Dealing with Misconceptions, Part 2

In my last post, I dealt with ways of handling student misconceptions in the moment. In this post, I will discuss how I follow through to make the course better, both from class to class and from semester to semester. I will draw on my Transition to Proof course as the primary example.

I am teaching Transition to Proof for about the fifth time this semester. At this point, I have a good idea of which proofs will cause students the most trouble, and I have specific ideas of what attempts I am likely to see. I have built up this mental cache of ideas by noting what sorts of proofs I have gotten in the past, keeping track of the activities and problems I have used with students, and remembering what kind of effect those items had on the students.

I use this information from class to class to make decisions about whether a misconception from one class needs to be dealt with in the next class. For instance, students had some misconceptions and general confusion around the logical terms "contrapositive," "converse," and "negation." One way I deal with misconceptions is to have multiple problems that center on the same topic, so that we see the same idea come up repeatedly in different ways. In addition, for the first time, I am trying to make use of this information by creating concept questions and using interactive engagement alongside IBL. 

Interactive engagement (IE) has been around for some time, and is probably best established in physics as a mode of instruction that produces significant gains in students' conceptual understanding. More recently, some evidence has emerged that IE has a significant impact in students' understanding of calculus as well (Epstein).

The concept questions I have created are short, multiple-choice or true-false questions that attempt to elicit students' misconceptions, so as to create a space for dialogue that leads students to confront the error in their thinking, and therefore come away with a more robust understanding. In the case of the logical terms, I created a few items that asked questions like, "Which of the following statements is true exactly when the statement, If A, then B, is true?" Or, "Which statement has the opposite truth value to, If A, then B?"

I have been using the questions by projecting them at the front of the class (and making them available for students to view as a Blackboard quiz or Google Form), asking students to answer each question, and then discuss their answers with a partner. Then we go through solutions as a class, usually by having a student explain their answer verbally, but sometimes we may draw a diagram to assist in the explanation. This has led to some good discussions, and has allowed us to zero in on specific issues students are having, without having to prove another theorem or proposition. It's relatively quick and focused on the issues students are having. 

In a recent class, there was disagreement about which choice was an equivalent expression of the definition of one-to-one. Because there were a number of students holding each opinion, there was a lively discussion among students, in pairs or small groups, attempting to decide which option was correct. The room was abuzz with mathematical discussion. As a whole class, I called on students and found three different answer choices that students thought might be correct. Eventually, a couple of students were able to remind the class of what we have learned about the contrapositive, and thereby convince the class of the correct option.

These kinds of discussions are exactly the kind of interactions that let students overcome misconceptions and solidify their understanding of key ideas from the course.

How else do you follow up to address student learning issues?

Dealing with misconceptions, Part 1: Seven ways to handle misconceptions in the moment

One reason that I like using inquiry-based learning is that I get to see students' thinking as we progress through the course. When I encounter students exhibiting some kind of misconception, part of me reacts in the moment to find ways to help students confront the misconception, and part of me is taking mental notes about the difficulties that students are having, so as to make use of them in the future. In this post, I take up the first part, the work that I do in the moment, as the thinking emerges.

Let me define misconception. For the purposes of this post, a misconception is any error that I judge to signify a lack of understanding of some aspect of the subject under discussion. I do not intend to count as a misconception things like copying errors, or simple algebra errors that do not change the meaning of the solution to a problem.

In the moment, there are many ways that I might deal with a misconception, and these are somewhat dependent on the misconception. The strategies below are written from the perspective of teaching proof-based class, but apply more widely than that. To wit, here are some strategies:

1. Open a space for peer review. In a group or whole class setting, it is my standing practice not to be the first to evaluate what a student says or writes. So if I see something amiss, I may turn the work over to the group or the class and just open a discussion with, "What do you think?" or "What comments or questions do you have?"
2. Refer students to a definition. One of the core practices of mathematicians is to use definitions. If I see that a student either applies a result not proven, or otherwise deviates from the definition, I will ask all the students to refer back to the definition, read it, and see how it applies to the issue at hand.
3. Ask for an example. If no one volunteers an example, I will ask students to come up with an example. I often tell them to work things out with an example on the side, so that they can see how what is in a proof makes sense (or not) compared with their example.
4. Walk through line by line. Sometimes, I have the students go through each line, read it, and explain the justification for it. 
5. Apply our understanding of the context. For an introduction to proof course, the context may be our concept image (i.e. our prior experience with the concept), for instance, having an idea of what "even" or "function" is supposed to mean. It also refers to applying our understanding of quantifiers. I may ask, "Is this a 'for all' or 'there exists' proof?" "Is this proof covering all possibilities, or is this one special case? Do we need to consider other cases?"
6. Ask a leading question. Here I may focus students directly on the issue. Have we proved this assertion here? Are we sure this is true?  
7. Provide an example to consider. If students are unable to come up with their own example of the statement being considered, I may provide them with an example and ask them to work it through side-by-side with the proof.

Note the move from general, open-ended questions to more narrow, pointed questions or tasks.

After dealing with the misconception on the spot, I try to make a note of what happened. My aim is to capture enough of the misconception so that I can make a decision about how to handle it in a future version of the course. That process will be covered in the next post.

What other strategies do you have? What did I miss?