Multiple choice questions in mathematics

It must be exam time. Discussions of multiple-choice test questions are in the air.

Terrence Tao has a nice post about multiple choice questions in mathematics (it is a follow-up to this post of his).

He writes about the pros and cons of giving multiple choice questions in a mathematics class. For example:

These quizzes give a misleading impression of what mathematical problem solving is, and how one should go about it. In actual mathematical research, problems do not usually come with a list of five alternatives, one of which is correct; often, figuring out what the potential, plausible, or likely answers could be, or even what type of answers one should expect or whether one should ask the question at all, is as important as actually identifying the correct answer. Multiple choice quizzes also tend to reward quick-and-dirty or sloppy approaches to problem solving, as opposed to careful, deliberate, and nuanced approaches.

Then he writes that such questions lead to:

overthinking a multiple [sic] quiz problem, searching for some subtle trick, degeneracy, or exception in the wording of the problem, or trying to play some sort of “metagame” in which one is trying to divine the intent of the examiner.

He then alludes to this wonderful scene from the Princess Bride.

This comment and the Princess Bride clip lead us to the next post about multiple choice questions.

In Ian Ayres’s article “The Art of SATergy” on the Freakonomics blog he writes about the process of second-guessing multiple choice questions in just the way Vinzzini did in Princess Bride.

He asks the readers to deduce the answer to a multiple choice question without knowing the question (treating it as the kind of “metagame” Tao mentions). The choices are:

a) 4\pi sq. inches
b) 8\pi sq. inches
c) 16 sq. inches
d) 16\pi sq. inches
e) 32\pi sq. inches

Read his post to check your answer and to read Ayres’ Vinzzini-esque justification (which was the same reasoning I used to find the correct answer).

[via God Plays Dice]

Finally, I would like to mention a product that my colleagues and I use once in a while. They are scratch-off multiple choice sheets. The name of the product is Immediate Feedback Assessment Technique, (“IF-AT”).

Each question on the sheet has four scratch-off multiple choice answers, one of which has a star beneath it. The students must scratch off choices until the star is revealed.

The benefits are that the students can see immediately whether they have chosen the correct answer and the professor can see how many tries it took for the student to find the correct answer. The students have a good time with this—especially if they are working in groups (“OK, I’m going to scratch off (b). Is everybody OK with that? OK? Are you sure?… [scratch, scratch, scratch]… noooooo!”).

I supply the pennies.