The evolution of students' understanding of mathematical induction: A teaching experiment.

摘要

This dissertation examines how students' understandings of proof by mathematical induction evolved during an 8-week teaching experiment. The design of the experiment was informed by a theoretical perspective that is a synthesis of two complementary theories: the Theory of Didactical Situations (Brousseau, 1997) and the Necessity Principle, Harel's (1998) theory of intellectual need.; This study provides an account of how the proof schemes and ways of understanding of a cohort of students progressed through three stages: pre-transformational, restrictive transformational, and transformational, as they worked through a series of proof by mathematical induction appropriate tasks. It also reports on the various didactical and epistemological obstacles the students encountered at each stage.; Harel's (1998) Dual Assertion and Harel and Sowder's (1998) proof schemes are used to explain the students' ways of acting in terms of two coexisting schemes, the students' ways of thinking and ways of understanding. The results of the study indicate that the students' conceptions of what constitutes a convincing argument changed in response to a series of shifts in the students' understandings of generality.