You may think you have explained the proof perfectly clearly. But chances are your students haven’t understood much at all!

This is the idea from the article ‘Lectures in Advanced Mathematics: Why Students Might Not Understand What the Mathematics Professor Is Trying to Convey’ by Lew et al.

The authors conducted a study where they recorded a lecturer presenting a proof, and they asked students to watch it. They asked the lecturer what were the key points he was trying to convey, and academics who watched the video agreed that he had made these points clearly. But even after several viewings, the students could not pick up on these points, even when asked the direct question “Did the lecturer make this point?”

The problem of understanding complex ideas, including proofs, in lectures is essentially a problem of cognitive overload. In order to make sense of a complex idea in a lecture, the student is being asked to do multiple things at once. Firstly, they need to listen. At the same time, they need to encode what is being said and written, which consists of three things:  (i) Understanding each point or idea,  (ii) Integrating new point with previous points., (iii) Integrating new point with one’s prior knowledge. Finally, they may need to also need to record the information (i.e take notes). That’s a lot of multi-tasking, and research shows that humans are really bad at it!

Here are the main ideas in the article:

1. Students prioritise written comments over oral comments, so make sure you write down the important stuff.
2. Students and lecturers have different ideas of the purpose of presenting the proof, so make sure expectations are clear at the outset.
3. Assess more than the formal, logical aspects of proofs.

The first point is problematic for many of us who write a proof line by line, and orally narrate why each line follows from the last. It’s therefore important to write down these holistic ideas. A colleague has a lovely idea of giving ‘thought-bubbles’ in the middle of proofs, which encourages students to stop and think about where the proof is going.

I was particularly fascinated by the idea that part of the problem is that lecturers and students have different intentions when it comes to understanding a proof. For students, understanding a proof means being able to clarify the ‘correctness’ of the proof, how each line follows from the previous lines. That is important to lecturers too, of course, but that is only part of the story. For lecturers, it is important to understand the proof holistically. What is the overall idea? For those of us that use electronic voting systems, we could ask holistic questions about the proof before we begin. We could also ask students to study the validity of the proof before the lecture. Then the lecturer can focus on the holistic ideas.

Another unspoken idea is that lecturers (consciously or subconsciously) think that whenever they are teaching a proof, they are adding to the student’s toolkit. That is, that they are not just introducing this proof in itself but introducing techniques that may be used in future proofs. Students are unlikely to have the sufficient background knowledge to make sense of this more general idea. I think we need to think carefully about how we help students ‘add to their toolkit’, but equally, we need to avoid isolating proof techniques into a separate course, which would perpetuate the idea that many students hold that proofs is a subtopic of mathematics. Another great idea to overcome this (from the same colleague as the thought-bubbles!) is to give students an ‘exit ticket’ at the end of a lecture, which might ask whether they had seen a similar proof technique before, and where.

How might we go about assessing more than the formal, logical aspects of the proofs? Well, for sure we shouldn’t be asking students simply to state and prove a theorem from memory, as this encourages only surface rote learning of the proof. Perhaps we could ask students to state in their own words what is the main idea of this proof? Or ask why a proof fails if a constraint is removed?

References

Article: How to Help Students Understand Lectures in Advanced Mathematics by Keith Weber, Timothy P. Fukawa-Connelly, Juan Pablo Mejía-Ramos,
and Kristen Lew

Article: Lectures in Advanced Mathematics: Why Students Might Not Understand What the Mathematics Professor Is Trying to Convey by Kristen Lew, Timothy Patrick Fukawa-Connelly, Juan Pablo Mejía-Ramos and Keith Weber