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


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 five studies. Fifth- and seventh-grade students serve as participants in the research.

In Studies 1 and 2, pairs of students are randomly assigned to condition and the manipulation will occur while student pairs study worked examples and solve practice problems in their mathematics classrooms. In Studies 3 and 4, classrooms will be randomly assigned to condition, and the manipulation will occur both in partner activities and in whole-class discussions. In Study 5, the classroom intervention will be "scaled-up" to more diverse classrooms in public schools as a first step towards assessing the generalizability of this teaching approach.

Studies 1, 3, and 5 will be on linear equation solving with seventh-grade students, and Studies 2 and 4 will be on mental math and computational estimation with fifth-grade students. In all studies, students will study worked-out examples of mathematics problems and answer questions about the examples. Treatment students will be shown a pair of worked examples illustrating different solutions to the same problem and be asked to compare and contrast the solution procedures. Comparison group students will work with the same examples, but will be shown each worked example separately and asked to think about the individual solutions. These studies will reveal whether explaining contrasting examples improves students' problem-solving ability, procedural flexibility, and conceptual understanding, keys goal in mathematics education.