1. Each lesson should revolve around a single task, prompt, or question
2. Each task, prompt, or question should be student-driven
3. The "resolution" of each task, prompt, or question must come from the students
Even though much of this sounds dramatic, I truly don't think it is too far removed from what is possible in even the most standards-driven schools. Thomas Romberg writes of a problem-based approach similar to this in which he sees students "studying much of the same mathematics currently taught, but with quite a different emphasis." I think this is a fitting description of what I propose here. I would push for much of this type of design to occur within a problem-based environment, but I don't think it is necessary. One could just as easily work with students to study, abstractly, a unit on exponential growth, trigonometric ratios, or any other traditional standards-based unit. This type of design is difficult and requires patience, but the impact on teaching, learning, and doing mathematics has meaningful impact.