Accelerating vs. Challenging

I have been thinking a lot about the difference between accelerating and challenging in mathematics. My prior blog posts clearly show that I am a big believer in acceleration. However, there is a big difference between doing things fast and challenging the brain to think deeply. I don't mean to say that gifted children should be challenged instead of accelerated. I mean that the process of acceleration does not seem to include enough hard problems, and this may become an issue later in the child's educational career.

How should I balance acceleration and depth of thinking? My daughter is only six. Hence, I am very careful about the amount of time we spent working on traditional coursework. We do about one hour of English and math every weekday. That is plenty for her. She moves so fast that I worry quite a bit about retention. The key here is that I want her to learn the fundamentals -- math and language arts -- while having enough time for independent exploration. She likes to do art, write stories, read about the universe and the human body, etc. Hence, I need to think carefully about how to challenge her without imposing much formal study time beyond what we do with EPGY.

Here is one strategy I have been testing. Assign one tough problem every week. Come up with a challenging problem that can be solved using what the kid already knows. The problem does not have to have an exact answer. That's not the point. The point is to teach him or her to work on something hard over an extended period of time. This means that you give the child the problem, make sure it is understood, and then leave him or her alone to think. Every few days, you ask if he or she wants to talk about the problem or need a hint.

Here are a few examples:

1. Compute the area of right triangle once it is understood how to compute the area of a rectangle.
   
2. Compute the area of an arbitrary triangle once you know how to compute the area of a right triangle.

Ans to 1: Make a copy of the right triangle. Stack it on top of the original along the hypotenuse.
Alas, you got a rectangle. Divide by 2 the area of the rectangle. You want the kid to come up with a formula, algorithm, etc. Don't reveal the answer. Give only small hints if they get stuck.

Ans to 2: Set the triangle on one of the sides and call that side the base. Drop a perpendicular from the base to the opposite vertex. Alas, you you two right triangles now. Use the answer to question 1.