appeared not to assist the deaf and hard-of-hearing students in solving the problems. Factors that may have contributed to this pattern of problem difficulty are discussed so that educators can better align mathematics instruction to the thinking of the deaf child.     

Cognitive Arithmetic Differences in Learning Difficulty Groups and the Role of Behavioral Inattention

The objectives of this study were to (1) evaluate group differences in specific arithmetic competencies among students with various types of learning difficulties (LD) and (2) to examine the influence of attention behaviors on possible group differences. Participants were a community sample of 291 third‐ and fourth‐grade students with reading difficulties (RD) and/or math difficulties (MD), or with no LD (51 MD, 66 RD, 89 MD + RD, and 85 No LD students). Students were administered computerized measures of cognitive addition, subtraction, and estimation (accuracy and response times), as well as other measures. Groups were compared with and without covariation for behavioral inattention and their interactions. Small sums addition, thought to reflect retrieval processes, estimation accuracy, and number knowledge, did not show differences between MD and MD + RD students, although both showed lower performance than the RD and No LD groups. Attention behavior had a variable impact, most typically making group differences larger, but did not alter the general pattern of group differences, except in the case of estimation.     

Examining calculator use among students with and without disabilities educated with different mathematical curricula

This study assessed if students with and without disabilities used calculators (fourfunction, scientific, or graphing) to solve mathematics assessment problems and whether using calculators improved their performance. Participants were sixth and seventh-grade students educated with either National Science Foundation (NSF)-funded or traditional mathematics curriculum materials. Students solved multiple choice and open-ended problems based on items from the State’s released previous assessments. A linear mixed model was conducted for each grade to analyze the factors impacting students’ self-reported calculator use. Chi Square tests were also performed on both grade’s data to determine the relationship between using a calculator and correctly solving problems. Results suggested only time as a main factor impacting calculator use and students who self-reported using a calculator were more likely to answer questions correctly. The results have implications for practice given the controversy over calculator use by students both with and without disabilities.     

Cognitive Strategy Instruction for Mathematical Word Problem-solving of Students with Mathematics Disabilities in China

This study investigated the impact of cognitive strategy instruction (CSI) on mathematical word problem solving of students with mathematics disabilities. A sample of fourth-grade students in a Chinese primary school was divided into a treatment group (75 students) and a comparison group (75 students). The sample consisted of students with mathematics disabilities only, students with both mathematics and reading disabilities, as well as average- and high-achieving students. Results showed that students at all ability levels (except high-achieving students) in the treatment group outperformed significantly their counterparts in the comparison group; the intervention effect was stronger for students with mathematics disabilities only than for those with both mathematics and reading disabilities. The present study indicates that CSI is a contextually and pedagogically appropriate model that has a strong potential to improve mathematical word problem solving.     

The mathematics skills of children with reading difficulties

Although many children with reading difficulty (RD) are reported to struggle with mathematics, little research has empirically investigated whether this is the case for different types of RD. This study examined the mathematics skills of third graders with one of two types of RD: dyslexia (n = 18) or specific reading comprehension difficulty (n = 22), as contrasted to a comparison group (n = 247). Children's performance on arithmetic fact fluency, operations, and applied problems was assessed using standardized measures. The results indicated that children with dyslexia experienced particular difficulty with arithmetic fact fluency and operations: they were 5.60 times and 8.54 times more likely than other children to experience deficits in fact fluency and operations, respectively. Our findings related to arithmetic fact fluency were more consistent with domain-general explanations of the co-morbidity between RD and mathematics difficulty, whereas our findings related to operations were more consistent with domain-specific accounts.     

生活化教学法在培智数学教学中的应用

在培智数学教学中 ,笔者应用生活化教学法进行以下五方面内容的教学 :用生活化教学法进行式题、应用题、图形表象的教学 ;对教材中本来就生活化的内容进行教学 ;用生活化教学方法进行主题教学 ,使数学贴近弱智学生的生活实际 ,从而提高数学教学的效果。     

Mathematics anxiety and working memory: Longitudinal associations with mathematical performance in Chinese children

The link between mathematics anxiety and mathematical performance in young children remains inconclusive. The present study examined the longitudinal associations between mathematics anxiety and mathematical performance (calculation and story problem solving) in 246 Chinese children followed from second to third grade. Multiple regression analyses showed that mathematics anxiety made independent contributions to mathematical performance beyond non-verbal intelligence, working memory, number skills, general and test anxieties. However, mathematics anxiety does not affect all children and all kinds of mathematical performance equally. Mathematics anxiety has a more pronounced impact on mathematical problems that require more processing resources, as opposed to simple arithmetic problems and straightforward story problems and children who are higher in working memory are more vulnerable to its deleterious impacts.     

The Effects of Paraphrasing Interventions on Problem‐Solving Accuracy for Children at Risk for Math Disabilities

The purpose of this study was to examine effectiveness of paraphrasing interventions on mathematics word problem‐solving accuracy in third grade children (N = 72) at risk for mathematics disabilities (MD). Three instructional conditions directed students’ attention through paraphrasing, via writing, to different propositions within word problems. Students were randomly assigned to one of four intervention conditions: paraphrase question propositions (restate), paraphrase relevant propositions (relevant) and paraphrase all propositions (complete) or an untreated control. A mixed ANCOVA indicated that paraphrasing relevant and complete propositions significantly increased posttest accuracy when compared to the control and restate condition. Results from the study provide support for the effectiveness of paraphrasing interventions that directs students to restate/paraphrase propositions of mathematics word problems relative to the control condition.     

Approach to mathematical problem solving and students’ belief systems: two case studies

The goal of the study reported here is to gain a better understanding of the role of belief systems in the approach phase to mathematical problem solving. Two students of high academic performance were selected based on a previous exploratory study of 61 students 12–13 years old. In this study we identified different types of approaches to problems that determine the behavior of students in the problem-solving process. The research found two aspects that explain the students’ approaches to problem solving: (1) the presence of a dualistic belief system originating in the student’s school experience; and (2) motivation linked to beliefs regarding the difficulty of the task. Our results indicate that there is a complex relationship between students’ belief systems and approaches to problem solving, if we consider a wide variety of beliefs about the nature of mathematics and problem solving and motivational beliefs, but that it is not possible to establish relationships of causality between specific beliefs and problem-solving activity (or vice versa).     

Piagetian cognitive functioning in students with learning disabilities

This study investigated the cognitive abilities as measured by the Inventory of Piaget's Developmental Tasks (IPDT) (Furth, 1970) of male and female fourth and fifth graders with learning disabilities. In addition, their cognitive abilities were compared to their reading and mathematics achievement, as measured by the Peabody Individual Achievement Test (PIAT) (Dunn & Markwardt, 1970). A two-way analysis of covariance revealed a significant mean difference in cognitive ability between students with learning disabilities and nondisabled students. No significant differences existed in performance between males and females. Multiple regression showed that a linear combination of reading and mathematics achievement yielded a significant positive correlation to cognitive ability.     

An examination of the reading performance of students with mild educational handicaps or learning disabilities

Reading performance is a primary concern in both regular and special education. Reading is also the dominant medium through which educators conduct lessons, or students acquire information in subjects such as science and mathematics. Reading performance can be measured in a variety of ways. The present study examined the performance of students with mild educational handicaps (n = 28) or learning disabilities (n = 38) on a combination of norm-referenced and curriculum-based assessment (CBA) approaches that used science content. Overall, there were no significant differences between the samples on measures of word recognition, but significant differences favoring students with learning disabilities did occur in comprehension.     

Strategic development of multiplication problem solving: Patterns of students' strategy choices

Low mathematics achievement is a persistent problem in the United States, and multiplication is a fundamental area in which many students manifest learning difficulties. This study examined the strategic developmental levels of multiplication problem solving among 121 elementary school students in Grades 3 through 5. A latent class analysis modeling was used to identify three valid groups representing different patterns of strategy choices for each of three types of multiplication problems. Findings indicated intra-group variability for problem-solving accuracy, for frequency of using different strategies, and for accuracy of executing direct retrieval/algorithm (DR/AG) strategies. Students demonstrated relative consistency in their strategy choices for solving the three problem types. Students who used DR/AG strategies most frequently showed the highest problem-solving accuracy and the highest accuracy of executing the DR/AG strategies. Students who most frequently relied on incorrect operations or who indicated they did not know how to solve problems demonstrated the lowest problem-solving accuracy among the three groups; the number of students in this group increased with problem difficulty levels. Implications are discussed in terms of identifying students' strategic developmental levels and providing differentiated instruction based on the identified levels.     

A further study of productive failure in mathematical problem solving: unpacking the design components

This paper replicates and extends my earlier work on productive failure in mathematical problem solving (Kapur, doi:, 2009). One hundred and nine, seventh-grade mathematics students taught by the same teacher from a Singapore school experienced one of three learning designs: (a) traditional lecture and practice (LP), (b) productive failure (PF), where they solved complex problems in small groups without any instructional facilitation up until a teacher-led consolidation, or (c) facilitated complex problem solving (FCPS), which was the same as the PF condition except that students received instructional facilitation throughout their lessons. Despite seemingly failing in their collective and individual problem-solving efforts, PF students significantly outperformed their counterparts in the other two conditions on both the well-structured and higher-order application problems on the post-test, and demonstrated greater representation flexibility in working with graphical representations. The differences between the FCPS and LP conditions did not reach significance. Findings and implications of productive failure for theory, design of learning, and future research are discussed.     

Empirical testing of a cognitive model to account for neuropsychological functioning underlying arithmetic problem solving

The efficacy of a cognitive-based arithmetic problem-solving model (Dinnel, Glover, & Halpain, in press; Dinnel, Glover, & Ronning, 1984) was tested using 989 students with learning disabilities. Comprehensive neuropsychological test battery information was used to predict composite arithmetic test performance as a means of examining the utility of this model. Results of this study offer support for Dinnel et al.'s (Dinnel, Glover, & Halpain, in press; Dinnel, Glover, & Ronning, 1984) model in accounting for arithmetic performance under continuous visual stimulus conditions. However, these data indicate a more complex neuropsychological underpinning to arithmetic performance in both visual and aural stimulus conditions. The neuropsychological aspects of arithmetic problem solving were discussed in relationship to this cognitive-based model.