Lately I have been using Inquiry Based Learning in my courses. Fundamentally, this is about engaging students in the mathematical process: requiring them to question, conjecture, argue, and critique. Perhaps the best short-hand is this: students do mathematical research at their own level.
More detailed information about Inquiry Based Learning can be found:
- The Legacy of RL Moore Project (in college mathematics teaching, IBL is a broader umbrella, held up in the middle by the Moore Method
- The Academy of Inquiry Based Learning
Of course, in the sciences, they are way ahead of us mathematicians. As I learn more, I’ll add more to the list.