10 - Proving Starting from Informal Notions of Symmetry and Transformations

Summary

In this chapter we consider the challenge of promoting students' ability to develop their own proofs of geometry theorems. We have found that students can make use of transformations and symmetries of geometric figures to gain insight into why a particular theorem is true. These insights often have the potential to form the basis for rigorous proofs. In the following classroom vignette, we see the excitement that comes from discovering an idea that seems to explain exactly why a theorem is true, followed by the realization that there is significant work to be done in order to develop a rigorous proof based on such an idea.

Classroom Vignette

Setting: A college geometry class using the Henderson (2001) text has been asked to work in groups to prove the isosceles triangle theorem (ITT). That is, given two sides of a triangle are congruent, prove that the angles opposite those sides are congruent. After about 3 minutes without much progress, the group of Alice, Emily, and Valerie burst into activity.

Alice: The book says to use symmetries.

Emily: Symmetries?

Valerie: That angle equals that angle —

Alice: Okay! Yeah! Yeah.

Valerie: And then this angle —

Alice: If you have, yeah! If you have, like, a bisected angle —

Emily: You do the angle bisector —

Alice: Yeah! And then this matches this [rotates her right hand from palm-up to palm-down across her triangle drawing] because it can lay right on top of it! [Moves her left hand to land (at word “top”) on palm-up right hand.] Because then you like rotate it.