Random Bits of Advice to a New Math Major

1. Reviewing lecture notes and memorizing definitions and theorems is actually really helpful to your learning.
2. Involve yourself in lectures! Talking in class boosts your self-confidence and your learning.

3. Start your homework as early as possible! This will get you better grades and give you a higher understanding.
4. Go to office hours and ask questions.
5. Be respectful to your professors, and remember they are human beings just like you. Give them the benefit of the doubt whenever you can.
6. Keep an assignment calendar and try to keep it updated as far in advance as possible -- even to the end of the quarter or semester if you already know assignment due dates.
7. Take care of yourself. Try to sleep, try to eat mostly foods you make, try not to overconsume coffee, try to figure out what supplements work for you, try to find some sort of physical activity you enjoy and do it a few times a week, try to take time away from math and do other things you enjoy.
8. Burnout is, unfortunately, real and crippling. See #7.

9. If you plan to take the GRE Math Subject Exam, start studying for it at least three months in advance, and heavily utilize your calculus and differential equations textbooks.

10. LaTeX all your homework.
11. If you don't get excited about math, don't attend grad school in math.

12. It's okay to make mistakes. One day when I was feeling particularly bad about getting a low score on a homework assignment (especially because I had been doing very well on my analysis coursework all quarter long), I walked into my professor's office to ask him about revisions. Once we had finished talking about the corrections themselves, he said “It's okay. I think you're feeling bad about yourself because you made a silly mistake or missed an important detail. I'm going to put this bluntly, and then I'm going to fix it: I don't care. So you made a mistake -- whatever. Welcome to being a mathematician!” and he shook my hand. Then I felt a little better.
13. It's okay if you don't get everything as quick as everyone else does. There is something to be said for being careful and precise.
14. Don't ever take more than 16 credits in a quarter. If possible, don't take more than three math classes at a time.
15. Don't listen to everything the seniors tell you, and don't listen to all fifteen things I wrote above. I got plenty of crappy advice from seniors when I was a new math major. Filter everything you hear through your personal viewpoint because most advice you hear is based on someone else's personality and how that shaped their experience.
16. You are not bad at math.