On 23 Mar 2006 11:33:53 -0500, email@example.com (Herman Rubin) wrote:
>Concepts and structure need to come EARLY, so the students >can know why, and not just how.
Not so; certainly not necessarily so, and far too sweeping a generalisation. You lose almost everyone if you pontificate. The young are generally not ready for theory simply due to the fact that they are very young, but might grasp it later when they have more detail to put to that theory.
I know and taught both fact and theory, depending on the age and the level. It's part and parcel of what you do. The order is important for reasons other than you suggest. You might read up on the general learning capabilities and capacity of different age groups[ as in Piaget's principles].
With those clearly showing exceptional talent, they can be prepared for competition level mathematics. Check out the olympiad competitions and others for that level of required competence. Others, by far the majority, need hands-on, "show me how to do it then leave me to do it." Others in between can have it one way or another; first practice then theory or vise versa. Some simply do not have the ability to assimilate both theory and practice, having sufficient problem handling simple examples one after another.