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Mathematical induction: cognitive and instructional considerations. (English)
Carlson, Marilyn P. (ed.) et al., Making the connection. Research and teaching in undergraduate mathematics education. Washington, DC: The Mathematical Association of America (MAA) (ISBN 978-0-88385-183-8/pbk). MAA Notes 73, 111-123 (2008).
From the text: The principle of mathematical induction (MI) is a prominent proof technique used to justify theorems involving properties of the set of natural numbers. The principle can be stated in different, yet equivalent, versions. The following are two versions common in textbooks: { indent=3cm \item {Version 1:} Let $S$ be a subset of $\Bbb N$ (the set of natural numbers). If the following two properties hold, then $S=\Bbb N$. } { indent=3.5cm\item{(i)}$l\in S.$ \item{(ii)}$k\in S,\text{ then }k+1\in S.$ } { indent=3cm \item{Version 2:} Suppose we have a sequence of mathematical statements $P(1),P(2),\dots$ (one for each natural number). If the following two properties hold, then for every $n\in\Bbb N$, $P(n)$ is true. } { indent=3.5cm\item{(i)} $P(1)\text{ is true.}$\item{(ii)}$\text{If }P(k)\text{ is true, then }P(k+1)\text{ is true.}$ } In this chapter, we explore students’ difficulties with MI when taught with the standard instructional treatment and we present results from our teaching experiments, which employed alternative instructional approaches. We begin by introducing a construct central to our work, the notion of a proof scheme. We then describe the standard instructional treatment of MI, pointing to its possible inadequacies. Drawing from our and others’ research on students’ difficulties with MI when taught with the standard instructional treatment, we demonstrate how many of these difficulties are indicative of students’ deficient proof schemes. Having described the standard instructional treatment and the related student difficulties, we proceed with an account of two independent, yet related, studies, [{\it G. Harel}, The development of mathematical induction as a proof scheme: a model for DNR-based instruction. Westport, CT: Ablex Publishing (2001)] and [{\it S. A. Brown}, The evolution of students’ understanding of mathematical induction: a teaching experiment. San Diego: University of California and San Diego State University (Diss.) (2003)]. After which, we present a synthesis of our results in the form of a three-stage model of students’ development of MI. We conclude with a summary and instructional recommendations.
Classification: E50 D70