Some Advice for Taking Your First Proof-Based Math Course

If you are like me, you did not particularly enjoy math in high school, so the idea of learning an entirely new type of math in university might be terrifying to you. I know this can be scary and overwhelming (it was for me!), so here are some things I learned when taking my first proof-based mathematics course. Hopefully you find this helpful.

This is mainly aimed at computer science students, since my course was an introductory discrete math course for computer science majors. And I’m not saying that you should listen to me, but this might help, who knows? It’s just some stuff I learned while struggling through the pain of trying to rewire my brain.

Also, if you’re a computer science student: induction is your friend, and it shows up literally everywhere, so you might want to get very comfortable with it and its very close cousin, recursion. Not being comfortable with recursion is probably a very bad idea.

Another note I’d like to put here is that it’s completely normal for math notation to be terrifying and cause your brain to want to shut down at first. Mathematicians like to act like their field isn’t terrifying, but it 100% is and I feel like they’ve lost touch with that fear (if they ever had it) because the mess of symbols actually has meaning to them now. With enough work, some of the notation will also stop being an abstract mess of symbols to you and will also start to have meaning. My professor put it this way: it’s like learning a new language. I tend to agree with that sentiment.

Anyway. Onto the actual list of bits of advice.

  1. Go to every class, or at the very least, watch every lecture. All of the ideas build on each other, and if you develop gaps in your understanding, things will just crumble eventually. (Also, it’s easier in my opinion to learn math by watching someone else do it than by reading a textbook, where a good chunk of the thought process will be abstracted away and invisible).
  2. Skipping assignments is also a terrible decision, for similar reasons.
  3. Sometimes, the prof might assume that you know some piece of background information and continue to teach the lecture as if you already know it. When you have that vague feeling, you should do what you can to figure out what that piece of background information is, then look it up.
  4. Discuss the problem sets with other students. This was both allowed and encouraged in my math courses, and it was often a lifesaver. (Just don’t copy from each other’s solutions if they’re graded assignments.)
  5. Sometimes I don’t understand things until I hear someone else explain them. Sometimes it takes me several weeks to understand something. I think this is normal.
  6. Did you understand the proofs the prof showed in lecture? No? Make sure you understand them eventually, because you will almost certainly make similar types of arguments in your assignment and tutorials. At the very least, it’s good to be able to recognize which ideas you can steal from lecture and tutorial proofs, and sometimes the process of trying to adapt them will actually lead to a point where things start to make sense.
  7. ChatGPT is pretty terrible at writing proofs, beware.
  8. I try to write all of my assignment proofs twice: once to work them out, and once to make them pretty for submission. I find this helpful because often my first attempt is messy/janky/hard to read, and I sometimes catch errors or missing details that way.
  9. This is just a thing I find annoying, but if you’re going to type math, I think you should use Latex or an equation editor or something. Relying on regular keyboard symbols to type math gets messy and ambiguous very fast.
  10. If you learn a definition in class and see it used more than once, you should probably remember it, because it will come up again in your assignments and you will have to use it yourself. Or at the very least, look up the definition before you use it.
  11. Generally, starting a proof with the assumption that the thing you’re trying to prove is true is a bad idea and things tend to go downhill from there.
  12. It’s a good idea to periodically go over stuff from earlier in the course, because sometimes, when you don’t get something, it’s actually because you’re missing some idea from earlier in the course that you accidentally glossed over.
  13. Stuck? Sometimes drawing pictures/diagrams helps.
  14. Lastly, I think “how is this useful??” is a perfectly valid question to ask in a math course. I feel like it’s easier to care when things don’t feel abstract and pointless. I spent a lot of time asking about applications of the content during office hours.