Most students seem to have considerable "test anxiety", and many talk themselves out of doing well before they even see the exam. If you do this, you will not be able to show what you are really able to accomplish. You need to try to gain some confidence in your own work, and realize that you can be successful, even on a difficult exam.
Here are some hints that you should consider. Most of these are things that you need to be done throughout the term, so don't wait until the day before the exam to begin this!
Review your notes each night. This critical part of the learning process often is neglected by college students who have been successful in high school without it. Reviewing your notes daily will help you to organize the material for yourself, and to locate areas where you may have questions. It is especially important if you spend most of your class time taking down notes and find it difficult to think about the material and write at the same time. Don't expect that your understanding will come during the lecture; learning simply doesn't work that way. You need to spend time thinking about the course outside of class. Unless they specifically plan for that, most students don't do this.
Come to class prepared. Do the homework before the next class, if you can, and be prepared with questions if not. Look over your notes briefly in the few minutes before class starts, so that you remember where we left off and what we were trying to accomplish. I will not go over homework problems that people don't have questions about, so if you want to know how to do the hard ones, you need to ask about them.
Form note-sharing groups. If you are having trouble taking notes and listening at the same time in class, you might consider forming a note-sharing group with two or three other students in the class. For example, if there were three of you in the group, each would take notes one class per week while the other two could pay full attention to the lecture. After the class, the note-taker would photocopy his or her notes and give copies to the other members of the group.
Understand the definitions first. The definitions and theorems are the key to understanding mathematics (not the procedures to follow). One can often do a computation without any real understanding of the material, so that is not a good measure of comprehension. Your gaol is to understand the concepts, so you need to consciously work at that. Most students find that translating between symbols and ideas (or vice versa) is one of the hardest aspects of mathematical thinking. One way to get practice at this is to take each definition we produce in class and write down how you use it in terms of mathematical symbols; if the definition already includes symbols or formulas, try to write them down in terms of words that say what the different parts of the formula are. This is a critical skill that you need to develop. I would be happy to look at your translations for you if you want advice on them.
Organize the course material as you study. When you go to review for exams or quizzes, you should think about other ways that the material could be organized. This will help you recognized the interconnections of the material. One good practice is to condense five or ten pages of notes into one page by selecting the most important items. This act of choosing helps you to organize the material in your head; you determine what is important and what is not. Without this ability, you will have way too much to study and will get hopelessly lost. Once you have gone through all your notes and condensed them, you can go back over the condensed ones and do it again. See if you can condense your notes down to one or two pages by the end of the term.
Don't forget to go over the homework and quizzes when you study. The quizzes and homework are not just for practice of things we did in class; they also should lead you to develop important concepts or procedures that I will expect you to know. A good practice, particularly when you are studying for an exam, is to look back at each problem from the quizzes and the homework assignments (particularly the ones that are not from the book) and try to give a one or two sentence statement that summarizes what that problem is about. If it's a procedure for doing something, make sure that it has been added to your notes.
Don't do all your homework by looking at the answers first. Many students look at the answer before starting the problem (or before thinking long enough about it), and then work toward that answer. Remember, however, that on exams (and in life) you don't have the answers to work toward; if all your experience is working with the answer already known, you will have trouble when you don't have this on the exam. When you do problems in preparation for an exam, don't look at the answers and don't use your notes until you are all done. Try to make the situation as close to an exam situation as you can. Choose four or five problems and give yourself half an hour to do them with no additional help. Pretend you are really taking an exam; that's the best way to be prepared.
Read through the examples in the notebook outside my office. I put copies of the best work there, and this is an important source of information for you. Make it a habit to go through this notebook after each assignment. I expect that you will know the right answers to the problems, even if you didn't get it right initially. If you aren't able to figure it out from my comments on your paper and the printed comments for the class, this is the place to go, or come and see me.
Before turning in a take-home quiz or problem set, read over the comments from the previous one. The printed comments that I make for you are an important source of information about what I expect on your written material. As you write up your answers for the next one, it is a good idea to go back and read what I said about the last one. Often these comments are about general problems that people are having, and you will be surprised at how many points you can gain simply by following my advice from previous problem sets or quizzes. Don't make the same mistake twice!
Don't leave your studying to the last minute. Many students seem to feel that they can learn ten weeks of material in one or two nights of study. Does that really seem feasible? The best way to study is to keep up with your work as you go along; that way you will not have to study before the exams, as you will already know most of the material, so you can spend your time brushing up on a few weak points rather than learning the basics. Consider this: if you are on a sports team, do you wait until a day or two before the game to start practicing? Try to apply that same attitude to your academic work.
Get a good night's sleep before the exam. Many students stay up too late studying the night before an exam. This is a bad idea, in general. If you had a big game to play the next day, would you think it was a good idea to stay up all night? Studies indicate the lack of even one or two hours of sleep can reduce your effectiveness and responsiveness by as much as 25%. Since most of you already will have a sleep deficite from staying up late perviously, this can be a serious problem. One study that tested sleep-deprived drivers showed that their reaction times could be as deminished as those of intoxicated drivers. That is, it was as bad for them as driving drunk! Do you think it would be good to take an exam while drunk? If not, be sure you get your sleep.
A second study showed that a sleepless night had an effect not only the day after, but again three days after (apparently the second day was OK), so be aware that if you have an exam on Friday, then staying up all night on Tuesday may affect that!
Another recent study indicates that sleep is a crucial component in memory; that the material one studies must be "locked in" by a period of sleep. Furthermore, after a certain time with no sleep, no new information can be retained. The old joke about "Can we stop? My brain is full." seems to be true; at some point, your capacity to add new information is seriously reduced until you experience actual sleep. And that's not just doing something else for a break, but actual, honest sleep. Don't neglect this!
Read through in-class exams before starting them. Most students start right in on the first problem when they get an exam. It is to your advantage, however, to take three minutes and read through the entire exam before starting in order to give you an idea of how much work there is, which ones are the easy ones, and which will take you longer. This is essential information if you are going to budget your time properly. Knowing what is coming is as important as knowing what you are currently doing. There is no reason why you have to do the problems in the order they were given. Some people like to do the ones they're sure of first, to get them out of the way, while others like to work on the hardest ones first, so they don't feel rushed on them later. Choose an approach that works for you.
Another reason to read through the whole exam first is that sometimes later problems can shed light on earlier ones. Don't neglect this important source of information!
Budget your time on exams. There is not a lot of extra time on in-class exams, so you need to budget your time wisely (that's one reason you should read through the entire exam first). If you end up spending 20 minutes on one problem, you may not have time for the other ones. Be aware of how long you have spent on one, and don't be afraid to stop working on it and go on if it is taking too long. Don't let one problem spoil your entire exam.
If you get frustrated, take a deep breath and regroup. I have seen students work themselves into a state where they can't even think anymore. Don't let this happen to you. If you start to feel overwhelmed, put your pencil down for a moment, close your eyes, and take a few deep breaths. The 15 seconds it takes to do that is well worth it if you can get back your composure. Remember, no one problem, or one exam, is worth the misery you make for yourself that way. Again, it's better to leave one problem blank than to let it block you from doing the rest of the exam.
If you've done all the problems you know how to do on an exam, and there's still time, make up some of your own problems. If you don't think the exam I've written allows you to show me what you know, then turn the exam over and write what you do know; show me what you can do. Don't just sit there in frustration, and don't just leave when there's still a half an hour left. Be careful, however, that you really can do the problems you write; nothing looks worse than if you can't even do the problems you choose yourself.
Pick up your final exams. I am surprised at how few students come to see their final exams. In the last few years, only a handful my students who did. I don't see how you can put the amount of effort and time you do into these, and then not care about the result. This tells me that most of you only care about the final grade, not about learning the material. The final exam is the culmination of the class, and the questions there bring together all the material in the course; I would expect that you would be curious about the things you didn't get right. Also, if you want a complete set of notes, the final exam should be part of that. In any case, it's an important source of information for your future that many of you are neglecting.