MoN brings up a great point - why worry about 'deep understanding' of self evident facts and then abandon that when things get a bit more involved?
I continually stress to DD that numbers are just instances of abstract thoughts as opposed to something in their own right ( I hope that wasn't too incoherent ) to try to get her to see the principles and not the mechanical steps.
We use pictures and diagrams a lot at home. Things like; x^2 + 2xy + y^2 boiling down to (x + y)(x + y) become easy to understand with some squared paper, a ruler and some colored pencils. Similarly, several different size cylinders ( like jars ), some string and a ruler help drum the constant Pi in.
The world around you becomes SO much more interesting when you begin to realize how much Maths and physics work together all around you - well it did for me at least LOL and so far, DD appears to learn things a lot like me.
I agree 100% with Wren about learning from being challenged - I never was as a school boy and I didn't really learn how to learn/study as a kid at all. Not a situation I want to see repeated with my DD at all.
