Introduction
I am always proud to be a Mathematics major in my undergraduate studies. The beauty of Mathematics — something only to be understood by real Mathematics lovers, is just so adorable. Nonetheless, many high school students do have some serious misunderstandings in this subject, and thus, put themselves into a wrong major and regret for good.
To be frank, I was not an exception from those misunderstandings. Now as a graduate from a 4-year Mathematics degree and a soon-to-be Statistics postgraduate student, I have to say that the content coverage and difficulty of studying Math were way beyond my expectation as a high school kid back then. So, I decided to write this article to tell you 5 facts that I wish I knew before I started studying for my Math degree. I hope this could give you some insights about what university Maths are really about, so you would know what to brace for in your next 3-4 years — if you still wish to go for it.
1. Math is all about Proofs
If you have ever wondered why (or even better, under what conditions) Taylor Series work, or how one could prove the Fundamental Theorem of Calculus you use in almost every Calculus class — congratulations you have got your first step correct by choosing to study Math because those are the problems we would like to deal with.
The most common misunderstanding of outsiders is that Math is all about sums and computations. In your high-school public exams, you might have come across various mathematical concepts that 'looks cool' to you — maybe you have worked hard to solve Ordinary Differential Equations, or you are just obsessed with the computation of eigenvalues and eigenvectors. What I am telling you here is that, these are simply not what university mathematics are about.
University Mathematics is more about rigorous proofs than tedious calculations work (we usually leave that to computer programmes, or even worse, engineers). That means the problem we ask is often 'why it works' instead of 'how it works'. Of course, computation is important in learning mathematics, but that is simply not enough for us, mathematicians — we are hungry animals with a thirst of knowledge. We seek the 'why' under every seemingly obvious 'truth'.
2. Nothing is Trivial
Let me ask you a very simple question that you must have learnt in high school calculus. If we have a sequence X(n) = 1 / n, what is the limit of X(n) as n tends to infinity? Obviously, the answer is 0 as one could easily observe the pattern of the sequence 1, 1/2, 1/3, ..., 1/n, ... — X(n) is getting 'closer and closer' to 0 as n increases and 'eventually' it 'reaches' 0 when n is infinity.
Now, I challenge you to prove it, in a mathematically rigorous way. Can you? I bet most high school students would not be able to do it. Maybe you would be able to explain how the sequence behaves intuitively, like with a graph or numerical simulations, but that's not rigorous enough for mathematicians.
This is because in Mathematics we rely heavily on definitions and axioms to write convincing proofs. There could be many seemingly trivial facts that we came across when we studied mathematics along the way, but none of them are actually 'trivial'. So, in order to use a particular fact, say maybe a theorem, it is important to prove or verify it under some predefined definitions and axioms.
So, back to answer my own question. How do pure mathematicians prove the above limit? The answer lies on a concept called Definition of Limits of Sequence (a.k.a. epsilon-N proofs). In short, we let any epsilon > 0, and we attempt to show that there exists a natural number N, such that if n > N, then |X(n)| < epsilon. Sounds confusing? Take it easy, that will be in your final mathematics exam in your first few semesters (both the definition and the proof)! Interested readers are referred to the nice textbook I used, written by Bartle [1].
3. The Struggling could be Painful
What I can tell you is that every mathematics major struggled hard at some point and doubted if they chose the wrong major. You could spend 5 hours brainstorming a single problem but only managed to write down the definition. You might be surprised how people around you being able to thrive while you struggled even in the easiest problems. You will also notice how the difficulty of your modules increase exponentially every semester.
The struggle is real and hard to deal with. For myself, I think patience is the key. I was once a very impatient mathematics student who gave up difficult practice questions pretty easily (and that did cost my grades back then). I was often disappointed when I underperformed and wanted to give up for something else.
What I realised from my experience was that, it takes time to digest and make progress. As Sheldon Axler mentioned in the introduction of his textbook that, there's no way reading a Math textbook like a novel, '...if you slip through a page in less than one hour, you're probably going too fast...' [2]. If you are doing mathematics in university, you're probably going to spend more time than others on working on problems, assignments and revisions. In my opinion, it worths if you really enjoy the process of thinking.
4. Math Research is Extremely Difficult
Every (ambitious) Math major once dreamt of being the best mathematician ever and received Fields Medal solving Riemann Hypothesis at some point of their study, but most succumb to reality eventually. This shows how tough it could be if you wish to pursue a research career in mathematics in the academia.
Admissions to Mathematics graduate schools are in general very demanding. Even for myself applying to Statistics Masters (which controversially I see Statistics as a branch of Mathematics here), I got all but one rejections from all the institutions I applied to over the world. The competition is just extremely fierce. High grades are essential but not sufficient for a nice grad school offer.
The process of pursuing postgraduate studies in Mathematics could also be overwhelming. You would be expected to specialise in some area (like Stochastic Processes and Probability, in my case), then narrow down your research interests (say, Optimal Transport), and finally focus on a niche topic of you interest (for example, Martingale Optimal Transport and Skorokhod's Embedding Theorem).
It's going to be an extremely tough way to go and I promise I would write more about my experience in the postgraduate studies in the coming few years.
5. Expect a Change in Career Path at some stage
Unless you are extremely talented and able to make important contributions to academia, it is not likely that you would stay in the field of mathematics forever after you graduate. In fact, I personally know many PhD graduates who decided to leave the academia after they finished their degree. So, you would definitely expect some kind of 'pivoting' career-wise at some stage in your life.
Since most mathematics graduates are usually proficient in basic statistics and programming as well, the most related and natural career choices for mathematicians are Data Scientists, Data Analysts and Financial Analysts. If you enjoy teaching mathematics, you might consider becoming a school teacher or a tutor as well. Some of my friends also work on IT-related positions like software engineers, which is good as long as your skillset is competitive enough.
As for myself, I worked as a Research Assistant in my gap year, researching on medical AI projects in a local university. It is not particularly math-related, but definitely trained my ability on independent research that could be beneficial in the future. One of the strengths of mathematics graduates is that they tend to be extremely fast learners who manage to adapt to different working environment.
Conclusion
So, these are some facts I wish I knew earlier before I started studying maths. If you are not discouraged by my words, I strongly recommend to have a go — it's gonna be a fun and tough journey.
References
[1] Bartle, R. G., & Sherbert, D. R. (1992). Introduction to real analysis. Wiley.
[2] Axler, S. J. (2020). Measure, integration & real analysis. Springer.
