We’re careening towards to the end of the semester in calculus, and I know I’m mostly posting about stats, but this just happened in calc and it applies everywhere.
We’ve been doing related rate problems, and had one of those classic calculus-book problems that involves a cone. Sand is being added to a pile, and we’re given that the radius of the pile is increasing at 3 inches per minute. The current radius is 3 feet; the height is 4/3 the radius; at what rate is sand being added to the pile?
Never mind that no pile of sand is shaped like that—on Earth, anyway. I gave them a sheet of questions about the pile to introduce the angle of repose, etc. I think it’s interesting and useful to be explicitly critical of problems and use that to provoke additional calculation and figuring stuff out. But I digress.
The issue is the deep meaning of change as we study it in calculus. We old-timers know that we should set up an equation for the volume:
and then differentiate both sides. With respect to t. It emerged in discussion that most students wouldn’t have thought of differentiating with respect to time, and that if you did, the right hand side would simply be . That is, they wouldn’t follow through on the implicit differentiation. Also, the idea of d/dt being a rate, or a speed, was apparently news to them, even though there have been a lot of speed and acceleration problems earlier in the course.
Having said that, they can implicitly differentiate the pants off anything. When they’re told—explicitly. They just can’t do it when it emerges naturally. Despite my efforts throughout the semester, they’re good at mechanical things and not so good at, dare we say it? Understanding.
Part of that is my failings as a teacher. I bet I’d improve if I did this a few more years. I knew in advance that understanding wasn’t mechanical proficiency, and that I preferred the former. But it is appallingly easy for them—and me—to take the easy path and value Just Being Able to Take The Derivative. Maybe my teaching is just good enough that I notice how shallow the understanding is.
But part of it is also the syllabus I’m tied to, the “cultural” expectations of what has to be in a calculus course. The tradition of the first semester course lets us say that we have to do all these theorems and techniques, and that students are already supposed to know all about functions from precalc. In fact, they don’t; and then there are a number of deep concepts in calculus itself (such as speed is derivative of any position with respect to time) to which we give short shrift.
And part of it, I worry, is that we demand that too many people pass calculus in order to become what they want to become, when in fact, what I really want is for them to be, I don’t know, functionate: able to handle quantitative relationships among quantities flexibly and fearlessly.