Perpendicular Thinking

As I contemplate this past academic year of teaching, I feel pretty good about my classes (and students!), although, as usual, I have some ideas for things I want to change next time. In the end, through all the ups and downs, it was a good year.

And yet.. the perfectionist in me wonders why I can never get the absolutely key, essential, critical, first-order, important concepts through to every single student such that they can demonstrate mastery of these concepts on the final exam. This year, there was one student who somehow made it through 3 courses (1.5 taught by me) that used a particular fundamental concept in many different ways and was unable to show any knowledge of this concept. Do I need to draw it for you? OK:


I have been teaching for >20 years, and the fact that a Science major came up with D as an answer instead of C or A/B shocked me.

Possible explanations for the student's bizarre wrong answer:

- I wrote an ambiguous, poorly worded exam question. Evidence against this: every other student in the class got this part of the question right, and there is no possibility that configuration D is ever correct, no matter how poorly worded the question.

- The student is "intellectually challenged" and doesn't understand even basic concepts that have been presented repeatedly during a year of classes by different instructors. This is not a very convincing explanation and, moreover, is not supported by other evidence for the student's abilities in these courses.

- The student was rushing, was extremely careless, didn't read the question, just jotted something down to make marks on the paper. Maybe, but it's hard to imagine any level of carelessness that could lead to answer D.

- The student was cheating, but did not cheat well. That is, the student glanced at another student's exam, but got the questions and answers mixed up.

No, I don't believe that. I don't believe the student cheated, but even if I am wrong about that, answer D is not even close to any answer to any question on the exam.

- The student knew the answer but experienced exam-stress brain-freeze. Perhaps in a calm moment, the student could easily have produced the correct answer, or at least not a totally wrong, physically impossible, not-even-worth-partial-credit-it's-so-wrong answer. During the exam, however, the student wasn't thinking straight and, for this question, gave what was perhaps the most incorrect answer possible other than leaving the answer entirely blank.

I do not totally reject that possibility, but the concept in question is not complex. The brain-freeze would have had to have been catastrophic. And yet, the only explanation I find even somewhat plausible is that, under stress, the student confused the elementary concept with another concept that was also discussed in these classes. Out of stress or carelessness, the student described this other concept (but even that not correctly).

I suppose a take-home exam with liberal time allowance is a way to alleviate exam-stress of this sort, but I have tried that format in the (distant) past, and know that in this particular course, students prefer to study, take a final exam, and be done. Fortunately, in this class there are many non-exam graded activities so that all but the most severe case of exam-stress doesn't result in a dire grade situation. [And of course, students with documented learning disabilities can take the exam in a testing center, in a quiet room, and in some cases are given extra time to complete the exam.]

When I think back on the most recent class that I taught, the incident described here is one of the things that sticks out most for me. I'm not sure why, except that even professors who have been teaching for a long time may wonder (especially after grading final exams): Didn't I teach them anything? (Or the variant: Didn't they learn anything?).

The answer may well be yes, but some of us are affected by the spectacular outliers: the exam answers that are so bad, they shock us. What does it mean that I am still shockable after all these years of teaching? That I haven't seen it all yet? That there are still teaching adventures -- good and bad -- to be experienced? I hope so.