I had an incredible calc 1 and 3 teacher who would start the class with a problem that we'd work on for 20ish minutes. usually it was something just out of reach with where we were topically, or something that required us to use like basic geometry (which many of us were rusty on), and then we'd talk about it, ask questions, and he'd segue into a lecture where he'd connect the dots with what we were studying. I vaguely remember one was essentially to prove herons rule which we only had the tools to do from vanilla geometry, then he showed how it could be done with calculus.

Lectures were almost Socratic. We were only graded based on exams and participation, which I think he kind of arbitrarily awarded based on how engaged he thought you were.

Brilliant guy, loved his classes. He'd studied under a couple members of the Bourbaki group and headed the math department at one of the top math schools in Latin America.

Anyway, I had a great experience with this approach and I really think it's the best way to teach math.

I currently teach high school math and have been trying a similar approach, and received push back from every experienced educator I've encountered, but have been receiving great reviews from students (especially ones that have low math confidence). This post was really encouraging to me, thanks for sharing!

The best math teacher I ever had throughout all my schooling structured lessons that way. There'd be an introduction to a new topic, with problems we'd get a chance to work through alone then aftwards we'd all go through them together. That was pretty much the extent of the lecture part. Then we'd get an exercise to work on for the last half of the class that covered what we learned. The teacher was always extremely patient and would help everyone who needed it. Sometimes explaining the same thing over and over to different people without ever becoming frustrated. There was optional(but highly encouraged) homework if we wanted more practice. The teacher would be available for an additional tutoring session to work on the optional homework if people wanted it. We weren't graded on it. All our marks came from exams.

That class was really the first time a lot of concepts actually started to make sense for me and I went from being a pretty below average math student to having some of the highest marks in the class.

The very best teacher I ever had was my calculus teacher, who was teaching it for her first year, and although claimed she had trouble with calculus, was a truly gifted teacher and an enthusiastic “math nerd.”

The shtick about being bad at calculus, whether true or not, translated into a highly Socratic and collaborative learning environment, where all the students were legitimately teaching each other. This was modeled by us “teaching” our teacher.

Thinking back, it really is incredibly remarkable what she accomplished. Not that standardized tests are the greatest measure, but in less than a year of calc instruction, a full classroom of smart but fairly unremarkable students all scored top scored on the AP Calc BC test (in other words, every single one of us tested out of 100-level college calculus).

This was a scenario where having a “large” (ie more than 1-5 students) class (I think there may have been about 20 of us) with a gifted teacher brought out a positive side to non-small classes that you don’t usually see. We all learned the material well enough to teach it, and we all had many teachers we could go to for help understanding a concept.

My algebra 2 teacher in high school did this. Class started with an assignment. You had to read the book, collaboratively work on the assignments, and then the last portion of the class he lectured, which was mostly answering questions. It worked really well.