Resources for students

General advice for students of mathematics

These are some things that I’ve learned over many years of doing and teaching mathematics, that I often wish I had learned earlier than I did:

Doing math involves a set of learned skills, just like playing an instrument or performing in a sport. People aren’t born with these skills; they must be developed through practice. While different people may require varying amounts of practice to develop different skills, the process is the same for everybody: the more you put in, the more you get out.
If math always came easy to you in the past, maybe it was because you weren’t really challenged. If at some point you find yourself struggling, it doesn’t mean that your abilities have run out, it might mean that now you are finally being challenged appropriately. This is a good thing, and means that you have the opportunity to grow your abilities more substantially.
There is a fundamental asymmetry in math: things that you understand seem easy and things that you don’t understand seem hard. Unfortunately this can make it feel intimidating when other people understand different things than you! This is called imposter syndrome and just about everybody experiences it at some point (it is even a subject of study in education and psychology). Try not to let it discourage you.
Related to the above, know that the people who speak with the most confidence about something do not necessarily know what they’re talking about more than anyone else.
Don’t pursue mathematics because you want other people to think you’re smart. If this is what motivates you, you’ll eventually have a bad time and burn out. You should pursue mathematics if and because you enjoy it. You will inevitably enjoy some parts more than others, but seek out the things that inspire you. There are lots of amazing things to learn and discover.
Asking for help isn’t a sign of failure, it’s a sign of self-advocacy and means that you are using the resources available to you.
Related to the above, come to office hours: it’s individual tutoring with the person who is teaching your class at no extra cost! Don’t be intimidated or feel like you’re wasting your professor’s time: we’re in this business because we like to help people understand things. This is the whole point of office hours and it’s satisfying. If no one shows up it’s boring.

Advice for students interested in graduate school

You don’t have to know for certain what area of math you want to specialize in, but it is good to have some possible ideas in mind. Spend some time getting what sense you can of the kinds of things that are being actively studied in contemporary fields (read survey articles, talk to professors, etc). Apply to schools that have departmental strengths in your possible areas of interest, preferably with several potential advisors.
A career in academia is not the only possible (or worthy) outcome of a graduate education. Many people who earn a PhD in math and go into an industry have benefitted from their experience in graduate school developing their problem solving skills and technical knowlege. So don’t feel like you have to commit to a single career path in order to consider graduate school.
Research experience as an undergraduate is not essential for graduate school, but it can be beneficial in a number of ways. Other things being equal, it makes you somewhat stronger as a candidate, but more importantly it can give you a sense of what the experience of doing actual research is like, which can be very different from the experience of learning math in a classroom setting. Consider applying for a Research Experience for Undergraduates (REU) over the summer of your rising 3rd or 4th year.
Letters of recommendation are one of the most important parts of your application. For this reason, it is important to get to know your undergradute professors well and let them get to know your particular strengths and interests. The more positive and detailed things they can say about you, the better. Make sure your recommenders feel they can write a strong letter on your behalf. Give your recommenders a clear list of your application deadlines well ahead of time (at least three weeks, preferably), follow up in advance of deadlines (a week or so ahead), and make sure you have confirmation that their letters have been sent (usually you’ll get a notification). Don’t forget to tell them about the final outcomes of your applications!

Chris Kottke, Associate Professor of Mathematics, New College of Florida

Resources for students

General advice for students of mathematics

Advice for students interested in graduate school