Education policy should aim to promote instructional methods that are easy for teachers to implement and have demonstrable, positive impact on student learning. Our research on comparison and explanation of multiple strategies illustrates the promise of this approach. In several short-term experimental, classroom-based studies, comparing different strategies for solving the same problem was particularly effective for promoting student learning. Thus, we developed a supplemental Algebra 1 curriculum to foster comparison in combination with explanation of multiple strategies. In a randomized control trial, teachers used our materials as intended, but much less often than expected, and student learning was not greater in experimental classrooms. Yet greater use of our comparison materials was associated with greater student learning, suggesting the approach has promise when used sufficiently often. These studies provide some evidence that easy-to-implement reforms can change teacher practice and improve student learning.
Comparison is a fundamental cognitive process that supports learning in a variety of domains. To leverage comparison in mathematics instruction, evidence-based guidelines are needed for how to use comparison effectively. In this chapter, we review our classroom-based research on using comparison to help students learn mathematics. In five short-term experimental, classroom-based studies, we evaluated two types of comparison for supporting the acquisition of mathematics knowledge and tested whether prior knowledge moderated their effectiveness. Comparing different solution methods for solving the same problem was particularly effective for supporting procedural flexibility across students and for supporting conceptual and procedural knowledge among students with some prior knowledge of one of the methods. We next developed a supplemental Algebra 1 curriculum to foster comparison and evaluated its effectiveness in a randomized-control trial. Teachers used our supplemental materials much less often than expected, and student learning was not greater in classrooms that had been assigned to use our materials. Students’ procedural knowledge was positively related to greater implementation of the intervention, suggesting the approach has promise when used sufficiently often. This study suggests that teachers may need additional support in deciding what to compare and when to use comparison.
Comparison is a fundamental cognitive process that can support learning in a variety of domains, including mathematics. The current paper aims to summarize empirical findings that support recommendations on using comparison of multiple strategies in mathematics classrooms. We report the results of our classroom-based research on using comparison of multiple strategies to help students learn mathematics, which includes short-term experimental research and a year-long randomized controlled trial using a researcher-designed supplemental Algebra I curriculum. Findings indicated that comparing different solution methods for solving the same problem was particularly effective for supporting procedural flexibility across students and for supporting conceptual and procedural knowledge among students with some prior knowledge of one of the methods, but that teachers may need additional support in deciding what to compare and when to use comparison. Drawing from this research, we offer instructional recommendations for the effective use of comparison of multiple strategies for improving mathematics learning, including (a) regular and frequent comparison of alternative strategies, particularly after students have developed some fluency with one initial strategy; (b) judicious selection of strategies and problems to compare; (c) carefully-designed visual presentation of the multiple strategies; and (d) use of small group and whole class discussions around the comparison of multiple strategies, focusing particularly on the similarities, differences, affordances, and constraints of the different approaches. We conclude with suggestions for future work on comparing multiple strategies, including the continuing need for the development of, and rigorous evaluation of, curriculum materials and specific instructional techniques that effectively promote comparison.