**The tragedy of mathematics teaching**

I often say that having to choose between sex or teaching mathematics, I much prefer teaching mathematics. The interesting thing here is not that I find it more pleasurable to teach mathematics than having sex. The interesting thing, which I never really took the time to explain, is *what* exactly in math teaching makes it so high on my scale of pleasure ? The answer to this question is easy: intimacy. Mental intimacy. (Which if you think about it is the primary ingredient of good sex)

### Part 1

I have two styles of teaching (with different feelings associated to them).

The first style is the 'large lecture room' style, during which I expose a course to a relatively large number of students, where 'relatively large' means anything above 10. In this style, I can be quite funny and entertaining, while being mathematically sharp and uncompromisingly precise. The experience feels to me like a representation, as if I was one of the actors of a play, the other actors being the mathematical objects that I manipulate. Being myself a part of the show is something I have learnt unconsciously over the years as I was noticing that the 'personality' of the narrator was playing a non negligible part in the impact of (non mathematical) talks I was listening. For instance, take your favourite TED talk, and imagine it being given by somebody else than the person who gave it. Unless the talk was highly technical with focus entirely on the subject itself, the 'personality' of the speaker was a part of the alchemy, seduction and interest. Incidentally, I keep using the word 'personality' but in fact it is a complex alchemy involving all possible non verbal forms of communications that can be perceived from the speaker.

There are cases where the speaker becomes somebody else on stage. It's a coping mechanism. Those people are shy or stressed or something, and they cope with the situation by adopting a completely different style than their natural style (like wearing a mask, really). Most of the time they don't even notice it, unless you show them a video of themselves. You know that the speaker is in this situation if they look slightly confused for fraction of a second if you interrupt them to ask a question. That's the time it takes them to drop the mask and re-become themselves to be able to interact with you. Every time this happens I hate the talk.

The second style, is a much more focused, and precise style that I adopt when I have a reduced number of students, ideally just one. In those moments to me the activity become virtually indistinguishable from hacking into a computer (in the sense infiltrating as to take remote control of). I used the term 'hacking' because it means about the correct thing for most people, but the analogy that is the most faithful to what I want to say is: imagine an advanced AI trying to take control of a part of the cognitive functions of a lower class AI without the latter noticing...

### Part 2

As I have already explained, regardless of the true 'nature' of mathematics, the activity of learning mathematics, is very close, in structure and in spirit, to the fact of building new computing processes in one's mind. As a student by learning a new piece of mathematics or working at making it more familiar, you are building new components in your mental and test/train their interfaces (to the rest of your internal mathematical world) and computational capabilities.

As a teacher, my job is to map the current structure, the current network, of processes and ideas in my students minds, evaluate their maturity and once I have a mental model of the way my students are configured inside, my job becomes to start the sequence of modifications that is going to move them from their initial state to a state where they can operate efficiently in a particular mathematical subject.

The initial mapping part is easy, it's just me probing the students. I achieve this by asking them a plethora of questions and paying very attention to the way they answers them, then inferring the state and internal organisation of their mathematical mind. The 'migration' is less straightforward but no more technically difficult. The idea is to help them updating/correcting their point of view towards some of their own internal mental objects and then have them, step by step, build the missing parts of their understanding and then help them to train/reinforce those parts until utilising them becomes natural. In practice, this means entering in a highly interactive process by which I use the entire set of previous observations to build a path for the students. Sometimes I get it wrong because I had underestimated the difficulty of a step (some students find some steps easier than other students), or need to modify the trajectory because I suddenly realise that my current plan conflicts with some other parts of their mental states, I then need, in real time, to adapt to the changing topology of their own minds.

In any case, it is not about doing the job of the students for the students. Remember that you simply cannot force a foreign mind to build a process, you can just direct them, guide them.

This method of teaching can only be applied to a very limited number of student at a time (and that's why I cannot use it when I have a larger set of students). But you understand why it feels like hacking (due to the amount of focus, interactivity and attention required from me, the teacher), and why I have a feeling of mental intimacy.

### Part 3

My teaching skills are not mathematics related. This means that I can use them for any subject in which I am knowledgeable enough to teach. Also, and as I had countless opportunities to notice, many good mathematicians do not have them, at all. Those people end up having teaching positions because they are good in their subject (as researchers, for instance), but lack any kind of teaching skills.

In the case of those teachers lacking teaching skills, they do not understand that their job is to stand next to the students wherever the students happen to be and help them to move, step by step, where the students should ideally be. They think that good teaching means exposing a perfectly logical and rational course. They evaluate their own performance as a teacher by how perfect the course was presented (and how well written the lecture notes are). They are the kind of mathematicians who mentally focus on 'the maths' when they teach, and completely miss the **social aspect of teaching**.

(update: I do not want to give the overall impression that I dislike or cannot give classical lectures. You can give a classical lecture that is going to be aligned with the need of not ignoring your students if you are aware of what you are doing. For instance talking to them at the break and get a pulse of what they think can make a big difference. This is a limited but already crucial form of interactivity.)

(A dramatic example of complete disconnection between a teacher and students was given to me once by a lecturer who took on himself to teach an advanced introduction to measure theory to his undergraduate students, for what he thought was good reasons, without realising that his students were completely lost, in part because nobody had told them about power sets before. The students looked heartbrokenly desperate for weeks trying to make sense to sigma algebras and measurable spaces before they eventually discovered that they simply could not understand anything because of a fundamental assumption the lecturer had made that turned out to be incorrect. Now, I don't blame people for making incorrect assumptions about their students, but how can you go on for weeks without noticing that your students are deeply lost without any sign of ever recovering ?)

When I teach tutorials, for instance, my only objective is: What can I do so that at the end of the hour my students are in an actual better shape than when they entered the room. I am not interested in throwing solutions at them, solutions that are simply going to be copied down in their notebooks while they think "I will have a look at that over the weekend". This is treating them like photocopiers; in which case we might as well just give them the printed solution and go for a drink. So now, I am not saying that a tutor solving exercises is a useless experience per se. I personally enjoy the experience and I have seen students really taking advantage of it; the problem which I think is not correctly addressed is the non negligible subset of students who come in because they have to but are put in a situation that is really not suited for their current state.

When I know that I have one hour with undergraduates, I really try to do something so that at the end of the hour if you scan their minds you would notice that I left them in a better shape than when they entered (even if it's a small increment). This is not easy, but it is achievable; you can even become good at it.

In practice, students have a limited (stronger in some students) ability to create their own internal teachers; an internal process whose job is to be the interactive entity which can evaluate the current state of their minds and somehow knows how to build a path towards proficiency. This is called knowing how to learn. Very few students use it correctly (the few who know discovered by accident). They others just 'practice' the material until they feel that their understanding is better, but this is often a waste of resources. Their increase of understanding, although real, is only a fragment of what it would have been had their learning been properly guided.

### Part 4

When I was a kid my dream was to have my own AI. I would have told it something like "Lucille, do you see that calculus book ? Do your magic and make me learn all of it by the end of next month. I have tried to start reading it myself but I am already lost at page 2". She would answer something like "Ok, no problem. This book is good but sometimes not very well written because the author favoured mathematical correctness over pedagogy, but don't worry I know what to do. Why don't we start with a game actually ? I give you a math statement and you give me the proof, or I ask for an example (or counter example) of something and you give it to me. Let us do that until you fail and then I will tell you which paragraph to read and then we will start the game until you fail again etc until the end."

She would have hacked into me, literally. And one month later would end the game with me having a quasi perfect understanding of calculus.

More realistically, the phenomenon of online courses (with more and more top universities putting their courses online for free) is one of the best development in education in the recent years. This gives to the student the ability to get away with having to be in the same room at the same time as the presenter, but beyond this, how feasible is it to somehow turn, says, the math undergraduate curriculum into a game. Ideally, we would tell first years "Welcome to university. Here is your login and password for the online learning game. If you can finish this game (and don't worry every step is easy), you can be pretty sure that you can pass the exams. By the way, don't use somebody else's credentials, because the program is going to completely adapt itself (ie: the learning paths it will build for you) to you."

With the amount of real talent and ingenuity that we know exists in the video game industry, I am sure that the project is feasible.