The Development of Mathematical Induction as a Proof Scheme - A Model for DNR-Based Instruction

# The Development of Mathematical Induction as a Proof Scheme - A Model for DNR-Based Instruction

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1 The Development of Mathematical Induction as a Proof Scheme: A Model for DNR-Based Instruction 1, 2, 3 Guershon Harel University of California, San Diego harel@math.ucsd.edu Appeared: Harel, G. (2001). The Development of Mathematical Induction as a Proof Scheme: A Model for DNR-Based Instruction. In S. Campbell & R. Zaskis (Eds.). Learning and Teaching Number Theory, Journal of Mathematical Behavior. New Jersey, Ablex Publishing Corporation (pp. 185-212).

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3 The Development of Mathematical Induction as a Proof Scheme: A Model for DNR-Based Instruction Abstract Students’ conceptions of mathematical induction in a standard teaching approach were found to be largely manifestations of deficient proof schemes, such as the authoritative and the symbolic non-quantitative proof schemes. On the other hand, when a fundamentally different instructional treatment of mathematical induction was implemented in an elementary number theory course taught to prospective secondary teachers, students’ conception of mathematical induction developed as a transformational proof scheme—a mathematically mature way of thinking . The alternative instructional treatment was guided by a system of learning-teaching principles, called the DNR system, which was developed and in turn implemented in a sequence of teaching experiments on the concept of mathematical proof.

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4 Introduction Undergraduate programs in mathematics usually do not include a course in number theory, but students are introduced to elementary number theory concepts in other courses, most often in a discrete mathematics course, which usually includes the concepts of mathematical induction (MI). MI is a prominent proof technique in discrete mathematics and number theory, where it is used to prove theorems involving properties of the set of natural numbers. Fermat, the founder of number theory, used a form of mathematical induction to prove many of his discoveries in this field (Boyer, 1968). Beyond its significance as a proof technique in mathematics, MI can provide a context to enhance students’ conception of proof, as we will see later in this paper. “There are unresolved problems concerning the teaching of MI which should benefit from a careful analysis” asserts Ernest (1984, p. 173) and adds that “there is … no systematic account in print of the teaching of MI, of the problems that arise, of the deeper issues involved or of the treatments given by textbooks” (p. 174). With the exception of Dubinsky’s (1986, 1989) work, little has been done in research on the learning and teaching of MI during the last two decades. One goal of this paper is to revive interest in this area. The research reported here is part of the PUPA project. 4 The general questions addressed in PUPA revolved around the development of college students’ p roof u nderstanding, p roduction, and a ppreciation: What are students’ (particularly mathematics major students') conceptions of proof? What sorts of experiences seem effective in shaping students’ conception of proof? Are there promising frameworks for teaching the concept of proof so that students appreciate the value of justifying, the role of proof as a convincing argument, the need for rigor, and the
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## The Development of Mathematical Induction as a Proof Scheme - A Model for DNR-Based Instruction

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