Riley Lark is my homie right now.
I’ve written a couple times about the usefulness of generating some cognitive dissonance, i.e. some sort of crisis of knowledge, to push students to think deeply. In both contexts I was talking about developing their skills with proof in particular, but actually I think shaking kids up / freaking them out / giving them something to look at that provokes them to question their assumptions is pretty much a great thing to do whatever you’re teaching. (It must be boring by now to also hear me say that I think proof should also be part of whatever you’re teaching, but there, I said it anyway.1)
Anyway, if it wasn’t completely clear what I meant by “cognitive dissonance” or “creating crisis,”2 this is what I meant.
Check out how in the last paragraph, the students have started to develop features of the language of mathematical rigor (“assuming that…”) spontaneously. No one told them to do that. But when your understanding is being shaken up, you don’t have a choice.
On the very same day that Riley posted this lesson description, he also started a conversation I’m very happy to see people having. He is leaving the full-time, full-pay job to hew an entrepreneurial path (designing SBG-supportive software). I did something like this as well. Like Riley, I’m not naturally comfortable with self-promotion, though the last three years have given me a lot of practice and really stretched me on this front. Anyway, Riley wants to hear your thoughts on the place of self-promotion in the math edublogosphere. I do too.
 Because I’ve been misunderstood before, let me just clarify that by “proof” I do not mean an overzealous exercize in rigor (mortis); I just mean that no matter what you’re teaching, everybody in the room should be talking about (even, accountable for) why they think it’s true. (In my humble opinion, class should never move on without this. Mathematics as a whole never moves on without this; why should math class be any different?)
 For the record, “creating crisis” is actually dy/dan’s vastly superior paraphrase of what I said. What I actually said was “giving them a crisis,” which sounds a great deal like “giving them a complex,” which is not what I had in mind.