My experience as a biology undergraduate turn product manager turn self-taught programmer turn Data Scientist has been quite a ride. In order to combat my ever pervasive, career-long imposter syndrome and fulfill my college-born desires to study math more, I’ve taken on an online masters in Pure Mathematics. This first semester of my masters degree has been an overwhelming amount of awe mixed with beauty and of course, crying. Being I just finished my first class in a higher level Mathematics (ever, as in I’ve never taken anything past Calculus 2), I’m proud I’ve made it out the other side alive and humbled. I’d like to share a few nuggets of insight I’ve gained through the course and hopefully relay something useful to those who might be considering pulling the trigger on their first graduate level math course.
Math at higher levels resembles philosophy.
I remember thinking to myself before the class started that “the logic in proofs are easy” and that “this class should be a breeze”. lolk. Hindsight is 20/20 and I’m naive, noted. When I was reading the textbook and experiencing my first proof, I was struck by how it somehow managed to be precise in its use of English to encapsulate the ideas necessary to show something to be true (which I think is strange in of itself) and also stayed rigid in a linear structure, always maintaining a stream of logic from one statement to the next. It was like witnessing a birth of truth. Starting from the assumptions of basic objects we then transform them with words into other objects which then eventually ended with a statement that somehow resembled a fact. Strange. It’s also odd that there is no emerging physical system to observe this behavior and conclude its truth. There’s no empirical test to be done, no experiment to be considered, no algorithm to put it through — just the contemplation and description of a nature of existence and truth. How utterly bizarre.
Proofs given in homework are no where near as difficult as the actual theorems provided in the text.
I remember studying the basics of Number Theory for the first time and feeling like I was not going to make it out of this class alive. The material changed from understanding the core methods used in proofs such as direct proofs, biconditional proofs, contradictions, etc., to actual mathy content. I remember reading the proof for the Euclidean Algorithm and nearly having a panic attack. Every single word emerging from the text seemed like absolute magic, and the reasoning behind each choice of logic was mired by the sheer complexity of the proof. I definitely wanted to cry — and did as true men cry in the face of math — after reading that proof and all I could think to myself after finishing reading it was “this is the end of me and my career interfacing with math” (dramatic, I realize). I thought it was all over, nothing left of me to be salvaged. I’d go back to my day job a failure and never to emerge into the beauty of this strange mathy world.
I then opened the homework and faced the first question “Describe the set $5 \equiv x \pmod 3$.” Really? That’s it? This is when I was ripped back into reality and faced the nature of the class. I’m not expected to rediscover maths from its foundations and bring to light theorems that took generations to build, millennia to discover, and obsessive geniuses to find. As I scrolled through the rest of the homework, I began realizing that there was a pedagogical goal where the questions are designed to build familiarity and deeper understanding of the material. Eventually it dawned on me that this is doable and the expectations I was building for myself were not realistic and not worth it. It was useless emotional distress. Don’t be me, don’t be a stress ball, have faith that the teacher and class are there to help you build your mathematical maturity and that they want to see the best in you.
Learn, use and abuse definitions and theorems.
This is the single item I struggled most with. Reading a chapter in a textbook filled with definitions and theorems is nothing like applying them to solve a problem. When facing the challenge of building a basic proof (i.e. apply a definition you just learned and do something trivial with it), I often found myself unable to expand on the definitions I just learned. I knew what divisibility meant, I knew what a function felt like, I understood how an image worked, but I had no idea how to use it. Why could I just invoke an even number, why could I just call something a function and use it as such?
Looking back, I faced two problems. First, I struggled translating the English definitions of these objects and theorems into symbolic form. Secondly, I couldn’t figure out why I could even invoke a definition or theorem and when to do so.
For the first issue, I blame myself for putting too much emphasis on the textbook and not looking elsewhere for materials that enabled me to use it. Not surprisingly, as a word of advice when studying, put priority on using what you’ve learned somewhere. For instance,
- Make 10 simple examples of the definition or theorem.
- Look at an example use of the definition or theorem and pay attention to what parts they use and under what conditions. Did all they care about was some part of the definition or theorem and threw out the rest? Under what conditions were they able to invoke something and what did they pull out of the definition or theorem? Most importantly, do you see where you can use this in a problem now?
- Look at the back of the chapter you’re working on and look for questions where you might be able to whip that definition out. Immediately try doing the problem and using the definition or theorem; be ruthless in using it.
Per the last note, who cares if you don’t get the problem right! Actually, if you’re a perfectionist and you have reservations on even trying the problem, plan to get the problem wrong and just use the definition or theorem. The goal is to get to the point where you can finally use these ideas, feel it in action and not be a passive observer of the author’s actions.
A secondary goal of that last note is to ground yourself in the messy reality you live in and remove yourself from school’s synthetic bubble. A school’s consideration of success revolves around a gray area of correctness where there are watchmen that make judgement on your actions, guiding your mind towards the rational, well discovered pastures of math. In the real world, where are the arbiters of truth you can rely on when discovering depths only few humans have seen, let alone understand? How many people can you reasonably trust with guiding you to the truth you seek? The void awaits you, and I suggest you begin to embrace it.
The second problem I faced were the questions of why I could even invoke a definition or theorem and when to do so. It seemed strange to suddenly make choices that if we ever saw something like $a=2x$ for some integer $x$, then we can just claim $a$ is even. Suddenly through some hand-wavy language, it had innumerable properties associated with it. Yet, where did this phantom of an even come from? Why was this even allowed? There was a disconnect between my understanding of logic (the disconnect is slowly dissipating in my explorations of the theorem proving language lean) and what it possible in a proof.
The question of when to use definitions or theorems is one that I’ve realized is the core element of math that won’t be escaping me anytime soon. You have all of these tools to tackle problems, and only some options fit the logical flow you’ve discovered. Which do you choose? When does it make sense to do so? Hopefully this becomes easier to recognize over time.
Embrace metaphor, analogy and drawing. Or die.
I’m convinced that if you’re not trying to connect the ideas in math to your life, things are going to stop making sense. I admit that topics can be difficult to draw in math, but search deeply for connections in your life to the ideas. Make them as weird as possible, get poetic about it, get philosophical about things. The deeper the connection with the topic, the less weird it’ll seem, and the more comfort you’ll achieve with the ideas. Math isn’t always something you can intuitively conclude to be true. It takes deliberate work to make it alive and sit within you. Do the work.
Find the right tutor.
My coursework is online and that puts me in a position where I’m rarely face-to-face with my professor, it sucks. The discussion hours never aligned with my schedule, and during the few times I did interact with him, I found that I couldn’t exactly ask the questions I needed to nor could I get strong feedback on the homework questions that I was struggling with. It’s the major gripe I have with online coursework. It also had side effects, I found myself in a compounding decline in understanding of the material given the fact that math curriculum is constantly taught from the ground up. After two weeks of feeling hopeless in Number Theory, I felt like I had no other option but to find a tutor that could help put the material into perspective for me. I’m so grateful I did as the insights you’re reading here were laboriously extracted from frank discussions with my tutors and my struggles with homework. I don’t doubt that I wouldn’t have learned what I know now without them. If you find yourself struggling and unable to get time with your teacher, bite the bullet and get a tutor.
It’s worth it.
Looking back, this course has been a true adventure that I’m so grateful to have embarked on. I remember when I first started doing proofs, it seemed that I could think for the first time; a lasting feeling to this day. Problems at work suddenly became a whole lot easier to reason about, and all the programming I did felt much more expressive. I admit, there were moments during the semester that were absolute hell, but the passion and respect I’ve developed for the subject is something that I’ll hold onto for life.
Thanks for reading.