Random Things I’ve Learned From Being a (Math) Student

One of my first memories from early childhood, was from 1st grade math class. I remember sitting on the little chair, with the math workbook in front of me. Just as I was done solving it, I’ve felt an intensive energy rush through my body.¹ The feeling was quite new to me, and I knew I wanted more challenges which will make me feel this inner flame again.

Fast forward 20 years, and here I am about to start my last year of undergraduate Math program. I didn’t invest alot of thought into what exactly I should study in university. It seemed like a good choice, as it was easy for me in highschool, while also opening alot of options for a future carrer. In hindsight, I had a great fortune not knowing exactly what I’m getting into. If I had known, I’m not sure I would have had the courage to make that choice again.

Here are some of the lessons I’ve learned over the last 3 years. I address them to myself as if I’m just starting out this journey:

First lesson - Have a firm belief in yourself

This lesson is the most initimate one for me. I find it to be at the core of why I’m here. The last 3 years had an enormous amount of worksheets, lectures and tests. There were fatigue, doubt in myself, jealousy, and pain of not being good enough. Long hours of trying to understand, even the most esoteric things (say hello to branches in complex analysis). I always knew I could do it. I felt it from within.² This belief gave me the wisdom and power to go through.³

I encourage you to find and strengthen that belief. For me, one thing that helped was to talk with supportive people who cared for me. It’s hard to doubt and challenge our inner voices. It feels like they are us. Talking with friends I trust holds a mirror against the raging sea of emotions. I can see myself clearly from their point of view which is always a more positive and competent version of myself.

Second lesson - People can get better

Well, daaah. Of course people can get better. On the surface, it seems obvious. But I think there is more going on here.

First, we all have limits. The most basic limit is of course that of time. Our life is finite, and we’re all going to die one day. A more subtle limit, is our natural ability to understand things. Here by natural ability, I refer to the amount of energy required to make progress in a certain topic. Sometimes real progress happens because our natural abilities are good enough for the problems at hand. But sometimes, it’s just not enough.

After roughly the second week of the first year - natural ability only got me so far. Almost every problem wasn’t solved on the first 10 minutes. But somehow with enough effort, I was getting better - Never as quickly as I wished, but still. I’ve managed to pass most of the exams in my first year, and I improved my skills and results afterward. I think about things completely differently today. Whether the ability to feel and write abstract concepts, or the depth and rigour I use when trying to reason through life.

This lesson was a big change of heart for me. Before university, even tho I wouldn’t say it, I felt as if there were hard limits on people’s abilities. Today I still think the limit exists, but it’s much more fluid and dynamic then I used to believe. I assume one of the reasons I felt this way - was that it constantly required me to admit the gap between where I was, and where I wanted to be. This was so painful at times, which led to me thinking that the limits on our improvment are mostly set in stone.

I don’t want to explore in depth exactly how people get better. There is one key element that I do find worth mentioning here - participate actively in whatever you’re trying to master.

Third lesson - Getting better is hard

This lesson is one that you’ll feel straight in the gut. It’s hard to explain with words the feeling of sitting for 2 hours infront of a problem set, and not being able to solve a single problem. You open the lecture notes, the recitations, skim for any idea to help you solve. You might sketch a couple of ideas, but non seem to fit. You try another problem. Same thing, you’re even more lost on this one.

I never realised how much time you spend like this, and how much mental effort it takes. Days in, days out, you have to show up. As a math student, you are like a professional athlete, but without the supporting team.⁴ But the fact the work is very hard, doesn’t mean it can’t be an absolute joy. There were moments of grace, when I felt like I was understanding and feeling the truth of the universe.

As time went by, these moments became rare and the work was mostly hard and grueling. There was a particular moment, while taking an ODE class. I remember thinking to myself: “I just don’t care about this”. This was the moment I realised I didn’t want to get better at math anymore.

So if you do choose to get better at something, the only way to do so in a sustainable way is to really enjoy the work in the deepest sense.⁵

Forth lesson - Change scopes and perspectives often

When I first started my journey with mathematics, I had a tendency to see things from the closest possible perspective. Every proof had to be understood down to the smallest detail. This had created 2 problems.

The first problem was that I lost sight of the story. A good proof has structure, and objects interacting with each other through logic. It has a certain flow to it which can’t be seen if we’re too attached to it’s specific details. It’s also clear that we can’t just see the big picture, as one little detail missing breaks the entire thing.⁶ I think you must develop a dynamic sense of perspective. Not only from small details to big story, but also of explaining and attacking the problem at hand. Some linear algebra feels natural with matrices, others much more with linear transformations.

The second problem is a controversial truth: not all things are created equal. The main logical step(s) we use in a proof, usually mean more then smaller details which makes a proof complete. This forces us to focus on the core of the problem ahead of us. This may be more obvious in music. Imagine an inexperienced player, say a child. What does his playing sound like? Every note has an accent on it, playing every beat with the same intensity. To the child, every note is equally as important as the ones which come after. The experienced player knows to play in a way which lights the overall structure of the piece, while still making us feel the small details.

We should aspire to do math as beautiful music is played. The big idea and it’s smaller details work together in harmony.

FIfth lesson - Embrace imperfection

Recently I went to a wine & cheese tasting a friend arranged. I met a friend there from a while back, and we made a small bet. I thought “We” was a second person pronoun, he correctly suggested it’s first person. As we looked it up and found that he was right and, more importantly, that I WAS WRONG - I felt annoyed. I really don’t like being wrong. This fact sometimes pushed me to the edge in my studies, making sure no holes were left behind which has resulted with a perfect answer. This is a good thing, as a valid proof indeed requires complete perfection.

But there is danger in the pursuit of perfection. Most of the time things are an absolute mess, or at least very far from complete. I was blind and inexperienced to find a solution to alot of problems I tackled. Even after seeing a solution in front of me, I just couldn’t understand some part of it. Feeling this way everyday has resulted in the stupid conclusion that I’m just not good enough for this thing.

Today I know a more gentle conclusion is appropriate: I must embrace imperfection. This of course doesn’t mean giving up on perfection. You must always aim to the highest perfection you can, math demands this from you. But you as a human, don’t have to be perfect. Trying to do things without fixiating on perfect results, makes the entire thing more productive and exciting anyways.⁷