[j2kun]

I have taught enough lectures to high school students (presenting "advanced proofs") and talked to enough geometry teachers who abandoned the two-column geometry proof to know that 1) is not true and 3) is not worth it. The problem is that people build up proofs like they're something to freak out about, or that the proofs that are presented are inherently mechanical because that's what students are taught. You can't expect someone learning to write proofs to be perfect any more than you can expect a first-time drawer to color within the lines. It's okay and should be embraced as an opportunity to reflect and improve. A proof is not complete just because you "got to the answer." It's complete when it's simple, elegant, and easy to explain to others.

I can and have explained beautiful proofs without the need for mechanical proficiency to ten year olds and mathphobes alike. Here are a few examples:

[1]: http://jeremykun.com/2011/06/26/teaching-mathematics-graph-t... [2]: http://j2kun.svbtle.com/things-mathematicians-know-proofs-ar... [3]: http://j2kun.svbtle.com/things-mathematicians-know-more-than... [4]: http://jeremykun.com/2011/06/26/tiling-a-chessboard/

The world is full of these cool problems and proofs. I could literally teach an entire course and do nothing but puzzles involving chessboards. That many teachers ignore these great topics is a problem, but it's certainly for a good reason (the myriad of other problems with high school education).