Rather than ask students to buy into the illusion that two-column proofs emerge spontaneously and fully formed from their brow, we are inquiring into how we mortals can better brainstorm and use our reference tools to create the sub-assemblies that we can use to build our rough draft proofs. Then we'll be better able to polish our final proofs and present our work.
This has meant that we are developing students' intuition that that these sub-assemblies are knowable and predictable. We call these our "major proof moves." Some of our major categories of major proof moves include:
- the relationships between parts & wholes
- a sense of bisectors and "half-ness"
- parallels and the results of parallels
- perpendiculars and their results
- right angles and their results.
It's working out surprisingly well.
Today we started experimenting with using these higher-order concepts to work on harder, multi-stage proofs. The kids were quite excited to be able to figure things out.