> Purposeful construction of misleading arguments strikes me, in > most instances, as misguided showmanship. There is a place for > such constructions at post-calculus levels, where proof begins > to become central to instruction. (I maintain that proof is > important, but not *central* until then.)
To me this statement is very strange. I was fascinated by misleading `proofs' when I was quite young, in elementary school. I remember reading a very good book by Dubnov about them. Some of them are quite understandable for children, and are quite interesting for them according to my experience. For example, the famous demonstration of cutting a 8*8 square into four pieces and making a 5*13 rectangle out of them. I often use this puzzle to challenge students who have never been taught proofs in a regular way and I think that it is a very good PREPARATION for a study of proofs.
Besides proofs of wrong statements there are also problems, where a wrong answer is very plausible. These are also very useful, according to my experience.
> The really interesting, and difficult, question is this: How > should we deal with students who invent their own "plausible, > convincing arguments" supporting "outright falsehoods"? > The question is interesting because students who are capable of > such things (almost always innocently, of course) are very > interesting people to teach. > The question is difficult because it means we must recognize the > outright falsehood (*whatever it may be*) for what it is. (So much > for the theory that knowing how to teach is more important than knowing > what to teach!) Then we must be able to deal effectively with the > situation. > I maintain that the most effective way to deal with the situation is to > present the "prover" with an example that clearly contradicts > what has allegedly been proved. And then step aside. > At least for a while. Noticeably longer than the "prover" likes.
I agree that it is a very interesting and difficult question. But for me this question is an organic part of a question which is still more interesting and important: how to teach students to prove. It is impossible to teach to prove without teaching to recognize fallacies. Also my experience tells me that just to ``present the "prover" with an example that clearly contradicts what has allegedly been proved, and then step aside'' is not enough. It works only with the strongest students. Most of them are not strong enough to manage this cognitive difficulty quite independently. What they need is a paradigm: what to value, what are norms of action in such cases, what are `rules of the game'.
Andre Toom Department of Mathematics email@example.com University of the Incarnate Word Tel. 210-646-0500 (h) 4301 Broadway 210-829-3170 (o) San Antonio, Texas 78209-6318 Fax 210-829-3153