Beginning proofs as groupwork

Today in Geometry students were taking their first crack at making up proofs. I have modelled this for them, walked them over the pons asinorum and tried to show them how we can use a deductive argument to see how our compass-straightedge constructions work. They have done a bunch of the classic SSS etc. problems and should know the triangle congruence theorems. They were ready, but I knew that it would be hard for them, so I wanted to have them work in groups.

I break my classes into groups nearly every class, if only for a short investigation. But I’ve been feeling unsatisfied, and recently returned to the book Designing Groupwork (Cohen) for some deeper insight. One thing the author stresses is the importance of assigning individual jobs to help break down students’ prejudged status structures. In an effort to bring the groupwork up a notch, and to be really clear and formal about what makes a mathematical proof “right,” I created both a scoring rubric for student proofs and a corresponding set of group member roles. I made up a list of simple proofs I thought the students could handle. The whole thing is a one-page (front-back) sheet, found here.

How it worked: I pitched the task to kids as their first real attempt at mathematical proof. They were actually sort of excited, and nervous. I told them their work would be scored and entered in the gradebook. ( I don’t usually do this, striving for a more righteous SGB, but sometimes it helps to raise the stakes just a tiny bit.) I provided one last model so they could see the level of clarity and justification I expected. Then I shuffled the cards and made groups of four. Each group got a sheet with one of the proof tasks highlighted. I put their names by the roles. And they got after it with the whiteboards.

All the groups were able to write a correct and clear proof, working together. One group in one section took a lot longer than the others, so kids had to do some other problems while they waited. The presentations were pretty good, though lacking in the class feedback department (which I encouraged, but didn’t require or formalize).

The whole lesson, with launch, group work, and sharing, fit into an 80-minute block just right. There are a lot of things left to prove on the list, and I plan to do the lesson again after Thanksgiving. I anticipate that they will be much faster, and more confident. Then, eventually, I’ll be asking them to come up with proofs without the group support, and I plan to use the same rubric (without the participation component) to assess them.

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We are all born into this world, and at some point we will die and that will be that. In the meantime, let’s enjoy our minds and the wonderful and ridiculous things we can do with them. I don’t know about you, but I’m here to have FUN.
-Paul Lockhart

In theory, theory and practice are the same. In practice, they are not.

-Yogi Berra


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