The goldilocks problem

In teaching students how to design and implement proofs, we start them with an extremely structured format. This structure is both a boon and an irritation, since it helps organize the ingredients of a proof, while at the same time making the format of a proof extremely predictable. We have the goldilocks problem of making the form predictable enough, but not too predictable.

If students want to prove something about all natural numbers, they introduce a name for a generic natural number, prove things about it, and then conclude that the results they've proved apply to all natural numbers.

We force them to indent the results they prove about the generic natural number, to emphasize that they are inside the "world" where name they've introduced is a generic natural number.

If they go further and implement a proof about an implication, of the form P implies Q, we have them (at least for a direct proof) assume P is true, indent some more, and then derive Q. Our rationale for this step is that if P were false the implication would be vacuously true, so they only have to worry about the case where P is true.

We expect students who become mature theoreticians will develop their own voice, and write clear, valid proofs that diverge from the proof format that we impose in our course, with its arbitrary order and indentation. We don't know exactly when they will develop this mathematical maturity (some of them may already have), so in the mean time we coerce them into mastering one format for proofs that they, and the course teaching staff, can agree works.

My challenge in communicating the proof format to them is to get most students to the point that they are nodding along with the proof format because they understand it, but stop before they are all nodding off. Goldilocks.