Another goal of this course is to exercise your ability to think, something that will benefit you even if you never use the material we covered.  Presumably we come to college with the purpose of developing our brain.  If the goal was to develop and tone our muscles, no one would question that a regular and sometimes strenuous physical exercise was necessary, rather than lying on the couch.  Brain power works the same way: it is increased by straining the brain through thinking, making connections, and yes, memorizing, and not by watching TV!

How to study: best results come from understanding what you are doing.  Understanding comes with a higher initial time investment, but it pays off later.

The first thing on a student's mind is usually solving problems given for homework.  HOW NOT TO DO HOMEWORK  is to look at the homework problem and flip back through the section in search of similar problems.  (You might as well admit it - you do it, don't you?).

Instead, you should do the following:

Have a pencil and scratch paper ready.  Mathematics is not read like a novel - it is more of a sing-along where you follow what you are reading with notes on paper that help you understand, perhaps by considering some examples or counterexamples or expanding ideas with some "what-if" explorations.

Start by reading the class notes pertaining to the section you are studying.  This will give you an idea which material and what kinds of problems I think are important. 

Then read the section in the book.  Identify ideas that were discussed in the lectures and work out the additional examples presented in the book if you need to reinforce these ideas or methods of solution. (I usually try to present examples similar to spirit in the book, but different).

Since the book is far more comprehensive than the lectures could ever be, there will be examples in the book that are unlike things that were done in class.  Those examples are good reading (and you should read them), but don't have such a high priority, so may be skimmed through if you don't have much time (and you usually don't, right?)

ONLY NOW that you have understood what the section is about should you start solving the homework.  If you have performed the steps above, this should not be difficult.  When you are done, it is a good idea to check your answer if the problem is an odd-numbered one and the answer is in the back of the book.   On occasion it will happen that a homework problem is unlike anything you have read in class notes or the book.  For example, it could be like one of the problems in the book that you skipped when reading through the first time - now go back and read it.  Or it may be something unlike those examples, in which case the problem relies on your experience and possibly uses some facts from previous sections.

Writing down the solutions on homework and tests:  in this course, we will be writing many proofs. The book outlines some good practices when writing them, and you should follow these. Usually, it's not a good idea to plunge into writing a proof right away: you should make a few notes aside to help you get to the gist of the idea. Once you see the gist of the proof, then write it in longhand. Your goal is to convince me that you understand what you are trying to prove and how to go about it, which often involves neither too little, nor too much writing. It is a good idea to reread your proof after you are done and check whether an informed reader who sees your text for the first time would be able to understand it.

Finally, write neatly and in an organized way.  Messy and unorganized papers annoy me and leave me with the impression that you don't know what you are doing.  Now, DO YOU WANT ME TO BE ANNOYED AND UNDER THE IMPRESSION YOU DON'T KNOW WHAT YOU ARE DOING WHEN I AM DECIDING HOW MANY POINTS TO AWARD TO YOUR EFFORT?

Preparing for a test: if you did the work assigned for homework, this should simply amount to reviewing.  Start several days before the test. Make sure you have all the basic stuff down (most of the tests usually deal with basic skills, only a few problems are more involved). Look at the review problems at the end of the chapter and work on them.  If some of them are not going so well, find the section from which they were taken and study it again.  Finally, and this is probably the single most important piece of test-taking advice that I can give you, as well as the most ignored one: GET ENOUGH SLEEP on the night before the exam.  Those extra hours of cramming beyond midnight are seldom worth more than having a clear and refreshed mind.

I don't even want to give any advice to people who start studying seriously the day before the exam.  This is a very ineffective method so don't be surprised if it gives poor results.

Taking the test: you probably know these tips already.  Do the easy problems first.  Since a lot of people do the problems in the order they are given, I try to arrange the problems so they go from the straightforward ones to the more involved ones (though sometimes the availability of space on pages interferes with this plan).  If you don't have a clue on how to even start a problem, skip it and come back to it later.  Also, it can't hurt to ask me a question about a problem during the test - the worst that can happen is that I tell you that I cannot answer the question or that this is a problem you should know how to do because we did it in class.