Rethinking Maths education: A perspective from Olympiad and higher education

Baptiste Serraille, PhD student at ETH Zurich

In the present text, I wish to express some thoughts about teaching and AI. These thoughts come from personal experience and the reader will notice the lack of references. My goal is to highlight some aspects of teaching mathematics that I think we should keep in mind, how they are being challenged by the arrival of generative AI and to propose some solutions to counteract the potential negative effects. I also decided to not talk about any of the other impacts of AI. In the next lines, LLMs are thus considered cost-free. I am talking about two experiences with teaching, namely university teaching and the volunteering work that I do for maths Olympiads and tournaments.

1. Mathematics as an experimental science

Mathematics is the part of physics where experiments are cheap.

Vladimir Arnol’d

Not anymore.

Ibrahim Trifa

In this section, I wish to explain in three different situations arising in a student’s learning in mathematics the positive impact that examples and experiments have. This might be obvious to anybody reading this post $\text{\textemdash}$ and appears implicitly in other posts $\text{\textemdash}$ ¹ but I believe that there is value in laying it down. Additionally, as this seems important for many of us and is also up to discussion, there is a need to write it and discuss it. Mirroring the take I have, we will go through examples.

Let me start by saying that most of the mathematics I have understood was through examples. The first example in our life is learning about addition, multiplication and division in elementary school. The “long multiplication” algorithm is very complex: it involves single digit multiplications, the aforementioned algorithm for addition, and the use of carries. The division algorithm is even harder. After a few tries (more like a lot) I mastered the algorithms and all of it was making sense. I could not, at the time, prove why the algorithm worked. It is only later that I could do it and by then it was a mere formality. On a similar note, it happens occasionally that I need to explain to some high school student the polynomial division algorithm and that I lack the time to do so properly. In these cases, I resort to picking two monic polynomials $A$ and $B$ with $\deg(B)=1$ and starting the division on a piece of paper in front of them. This is usually enough to give them a taste of the actual result and we can then continue our discussion. I have no intent to give them a proof of the above algorithm for polynomials as they would see it later at university; but I am happy to give them just enough intuition.

The opposite situation also holds: if a student is confused by the proof of the polynomial division algorithm, the best remedy is for them to try it on multiple pairs of polynomials. Once they are convinced it works, they can re-read the proof.

Let me give the last situation I thought of. In my daily life I am reading through articles, lecture notes, listening to talks or having a discussion with a colleague and trying to learn from all those sources. Regularly, I come off after those situations thinking that I got it. However, when faced with the smallest, easiest application, I stumble and fail. This means that I had not actually grasped fully what I thought I did. It is only after some back and forth between exercises and the material that I am content with my degree of understanding. This shaped my learning experiences, I know that whenever I think that I understood a set of lecture notes this is probably a lie I am telling myself and that I need to be more careful. I might sometimes prefer rougher lecture notes, with lacunary proofs to force myself to fill them and get the most out of them.

In summary, I described three situations: the first one where we start with examples and they almost serve as proofs, the second one where we learn the proofs and understand them only through examples, and the last one, more blurry, where the examples show us that we did not understand. It is the last one that worries me the most with regards to LLMs.

2. University teaching

The short walk from the elevator to my office goes in front of a row of tables where students $\text{\textemdash}$ escaping the sunlight $\text{\textemdash}$ are working and throughout the two years that I have been using this path I could see the increase of computers showing a prompt rather that a pdf of a standard internet search, I also noticed a slight decrease lately. This was the first time that I realised how big of an impact LLMs were having. It is now apparent that we have to adapt, as teachers, to LLMs and to the impact they have on how students are working. I wish to develop here a thread of consideration on the matter with regard to university teaching with an emphasise on mathematics.

2.1 A focus on normal classes

I will start by stating my biggest fear. I fear that students experience the last point of the first section only too late, or never, and that it will result in a shallow understanding. Testing our new knowledge against examples is something that happened to all of us because we had no choice and now we are purposely recreating it because we know its benefits. I worry that with LLMs it is much easier to go through a series of exercise without actually doing them. Last year, during the topology course, I would upload the solution of week $n$ on week $n+1$ only (and $n+k$ in case I forgot to do it). None of the exercises were original but students needed to do some work in order to find their solutions on the internet. The exercises were not graded but we felt like delaying the upload of the solutions was helping them. Nowadays, the ‘lazy’ $\text{\textemdash}$ it will be apparent later why I put this word in quotation marks $\text{\textemdash}$ student can just prompt an exercise and get a solution, read it, lie to themselves about having understood it, and move on to the next exercise. I fear that this student will not realise the laziness and the negative impact of this approach and will only see the time saved. In practice there is a duality between short term and long term learning, we just hope that the current short term optimisation to pass exams leads to long term gains for the students. This is, I believe, mostly the case in the current system. The arrival of LLMs is reshuffling the cards and we need to adapt to it. On a separate note, let me point out the fact that LLMs feel particularly good and convincing to read. This is another pitfall of its use and adds to the previous worry.

We shall now dive into the psychology of these $\text{\textemdash}$ as we called them $\text{\textemdash}$ lazy students. University students $\text{\textemdash}$ and in general young adults² $\text{\textemdash}$ suffer from a lot of pressure. They might be moving out from their parents’ home and starting their adult life. They have to adapt to the increasing workload and difficulty of university classes. They might work on the side to pay for their housing. In some countries, they even have to pay outstanding fees to attend a university. They have to think about their future job and some of them might start to think about the future in general and gain some political consciousness (for the better or for the worse $\ldots$ ). In short, they face a lot of changes and pressure and as a consequence they can feel pushed to go to the shortest and fastest solution to pass their exam or do their homework. They now also have the opportunity. I believe that it is our duty to help them manage their use of LLMs. Of course LLMs are also new to us and, as such, we will make mistakes but it does not and should not prevent us from trying. We should also exchange about our positive and negative experiences in making our students work better with LLMs. I propose here a few points that I believe can be easily implemented, in order for students to focus on long-term deep understanding, rather than the short term.

Having a discussion about LLMs with our students. This is definitely the first thing we need to do as they are the ones affected by it. They will then think about it and decide whether we are correct or not.
There are situations where using an LLM is cheating and students need to understand it. Of course it is not always akin to cheating but in a myriad of situations it is. A very nice heuristic to keep in mind in order to understand this is to replace mentally the spot taken by the LLM by a stronger student. Trying to get a lesson explained by a stronger student would not be considered as cheating, similarly chatting with an LLM about the lesson should not be considered as cheating. However, looking at the solution of the exercise above the shoulder of a stronger student does feel like cheating and similarly, prompting an LLM for the solution of an exercise would qualify as cheating too.
Bonus points for work done in exercises classes. Maths classes have exercises classes that student do not always attend. In many classes, at least at ETH, there is a system of bonus points for handing in some homework. All of this is important and we could give incentives for the students to go more in the exercises classes. A system of bonus points for doing some exercises in class would be a good start. Points could then be awarded for trying the exercises, not even solving them.
Harder exams. Giving harder exams with rather different exercises from the homework could force the students to really understand the material instead of going superficially through it. Of course, the grading system and passing grades should be dealt with appropriately. The target audience would be the maths students.
More personal interaction? This one is harder to implement but I believe that one of the reasons that students use prompts is that it is quite hard to talk to a teaching assistant or professor, they are busy and have a lot of students. Prompting might palliate a lack of support from our side.

2.2 A focus on bachelors’ and masters’ theses

I want to dedicate a specific section to bachelors’ and masters’ theses. This will be my strongest and most radical claim. I claim that a student working on a thesis should not use any LLM at all, apart from maybe correctness of the language afterwards and the generation of pictures. I will share with you some of the things I said to my last bachelor student on our first meeting.

My first point is that when doing something for the first time, we should not be doing it with the help of an LLM. The reasoning is the following. One needs to understand fully what the exercise takes; what it is to read through a maths book on your own; what it is to fail to understand some passage and how one can get around the obstacles to eventually understand; what it is to be hopeless when none of what we are reading makes sense anymore. Only then it becomes profitable to use LLMs. When a student writes a thesis, it is usually their first time reading through an article and as such they should do it without the help of an LLM. Anything else would miss the point of a thesis in the first place.

Additionally, there is nothing that an AI can do that the advisor cannot do. Indeed, if one wants to understand something it is wise to let it sit in their mind before prompting it, by this time a meeting with the advisor would have taken place and the question can be asked to the advisor directly.

There is fundamentally a social element in how we operate in maths and our students need to learn it. We are not just cracking problems, but also discussing them with each other. This might be hard, for example asking a question we fear is stupid to a colleague might not be easy. This is nevertheless an important component of our work. It might be very tempting for a young student to prompt a “stupid” question instead of asking it to their advisor by fear of looking dumb in front of the person who would later grade them. On the contrary, I would argue that if there is a place where they should do it is with us. We should make it a safe environment for them to experiment with it. If they do not feel like doing it now, how would they do it later, whether in or outside of academia? We definitely do not want to live in a world where our fear of judgement directs us to LLMs.³

3 Maths Olympiads and tournaments

I spend some of my free time organising and helping with various maths Olympiads and tournaments. I will explain here what has changed and what has not changed in the past years.

Most of us have seen the highlights of some AIs getting gold at IMO. This is an outstanding achievement but not much has changed in our ways, at least for now. IMO is about students solving problems alone and with limited time at their disposal. This means that when planning a test for my students I still need to make sure I can solve the exercises by myself. This is crucial as I am trying to figure out what is doable by them. When preparing a class, I still need to solve the exercises by myself. How would I manage to teach them how a human could solve some exercise if its solution is given to me by an LLM? Even before the advances of genAI I had access to solutions of the exercises, and I would not look at them because it would miss the point of my job. I would also miss the enjoyment of solving the exercises. The biggest issue is cheating. In order to cheat before, one needed a strong friend willing to cheat and that could solve the exercises quite easily. There are not that many people. Another possibility to cheat would be to bribe or get the answers from the teachers or leaders, this indeed happened multiple times at IMO. However, the effort needed and the amount of people involved are now lower, this facilitates it. This year at IMO, there were metal detectors as well as internet jamming in the exam room, this will eventually be the standard for this kind of competition.

I am also part of the organisation of a maths tournament whose aim is to introduce high school students to research. The pitch is simple: teams of high school students are given numerous problems, for a few months they have to tackle some of them, and they have to produce a written solution. On the site of the tournament the teams gather and some of their papers are selected via a draw. They have then to give some reviews $\text{\textemdash}$ similar to a referee report $\text{\textemdash}$ of the other teams’ papers, present their own papers to everybody, and a person from each team will eventually go to the board to discuss the solutions. The idea of this last part is to recreate how an informal discussion between mathematicians would go. They are graded during all those steps by a team of jurors. This is overall very interesting and complement nicely what maths Olympiads teach in my opinion. If the reader is interested in these competitions, they should know that the first in the genre is named ITYM, that there are two consequent national versions in France and Morocco named TFJM and MTYM as well as a European version, ETEAM. Each of those tournaments have different rules and/or spirit but they abide to the same global logic.

The only rub is that with the advancement of LLMs and with the amount of time given to the students, we have seen an increased number of solutions that are AI-generated almost entirely. It used to happen that the supervisors of the teams $\text{\textemdash}$ usually maths teachers or researchers $\text{\textemdash}$ helped their team with the solution (which is also forbidden) but the rise of LLMs has led to team to use AI-generated solutions even when their leaders were doing their job correctly. In all of those cases, this rendered extremely poor debates, as the team whose solutions where AI-generated usually did not understand what was going on, and this was an overall very frustrating experience for everybody involved.

One could argue that if researchers are using LLMs to solve maths problems then students who want to learn about research might be allowed to use it too. However, according to the principle formulated previously this would be similar to using their leaders to do the maths which is obviously forbidden. I want to also highlight that from the point of view of the juror that I am, there is also nothing more discouraging than volunteering and taking some of my free time to read AI-generated solutions. The only reason why I am doing this free work is because of the human aspect in teaching and learning. If not for the students I would not be coming here. If this issue is not tackled, the risk is going to be the shortage of jurors and consequently the death of a competition which aimed at teaching cooperation, exchanges, and connecting high school students with researchers.

Acknowledgment

The opinions expressed in this text are my responsibility only but I would like to acknowledge many people with whom I had inspiring discussions about this topic. I would like to thank Dustin Connery-Grigg, Ipsita Datta, Sahar Diskin, Segev Gonen Cohen, Simon Machado and Francesco Morabito. I would like to thank Ibrahim Trifa whose joke I enjoyed so much that it became the start of this essay. I would like to finally thank Simon Machado for encouraging me to lay my thoughts on paper.

The reader will probably agree that an undeniable positive aspect of genAI is me trying to learn how to use the em “dash”.
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Not me, 25, talking about young adults like it is long-gone epoch for me.
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One might think that I am talking here about an aspect of AI-usage that is more social than related to teaching. I believe however that, despite its social nature, this aspect is part of the mathematical education. ↩︎

Received 16 August 2026.