Visualization Techniques

This is an ugly phrase to describe what I think is an important aspect of learning math. A lot of math skills go beyond reasoning, logic, and memorization. Sometimes a student needs to be able to visualize a concept or a relationship in picturesque terms. Here is an example.

Suppose you wanted your students to understand why cos(theta) and sec(theta) are even functions, while the other four trig functions are odd. There are several ways to show them, but an especially effective way is to get them to imagine two radii of the unit circle sweeping away from theta = 0 at the same time, so that the first angle is always the opposite of the other. They should be able to grasp more or less directly that the cosines of the two angles will always be identical, and therefore cos(-theta) = cos(theta), ie, the cosine function is even. The same visualization technique will illustrate why the other trig functions must be odd.

Hence the value of math teaching software like fathom and sketchpad, which allow you to make the visualization precise and immediate. However, I think that good teachers need to be able to think of visualization tricks on the spot, to explain them in words, so that the students have to construct the visualization in their heads from time to time.