In the geometry course: by discovery (of proofs). Very Socratic. I just looked at the transcript of a class on circle theorems. It covers 24 pages, including the student comments that got posted. The typical length of the teacher's contributions is 1-3 lines. There are only a few that are longer than that, and looking at them they are typically of the form [three lines of summary of last bit]"Any questions?"[two lines of intro of next bit]. The teacher asks lots of questions - here are a few random examples from one transcript:
Let's begin with the circle; now what might we draw?
Why?
Wait, aren't all quadrilaterals cyclic?
Let's see how this helps us understand cyclic parallelograms. In a parallelogram, what do we know about opposite angles?
and puts in diagrams. He posts a selection of student answers, typically only the correct ones - no idea how many incorrect ones there are, but he does sometimes say things like "Some of you are saying... But...". Presumably he also selects the suggestions that take him in the direction he wants to go! There are TAs behind the scenes, and I'd hope they are helping the students who are not submitting answers that are what the teacher wants (whether they're wrong, or whether they're just not the desired direction) but of course I haven't seen that.
