# Multiple Mark Questions

I’ve recently become a fan of the “mark all that apply” type of question my classroom response system facilitates. I call these “multiple mark” questions in my book. Here’s one I used in the linear algebra course I’m teaching this fall.

This question is adapted from one of the questions written by Project MathQuest out of Carroll College. Their version of the question wasn’t a multiple mark question. Instead, it included a fifth option, “More than one of the above are possible.” While that option makes the question more interesting and more challenging for the student, it also yields inconclusive data about student learning since students submitting that response may have different ideas about which of the four options are possible. That’s not all bad, of course. Given my interview with Kelly Cline, one of the PIs for Project MathQuest, I can imagine Kelly leveraging that ambiguity into a productive classwide discussion of the question.

However, I decided to turn this question into a multiple mark question by adding the instruction, “Mark all that are possible.” As you can see from the results, 20 of the 20 students present that day indicated that option 3 was possible, 19 of the students indicated options 1 and 2 were possible, and 14 of the students felt that option 4 was possible. This was very useful feedback for me, since I could quickly tell that the class was in agreement on options 1 through 3, but option 4 deserved some further discussion.

I’ll admit, however, that I got a little tripped up on my own question logic here. As it turns out, all four options are possible, which was not my intent when I included this question in my lesson plan. Option 2 is only possible if the third plane intersects the two overlapping planes and option 4 is only possible if the three planes are parallel because they are in fact the same plane. The way I’ve worded the question, these wrinkles aren’t addressed, making all four options possible. As a result, the question doesn’t do a great job at uncovering student understanding of these wrinkles.

Here’s the question I should have asked instead:

Suppose you have a system of 3 linear equations in 3 variables. Which of the following conditions would guarantee that the system has an infinite number of solutions? Mark all that apply.

- All three equations represent the same plane.
- Two of the equations represent the same plane.
- The three equations represent planes that intersect along a line.
- The three equations present parallel planes.

With this question, only options 1 and 3 are correct. With option 2, it could be that the third plane is parallel to but distinct from the two overlapping planes, yielding no solutions instead of infinitely many solutions. With option 4, it could be that the three planes are not the same plane, again yielding no solutions instead of infinitely many solutions. This wording of the question puts the special cases in their proper places.

Have you used multiple mark questions? Do you find them more difficult to write? How do your students respond to them?