Hey and welcome to my new blog, where I’ll discuss mathematics I like, study skills, and the lifestyle habits that keep me productive, organized, and stress-free on my academic journey.

Mathematics is a famously difficult subject to study at university, so I’m going to kick things off with some general mindset tips for fellow undergraduates that either I’ve found helpful, or that I believe lacking in this attitude has held back some of my peers. I’ll flesh out some of these ideas in later posts.

#### 1. Have a growth mindset

One of the big themes in pedagogy right now is the difference between a fixed mindset and a growth mindset. Traditionally, it was believed that intelligence is some fixed characteristic of a person, and that eventually, you will reach the limits of your intelligence. According to a new school of educators and psychologists pioneered by Dr Carol Dweck, this is not the case: students who believe that their intelligence can be expanded through diligent study and practice tend to be more successful. Maybe you did and maybe you didn’t find A-level mathematics too demanding: at some point in your mathematics degree, you will find things difficult and doubt yourself. Believing that you can stick with it and expand your mind to eventually understand will give you the motivation to work hard and conquer your obstacles.

#### 2. It’s about concepts

In A-level mathematics, a lot of attention is given to computation, especially testing your ability to solve integrals. The good news is that in the real world, computers can solve many difficult integrals in seconds. Did I say good news? That’s also the bad news: no one is going to employ you to solve standard integrals for them. It is far cheaper and faster to type the integral into a computer system such as Wolfram|Alpha. Although computational skill is important, both for building intuition and getting good grades in exams, the thing that makes a mathematics graduate valuable is their ability to solve difficult problems, and reason clearly about difficult mathematical concepts. So don’t skimp on understanding the theory! Learn the definitions inside out, and have at least a sketch of the proof of each theorem in your mind.

#### 3. Be uncomfortable with not understanding

If something doesn’t seem right, or a lecturer takes a step in a proof that goes way over your head, avoid the temptation to suppress the feeling of discomfort that arises. It’s probably a clue that there’s something you’re missing, and figuring out the solution, perhaps by asking the lecturer, or reading up on it from another source, will resolve that tension in a much more satisfactory way. If you suppress the feeling every time you don’t understand, your understanding will be full of holes, and that can stay with you for a long time, especially if these concepts come up again and again in future classes.

#### 3.5. Be comfortable with not understanding

That said, there will be things you don’t understand, for whatever reason. Finding a peace that allows you to make progress despite not understanding some details is crucial for staying sane and keeping on moving forward. The point is, though, you’re not suppressing the feeling — you must have the intention of understanding it at some point in the future, and always be on the look-out for that key insight that will clear things up for you.

#### 4. Be comfortable with being stuck

This is a big one. In A-level, you don’t *really* get stuck. You may have struggled with a technique or two. But it’s still a case of a) try a problem for a few minutes b) see if you got it right. Rarely, if ever, will you have been expected to be stuck on a difficult problem for hours. And this is a problem I’ve seen from many of my peers. They don’t know how to be stuck. In fact, they can’t bear to be stuck. They get stuck, and immediately ask for help, or give up, or assume the problem is unreasonable. Being stuck can be uncomfortable, but don’t give up. Think about the problem, and attack it from many angles. Be creative. But also, do it in bursts. If the problem is proving difficult, get a cup of tea. Still stuck? Go for a walk, and come back. Even better, sleep on it — you might be surprised how good your brain is at finding solutions in your sleep. Being stuck is painful at first, but eventually making progress is one of the most gratifying experiences you can have in your academic life.

#### 5. Find time to focus

This shouldn’t be that surprising, but you can’t solve difficult maths problems or learn difficult concepts if you’re half watching TV, instant messaging, Facebooking, or whatever. Your brain just can’t do both things at once. So try to seperate your time so that you can focus entirely on mathematics during some periods, and entirely on chilling out in the others. This also includes separating private study from group study. Learning with friends is great and effective for sharing ideas, but the deep understanding of these ideas comes from private contemplation. I tend to work alone in the morning and with friends in the afternoon.

#### 6. Immerse yourself in mathematics

Read maths blogs, read maths books, learn to write mathematically (especially learn LaTeX typesetting), watch maths lectures online, and so on. Learn about the lives of famous mathematicians, and let them inspire you. Mathematics is seen by many outsiders as a dry subject, but the reality is that mathematical culture is vibrant, and there’s always new interesting things people are discovering and sharing.

#### 7. Get good linear algebra and calculus skills

These are the two gifts that keep on giving. Learn these two theories, and learn them properly — not just how to do computations with matrices and derivatives, but the more abstract theoretical underpinnings of both. Learn about abstract vector spaces and linear homomorphisms, and the techniques of real analysis. These skills will serve you very well throughout your degree, and in any future mathematics you do.

Well that’s all I have for today. Please leave a comment if any of these tips resonate with you, or you have solid mindset advice of your own to share, subscribe to the feed at thetangent.space/feed.

Links relating to today’s post:

- https://terrytao.wordpress.com/ is a blog by one of the world’s foremost mathematicians today. His sidebar has links to dozens of other maths blogs, which makes it a great way to dive in to mathematical culture.
- http://calnewport.com/books/deep-work/ is a book about how to find time in your life to do work that requires deep concentration.
- https://www.brainpickings.org/2014/01/29/carol-dweck-mindset/ more information about growth vs fixed mindsets.

Thanks for this Sam. A really useful post for those who are about to set out on a mathematics degree, and useful for those already on one but who may have lost their way.

Great article Sam!