Inspiring Exercises for Undergraduates
نویسنده
چکیده
There are two ways to learn mathematics: to discover (research) and to rediscover. Inspiring exercises are used to guide students at all levels to rediscover the essential meaning of various individual pieces of mathematics. Their r^ ole has become all the more important at a time when Bourbakisation is the dominating fashion. While mathematics is decorated with a thick layer of cosmetics, such as utmost generalities, excessive formalism and rigour, subtle proofs, etc, its natural beauty (intuitive ways of thinking, simplicity of ideas, etc) is obscured. From a pedagogical point of view, to say the least, it would be most desirable if a theorem, or an idea, can be fully explained by, hence rediscovered from, one or two simple examples. These kinds of illustrative examples actually exist everywhere, and at all levels. We can use them as inspiring exercises. In this paper, we give ve sets of examples, beginning with a simple one on Abel's Identity, followed by examples on Hensel's Lemma, Finitely Generated Abelian Groups, Baire's Category Theorem and the Weierstrass Preparation Theorem. Solutions, hints and discussions are provided in the last section. Look at the following stairway and rediscover the Abel identity: This is true in any number system (ring); the stairway suggests a proof.
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