In article <firstname.lastname@example.org> Michael Rogers, email@example.com writes: >The two-column "method" focuses one's attention on the second >phase of proving a theorem. It forces one to clearly show why >your assertions are true. One should also be forced to do this >in the paragraph style. One problem with the two-column style >is that it focuses so much attention on the second part of >proving, the write-up, that students do not realize that they >are supposed to still do the first part. And as you said, >without the first part, you are likely to fail to find a proof.
Another problem with two-column proofs is that they offer no indication of what are the main ideas, the important steps, in the proof. While from the point of view of formal logic, all steps in a proof are equally important, from the point of view of mathematical understanding, usually a proof has one or two key ideas. This is where the AHA! resides, if there is one.
For this reason, I think it is a very useful exercise to ask a student to summarize a proof in one sentence. It is also useful to take a two-column proof, examine it and then mark one or two lines which contain the main ideas.
Paragraph proofs have the advantage that they are actually written as communication as well as formal argument, but there is a danger that someone new two proofs will not realize the formal structure that is supposed to underly the reasoning in the proof.
Jim King Dept of Math University of Washington Seattle, WA 98195