The purpose of this chapter is to describe how computer-based tools can help students in the doing and learning of mathematics, and to provide specific examples that illustrate the way in which well-designed technologies can support mathematical discovery and understanding. I begin with an example.

Task. Take the three vertices of a triangle ABC and reflect them each across the opposite side of the triangle to obtain a new “reflex” triangle DEF (convince yourself you always do indeed get a new triangle!). Repeat the process. Most people who have seen this problem conjecture that the reflex triangle, after several iterations, converges to being equilateral. But I thought it was a perfect problem for The Geometer's Sketchpad, which effortlessly allows an arbitrary triangle to be iteratively “reflexed” and its measurements to be computed. I dragged vertex A after producing DEF and quickly realized that the equilateral conjecture was false, for I could produce a DEF that was a straight line! And more: DEF seemed to change in a very chaotic way as I continuously dragged vertex A. When, if ever, would the figure become equilateral? When would it not? How did the “function” behave?

At this point, I realized that I needed some kind of measure of the degree to which the triangle had become equilateral, especially since the reflex triangles were getting increasingly large–exploding off the screen–as the iterations increased.