| 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 compare and evaluate the procedures of others. Despite the increasingly wide adoption of reform pedagogy and its intuitive appeal, most research evaluating the effectiveness of this approach has been descriptive, with few controlled empirical evaluation studies. I am in the final phase of a large federally-funded project that rigorously explores the role of comparison in mathematics learning. In several controlled, experimental studies, conducted in elementary and middle school mathematics classrooms, my colleagues and I have demonstrated that students who learned by comparing and contrasting alternative solution methods made greater gains in flexibility (as well as conceptual and procedural knowledge) than those who studied the same solution methods one at a time. In my talk, I will be speaking about my research in the area of algebra, with particular emphasis on (a) what I mean by flexible problem-solving, (b) what I mean by comparison, and (c) what teachers can do to make use of comparison to improve students learning of mathematics. |