# Flexible Clicker Questions

I thought I would share a story about a clicker question I used yesterday in my linear algebra course. Although not all of my readers will follow the mathematics, I hope they’ll all appreciate some of the pedagogical ideas mentioned below.

This semester I’ve brought a bucket into class with me every day. At the end of each class, I encourage my students to write questions they have on the day’s material on slips of paper and drop these questions in the bucket. This gives me useful feedback on a regular basis regarding what my students find confusing. I try to respond to a couple of the more common or interesting bucket questions at the start of the next class.

It occurred to me a few weeks ago that a given bucket question might be of use to a few students but not all of my students, making it less useful to review during class. So I’ve started trying to turn bucket questions into clicker questions, so that they provide me with a better sense of how many students share a particular student’s confusion. I’ve been pleased with how this has worked out. For example…

On Monday, a couple of my students asked related bucket questions that I turned into the following clicker question:

Is it possible for the standard matrix of a linear transformation not to have an eigenvalue?

- Yes – High Confidence
- Yes – Low Confidence
- No – Low Confidence
- No – High Confidence

You’ll notice that I’ve include confidence level in the answer choices. I’ve been doing this regularly this semester for questions with only two answer choices (Yes/No, True/False). I find that knowing, say, 65% of my students answer a True/False question correctly doesn’t provide me with very useful information on their understanding since half of the students with no idea about the question are likely to answer correctly anyway. Including confidence level provides me with a better sense of how difficult my students find a question.

After my students had a chance to think about this question independently, they voted:

As you can see, the class was almost evenly split on this question, with the majority of them not very confident in their answers. This told me that we should spend more time on the question, so I had them discuss the question in pairs and re-vote. During the pair discussion, one of my students, let’s call him Jack, asked, “When you say eigenvalue, you mean just real eigenvalues, right?” That kind of gave away the question, since I was asking this question to see if students would remember that eigenvalues can be complex numbers. We saw a linear transformation in the previous class that had no real eigenvalues but did have complex eigenvalues, so this should have been on their radar. However, we also saw a useful way to visualize the effects of real eigenvalues. That method doesn’t work for complex eigenvalues, so it’s likely that some students weren’t considering complex eigenvalues when they answered this question since we didn’t have a tool for visualizing them.

Here are the results of the second vote, “tainted” by Jack’s question:

As you can see, most of the students went with Jack on this one, asserting that the standard matrix must have (possibly complex) eigenvalues and feeling confident in this assertion. This is a reasonable assertion because every matrix of size *n* x *n *must have exactly *n* eigenvalues, counting complex ones and allowing for multiplicity. That was an assertion I made a couple of class sessions ago.

So far, so good. I was disappointed that Jack had “spoiled” the question, but it still helped make the point I had intended it to make. Then one of my other students, let’s call her Juliet, asked, “What if the standard matrix isn’t square [that is, what if isn’t *n* x *n*]? Then it wouldn’t have any eigenvalues, right?” Good point, Juliet. I had commented on Jack’s question that the clicker question should be read as stated, so that complex eigenvalues should be allowed. Juliet essentially called me on that, noting that the clicker question doesn’t specify if the linear transformation in question has a square standard matrix. In fact, linear transformations need not have square standard matrices and non-square matrices don’t have eigenvalues, so the correct answer to the clicker question is “Yes.”

This question ended up working better than I had hoped since the fact that non-square matrices don’t have eigenvalues wasn’t clear to the students based on past class sessions. In fact, several students had asked about that issue in their bucket questions at the end of the last class session. Juliet’s question presented us with a great opportunity to clear that issue up, and I was able to enlist a couple of students (including Juliet) in helping me prove the result about non-square matrices at the chalkboard. As a result, what started as more of a recall question (Do students remember that every square matrix has eigenvalues?) turned into more of a concept question (Do students understand where eigenvalues come from well enough to argue that non-square matrices can’t have them?).

What are the takeaways from this story?

- Clicker questions need not be written as well as exam questions. This question had two wrinkles to it–one that I planned and one that I didn’t plan. That would have been a minor disaster on an exam question, but it worked very well for a clicker question since it helped me surface not one, but two, student misconceptions and helped generated useful class discussion about those misconceptions.
- Handling a clicker question like this took some agility on my part. I probably could have done a better job facilitating this question (particularly in my response to Jack’s question), but it certainly helped that I was willing to react on my feet to the discussion as it emerged.
- Writing good clicker questions can be as easy as adapting questions posed by students prior to class. My question bucket has yielded several useful and engaging clicker questions just in the last couple of weeks.

Have you asked any clicker questions that didn’t go as planned but turned out to be useful nonetheless?