Walter Whiteley's story about his students not being convinced by proofs in the face of unexpected results reminded me of a story told by Joel Teller of the College Prep School, a prestigious private school in Oakland, CA.
Years ago, while experimenting with alternative forms of proof, Joel asked students on an exam to provide a "convincing argument" for some basic geometric theorem. He wanted to leave open the possibility for some students to write "paragraph" or flowchart proofs, but was shocked to find that NO student wrote any sort of formal proof (much less the expected two-column proof). When he asked his students about this, they responded indignantly that they all knew how to write a formal proof of the basic theorem, but since he had asked for a convincing argument, they thought he did NOT want a traditional proof, since no one would ever be convinced by one of those.
While I agree that students need experience in logic and systematic argument, I doubt that proofs provide this for the vast majority of high school students. In fact, I am convinced that they can do much more harm than good, by focusing attention away from more fruitful (and useful and accessable) mathematical ideas. As with much of traditional mathematics instruction, our attempts to teach (geometric or algebraic) proofs tend to fail because we start with the abstract before students have enough experience with the concrete. We start with minute details like the definition of betweenness or the commutative property, and try to erect a delicate intertwined structure, instead of routinely asking students to justify, conjecture, explain, or convince in informal terms that they can all understand.
My objection to proofs (at the high school level) is that they not only fail to teach cogent argumentation, but prevent large numbers of students from going on to learn further mathematics, and falsely convince many that they are not "good" at math. How many adults do we know that hit a wall in high school geometry because of proofs? I'm thinking of students who might never choose to study formal mathematics, but would nevertheless be interested in studying science, engineering, or any other technical field. I have students in my (non-AP) calculus class who cannot "do" fractions without a calculator, but are pros at understanding what derivatives are and how they relate to graphs and applications. A non-traditional curriculum has allowed them to reach this point. Imagine how they perceive their own mathematical ability! Isn't that what teaching high school mathematics is about -- opening up opportunities? I tend to think that any mathematical topic that weeds out students at such an early age needs to be reevaluated. Is it so important? If so, when is is appropriate to introduce? What alternative methods of instruction are available? Who would suffer if it were delayed, and who suffers if it isn't?
Carlos Cabana, San Lorenzo High School, San Lorenzo, CA