Thirty-four matched pairs of sixth-and seventh-grade students were selected from 358 participants in a comparison of an explicit concreteto-representational-to-abstract (CRA) sequence of instruction with traditional instruction for teaching algebraic transformation equations. Each pair of students had been previously labeled with a specific learning disability or as at risk for difficulties in algebra. Students were matched according to achievement score, age, pretest score, and class performance. The same math teacher taught both members of each matched pair, but in different classes. All students were taught in inclusive settings under the instruction of a middle school mathematics teacher. Results indicated that students who learned how to solve algebra transformation equations through CRA outperformed peers receiving traditional instruction on both postinstruction and follow-up tests. Additionally, error pattern analysis indicated that students who used the CRA sequence of instruction performed fewer procedural errors when solving for variables.Abstract thinking requires a person to work with information that is not readily represented at the concrete or pictorial level (Hawker & Cowley, 1997). To work with abstract information is to understand theoretical properties and think beyond what a person can touch or see. On a practical level, an ability to work with abstract concepts allows one to work with predictions of what may happen and expectations about what is happening elsewhere. Abstract knowledge may also be considered a conscious awareness that a symbol stands for some Requests for reprints should be sent to Bradley S. Witzel, Ph.D., Winthrop University, Richard Riley College of Education, Department of Curriculum and Instruction, Rock Hill, SC 29733. Electronic inquiries may be sent to witzelb@winthrop.edu.

Students with learning disabilities struggle to acquire essential mathematical concepts and skills, especially at the secondary level. One effective approach to improving secondary math performance supported by research is the concrete-to-representational-to-abstract (CRA) sequence of instruction. Although CRA is an evidenced-based instructional practice, it is somewhat complicated for teachers to implement in their day-to-day instruction.To better assist implementation of the CRA, an instructional approach with the mnemonic CRA-MATH was developed as a flexible roadmap to help math teachers better address student deficits using the CRA sequence of instruction. The purpose of this article is to describe the seven-step strategy used to guide curricular CRA modifications in secondary mathematics.

FIGURE 2. Proposed motivational model. Note. Teacher response is the same as that in Figure 1.

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Since 1975, researchers have conducted interventions to improve the word-problem performance of elementary school students facing mathematics difficulties. The current study reports a meta-analysis of 52 studies that examined the effect of these interventions. We estimated multivariate, random-effects models (REM) with robust variance estimation (RVE) with and without outliers. Results showed a large, positive, and significant mean weighted effect size ( g = 1.01 for the model with outliers; g = 0.81 for the model without outliers). Findings of meta-regression analyses showed several moderators, such as sample composition, group size, intervention dosage, group assignment approach, interventionist, year of publication, and dependent measure type, significantly explained heterogeneity in effects across studies. A sensitivity analysis showed these results were generally robust to outliers. We offer possible explanations for the findings and discuss study limitations. Finally, we propose recommendations for future research and classroom practice.

We quantitatively synthesized findings of quasi‐experimental and randomized control studies of interventions designed to improve the mathematics achievement of secondary school students with mathematics difficulties (MD). We identified 45 studies (49 interventions) published between 1978 and 2020 and classified interventions into five categories: technology‐based interventions (TBI), schema‐based interventions (SBI), use of visual representations, cognitive‐based instruction, and other. We used robust variance estimation (RVE) to address dependency in effect sizes (ESs). The cumulative effect size across 139 outcomes was moderately large (g = 0.52), with a substantial amount of between‐study heterogeneity. A multiple meta‐regression analysis showed several significant moderators of interventions' effectiveness, such as content domain, intervention length, and dependent measure. We discuss limitations and implications for research and classroom practice.

Making sense of fractions is critical for building the mathematical competence of upper elementary students with and at-risk for a mathematics learning disability. The primary aim of this study was to investigate the effectiveness of teaching students with and at-risk for learning disabilities an intervention in which they learned to construct written arguments to develop their fraction knowledge. We sequenced the intervention using the 6 stages of Self-Regulated Strategy Development (SRSD). The effects were tested using a pre-posttest cluster-randomized controlled trial in which 10 teachers were randomly assigned to treatment and control conditions. Measures included a far-learning fraction test and proximal writing measures that assessed quality of mathematical reasoning, total number of rhetorical elements, and total words written. Teachers in the treatment condition received 2 days of professional development before implementing the six-lesson intervention, 3-4 times per week for 30 -45 min. Clustering effects were corrected to examine differences in pretest-to-posttest change scores using individual scores for students in the SRSD group (n ϭ 28) and control (n ϭ 31). Results favored students in the SRSD condition from pre-to posttest on fraction test (g ϭ 0.60); quality of mathematical reasoning (g ϭ 1.82); number of rhetorical elements (g ϭ 3.20), and total words written (g ϭ 1.92). Special education students in the SRSD condition (n ϭ 16) demonstrated greater gains in fraction scores from pretest to posttest compared to their nondisabled peers (n ϭ 12, g ϭ 1.04). The findings from this study support the genre knowledge hypothesis of writing-to-learn. Educational Impact and Implications StatementThe authors examined an explicit strategies instructional approach in which argumentative writing was used as a writing-to-learn activity to teach fractions. Children with and at-risk for a learning disability in 4th, 5th, and 6th grades were taught two learning strategies for constructing an argumentative paragraph in which they developed their fraction knowledge, addressed mathematical misconceptions about fractions, and worked individually and together with peers to develop their fraction and genre knowledge. The results show argumentative writing can be used as a learning activity for improving students' fraction learning, as well as the quality of their mathematical reasoning.

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