1. Transition to Proof
CUPM: Proof is not something that can be taught in a single “bridge” course. It must be developed in every course.
Me: Proof is part of mathematical communication, explaining how to connect a new mathematical insight to what is already known.

2. Difference between finding a proof (ie a personally convincing argument) and communicating a proof.
Importance of efficiency in proof, but never at the expense of clarity.
Importance of analyzing one’s own proof and those of others for “correct, clear, concise”.

3. You prepare students by giving them lots of practice in communicating mathematics.
Communication of mathematical proof has it own rules. Two of the dominant characteristics are the use of precise definition and logical implication. But these cannot be taught in the absence of meaningful proofs.
Difference between finding a proof (ie a personally convincing argument) and communicating a proof.
Importance of efficiency in proof, but never at the expense of clarity.
Importance of analyzing one’s own proof and those of others for “correct, clear, concise”.