Here is some advice I give to students in my mathematics classes at all levels.

Get the book!

If your course is built around a text, get it and read it as you work through the course. You should be reading the text and working through examples and problems several times a week. Students in my classes who have tried to get by without getting their own copy of the book, depending on friends and the library, have done substantially worse because they studied from the book less and learned less from it. All of us faculty try to use books that will continue to be useful for you after the course is done, both for reference and for additional learning. So get the book, and use it regularly and well.

The best way to master mathematics is to maintain a constant habit of doing mathematics.

This means, in particular, that for each math class you are taking, you should schedule at least one hour at least three days a week to work on that class. Spreading your work out over the week is much more effective that trying to cram all of your studying for each class into one night of the week. (If you are contemplating staying up all night before class cramming, you may as well sleep without studying—it is just about as effective and gets you more sleep.)

Don't be afraid to be confused.

Learning mathematics is a process which often involves first developing and refining a confusion and then resolving it. This process is a struggle. But if you aren't struggling you probably aren't learning, so endeavor to learn to struggle well. By developing and refining a confusion and then resolving it, I mean:

Read the text, go to class, work the problems, and get confused. A course which offers you something to learn should offer you some good confusions.
Don't be lazy in your confusion. When you have developed a good confusion try to understand exactly what you are confused about. Here are some good things to ask yourself when you are faced with a confusion, whether it is an exercise, a difficult passage in the text, or a difficult passage from the lecture.
  • "What is the most concise and precise way I can describe my confusion?" The answer, "I can't do problem 17," is not useful. Can you read problem 17? Do you understand all of the terms? If not which ones? If so, then you understand quite a bit about problem 17; what exactly is your confusion?
  • "Is there anything like the problem at hand—perhaps something more general or something more specific—which I have already come to understand?" Try to think about all that you have seen, in or out of the course, that reminds you of this problem. Maybe you remember something which is similar but not the same, and maybe some variation of the solution of the old problem will work for the current one?
  • "How does this issue relate to the context in which it arose?" What happened just before (in your reading or in the lecture) you became confused? Are you sure you understood that? How does this relate?
  • "Can I change the confusion at hand in some way to make it tractable?" Maybe you can't solve the problem at hand, but you could solve it if only you changed it a bit. Look at the solution to the changed problem. Maybe it can help you understand the problem at hand.
  • "What is given?" What, exactly, do you have to reason from?
  • "What is desired?" What, exactly, do you need in the way of an answer?
  • "Can I make a good guess? Why is it good? And how can I check it?" Guessing is under-rated. If you can make a guess and check it, then you are possibly all the way done with the problem. Even if, after checking, you see that your guess is wrong, it may still be the case that thinking about your guess helps you to think more clearly about the problem.
Don't expect to solve every problem right away. If you are stuck on a problem, leave it for a while and then go back to it. If you are still stuck, do something about it: Search the examples and the exposition in the text for the answer. Try the problem again. Work related problems which you do know how to do with an eye toward discovering what you are missing. Try the problem again. Ask in class. Try the problem again. Then read the text again. Ask your classmates. (Study together at a regular time. Share your confusions!) Read the text again. Ask your professor in or outside of class. (But be prepared! A well-cooked confusion is often the beginning of a very productive discussion, whether in or out of lecture.)

Seek lots of different kinds of help.

Talk to your classmates. Go to the clinic. Talk to your professor. Speak up in class. Work together with your classmates, but do not copy from them.

Be organized.

Get a daily planner. Schedule your work-time a week in advance if you can. Evaluate your effectiveness regularly. Schedule more work-time (or different hours) if you are not getting enough done. There is no one correct prescription for how much time you should spend outside of class. However, if you don't come to class with a refined confusion and if you don't resolve your new confusions when you leave class, then you will probably learn very little. So time outside of class is of great importance. For many students, spending twice as much concentrated time outside of class as is spent in class is too little.

College mathematics exams are different from most high school exams.

For one thing, we often want to see how deeply you understand the material, not just how well you do routine problems. Explaining a line of reasoning is often more important than the answer itself. Often problems will require that you have synthesized the material, not just digested discrete chunks of it. When working a computational problem, you must be able to do the calculations, but you must also understand the ideas behind what you are computing.

Send questions, comments and complaints to Thomas Hunter
Last modified: 12/22/2010 by Thomas Hunter.