Grading those exams is painful because the class uses online math homework, so for many, the first time they have to write a proof for another human is on that exam. Each answer you have to read closely to see if they understand the ideas but can't explain them well or if they just wrote random words on the page.
Damn, my most sincere wishes of good will toward you should manifest in some sort of alcohol in front of you about now. Not to mention that this format just feels wrong. Making an exam on something that most people never attempted before must look to students completely arbitrary on top of that. Like a medical school that ends with a pie eating contest to determine who gets to become a doctor.
Anyway, you and I should be thankful that we've had something of a nonstandard education in math, and do our best to help others see what we see.
The thing is that I wasn't all that aware about the gap. But I do agree, the least we can do is to show that maths is not about learning some schema but about actually deducing solutions and formulating proofs. Hell, now I feel like I should rewrite most of the complex numbers primer that I made since I have assumed that the reader is prepared to do some heavy lifting instead of (at least that's how it seemed to me) being 'spoon-fed'.
On that note, it's probably for the best that I didn't share most of those Maths for Project Euler sheets. After about second part I have started shortening proofs and fell into the mathematicians mindset of "but that's a trivial step, removed". XD