Asklemmy

it takes 162 pages to formally prove that 1+1=2

This is ridiculously backwards, Whitehead and Russell's motivation for writing the PM was to come up with a set of axioms and deductive rules that the entirety of mathematics could be derived from. When they worked out their proof that 1 + 1 = 2, it didn't tell the world that now 1 + 1 = 2 is now officially a fact, it told the world that the logic and axioms they built were enough to be capable of deducing some very simple facts that we've already been confident are true. The hope was that maybe if we keep working at this and modifying our rules when need be, we'll be able to get a set of axioms and inference rules that are sufficient to determine the truth of any mathematical question. Calling that a proof that 1 + 1 = 2 would be saying their brand new theory was somehow more valid and more fundamental than addition of natural numbers.

A few years later Gödel came along and completely obliterated any hope of a project like that succeeding, and today literally no one thinks of the PM as more than a historical curiosity. (If you actually wanted to prove 1 + 1 = 2 from first principles today, you'd use the Peano axioms for the naturals: S0 + S0 = S(S0 + 0) = SS0, done.)

That's a tangent from the actual topic but I feel compelled to call it out.

Getting back on track, probably 90% of the points I give on exams are for partial credit, because there need to be distinctions between having no clue, knowing where to start and getting stuck, understanding essentially every meaningful step but then writing 1 + 1 = 3 to wrap up, etc. I'm grading on both their ability to solve problems and their ability to communicate their ideas. Both are equally important.

This is very controversial, but I don't go out of my way at all to worry about cheating. I don't want to play policeman and teach with the mindset that my students are potential criminals. Even if I'm 99% sure a student is cheating, if I'm in the profession long enough I'll eventually hit that 1% where I'm giving a decent student an undeservedly hard time. I'm not paid anywhere near enough for it to be worth having a more adversarial relationship with my students.

I had a student earlier this month where it looked like he probably snuck out his phone for an exam. I just wrote a note on those problems that I couldn't follow his work and wasn't comfortable giving points for work I don't understand, please walk me through your solutions for the points back. I told him this verbally as well when I handed it back to him as well. He never took me up on that, but it feels more humanizing than just calling him a cheater. I think OP is getting at something similar, but I think there's value in not phrasing it in an accusatory way.

Being somewhat sympathetic to OP though, there is a sense of feeling insulted when a student puts very little effort into pretending they're not cheating. I try not to take it as an affront to me personally and imagine that they do the same for all their instructors, but I do feel kind of peeved sometimes.