I have a student who doesn't do well on tests. He has completed geometry, and will be studying intermediate algebra. I have decided to explore calculus with him.
I hope that, by learning a little of something that his other fellow students know little or nothing about, his self-esteem and interest in the subject can be kindled.
My introduction consists of attempting to define and determine the slope, or inclination if you will, of a curve at a particular point. We use y = x^2. We take two points on the curve and determine the slope of the line that joins them. Then we move one point close to the other.
We say that the slope of the curve at a particular point is the unique quantity approached as one point moves closer to the other, taken as static.
This gentleman may never even take a calculus course. But I feel the slope of a curve at a particular point is not difficult for him to grasp, in the context of math he has already taken. He may also be able to use his insights gained from this introduction to graph equations.
At any rate, this raises the subject of enrichment vs teaching to the test. Is it possible that enrichment can raise the level of cognition, and spark greater interest in the subject, thus actually enhancing performance on tests? One of course must have a varied tool kit to approach the needs of different students.