Papers by Robin Schumacher
The Elementary School Journal, 2017
The goal of the present study was to describe fraction-calculation errors among 4 th-grade studen... more The goal of the present study was to describe fraction-calculation errors among 4 th-grade students and determine whether error patterns differed as a function of problem type (addition vs. subtraction; like vs. unlike denominators), orientation (horizontal vs. vertical), or mathematicsachievement status (low-vs. average-vs. 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 two instructional settings that focused on two different types of fraction understandings: core instruction that focused on part-whole understanding vs. 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 only participated 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.
Journal of learning disabilities
In this article, the authors summarize results from 5 randomized controlled trials assessing the ... more 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.
Developmental Psychology, 2012
The purpose of this study was to investigate the contributions of domain-general cognitive resour... more 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 ϭ 279, mean age ϭ 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.
The Elementary School Journal, 2016
Journal of Educational Psychology, May 1, 2014
This study investigated whether individual differences in working memory (WM) moderate effects of... more This study investigated whether individual differences in working memory (WM) moderate effects of 2 variations of intervention designed to improve at-risk 4th graders' fraction knowledge. We also examined the effects of each intervention condition against a business-as-usual control group and assessed whether children's measurement interpretation of fractions mediated those effects. At-risk students (n ϭ 243) were randomly assigned to control and 2 intervention conditions. The interventions each lasted 12 weeks, with three 30-min sessions per week. The major focus of both intervention conditions was the measurement interpretation of fractions. Across the 2 conditions, only 5 min of each 30-min session differed. One condition completed activities to build fluency with 4 measurement interpretation topics; in the other, activities were completed to consolidate understanding on the same 4 topics. Results revealed a significant aptitude-treatment interaction, in which students with very weak WM learned better with conceptual activities but children with more adequate (but still low) WM learned better with fluency activities. Both intervention conditions outperformed the control group on all outcomes, and improvement in the measurement interpretation of fractions mediated those effects.
Journal of Educational Psychology, 2015
Exceptional children
The purpose of this analysis was to examine achievement gaps on fractions for very-low-performing... more 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 similar to 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 ach...
The Elementary School Journal, 2010
The purpose of this study was to assess the effects of schema-broadening instruction (SBI) on sec... more The purpose of this study was to assess the effects of schema-broadening instruction (SBI) on second graders' word-problem-solving skills and their ability to represent the structure of word problems using algebraic equations. Teachers (n = 18) were randomly assigned to conventional word-problem instruction or SBI word-problem instruction, which taught students to represent the structural, defining features of word problems with overarching equations. Intervention lasted 16 weeks. We pretested and posttested 270 students on measures of word-problem skill; analyses that accounted for the nested structure of the data indicated superior word-problem learning for SBI students. Descriptive analyses of students' word-problem work indicated that SBI helped students represent the structure of word problems with algebraic equations, suggesting that SBI promoted this aspect of students' emerging algebraic reasoning.
Journal of Learning Disabilities, 2013
The purpose of this study was to compare subgroups of students with various forms of learning dif... more 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.
Journal of Experimental Child Psychology, 2012
The purpose of this study was to assess whether understanding relational terminology (i.e., more,... more The purpose of this study was to assess whether understanding relational terminology (i.e., more, less, and fewer) mediates the effects of intervention on compare word problems. Second-grade classrooms (N = 31) were randomly assigned to one of three conditions: researcher-designed word-problem intervention, researcher-designed calculation intervention, or business-as-usual (teacher-designed) control. Students in word-problem intervention classrooms received instruction on the compare problem type, which included a focus on understanding relational terminology within compare word problems. Analyses, which accounted for variance associated with classroom clustering, indicated that (a) compared with the calculation intervention and business-as-usual conditions, word-problem intervention significantly increased performance on all three subtypes of compare problems and on understanding relational terminology, and (b) the intervention effect was fully mediated by students' understanding of relational terminology for one subtype of compare problems and partially mediated by students' understanding of relational terminology for the other two subtypes.
Journal of Educational Psychology, 2014
This study investigated whether individual differences in working memory (WM) moderate effects of... more This study investigated whether individual differences in working memory (WM) moderate effects of 2 variations of intervention designed to improve at-risk 4th graders' fraction knowledge. We also examined the effects of each intervention condition against a business-as-usual control group and assessed whether children's measurement interpretation of fractions mediated those effects. At-risk students (n ϭ 243) were randomly assigned to control and 2 intervention conditions. The interventions each lasted 12 weeks, with three 30-min sessions per week. The major focus of both intervention conditions was the measurement interpretation of fractions. Across the 2 conditions, only 5 min of each 30-min session differed. One condition completed activities to build fluency with 4 measurement interpretation topics; in the other, activities were completed to consolidate understanding on the same 4 topics. Results revealed a significant aptitude-treatment interaction, in which students with very weak WM learned better with conceptual activities but children with more adequate (but still low) WM learned better with fluency activities. Both intervention conditions outperformed the control group on all outcomes, and improvement in the measurement interpretation of fractions mediated those effects.
Journal of Educational Psychology, 2013
The purposes of this study were to investigate the effects of an intervention designed to improve... more The purposes of this study were to investigate the effects of an intervention designed to improve at-risk 4th graders' understanding of fractions and to examine the processes by which effects occurred. The intervention focused more on the measurement interpretation of fractions; the control condition focused more on the part-whole interpretation of fractions and on procedures. Intervention was also designed to compensate for at-risk students' limitations in the domain-general abilities associated with fraction learning. At-risk students (n ϭ 259) were randomly assigned to intervention and control. Whole-number calculation skill, domaingeneral abilities (working memory, attentive behavior, processing speed, listening comprehension), and fraction proficiency were pretested. Intervention occurred for 12 weeks, 3 times per week, 30 min per session, and then fraction performance was reassessed. On each conceptual and procedural fraction outcome, effects favored intervention over control (effect sizes ϭ 0.29 to 2.50), and the gap between at-risk and low-risk students narrowed for the intervention group but not the control group. Improvement in the accuracy of children's measurement interpretation of fractions mediated intervention effects. Also, intervention effects were moderated by domain-general abilities, but not whole-number calculation skill.
Developmental Psychology, 2012
The purpose of this study was to investigate the contributions of domain-general cognitive resour... more 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.
The Elementary School Journal
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Papers by Robin Schumacher