## Using Contrasting Examples to Support Procedural Flexibility and Conceptual Understanding in Mathematics

### Dr. Jon R. Star, Harvard University

Dr. Bethany Rittle-Johnson, 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 seventh-grade 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 worked-out 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' problem-solving ability, procedural flexibility, and conceptual understanding - key goals in mathematics education.