Ok, I'll put in my two cents. Why do mathematicians do proofs?
1. It's fun. (no, really) It's just problem solving taken to another level. There's the same kick from successfully proving something rigorously that there is from solving a tricky problem.
2. There are lots of things that *seem* right for hundreds of cases or more and are in fact false. E.g. (n+1) | (2^n - 1) iff (n+1) is an odd prime. This is true until. The first counterexample is at n=340.
3. Proofs lead to new conjectures. Knowing *that* something is true is all well and good. But if you know *why* it is true, you can use that to make more predictions about other things that might be true for similar reasons.
The above are all things we want to instill in our students, no? The love and excitment of solving problems, the need to be truly (mathematically) convinced of something before believing it, and the desire to do experimentation -- search for new mathematical results from old ones. This is why we have even elementary school students justify their answers and argue/defend their positions. It's the start of proof.
High school (maybe even middle school) is a good time to start formalizing the argument process, but I would argue that the two column proof is not the way to go... It doesn't really lead to andy of the 3 positive aspects of proof, and it's not the way "real mathematicians" write proofs.
Two column proofs are fairly sterile, they leave little room for creativity (and hence the joy that comes from problem solving). Often students are simply asked to prove a result -- so they already know it's true because they're beings asked to prove it. A better way is to have them *find* a result through experimentation, and then want to prove it because they're already convinced themselves that it's true. And two-column proofs rarely lead directly to another result.
So, there you have it. My opinions are, of course my own and not necessarily those of EDC or the Connected Geometry project (though we are *not* including 2 column proofs and we *are* including lots of other argumentation....), blah, blah, blah
-- -- Michelle Manes Reasearch Assistant, Connected Geometry Project Education Development Center