Okay, I was afraid of this. Students need to be told some things. I don't like the question "Let's begin with the circle; now what might we draw?" Why would it occur to someone that a quadrilateral is something you might like to draw in a circle? Or has some prior activity in the course, a previous class, reading a book, or problem solving, prompted the student to know that a quadrilateral is in fact something a geometer might like to draw in a circle. Of course this is just one short excerpt (and I latched onto that one question).
The examples I gave were random, non-consecutive questions - there's a lot of missing context! (I would just send you the whole transcript, but they are strict on not sharing their materials, reasonably enough.) The "what might we draw?" question was in the context of trying to answer the question "can we draw a cyclic trapezoid" and was eliciting the answer "a pair of parallel lines".
I just worry that they might be taking it too far. Don't they sometimes lecture on some pieces of theory? How do students get to know what the definitions are, what axioms there are, etc? Do they read some theory from a book outside the class?
Yes, they read from the book outside class, as I said in my very first post. However, they don't necessarily read the chapter that goes with the class before the class. Definitions and axioms/results are introduced in the class, one at a time, results normally being derived in class.
Lecturing doesn't work so well in text form - it's kind of like writing a book :-)
Human civilization has advanced because, while some people invent things, discover things, solve problems, etc, the crucial thing is that these things are communicated to others and propagated throughout society. Humanity wouldn't get far if everyone had to reivent the wheel. It's true that the typical school system is way too spoonfeedy, and there needs to be much more invention, discovery, problem solving, generally thinking creatively, in the education process, but if this is taken too far, it doesn't work. Their needs to be a certain amount of simply telling people stuff.
So, should I be concerned AoPS might have the balance wrong in their approach with their courses?
I agree with your general point, but personally I don't think you need be concerned about AOPS - though obviously, you'll make up your own mind when you try it! What AOPS covers so well is the ground between unguided discovery and lecturing: what we might call guided discovery. Students may end up feeling they discovered the standard results for themselves, but if you look at what happens, it was carefully planned: they were prodded just as much as was necessary to take each step. I think this is crucial in mathematics: it gives them a chance of not simply memorising the standard results, but rather, reconfiguring their brains such that the standard results become obvious. For this, giving a proof is not sufficient (at least, not for fairly inexperienced mathematicians): it helps to be led through the proof with some discussion of why each crucial step is a reasonable step to take. For example, in geometry, knowing which lines it might be useful to add to a diagram, and why, is often the hard part. I haven't seen anywhere else do anything like such a good job of getting this kind of thing over.
