Advice
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:
- Develop:
- 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.
- Refine:
- 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.
- Resolve:
- 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.