Sunday, March 22, 2009

Musing on Multiple Choice

I was reading back-posts on Dave Marain's Math Notations blog.
From 2008, he has Figure Not Drawn To Scale!

In the circle at the left, O is the center, A, B and C are on the circle and OABC is a parallelogram. If AB = 6, what is the length of segment AC (not drawn)?
(A) 3√2 (B) 3√3 (C) 6
(D) 6√2 (E) 6√3

Dave also includes "POINTS TO PONDER:"
  1. Is this an appropriate standardized test question?
  2. Are you an opponent of multiple choice (aka, "multiple guess") questions. Why?
  3. We can also say much about the issue of drawing figures that do not appear to be what they are? Is it just the testmaker's way of misleading or trapping students or is there a valid purpose to this?
I think this an excellent question, but that depends on the test. It might be inappropriate for the PSAT, just right for the SAT I, and too easy for more advanced levels. It has a good balance between the knowledge required and the amount of critical thinking necessary. You need to twig on the parallelogram and it's properties, that it's also a rhombus (at least by properties, if not by name), that the triangles are congruent and equilateral and finally find the length of the line in question as two heights. All fairly easy steps if one understands the material.

This question is, in my mind, perfect for demonstrating that critical thinking alone is not enough. A student needs to memorize/ understand many properties and theorems as well as master critical thinking skills and all that 21st century garbage. I say "garbage" in the nicest way -- I actually refer to the attempt to teach the latter without the former. I am firmly on board with "Poor Elijah" and others who deride the "transformation to 21st Century thinking" if it doesn't include knowledge and facts for the students to be critically thinking ABOUT.

Dave's second question about the utility of multiple choice (despite the "clever" dig about "multiple guess") leads me to point out that the utility of ANY test is a function of the test creator's abilities rather than any inherent properties of the style itself. A good teacher can make a good test, whether it be MC or short answer or extended answer.

I think multiple choice is actually the hardest style to do well. The problem is that you need to do all of your thinking, grading and question-testing before you put it together and administer it. It's quick to score though and that makes it easy to give immediate feedback - use it for homework checks. I know a bunch of college professors who love it because it allows for much analysis and statistical scull-sweat. In general, I'm for it. Cheating is an issue, but a slightly randomized set is pretty easy to do.

My principal hates them because he thinks they are "inauthentic" but I will ignore him until he learns to differentiate between "their" and "there," between "then" and "than" and between "percent of" and "percent more than."

Lastly Dave asks about the "Not drawn to scale" thing. I love it. I can't draw worth a damn and I'm an engineer - go figure. All of my diagrams on the board are "not to scale." I think it's good "critical thinking" to learn to determine what is said or stated and then determine the next steps. Too often, a student will haul out a calculator and estimate/guess-and-check and assume he's done while assuming information that isn't true or that needs proof.

This problem above loses three out of five essential ideas if the diagram is perfect, downgrading it from "hard" to "easy." That's fine if you are solely after the height of an equilateral triangle. I like this one better.

That's all for now. Have a lovely snowy day.


  1. I agree with you here, and I would like to ask you a question. You refer to a principal who has difficulty with grammar. I have noticed that mathematics teachers tend to be much "cleaner" with the mechanics of language than teachers of other subjects. Have you observed this as well? Why is this, do you think?

  2. I think that those with a mathematical mind care about details more than many others. Once we get the rules, I think we do tend to be more pedantic about following them.