Preparing for Exams
Here I'll outline the strategies that I believe work best in preparing for exams in mathematics. To be clear, this is based primarily on my own experience and what I have found to be the most effective for myself. I recommend taking the outline here and tweaking it to most effectively meet your needs. Here are the steps to my method:
List the Topics
Make a list of all the major topics you are responsible for. It can be very high-level and overarching or more detailed as you see fit. Make sure your list is comprehensive (check with the instructor beforehand).
Get the Basics Down
Write down (as a list, or using flashcards) all of the definitions and theorems you need to be familiar with in precise and technical terms. If a problem asks you about some definition or theorem and you don't know it precisely, you likely won't solve it. In other words, here you should memorize what you really need to have down for the exam.
Systematically go through practice problems from each of the major topic areas in order to diagnose your comfort level with each topic. Use the textbook, old assignments, material from your instructors, and the internet to source these problems.
Focus on Weak Spots
Do heavy conceptual review (via YouTube, the textbook, recorded lectures, other online resources) on the weaker topics, review solutions to challenging problems in those same areas. [This is my favorite place to use group review sessions to get some explanations and pointers from peers or the professor/TA.]
Revisit practice problems from those areas and re-evaluate your comfort level with them.
Rinse and Repeat
Repeat steps 4-5 as needed and as time permits.
Shore up knowledge in your more comfortable areas by doing any needed conceptual review and going through more practice problems as needed (for proof-based courses, also reviewing proofs of major theorems). You should ensure you feel comfortable with the problems and could reproduce solutions in isolation if needed.
You've studied hard at this point and it is important to go into exams with a clear mind and free of nerves and stress. Take a walk; drink tea; listen to music; do whatever you need to do in order to make sure you are mentally poised for the exam. Make absolutely certain to get a good night's sleep before the exam!
Most of the advice below applies to timed exams. For take-home exams or ones without time constraints, I think the advice below may still be helpful, but perhaps less directly.
Time management is key: determine at the start of the exam how long you can afford to spend on each question, and keep track of your progress both after you finish each problem and as you're working through them. Re-evaluate the situation repeatedly to keep the optimal strategy in mind.
Do the easy problems first and quickly, save the hard ones (proofs especially) for last. This will afford you the most time to work on the most difficult problems.
Do not spend too high a percentage of your time on any one problem, unless it is the last one you're doing (in which case it should also be the most difficult one).
Most exam questions will not be unreasonably difficult, considering time constraints. Keep this in mind when writing up solutions. If your answer seems too complicated, odds are that it is (which doesn't necessarily mean it's wrong; maybe there's a simpler way to express the same idea).
Even if the exam doesn't start well and you're behind on time, stay calm. There is still opportunity to make it up, and stressing out will not help you do the best you can with the time you do have remaining.
Keep in mind, exams likely emphasize the major points of the course. Odds are that solutions will involve major over-arching principles or main theorems. It is not likely your solution needs to be very nuanced and repeatedly reference more minor results.
What I Wish I was Told Sooner
There is a lot to put here, but I will try to keep it brief.
Mental and physical health are essential to succeed in academics. Make them a primary concern; do not put school work above them unless absolutely necessary.
Grades are not everything. Regardless of how they turn out, life will go on and you will be alright.
Make sure to strike the appropriate work-life balance (in college especially).
Don't allow yourself to fall behind; catching up is way harder than keeping up.
Go to lecture and discussion. Attend office hours. Just do it. It is worth your time.
Read the syllabi for your courses carefully. They should inform how much to care about your homework vs exams, where to find resources for the course, and so on.
Math is hard; it is alright if you don't understand something the first time you see it.
If you have a question, ask it. Remaining silent will only hurt you in the long run. Furthermore, it's unlikely you're alone in your confusion.
Have faith in yourself. If you want to succeed, you have to first believe that you can.
Set a schedule and keep to it as best you can. This will help to make you maximally efficient in work and afford you the most time to do whatever else you desire.
Make a to-do list and keep it updated. They help to organize so much.
Engage in career planning as soon as possible. Lay yourself a roadmap for how to land the career you want and start taking the necessary steps as early as you can.
Start investing as soon as financially possible. Growth compounds and the sooner you start, the better.
Pursuing an Advanced Degree in Math
This section is primarily aimed at younger students who want to study math beyond what is typically seen in undergraduate material. Admittedly, I do not have experience in advising students beyond this level, but I still think I can possibly provide some valuable perspectives.
The first thing to understand is that the journey will be a long and difficult one. If there's one trait that's needed to find success in studying advanced math, it is mental fortitude. There will be times when you will find material difficult, when professors or other superiors will demonstrate a lack of faith in your abilities, and when the work will feel like too much. It is important to respond well, and to use those hurdles as motivation without letting them overwhelm you. The simple truth is that if you cannot cope with that adversity, it will be very hard to find success in studying high level math.
I think it is also important to adopt a particular mindset about math. You should hold a dedication to seek understanding, and your focus should not be on grades alone. In my eyes, it is vital to hold a genuine intellectual curiosity about the content, because to reach this advanced level it is important to explore the subject on your own and to have it be self-motivating. Internal motivation to study math is key because it will require hours and hours of reading textbooks, attending lectures, watching videos, and researching online. It will take work to seek out research opportunities and find problems which are accessible to students before graduate school. It will be easy to get burnt out if the motivation doesn't come from within.
Finally, about math. Take courses from a number of different areas and try your best to get some breadth as an undergraduate. It is very tempting to get very deep into one research area because this will likely be beneficial for the sake of graduate program applications, for example, but I don't think this is the optimal approach. Doing research is important, but if you're going to go to graduate school regardless, having a wide base of knowledge is probably more important for your future career, and also means you will probably more quickly find which sub-fields of math are most appealing to you. In particular, I would recommend taking at least one advanced course in all of the following subject areas: Linear Algebra*, Differential Equations, Real Analysis*, Complex Analysis, Abstract Algebra*, Topology*, Discrete Math and Combinatorics, Set Theory and Logic, Number Theory, Dynamical Systems, Probability Theory, and Numerical Analysis. It's fine to miss out on a couple of those (the asterisks indicate the truly essential ones), but you really should take as many as possible. If it seems like a lot, it's because it is: about 60% of all the classes I took in undergrad were math courses, and I still missed out on several of those topics (but saw others I didn't list too).