I got a message recently from a former student, Tigh Bakker, who is now starting out as a practicing teacher. He sent me a thoughtful question, and seemed happy to have me share it here in hopes to generate a bunch of ideas.

Here is an excerpt of Tigh’s message to me:

I am currently directly in the middle of teaching proofs to my high school students, and all of my “high flyers” seem to grasp it, but a lot of my lower students are stumped and performing poorly. I know you don’t teach the lower level of Geometry that I do, but I was wondering if you have any tips/suggestions to helping students understand how a proof flows. Most of what we are doing now is simple triangle proofs using congruences such as ASA, SAS, SSS to then use the notion of CPCTC (corresponding parts of congruent triangles are congruent). Our book is called “the CME project” developed by Pearson and is rather open ended (I like to think of it as the high school version of how you teach). Knowing that you teach in an inquiry based fashion, how do I get my lower students to delve into the material and really work at learning some material without my guidance? Any tips at all would be great.

I sent him a few ideas yesterday. I don’t claim these are original to me.

This is indeed challenging. Getting students to change their mindset is hard. I imagine self conscious middle school and high school students would be tougher to convince.

I also struggle with the students at that margin. For some, there is safety in group work. But you don’t want a bunch of struggling students to latch on to each other and then drown together. As to the “flow of a proof”, here are some ideas. I have no idea of how they might work!

- Give them a proof written clearly, but cut up into individual sentences. Have them arrange the sentences in the correct order.
- Give them examples of poorly written proofs (not their own), and ask them to critique.
- Show them examples of various versions of a single proof. Range from good to bad, lots of style variation. Have them do some sort of ranking or vote on an order of best to worst. I hope that a subsequent class discussion on what makes a proof good or bad would help focus their attention.

Does anyone else have ideas for Tigh to try? Drop some in the comments!

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I think you’ve given three good pieces of advice. Here are a few other ideas off the top of my head:

1) Take a proof, cut up the lines as you suggest, but then cut apart the statements from the reasons. Have students work to put the pieces together properly. This can be a great group activity, especially when you start adding superfluous statements and reasons.

2) Another fun activity is to give students a “Theorem” and a sufficient, but unnecessary, set of “Givens”. Which can be removed?

3) Have students create their own proof problem. Exchange with peers, try to prove, and share the results.

4) It’s hard at first, but I’ve found that transitioning students to “narrative” style proof (as opposed to statement/reason) helps free them up a bit.

Good luck to your former student!

I also think you have good suggestions. I will add this:

“If you are doing the ‘Two-Column Proof’ method of proving, stop.”