Billy Levin

Revisiting Mathematics

I spend a chunk of my free time (almost) every day studying maths. It is usually the first thing I do once I wake up. I sit at my desk, play some peaceful music, light a candle if it’s cold or dark, and joyfully learn things and work through problems on a piece of paper. I am not bogged down by any sense that I need to learn at a particular pace; any frustrations at not being able to figure something out are ephemeral. However, this was not my first foray into maths, and previous attempts were not so joyful.

University

I have a bachelor’s degree in maths, which I finished around 7 years ago. I chose maths as my field of study because I wasn’t yet sure what I wanted to do in life; I thought studying a general subject that I’m good at and somewhat enjoyed made sense in order to keep my options open while I figured that out. I was not in love with it and did not have any special talent.

I did not at all make the most of the resources available to me at university. Instead, I studied just enough to get through it. At the time, I was of the opinion that “tryharding” was inefficient and to some extent cringeworthy — a ridiculous way to think!

During my studies, I’d occasionally stumble upon and recognise mathematical beauty but I never attached myself to it. It’s obvious in hindsight that these experiences would’ve been both more frequent and more resonant if I’d put more effort into understanding the subject more deeply.

Not immediately, but as time went on, I started to feel regretful about my approach to university.

Deciding to Revisit Maths

After a few years of quite happily working in and studying software development, I began to feel a yearning to revisit maths. I think this was mostly due to my regret about not making the most of university. Plus, I had the feeling that it would help me become better at writing software due to its overlap with computer science plus the more nebulous rationale that it would improve my “logical thinking”. At this point, I wasn’t looking at maths as a hobby so much as an opportunity to be productive.

At around the same time, I’d been hearing more and more good things about Math Academy ↗︎ (fellow UK readers will notice a clear misspelling in their company name!). After being tempted a few times, I finally decided to take the plunge.

A False Start

It was pretty easy at first because I was mostly re-learning dormant concepts that I’d previously understood, and so it was just a case of “waking them up”. I was immediately addicted and was spending a lot of time working through the curriculum. After three months of this, I completely burned out.

I had reached a wall where many of the topics were now ones that weren’t so “automatic” for me; I could no longer whiz through questions after briefly skimming the explanatory material. It was uniquely frustrating to read words and concepts that I’d once been familiar with, to know they’re in my brain, but to not be able to properly grasp onto them. Since I felt like I was already playing catch up, I found these stumbling blocks really demotivating.

At its very worst, I guessed several answers without even attempting to work through the problems at all, simply so that it would be “done”. It was at this stage that I asked myself “what am I even doing?” — I first paused and then subsequently cancelled my membership.

The Purpose of Learning

After some time away from maths, and finding that I was missing doing it, I realised that I actually didn’t want to study maths for any pragmatic reasons. My initial desire for it to be productive was because I felt behind. If I’d both tried harder in the first place, and stuck with maths after university, I’d be a lot “further ahead” than I am now. This made everything feel more urgent, and this contributed to me burning out when progress began to slow down.

After thinking back, I noticed that maths had never been “useful” to me, and yet I’d still enjoyed it at times when I was properly engaged. So why did I think it needed to be productive now? Instead, I wanted to learn it for its own sake — for the joy of working in the abstract, of combining multiple concepts into elegant solutions, and of following my curiosity without feelings of guilt for not being “productive”. A recent example of this was learning about the Cayley-Hamilton theorem and its implications. When I saw how some expressions involving matrices with large powers could be greatly simplified without laborious computation, knowing whether it has any real-world applicability would’ve made no difference to my excitement.

This realisation is what got me back on the horse. Before, I’d get frustrated when I didn’t understand something straight away due to some misguided sense of urgency, whereas now I take it as an interesting opportunity to learn something properly with the understanding that I’m not working against the clock. It’s a relief to have figured out what I actually wanted to get out of maths — whatever it decides to give me — and it allows me to fully enjoy the process without conflicting feelings.

This is not to say that maths will never be useful to me in the productive sense: there are of course many areas of computing where it overlaps; I’m particularly interested in software correctness; and I do put some credence into the whole “logical thinking” thing. It’s almost a certainty that at some point I’ll study type theory, category theory, formal methods (already am!), etc. These will still be fun, but I’ll want to also be intentional about any concrete goals I’m working towards.

While the mechanics of effective learning seem to be consistent, I think that the reasons you’re learning are important to think about so that you can approach it with the correct mindset. For example, if I need to know something technical for work by a given deadline, and I’m struggling to understand, I’m going to want to treat it with more urgency than I would for a low-stakes maths topic that I’m in no rush to figure out.

The Present

At the moment, I am going through the “Linear Algebra” course on Math Academy and “Algebra: Notes from the Underground” by Paolo Aluffi (if you’re interested, I found this book through Ludwig’s blog post on maths books ↗︎). For as long as I’m still using Math Academy, I’m restricting myself to one textbook at a time. Unfortunately, I do have other important things that I need to dedicate some of my free time to.

Regarding Math Academy, one of the things I like most about it is that it handles all of the “meta” around learning for me: it manages my knowledge graph, assigns me lessons based on completed prerequisites, and figures out when I need to do spaced retrieval practice for a topic. It’s really nice to be able to just focus on the maths. When I’m finished with Math Academy, I’ll have to introduce a bit more effort and rigour into my studying to ensure I can keep up with new, difficult topics; but I’m pretty okay with revisiting concepts I’ve forgotten since I’m in no rush.

How I use Math Academy has changed a bit since my initial attempt:

The Future

In the short term, I’m going to go through all of the Math Academy courses that interest me, plus textbooks. My goal is to learn the content well and enjoy the process of doing so! I’m still mostly re-learning (or learning properly for the first time) mathematical subjects that I’ve seen before in university. I think it’s a good sign that I’m still finding this engaging and fun, because surely when I venture into parts unknown it’ll be even more interesting!

In the long term, I’m excited to say that I have absolutely no idea what path I’m on. Life is (hopefully) long and I will simply follow where my curiosity takes me in the coming years and decades.