[Micanthus]

The best professor I ever had combined the two approaches. She would lecture at extremely fast pace for a few minutes, then select a student to do a related problem on the board. If the student got stuck, they could ask for help from the class and if the whole class was stuck we could ask for help from the professor of course. The in-class work wasn't graded so it was pretty low-pressure once you got used to it. After the problem, the professor went back to lecturing and repeat.

But I could also easily see this backfiring in certain ways. A couple other things that enabled this to be a positive experience:

1) It was a higher level math elective, everyone in that class chose to be a math major and chose to take that particular elective. This avoids having students who simply don't care, which often ruined similar situations in high school in my experience.

2) It was a relatively small class, I think about 15 students, in a relatively small university. Because it was a small school, I already knew most of the other math majors, and the class had a more personable and trusting atmosphere. Even if you didn't know a person, and thought they were stupid or annoying, you didn't want to burn that bridge because you were almost guaranteed to be in a class with them again later. Because of all this, making a mistake was no big deal, even the shyer less-confident students didn't seem particularly nervous.

3) Class attendance and participation was not graded. This made it very clear that doing the problems was for our benefit, it wasn't busy work.

4) You didn't volunteer, and it wasn't an option to sit out. The professor literally was going person-by-person based on how we were seated, starting from a random person each class. This way it wasn't an oppurtunity for know-it-alls to show off, and less motivated and shy students couldn't just disengage

With a slightly different setup it sounds like torture. But as it was, it kept the class engaged and you could immediately apply what you were learning in lecture to cement it.

That was the hardest math class I ever took (discrete and combinatorial mathematics) because--for scheduling reasons--I'd gotten an exception to take it before I'd met the the prereqs, which meant I was a freshman taking this 3000-level elective in parallel with calc 2 lol. If it was with a worse professor I definitely would have failed, I was not at all prepared. I've taken classes with a completely "flipped classroom" and I just don't have the motivation to do well in classes like that, but I also can't focus enough on lectures with no engagement (at least I couldn't back then before I started ADHD medication)