Saturday, November 12, 2011
I threw them a curve
Most math teachers would agree that we want two things from our students: (1) correct solutions to math problems and (2) an understanding of those solutions. Of course, some students are perfectly happy with mere technical facility: Please teach us the algorithm so that we can turn the crank on it, generate correct answers, get our college credit, and get the hell out of here. They balk when we probe for conceptual understanding. Other students, naturally, claim a profound knowledge of the conceptual underpinnings of the subject matter but lament their difficulty with the merely technical and computational aspects. Will the twain ever meet?
Course grades in math classes tend to be based mostly on the demonstrated ability to compute accurate results. It's more difficult to probe for evidence of their conceptual grasp. Occasionally, however, I give it the good old college try. Here's a graph I presented to one of my calculus classes. I asked my students to look at each of the points indicated by the red dots and make some judgments about the function and its first two derivatives.
My students had a little table to fill in. The instructions said, “Fill in the table, using +, –, 0, or DNE (for positive, negative, zero, and “does not exist,” respectively) for f(x), f ʹ(x), and f ʺ(x) at the indicated values of x.”
A small panic ensued. “Where's the formula for the function, Dr. Z?” “How can I compute derivatives if I don't have the formula?” I counseled them to calm down and consider that I wasn't asking for numerical values—yes, quibblers, except for 0—and that actual computations were unnecessary.
Consider, for example, the point corresponding to x = −1. The value of f(−1) is pretty clearly 5, hence positive. The point is also a local maximum, so a tangent line at that point would be horizontal; the slope is therefore 0 and that's the value of f ʹ(−1). Finally, the curve is concave down in the vicinity of a maximum, so f ʺ(−1) is necessarily negative.
No need to panic.
The trickiest case (if “tricky” is even the right word) is probably x = 3.2 (or thereabouts). It's approximately midway between a local maximum and a local minimum, suggesting that it must be at or near a point of inflection, where the concavity changes and the second derivative must be zero (or nonexistent). That takes a little discernment. In most cases, however, the answers should be evident to any first-year calculus student with a genuine understanding of the significance of the first and second derivative.
At the class's post-exam discussion of the results, the reviews for this problem were decidedly mixed. When pressed slightly, there was a grudging consensus that, “Oh, yes, it's clear now,” but my more computation-driven students remained unmollified. They preferred to demonstrate their differentiation chops on actual formulas using the rules they'd memorized.
The experience triggered an odd recollection with me. I remembered my grandfather at the dinner table, finishing off a meal my grandmother had prepared with a recipe she had never used before. She was eager for his verdict:
“Was it good?” she asked. “Did you like it?”
My grandfather nodded his head.
“Yes, thank you. It was very good. But don't make it again.”
A few of my students may despair, but I'm keeping that calculus problem in my recipe box.