I agree with ColinsMum-- the explaining IS the part that must happen in order to do higher math.
There's not a right "method."
But there is a right way to communicate what you are using as a method.
It has to be valid as a set of instructions for another person in order to be correct, basically.
But it's the intuitive jumps that lead to the higher math being discovered in the first place. The explanations are how you prove to others that the jump was correct, and the road map that lets others follow in your footsteps.
The intuitive jumps are important, sure. But they only give you conjectures, which are worth very little without proof. The explanations are how you satisfy
yourself that what you believe is true. Otherwise there wouldn't be a million dollars riding on each of the Millenial Problems, and Andrew Wiles needn't've bothered.
Don't let your kids think they've finished when they've got the answer in the back of the book - really honestly, if they can't explain, they've hardly started.
