I am also interested in student-driven pedagogical methods. I have been involved, first as a student and later as a teacher, in the Boston-Area Math Circle, an extracurricular program in mathematics for students in primary and secondary school. Unlike traditional math classes, the Math Circle classes are driven almost entirely by the participation and direction of the students, with the instructors mainly providing questions rather than answers.
Here is a pdf of an article that appeared in Focus Magazine featuring a new solution to a familiar problem by the Math Circle students, to which I contributed. The picture depicts Math Circle students; I'm standing third from the left. The students on the floor form an example of an integer-sided triangle.
Returning to the Math Circle as an instructor while I was in college, I taught semester-long courses on Paul Sally's Four-Numbers game, group theory and the Rubik's cube, the Cayley Klein geometries, and the building the real numbers. Here are notes on the Four-Numbers game 1 2 3 4 5 and the real numbers 1 2 3 4
I try to adapt some of the approaches to teaching I learned during my time in the Math Circle to my current teaching.