Shirali, Shailesh
(2020)
How To Prove It.
At Right Angles (7).
pp. 4852.
ISSN 25821873
Abstract
The topic of ‘proof by induction’ is now a standard part of the syllabus of mathematics at the 1112 level. Most students consider it a ‘scoring topic’ – they generally master the mechanics of proof by induction quickly, as the proofs follow a standard trajectory and are easy to mimic.
My experience as a mathematics teacher, however, suggests that the vast majority of students do not grasp what proof by induction is all about. While they are able to mimic all the required steps, most of them do not grasp the essential logic of such proofs. Indeed, to a good many students, these proofs give the impression of circular reasoning! For someone steeped in the culture of mathematics, it is not easy to understand why students find it so difficult to grasp the essence of such proofs. Is it because the topic is taught in haste, with insufficient time spent on the subtleties involved (and there surely are many subtleties involved)? Or is it because proof itself is inherently a difficult topic? I suppose that a great deal more research is needed to understand the core of the difficulty. It would be well worth it for teachers themselves to undertake such research, rather than wait for experts to take up the task.
In this and the following episode of How to Prove It, we shall dwell on some critical aspects of induction proofs (aspects which possibly are not emphasised strongly enough) and study a few examples that show how valuable and versatile it is as a proof technique.
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