> ...at all but the most rigorous level, one > can construct plausible, convincing arguments of outright > falsehoods. I think one should try to confront students with such > arguments sufficiently often to at least make them suspect the > existence of levels of reasoning beyond those they are being > challenged to master.
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.)
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.