Putting two things together (I think): Analying invalid arguments is certainly important, but the most important invalid arguments are those produced by the students themselves. The ones suggested by Ted and Andre are useful "puzzles" to help advance their thinking (and in fact I myself use some of them), but students are perfectly capable of generating invalid arguments all on their own power. The advantage in addressing the students' fallacies is that since they reflect exactly how the students think, it may more likely change their views.
And one of the best ways of eliciting their faulty reasoning is to give them statements about whose truth there is a question. Too often students are only asked to prove "true statements", which tends to short-circuit proof-making. [As I once heard a student say, "The book is asking me to prove it. If weren't true, they would ask me to prove or disprove it. Therefore, it must be true. I'm willing to accept it."] In this way, "proofs of wrong statements" become more than an adjunct; it becomes part of the core of having students explore mathematics. Gary
At 18:47 -0500 3/13/97, Ted Alper wrote: ... >Andre adds: > >> I want to continue that proofs of wrong statements are just >> one kind of useful ways to puzzle children. [...] >> But all this is just a preparation. What should >> come next is training in solving problems right - that is so >> that to avoid all these `miracles'. That is why it is so >> important for children to solve problems where it is clear how >> to check the answer and to find out whether it is right or wrong. > >I find nothing to disagree with here. "Proofs of wrong statements" >are certainly not the main course! They are an anjoyable and useful >supplement, though, and particularly relevant to this thread of >"plausibility arguments".