>I think that all this argues against your seeming position that proof should be not there at all before 8th grade. Or if you're OK with before, then how much before? 7th, 6th, what?
Well, I wasn't arguing that point at all, though it seems wrong to me to introduce minus times minus with either a proof or to justify it simply by proof. In fact, you can only justify by proof by making prior determinations that are not subject to proof.
As far as the "teaching proofs" in grade school, its an interesting idea that I have no preconceived stance on. If I wanted to develop a stance I'm sure I would be biased away from basing proof on algebra before more traditional logical concepts were developed. But that's just off the top of my head. I really don't know. Mere symbol manipulation doesn't make a proof unless you understand and believe in symbol manipulation that way. Kids are fine with games with formal rules (checkers, chess, others) so I've no doubt they can also play symbol manipulating games, as a game, sometimes to get a prize.