Robin F. Schumacher

Contributions of domain-general cognitive resources and different forms of arithmetic development to pre-algebraic knowledge.

The purpose of this study was to investigate the contributions of domain-general cognitive resources and different forms of arithmetic development to individual differences in pre-algebraic knowledge. Children (n = 279, mean age = 7.59 years) were assessed on 7 domain-general cognitive resources as well as arithmetic calculations and word problems at start of 2nd grade and on calculations, word problems, and pre-algebraic knowledge at end of 3rd grade. Multilevel path analysis, controlling for instructional effects associated with the sequence of classrooms in which students were nested across Grades 2-3, indicated arithmetic calculations and word problems are foundational to pre-algebraic knowledge. Also, results revealed direct contributions of nonverbal reasoning and oral language to pre-algebraic knowledge, beyond indirect effects that are mediated via arithmetic calculations and word problems. By contrast, attentive behavior, phonological processing, and processing speed contributed to pre-algebraic knowledge only indirectly via arithmetic calculations and word problems.

Cognitive and Mathematical Profiles for Different Forms of Learning Difficulties

The purpose of this study was to compare subgroups of students with various forms of learning difficulties (< 25th percentile) on cognitive and mathematics characteristics. Students with mathematics difficulty (MD, n = 105), reading difficulty (RD, n = 65), both (MDRD, n = 87), or neither (NoLD, n = 403) were evaluated on an array of cognitive measures (e.g., working memory and language) and on mathematics measures of foundational numerical competencies, computation, and problem solving. Results revealed expected level differences among groups in both domains: NoLD outperformed RD, and MD outperformed MDRD. Profile differences were noted among pairs of subgroups on cognitive measures. On mathematics measures, profile differences were noted between RD and other subgroups, but not between MD and MDRD subgroups. The most discriminating cognitive measures were processing speed and language; the most discriminating mathematics measures depended on the subgroups being compared. Results were further evaluated according to more severe (< 10th percentile) criteria for MD and RD, which generally affected level differences more than the profile patterns. Results have implications for understanding comorbid MD and RD and for conceptualizing core deficits in MD.

Inclusion Versus Specialized Intervention for Very-Low-Performing Students What Does Access Mean in an Era of Academic Challenge?

The purpose of this analysis was to examine achievement gaps on fractions for very-low-performing students as a function of whether they receive inclusive fraction instruction or specialized fraction intervention and with the shift to Common Core State Standards (CCSS). In three randomized control trials conducted in 3 consecutive years, 203 students who scored at or below the 10th percentile in mathematics (mean standard score ~75) at the start of fourth grade were randomly assigned at the individual level to 12 weeks of inclusive fraction instruction or specialized fraction intervention. In Year 1, the fourth-grade mathematics curriculum was guided by initial state standards; in Years 2 and 3, the state was transitioning to CCSS. In each of the 3 years on each measure, results indicated significantly stronger learning and markedly smaller post-intervention achievement gaps for specialized fraction intervention than for inclusive fraction instruction. Yet, the size of achievement gaps grew over the years in both conditions, as CCSS increased the depth and challenge of the fraction curriculum and produced differentially stronger learning in not-at-risk classmates. Implications are discussed in terms of the provision of services for students with learning disabilities in the era of CCSS and the meaning of access to the general education curriculum.

Fraction Intervention for Students With Mathematics Difficulties: Lessons Learned From Five Randomized Controlled Trials

In this article, the authors summarize results from 5 randomized controlled trials assessing the effects of intervention to improve the fraction performance of fourth-grade students at risk for difficulty in learning about fractions. The authors begin by explaining the importance of competence with fractions and why an instructional focus on fractions magnitude understanding may improve learning. They then describe an intervention that relies strongly on this type of understanding about fractions instruction, and they provide an overview of the intervention’s overall effects. This is followed by an overview of 5 intervention components for which the authors isolated effects. They conclude by discussing some of the lessons learned from this research program.

Life on the Number Line: Routes to Understanding Fraction Magnitude for Students With Difficulties Learning Mathematics.

Magnitude understanding is critical for students to develop a deep understanding of fractions and more advanced mathematics curriculum. The research reports in this special issue underscore magnitude understanding for fractions and emphasize number lines as both an assessment and an instructional tool. In this commentary, we discuss how number lines broaden the concept of fractions for students who are tied to the more general part–whole representations of area models. We also discuss how number lines, compared to other representations, are a superior and more mathematically correct way to explain fraction concepts.

Error Patterns with Fraction Calculations at Fourth Grade as a Function of Students’ Mathematics Achievement Status

The goal of this study was to describe fraction-calculation errors among fourth-grade students and to determine whether error patterns differed as a function of problem type (addition vs. subtraction; like vs. unlike denominators), orientation (horizontal vs. vertical), or mathematics-achievement status (low-, average-, or high-achieving). We specifically addressed whether mathematics-achievement status was related to students’ tendency to operate with whole-number bias. We extended this focus by comparing low-performing students’ errors in 2 instructional settings that focused on 2 different types of fraction understandings: core instruction that focused on part-whole understanding versus small-group tutoring that focused on magnitude understanding. Results showed students across the sample were more likely to operate with whole-number bias on problems with unlike denominators. Students with low or average achievement (who participated only in core instruction) were more likely to operate with whole-number bias than students with low achievement who participated in small-group tutoring. We suggest instruction should emphasize magnitude understanding to sufficiently increase fraction understanding for all students in the upper elementary grades.