[gizmo686]

I was a student from a small private school that literally wrote their own math book, so I have no idea how generally applicable this is. As you suggest, a common technique my teachers employed is setting us loose on problems we did not yet have the tools to easily solve (but which were within reach). We would typically work in small groups, and if necessary the teacher could speed up progress by dropping us hints. We inevitably (in the beginning) would come up with week/non-rigourous solutions, which would often lead to debate as a class, pushing us to formalisation. As far as learning new techniques/generalizations/ETC, we would almost always 'learn' them after we have already been using them.

One thing I noticed during in math classes is that I don't really need to know anything. For me, and most of my classmates, most of formulas could be easily derived from simple and intuitive principles. For example, almost no one in my class actually 'knew' the quadratic equation, or the common trig values (ie. sin(30)). What we did know was how to quickly find those if we needed them.

As to your question of a standard pace, groups tend synchronize themselves. If every comes in at a similar place, and you have alot of group work and full class collaboration, then the slower students will gennerally still be able to follow the groups discovery trail, even if they do not contribute as much. The important thing here is you make sure that students are comfortable to ask questions, and that you do not have a few students dominate the discussion such that they loose the rest of the class.

Again, that comes from the perspective of a student at a school where the teachers had a lot of leeway in how and what to teach.

[nileshtrivedi]

Would it be possible to get that math book they wrote and perhaps the teachers' notes to go with it?