To this day, I'm still trying to understand why my freshman calculus class (proof-based), was as effective as it was.

It was 30 students, 1 lecturer, and a whole bunch of black board panels.

I think part of it was that it felt like a conversation, as well as a game. He would lay out the pieces (assumptions, definitions), and then point us in some direction ("now how would we show X?") We'd throw out ideas if we had any, and he'd either rebut us or nudge us in the right direction. I was 100% engaged in that class - no checking of phones or surfing the net - and it was just myself and my notebook, unlike in other classes that were slides-based and I picked up the bad habit of zoning out when something familiar was being covered. You can't do that in a conversation! The material and strategies I learned in that class completely built the foundation for my math major.

I am still not sure if math is one of the few topics you can take this approach for, slides are the devil, or if the lecturer was just secretly brilliant.

I started reading How to Solve It by Polya (as one of my interests is education, particularly math education). Your professor's technique sounds like something straight from this book. Present a problem, ask the students for the next step. Guide them towards the solution (reference similar problems, maybe be explicit if they're way off).

By engaging them in a dialogue you keep them engaged and focused on the material (or at least more of them than in a straight "speak at them" lecture). And by giving them a chance to answer for themselves, you're encouraging them to think (rather than merely commit things to memory or similar).

Yeah, the "master & his student" approach is unparalleled in terms of effectiveness.

The only problem is I can think of many specific times in my life where another student asked a question, and I had the thought, "Fuck, that's a good question, I didn't even think of that," and I learn down a path that I had simply applied false assumptions to without realizing it.

Actually I think more effective is the 'new student and the student-who-recently-completed the course' because the old student remembers his misconceptions. Whereas the professor suffers from the curse of knowledge:

https://en.wikipedia.org/wiki/Curse_of_knowledge

I'd add a computer of some sort to that, probably a laptop

but basically I agree 100%

As a professor, I can say most professors don't like lecturing, but for some classes the alternative is to have small interactive classes, which isn't financially tenable for most universities for all classes.

So my reaction to these sorts of studies is "ok, then should we increase tuition to make up for splitting this 200-person class into 20 sections?" Even classes of 20 are too big.

I would like to change something about classes that still have a big lecture component but I'm not sure how to do that optimally. I think some of them could be done in an online format with videos, but I'm not sure if that's the best way either. Maybe if some small group discussions could be added?

Bloom's Two-Sigma strikes again: https://en.wikipedia.org/wiki/Bloom%27s_2_Sigma_Problem