In the first semester of 2025-2026 I am teaching Advanced Algebra. In the first semester of 2024-2025 I gave a lecture course in Oxford (about how information theory enables proofs of central results in some fields within pure mathematics), and between 2018 and 2022 I was involved in small-group teaching for Trinity College, Cambridge.
I have found that teaching can be an extremely uplifting experience (both for the teacher and for the student) when performed well, and to increase the number of people feeling that way I have decided to collect here a few attitudes that I have found most useful to take in my teaching style to help students get closer to their greatest potential when I talked about maths with them over the last few years. I will focus on five essential points, but might eventually write a longer version.
Considerably emphasize the central mathematical reasoning skills.
I think that it is important that my students understand the ideas and proof techniques of a course as thoroughly as possible. In particular, they should be able to understand why small modifications of results of proofs seen in the lectures would or would not work: for instance, why a statement still holds under weaker assumptions if it does, or what a simple counterexample or class of counterexamples would be otherwise. Even more importantly than what I have to say about the results themselves, I aim to ensure that the student eventually acquires the skills necessary to ask a high number of simple questions, and to efficiently locate simple answers to these, when such answers are indeed simple. As a way of encouraging these habits in the students, I emphasize that learning them will save the student from doing a large amount of rote memorisation, as many will tend to naturally remember such personal processes of understanding far better than they would memorise a list of proofs externally presented to them. Correspondingly, I frequently ask the student about various changes in the results that we aim to prove, although the end goal is that the student eventually arises with these suggestions by themselves. Likewise, I encourage the student to note analogies when they arise, and ask themselves how strong the analogy is, and whether and how it could be used to unify statements or proofs.
Motivate the contents as much as possible, at various scales.
I find that while lecturing undergraduates, it is worthwhile to begin by motivating the contents of the lecture, in terms of both their relation to mathematics already known by the students at that point, and (if possible) their relation to non-mathematical subjects and real-life applications. Both will by themselves help to raise the curiosity of the student, but doing the latter also keeps the attention of students who are highly able to do mathematics but are primarily motivated by applying the corresponding skills outside of academia once they graduate, which hence leads to a class that is on the whole more enthused than it would be had I only done the former. One feature of course material which I found enticing as a student was having not only the course as a whole be properly motivated, but having a motivation even at the scale of small steps in a proof: why can this particular step be anticipated as potentially worthwhile to make, rather than some others ? I strive hard to produce such local motivations. Another similar technique (which may be slightly artificial, but which I have found to work for me) is to tell a small story about how a proof can arise in a fairly systematic manner from a rather small class of simple prompting questions that the student can ask.
Ensure that the student does not underestimate the research opportunities already available at their level.
I tend to explicitly tell the student when they do good work. When a student furthermore shows interest beyond merely mastering the material, I ask the student whether they would possibly be interested in doing research. One thing that I have realised is that a very large proportion of students (even of very good students) routinely appear to assume that essentially no researcher would be interested in working with them, when the reality in many cases is that researchers would very much like to find additional students, provided that the level at which they do mathematics and their motivation are both sufficiently high. In particular, whenever I find a promising student I try to find the time to sit with them, discuss their research interests, and encourage them towards some concrete first steps, such as reading the basic literature on the corresponding topic (this is, of course, easier in my area, combinatorics, than in some others) and then considering writing to an expert in the topic once they have done so. As this discussion between the student and I may well in some cases be the only such discussion that a student may have in a whole semester or even year, having that conversation is an easy way to have a large positive effect on the student's future.
Remove unnecessary obstacles.
I aim to go through mathematical interactions in a way that is as relaxed as possible (while still being enthusiastic about the material). I take the view that it does not help students to make them feel that the material will be difficult for them to grasp, over and above the fact that they will have to spend a certain amount of time working in good faith to become familiar with enough questions, examples and ideas. In particular, this effect will be even worse for disadvantaged students. One thing that I aim for when talking to students regularly is to gradually build up the confidence of the student, for instance by slowly increasing the difficulty of the questions that I ask the student. I like to stress how one kind of task can be made easy in a systematic way when this is possible, even though it of course does not originally look that way.
Underscore how the relevant area is not a finished subject.
One aspect of combinatorics which I had found particularly attractive as a student was that it was easy to formulate a large number of open problems that I could easily understand at the time. Although combinatorics is sometimes cited as the prime example of an area that lends itself to that, there are several others that do. I favour regularly mentioning such problems when teaching even first-year or second-year undergraduate courses when these problems can be found, for several reasons. The first is to remind the student that the area still has a lot to offer that is not yet discovered or not yet in finished (or, frozen, one might say) form, in sharp contrast with the appearance of a lecture. The second is to communicate to the student that being able to contribute to mathematics is not such a remote prospect for them. The third (which can alternatively be achieved by occasionally setting to the students a question with a known answer but clearly labelled as very difficult) is to offer to the student the possibility of a small experience of research lasting an hour or so in a way that they know in advance will not encroach on their confidence if they do not get anywhere (as they are expected to). In a similar spirit, I also like to mention recent developments from the last few years, especially when the pace of progress has been fast.