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2006Research Conference | June 15–16

This conference highlighted the work of invited speakers, independent researchers who have received grant funds from the Institute of Education Sciences, and trainees supported through predoctoral training grants and postdoctoral fellowships. The presentations are those of the authors and do not necessarily represent the views of the U.S. Department of Education or the Institute of Education Sciences.
Hyatt Regency Washington on Capitol Hill
400 New Jersey Avenue, N.W.
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Does Using Contrasting Cases Increase Problem Solving, Flexibility and Understanding? An Experimental Study on Early Algebra Learning

Bethany Rittle-Johnson, Vanderbilt University
Jon Star, Michigan State University
Howard Glasser, Michigan State University
Kosze Lee, Michigan State University

Abstract: Encouraging students to share and compare solution methods is a key component of reform efforts in mathematics (e.g. NCTM, 2000), but experimental studies that more conclusively demonstrate the benefits of sharing and comparing ideas for student learning are largely absent. In this study, we experimentally evaluated a potentially pivotal component of this instructional approach that is supported by basic research in cognitive science: the value of students comparing multiple solution methods. Seventy seventh-grade students were randomly assigned to learn about early algebra by either 1) comparing and contrasting alternative solution methods (i.e. contrasting cases) or 2) reflecting on the same solution methods one at a time. All students worked with a partner in their regular mathematics classrooms to study and explain worked examples to algebra equations such as 2(x - 3) = 8. Students in the compare group were more accurate and more flexible in their problem solving, and showed comparable gains in conceptual knowledge. In particular, comparison seemed to facilitate attention to and adoption of non-conventional methods.

These findings provide direct empirical support for a common component of reform mathematics teaching. Simple exposure to multiple ways does not maximize learning, underscoring concerns that some teachers' attempts to implement reform pedagogy have resulted in simple "show-and-tell" of student methods without discussion or comparison of the methods. Future research will extend this work to a new task (computational estimation) and to use of teacher-led discussion. Overall, it seems to pay to compare.