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.