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:
- 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?"
- 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.
- 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.
- Walk through line by line. Sometimes, I have the students go through each line, read it, and explain the justification for it.
- 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?"
- 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?
- 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?