- Rarely limit our study to just triangles. I'm looking at you, triangle congruence and triangle similarity. You think that AA is a shortcut for similarity? Look at this square; look at this rectangle. Why should triangles be different?
- Ask kids to prove things that they don't have the tools for proving. For example, ask kids to offer a convincing argument that all these triangles are of equal size.That'll create the need for similarity (or the side-splitter theorem, or parallel line theorems) in a jiffy.
- Vary the problems that we work on between the visually obvious and the visually unintuitive, because one of our major jobs is to understand the relationship between the way we see things and mathematical argument.
- Treat definitions like theorems. (Thanks, Justin + Lakatos!)
- Two-column proof, or axiomatic proof, is an abstraction of proof. It is not a natural or normal way to communicate mathematically. The few cases I know of mathematicians working from axioms are remarkable for this peculiarity. (Most often, mathematicians are studying the properties of these formal systems, rather than working within the formal system itself.) So care has to be taken to put formal proof in its proper context, lest kids walk out of the year thinking that mathematicians sit in a room and pump out theorems from axioms all day.
I'd love to hear your thinking about these, as Geometry is a course that I still feel slightly uncomfortable with. See you in the comments!