Facilitating the Transition from Empirical Arguments to Proof

Abstract

Although students of all levels of education face serious difficulties with proof, there is limited research knowledge about how instruction can help students overcome these difficulties. In this article, we discuss the theoretical foundation and implementation of an instructional sequence that aimed to help students begin to realize the limitations of empirical arguments as methods for validating mathematical generalizations and see an intellectual need to learn about secure methods for validation (i.e., proofs). The development of the instructional sequence was part of a 4-year design experiment that we conducted in an undergraduate mathematics course, prerequisite for admission to an elementary (Grades K–6) teaching certification program. We focus on the implementation of the instructional sequence in the last of 5 research cycles of our design experiment to exemplify our theoretical framework (in which cognitive conflict played a major role) and to discuss the promise of the sequence to support the intended learning goals.