Advice for mathematics students

About to embark on a mathematics degree? Here are some tips to help you succeed, mostly by avoiding my mistakes.

Photo by ThisisEngineering RAEng on Unsplash

JB: I originally wrote and published this in April 2022, shortly before I moved my blog over Substack.

Mathematics is a wonderful subject to study at university, but it is not without its challenges. I am sure you need no reminders of the challenges that maths brings! But in case it proves useful to any mathematics students who read this, I thought I would combine my existing advice into a single post. At the very least, I want to lead you away from the mistakes that I made while studying maths at university.

I did not do very well at university, and so I want this post to additionally provide encouragement to current students who are struggling or feeling unhappy. I did quite well after graduation, although things seemed pretty bleak during my maths degree. Just know that things can get better, and that you are not alone in finding this time difficult. Hopefully I can give you some helpful pointers in this post!

Wait a second, I did not do very well at university? Then why should anyone listen to what I have to say? That is a fair, if slightly harsh question. After seeing others who did succeed, and reflecting on my own mistakes, and then correcting those mistakes with further study and work, I hope I can give a better guide than others who might have sailed through and got straight As on every paper and exam.

Make friends. Don’t “lone wolf” your assignments. I really enjoyed maths at high school. I discovered in the two years before university that I was finally good at something useful, and I was fortunate enough to find a lot of the homework fun and easy to understand. I read ahead in my textbooks, curious to know more about this fascinating subject. It seemed a natural fit for me as a university subject.

University-level mathematics was a very different experience. You aren’t supposed to find every assignment fun and simple. Unless you’re a super-genius, you’re not likely able to do it all by yourself. I didn’t make an effort to make friends in my class, and struggled to complete nearly every assignment I was given. I was not prepared for my exams, and did worse in each successive year.

The students who got first-class degrees in my class all understood the need to collaborate on assignments, and formed themselves into groups of friends. They weren’t best friends, and didn’t understand everything straight away, but they worked together very effectively on assignments. Mathematics at university requires collaborative effort. Five happy brains are far more effective than one.

You don’t need to make five new best friends. You merely need to find a group of people in your class who are open to collaborating on assignments, without necessarily just giving you the answers. You should contribute just as much as any other students in your little group. I always come back to this story I saw about students building a shared doc of questions and answers over time.

I was deeply unhappy at university, and I think that part of it is that I have always found it difficult to make friends. I still struggle with this today. I would look on the groups of other students and feel sad and envious at being by myself. If I had approached them and simply asked to join them, I might have taken a step towards a much better score on my degree. But I didn’t, and I still regret it.

Mathematics is supposed to be challenging. Don’t be discouraged. A corollary of the above point is that you shouldn’t feel like you don’t belong in a mathematics course at university just because you find the concepts or the assignments difficult to understand. It is meant to be a challenging degree. Newsflash: That is why mathematics graduates are so highly sought after in the employment market!

If you find yourself unable to start with an assignment, then take a deep breath and look through the material you’ve been given in the course so far. In my experience, there is generally something in there to give you a clue, no matter how small. And if you haven’t made friends yet, go back to the previous point and try to make some friends in your class and collaborate on the assignment.

Don’t forget that there are tools on the Internet to help you with mathematical problems if you’re struggling. Wolfram Alpha is a popular tool for solving equations, but you can also look for free videos and lectures on YouTube. I can’t provide a prescriptive guide for any given problem, but I can give you advice for solving maths problems: prevail, and keep looking at them from new angles!

My favourite method of working on mathematics is in time-boxed sessions. That means taking 25 minutes to work exclusively on a problem without any distractions, taking a short break, and then starting another session. Before I know it, several hours have passed, and those hours are much more worthwhile than if I had absent-mindedly checked my phone while pretending to work.

If you can’t shake the feeling of discouragement, then try to remember how far you’ve gotten so far, and then channel that into pride at having survived in a notoriously difficult degree. Do you know how rare it is to have the mathematical ability to do a degree in this subject? It is incredibly rare! And you are among that cohort of people, so feel proud of yourself, and definitely don’t feel discouraged.

Learn programming languages. Try to build things. I struggled through most of my mathematics assignments at university, but I consistently enjoyed the code modules we were given to take as part of the degree. When I was allowed to use Maple or Matlab to investigate a problem, I did, and when I had the opportunity to write my answers in LaTeX, I did. I have an intrinsic love for building things.

It should not be a surprise, then, that I have become a software developer in my career after university. My mathematics skills have been very helpful in my job, but certain programming concepts have helped me in my postgraduate study too. Scary-looking sigma syntax for sums are just one way of describing an iteration over many values of a variable. This is how code can help with mathematics!

Start with exercism.org or codecademy.com if you are unfamiliar with code or programming. Just give it a try, even if it seems scary. If you are good enough at maths to do it at university, then trust me as an experienced developer that those skills put you above most software developers, and I am not exaggerating. Code will help you to understand maths, and it might end up being your job one day!

As an experienced developer, I can’t overestimate how powerful you will be if you possess programming skills and a mathematical ability. They compliment each other in what is called a “talent stack” by bestselling author Scott Adams. You will be in a much stronger position no matter what job you end up in, because you will have a trained ability to understand, formulate, and solve problems.

I recently took a greater interest in functional programming and started using Haskell as my second programming language. It is a notoriously tough language to learn, but it is full of beautiful concepts that translate directly from maths. In particular, category theory is a great use of mathematics in the programming world. Well-written functions are simply mappings between categories!

Attempt to learn concepts by writing useful summaries of them. This may be one of the most important pieces of advice in this post. No matter what concept you are struggling with, attempting to teach it to yourself by writing a summary of it is a great way to improve your understanding. Simply take a concept and try to write a short summary of it, either in layman’s terms or technical terms.

In mathematics we often deal with a certain type of syntax, so I found it helpful to break complex bits of maths into bits of code, or explain them in plain English. Literally writing this down onto paper might seem silly, but it is surprisingly helpful to clarify your ideas. It should not be a shock that this method is very transferable to an infinite variety of subjects, and I have used it in my job!

This is especially good if you find yourself stuck on a problem and out of ideas. Go back to the lecture notes, find the underlying topic, and resolve to understand it before going back to the problem. Open up an email and pretend to write an email to a friend explaining each concept. This might feel silly, but it moves you from the feeling of “ugh, assignments” into restructuring the concept in a simple way.

Think hard about how to best fit a concept into different lengths of text: one page, one paragraph, one sentence. Diagrams can be especially helpful if you are a visual learner. Look up videos on the Feynman Technique, which is essentially the same thing, and try out this video on studying smart without burning out. Tim Ferriss’s blog is a veritable treasure trove of tips on speed-learning many different topics.

Imagine how useful it will be to have a list of such summaries when you come to your final exams. Collate these over time, improving them where necessary, and putting the priority on finding areas you don’t yet fully understand. This is an excellent study system that will be a more powerful time investment than getting frustrated at assignments. There is very little harm in giving it a try!

Don’t put up with unhelpful lecturers and professors. Upon entering an MSc course in mathematics at the Open University, the biggest difference I found from my experience in an undergraduate degree at the University of Edinburgh was the helpfulness of the instructors. At the OU we had no lectures, just extensive notes and occasional screencasts. Instructors were encouraged to answer our questions.

Because of the nature of remote learning, it should be obvious why OU instructors needed to be helpful if we came to them with questions. We had no other way of getting feedback! At Edinburgh I remembered being treated as a nuisance, and professor office hours seen as a chore. It was common to not feel welcome in these sessions, with louder students dominating the discussions.

Now I admit that this is a bit unfair of me to say, since there is more I could have done to improve my studying experience, and there were some professors who saw teaching as a pleasure and were glad to receive student questions. But I believed, falsely, that grumpy unhelpfulness was the norm in mathematics, and I simply got used to it, not making the full use of my professors’ time as others did.

My advice is to push past any feelings of awkwardness when asking your instructors for help with questions. You are there to get a degree, and they are paid to help you to do that. You shouldn’t be rude or combative, but persist if you still don’t understand a topic and need help. Don’t put up with any rudeness or unhelpfulness. Respectfully push past it and get the help you are paying for.

Don’t forget to remain respectful of your professors, since they have earned their positions and are authority figures in the university. But learning to push past feelings of awkwardness and discomfort is a skill that will pay dividends in your career after university. Persist with problems and don’t remain satisfied with going halfway. This is stressful, especially for polite people, but it is worthwhile.

Mathematics is about curiosity. Don’t stifle it. When I became used to having no friends and facing unhelpful lecturers, maths stopped being the fun subject that filled me with energy, and became a drag, reminding me every day that I wasn’t the smart student I thought I was. I skipped lectures and spent more time on my laptop, becoming obsessed with other subjects, which only made things worse.

When I received my disappointing final degree result, I decided that maths was a closed chapter in my life, and moved on, feeling heartbroken. It wasn’t until a few years later that I rekindled my interest in maths and applied to the OU’s MSc course. It was challenging to face maths head-on once more, but I knew I needed to redeem myself as a personal challenge. It reawakened the curiosity inside me.

My advice here, then, is to try to notice your curiosity for maths slipping. I firmly believe that mathematics is a subject for curiosity and exploration, open to every student with an aptitude for it. So if you do notice that slipping away from you, then attempt to figure out what is causing it and reverse it. Curiosity is what gives joy to mathematics, and losing that joy might cause you to suffer in your work.

For me, time away was a costly but effective way to refresh my perspective. But that is not practical for everyone, especially those in the middle of degrees. A more practical approach might be to go back to basics and try to find joy in getting a better grounding of the work you’re not enjoying. This way you strengthen your studies and, as a bonus, feel like you’re gaining a fresh perspective.

I put this as the final point in this post because I think it is important to end on a note of curiosity, exploration, and joy. I love that childlike feeling of wonder that I get whenever I open a new mathematics textbook and come across new ideas. It’s important to capture those feelings of awe and wonder wherever we happen to find them. I hope you find and embrace those feelings in your maths studies.