Today’s topic in calculus was concavity: specifically, what f”(x) tells you about f(x). Students buy the fact that the sign of f’ gives you the increasing/decreasing nature of f. They can wrap their heads around that pretty well. However, the second derivative is a bit more abstract to think about. I talk about the slopes of the tangent lines increasing and what that would do to the curvature of the graph. But, ultimately, most students choose just to memorize that the sign of f” gives you the concavity of f. And then they go merrily on their way, working the problems from the textbook.
Today, I had a student that was not going to just take my word for this fact. He kept asking, “But why?” And he kept challenging me to think of different ways to explain it to him. I think he finally wrapped his mind around it.
I love, love, love the fact that he was not willing to move on until he could convince himself of this mathematical truth. And I love that he wasn’t scared to keep telling me that he didn’t quite understand…yet.