Project Overview
Using Contrasting Examples to Support Procedural Flexibility and Conceptual Understanding in Mathematics
Dr. Jon R. Star, Harvard University
Dr. Bethany RittleJohnson, Vanderbilt University
Dr. Kristie J. Newton, Temple University
Funding provided by the Institute for Education Sciences and the National Science Foundation
Recent reform efforts in education are motivated by endemic problems with students gaining only inert knowledge  rigid, inflexible knowledge that is not accessed or transferred to solve novel problems. As both international and national assessments indicate, mathematics is one of the most critical domains for overcoming the inert knowledge problem. Too few mathematics students have the ability to flexibly solve novel problems; such flexible problem solving requires that students integrate their conceptual knowledge of principles in the domain with their procedural knowledge of specific actions for solving problems. Current "best practices" in mathematics education seek to promote the development of flexible knowledge through the use of classroom discussions, where students share procedures and evaluate the procedures of others. Despite the intuitive appeal of such educational approaches, the psychological literature is inconclusive about the benefits of this type of reform pedagogy. In this project, we rigorously evaluate a potentially pivotal component of this instructional approach that is supported by basic research in cognitive science: the value of students explaining contrasting examples.
We compare learning from contrasting examples to learning from sequentially presented examples (a more common educational approach) in several studies. Fifth and seventhgrade students will serve as participants in the research, which occurs in the mathematical domains of algebra equation solving and computational estimation. In all studies, students study workedout examples of mathematics problems and answer questions about the examples. Treatment students are shown a pair of worked examples illustrating different solutions to the same problem and be asked to compare and contrast the solution procedures. Control group students work with the same examples, but will be shown each worked example separately and be asked to think about the individual solutions. These studies together explore whether explaining contrasting examples improves students' problemsolving ability, procedural flexibility, and conceptual understanding  key goals in mathematics education.

For more information, contact:
compare@gse.harvard.edu
Participating institutions:
Harvard Graduate School of Education
Temple University
Vanderbilt University
Funded by:
National Science Foundation
Institute of Education Sciences
