[loewenskind]

>You come across patterns like that by doing the tedious work the hard way and looking for short-cuts.

This doesn't work for me. After a little bit of tedium my mind quickly switches through "just fight your way through this" and no more insight will be gained. All the math shortcuts I know I either read about or discovered while playing with numbers on my own time without any homework deadlines.

[psykotic]

Yes, an important point is that everyone learns differently. Aimless computation is the furthest thing from my mind, and I agree that most people wouldn't notice these patterns if left to their own devices. Computation as an accessory to constructivist learning guided by teachers is what I'm suggesting must be part of a solution.

I'm skeptical of curriculum reform that seeks to excise an important source of learning for a sizeable subset of students.