I Had to Teach It Before I Could Understand It

Until university, I studied mathematics in Arabic. Then I arrived at university and it was in French. The equations looked familiar, which made the change seem smaller than it was. I could often recognize what a professor was doing while still trying to understand what was being said about it. A French term would send me back to an Arabic term, and from there I would try to reach the idea. Sometimes I got there. Sometimes I got close enough to finish the exercise.

I kept going. If I could follow a lecture, use a formula, write the code and arrive at the expected answer, I took that as understanding. This is one of the convenient lies education teaches us. I became good at jumping over the things I could not quite explain, and the correct answers gave me little reason to go back.

The gaps were difficult to see because I had learned around them. Basic ideas still belonged to the Arabic in which I had met them; advanced ones arrived in French, depending on connections I now had to translate to retrieve. A familiar word could be just slightly wrong in my head for years. Later, something would fail to make sense and I would have to follow it all the way back to that small misunderstanding.

I wrote about that experience with language separately. What matters here is that my response, eventually, was to start again. I began studying mathematics in English from the ground up.

It was embarrassing sometimes. I had used these ideas, passed exams on them, built things with them. Going back to an elementary explanation could feel like admitting that the work above it did not count. But nothing you do not understand is beneath you. It is underneath you. I needed to know what I had actually been standing on.

English gave me a continuous route through the material. I could find an elementary explanation, a textbook, a proof, an implementation and a current paper without changing languages in between. I could stop on a definition instead of moving at the speed of a course. If two explanations seemed to describe different things, I could stay with them until I understood why. A rule I had memorized could become a response to a problem; a proof could become a sequence of choices I could follow, rather than algebra I was meant to accept.

I did not come out of that thinking English owned mathematics. It was useful to me because of the material available through it and because I could choose the pace. It helped me reconnect things I had already learned in Arabic and French. I was putting them on the same table for long enough to see how they fitted.

Then I started giving talks, making content and guiding people through machine learning in Darija. I knew what it was like to spend part of a lesson fighting the language before you could reach the subject. There were people I could make that easier for. That was the reason I began.

Preparation was where I started finding trouble. I had perfectly serviceable English sentences for gradient descent, generalization, attention and kernels. I could recognize them in a paper and use them in conversation. Explaining them in Darija meant deciding what I could leave out and what would make the explanation wrong if I left it out. Knowing the usual sentence was suddenly not much help.

An audience made the problem harder to ignore. When I studied alone, I skipped the same step each time and stopped noticing the skip. Another person had no reason to follow that route. They could ask why we normalized something, why a distance should count as similarity, or what would happen if we removed a term. An apparently elementary question could reach a place I had avoided for years.

Sometimes I needed a better example. Sometimes I was trying to use an example to get around a condition I had not understood. There were explanations I could follow on paper without being able to say why anyone had chosen that approach. A listener’s confusion could show me that I had answered the question I knew how to answer instead of the one they had asked.

Darija helped because the familiar prestige of the English phrase did not come with it. Something that sounded authoritative in a paper could sound ridiculous when I tried to say it plainly. The person listening was allowed to remain unconvinced. I had to go back to the derivation, or look for the assumption I had taken for granted. This was useful teaching work, and a lot of it was work on my own education.

Guiding people over time also made me put the subject in an order I could defend. My own route had been a mixture of courses, tutorials, code, papers and things learned much later than I needed them. I could not simply hand someone that route. Where should they meet optimization? How much probability did an explanation of a loss assume? When did linear algebra need to become something more than operations on arrays?

Some technical terms could stay in English. I cared much more about whether the logic between them had been explained. Preparing a talk made me follow an idea back to what it needed; building a sequence of talks made me notice dependencies I could overlook in a single explanation. That is where my foundation in deep learning became more coherent.

It also changed the questions I wanted to work on. “The model learns features” was useful until I had to explain where the feature was. “Attention finds what matters” left me asking what computation I was calling attention, and what sort of explanation its weights could give. “A kernel measures similarity” became a question about why a particular function should deserve that interpretation. I could implement the methods and still be dissatisfied with what I could say about them.

The dissatisfaction stayed after the talk was over. I would follow a question through the equation and its assumptions, and sometimes find that the part I wanted explained was a question I could work on. I did not suddenly stop valuing implementations or experiments. I began wanting an account of what they were doing that I could stand behind when someone asked me to explain it.

The child who had once built years of knowledge on almost-correct meanings had become an adult unwilling to leave an almost-correct meaning alone.

I still use the formal vocabulary, write in English and work through the equations. I also still find places where I have let a familiar phrase stand in for understanding. Having to teach gives me a reason to notice them. I can think I know a subject quite well until I have to choose where an explanation should begin for somebody else.

I thought I was giving talks in Darija because I had knowledge to share. I did not know that Darija would ask me to earn it again.