**When thinking is too much trouble**

My exams seldom contain surprises, but my students' answers do. Since I'm a firm believer in keeping track of student progress with frequent quizzes, I telegraph my punches. Students have plenty of opportunity to discern what facts, techniques, procedures, and calculations I deem the most important. (They also—most of them—learn the importance of regular attendance so as not to miss these pre-exam rehearsals.)

Of course, some students take it too far. These are the students who have had the unfortunate educational experience of intensely patterned teaching to “the test.” These are also the students who badger me for “practice tests” in advance of each exam. What do they want? Problems that are

*exactly*like the ones they'll encounter on the exam.(I presume I'm permitted to change the numbers a little bit.)

What surprised me most this past school year was the discovery of this tendency among my calculus students. I was used to seeing it in my lower-level classes like algebra, but in multivariate calculus? An example of the behavior of the template-driven student will suffice. You'll see the problem, even if the terms are mysterious.

On a quiz I asked multiple questions about the gradient of a function of two variables. In part (a) I asked them to compute the gradient and evaluate it at a given point. In part (b) I asked them to use the gradient to compute the directional derivative in a given direction. In part (c) I asked them to calculate the greatest possible value of the directional derivative. In part (d) I asked them to find the direction in which the greatest possible directional derivative would occur. Pretty standard stuff.

On an exam I asked my students to (a) compute the gradient of a function of two variables and evaluate it at a given point. No problem. In part (b) I asked them how large the directional derivative could be? Several students were thrown for a loss. They wanted to compute a specific directional derivative, but instead I was asking them for its maximum possible value. There was no way they could do what they wanted to do because I had not provided a direction,

*so they made one up*. They had memorized the pattern in the quiz and insisted on replicating it exactly on the exam. Since I had, in effect, swapped (b) and (c), they were deeply perplexed and forged ahead with the moves they had learned by rote.

Embarrassing! It wasn't a very large number of students, but I had been hoping they had been weaned away from this tendency by the time they arrived in the calculus III class. I learned otherwise.

## 3 comments:

In part (b) of your quiz problem I think you meant to say "...compute the directional derivative in a given *direction*."

Edit: "the directional derivative in a given derivative [sic]".

Indeed. Thanks. All fixed now.

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