I’m spending the afternoon grading student papers from the Foundations course, and realizing that it might have helped to separate out the process of mathematics a bit more. We gave them a take-home assignment to write up a proof that we discussed in class, in detail. The issue is, though, that our classroom discussion included a lot of exploration and kind of side-conversations, which have worked themselves into the submitted proofs in interesting (in the not-great sense) ways.
There’s a great course in the Budapest Semesters in Mathematics program called ‘Conjecture and Proof,’ which combines a proof-writing class with a problem-solving class. (But it’s really focused on the problem solving.) In Foundations we’ve been striving to get across the importance of rigor and proof, while teaching basics of proof-writing and techniques of proof. Inevitably, though, such a project has to be mixed with some problem solving alongside: students need to write proofs that they haven’t seen before, and that involves solving problems. So we’ve ended up a bit reversed from the C&P structure, which places a lot more emphasis on the problem-solving than the proof writing. (And I feel a bit like we’re falling on the wrong side of history in this sense…)
So what we’ve ended up doing is giving some problem-solving techniques in class, described a bunch of proof techniques, and haven’t been getting back very proofy proofs; a lot of them are including these exploration steps which aren’t a part of a formal proof.
To deal with this, I’m imagining breaking out the problem-solving process into four distinct parts: Exploration, Conjecture, Testing, Proof. This allows focusing on the specific skills important at each stage of the problem-solving process. Exploration involves all of the getting-oriented techniques that are important for turning a blank page into a set of useful ideas. Conjecture is the process of identifying patterns in the problem (which should lead to a solution). Testing is the process of looking for holes in the conjecture (aka, counter-examples), to make sure you’re not too far off track. And finally, proof is the process of condensing the ideas obtained into a rigorous proof. (And maybe a final streamlining step after the first draft of the proof is obtained, too…)
There’s a lot of feedback in the teaching of the process, though; you need to understand some techniques of proof to guide exploration, for example, just as you need to build some exploratory skills to find new proofs. But I think it could be a useful teaching framework, which would at least separate out evaluation of exploration from the writing of rigorous proofs.