As I have mentioned before, our school has AP Calculus BC as a second year course in high school calculus. One of the main reasons we do this is that we feel it creates a space where we can explore, we can dig in to deep problems, and we can give our students the opportunity to really reflect in a way that the pace of a one-year (or 1.25 year) course does not allow. I have heard students say things to each other like ‘Last year, I knew how to solve those problems, but I really did not know why I was doing it that way.’ The pace of the AP curriculum creates such pressures that students too often can fall into that trap of just learning *how* to do something. One of the tangible joys of teaching the BC course under these conditions is that I can routinely take days like today where I just toss a problem set at my students and listen to them think. The problems bounce around – sometimes they are calculus based problems, but more often they are not. This morning our problem set started with this question:

Find the point of intersection of the lines tangent to the curve if those lines are tangent at x = 1and x = 5

This is not a challenging problem in theory for my students in this course. However, the algebra is a bit snarly and unpleasant. They debated amongst themselves and seemed happy at the end about their work. I stepped up and did the problem on the board, with their guidance, and we arrived at the conclusion that the x-coordinate of the point of intersection was x = 3. Hmmm, I wondered aloud whether there is something going on here that x = 1, x = 5, and x = 3 seemed to have a nice relationship. I quickly examined the boring old standard parabola at the origin facing up and we saw the same thing. One of my students commented that this was *probably* generally true but these examples certainly did not prove this in any way. I sat back down and got to my work and a few minutes later a student named Richard called me over to show me that he had confirmed that this would *always* be true by replacing the 1 and 5 from my problem with *p* and *q*. His answer, instead of 3, was the average of *p* and *q. *

I cannot accurately convey how pleased I am that he was persistent enough to do this. It would have been the easiest thing in the world to accept this conclusion and to move on to the next problem *especially* since his neighbors had moved on. So he removed himself from a stimulating conversation with his friends to satisfy his curiosity (and maybe to prove something to himself about his ability to push through) about a problem that likely will not have any future impact on his calculus grade. What *will* have an impact is his increased belief that he can solve a snarly problem. Pretty pleased, I must say that my Friday morning started off this way.