As an undergraduate I was given some advice on studying that I found very useful. I want to share this with you.
It is important to remember that Mathematics is a tough discipline. On the one hand, there are only right and wrong answers. On the other hand, there can be several approaches that yield the answer. This is what makes the subject so interesting.
To a lot of students a professor working at the board is a magician, in some cases a really boring one. Students are led to believe that the instructor possesses some magic power that makes math really easy for him or her. This could not be further from the truth. While your professor may have a natural inclination for the subject, he or she has put in years of hard work to become proficient.
An analogy
I have always liked the sport analogy. There are several people who like to play sport. Some are more talented than others, and some have a natural inclination toward a specific sport. No sportsman who has achieved even a reasonable amount of success, has done this based solely on ability. They have put in several hours of practice, watched and learned from others, talked to their coach, played in competitive games, and lost and won.
Doing mathematics is the same. There are rules of logic to which must adhere. You have to practice the examples, learn from your book and the instructors, do homework, take tests, and pass and fail.
How much work?
In order to get the most out of your class and to perform at your best you need to put in a certain amount of time working outside of class. There are no hard-and-fast rules, but there are good rules of thumb.
Rule of Thumb
For every credit-hour that you take, you should put in at least an equal amount of time outside of class. If the class is part of your major, then you should probably put in at least one-and-a-half to two times as much work outside of class.
So, if you are taking a 4-credit class that is part of your major, you should put in about 6 — 8 hours of work per week outside of class.
What to work on outside of class.
The best way to study for a course is to take the following steps
Reading
Read ahead. If you going to cover a section or chapter in the next lecture, open the book and read that chapter or section before class. You don't need to understand every detail. Reading the book will help you follow the lecture and it will help you ask questions based on things you really don't understand. If you haven't read the book, you may find the amount information presented in class overwhelming. Remember that your instructor is an expert, not a textbook. When you ask a question from your instructor, he or she can give you additional insight that the textbook may not provide.
Read after class. Take a look at the book in the evening after class. You will find this second reading, combined with the lecture notes, really helps you understand the ideas.
Look at the examples in the book. A lot of mathematics works by analogy. You need to have a lot of examples at hand before you can really understand the connections. You will also need examples when you think about proofs and counterexamples.
Read critically. Question the hypotheses of a theorem. Think about why a certain condition has been imposed. Ask where each hypothesis was used in the proof. Look for typos, mistakes in calculations, and in the solutions at the back of the book.
Homework
Do homework. Do every problem that you are assigned. If you have time, do a few more. Do not look at the solution first and then try to work out the problem. It is much harder to work out a problem to which you you do not the answer. You will only gain insight and ability if you work a problem blind. It's also okay to do a calculation, and get stuck. Coming back to the sport analogy, a harder work-out builds strength and stamina.
Mathematics has its own language. It is a subset of English, but with words having very specific meaning. Try to communicate accurately and precisely. Avoid vague terminology and heuristics. Learn the definitions. There are usually only a few ways in which results are stated. When writing try to emulate the writing style of the book or the writing style of your instructor.
Office Hours
Use office hours wisely. If you have read the book, and worked the problems, you will find that you get the most out of office hours. You will be able ask specific questions that help you finish a problem, and you will understand the response. Your instructor is not a personal tutor and they are not meant to help you cram for a test.
During class
Take notes. You will not remember what was said, or how a calculation was done. Something that seems perfectly clear during class has a habit of being very blurry afterwards. Even if you lose the thread of what is going on in class, keep writing. Later on you can talk to classmates and the professor to figure it out. If the professor is going too fast, you should ask them to slow down.