notation. Learn the definitions right away, not just before an exam. You shouldn’t have to keep looking up definitions while writing your homework proofs.
2. Sometimes unfamiliar notation is really hard to understand. Suppose that you are reading a homework problem; that is, you have copied out on paper the statement of the problem. And suppose that the statement of the problem makes no sense to you whatever. Any graduate student will tell you that this happens all the time: you want to work on a problem, but you don’t quite even understand what the problem says. What you should do is at least write out the statement of the problem. (If you have already done this, do it again. You’re trying to make an impression on your mind.) And if you have any ideas (even patently silly ones), play with them a little. Then stop working on it and do something else. The next day, after you have slept, it is very likely that you will understand the problem at least a little bit better. Your unconscious mind does some of the work while you sleep. (This is one reason why you should always start your homework as soon as possible. At least acquaint yourself with the problems, even if you don’t solve them right away.)
3. You cannot learn to prove a theorem by watching a professor prove it on the blackboard, even if you take notes. Unless the professor makes a mistake (and this does happen occasionally), any proof done by a professor on a blackboard will look easy and seem to make perfect sense. (The professor, like a professional magician, has usually practiced the trick before dazzling you with it.) The test of whether you understand a proof is not whether it seems to make sense to you, but whether you can prove the theorem yourself. See whether you can reproduce the professor’s proof without looking at it. If you’re really feeling enthusiastic, try to prove the same theorem in a different way. Even if you don’t succeed, this is great mathematical exercise.
4. Most theorems can be proved in several different ways. Your proof of a theorem and my proof of the same theorem need not be the same in order for both to be correct. (In fact, this is one of the most important and enjoyable features of mathematics: that there are many, many ways to arrive at a given mathematical truth.) There are some theorems, though, which are hard to prove unless you remember a particular trick. In this case, you should definitely leam the trick. Memorize it. Practice until you can do it easily, like a card trick. Mathematical tricks are very useful, and you should know as many of them as possible. Your bag of tricks is part of your mathematical toolbox. As Feynman says somewhere, don’t despise tricks; make them your own.
5. The following experience is very common among students of mathematics. You stew over a problem for hours or even days, unable to see how to do it. Then, later, you’re doing something else and not even thinking about mathematics. Suddenly you see the solution in your mind, and it’s obvious.
This happens to everyone, beginners and professionals alike. The time spent stewing over the problem was not really wasted. It helped you to find the “obvious” solution.
What is “obvious” depends on what you know. In mathematics, everything you already understand seems easy, and everything you don’t understand yet seems impossible.
A variant of this experience occurs as follows. A professional mathematician is reading a mathematical paper in a scholarly journal. One paragraph contains the sentence: “It is obvious that 0.”
The mathematician works for three days, and finally proves that x = 0.
“Oh, yes,” says the mathematician, without conscious irony. “That is obvious.”
You should not use the phrase “it is obvious” in your proofs, even though you may sometimes see this phrase in print.
6. Sometimes you, the student, may feel that everyone else in the class understands the mathematics in this course easily, and that you alone are confused. This impression is almost certainly false. In any case, if you are confused about some detail of a proof that is being presented, you should ask a question immediately. Since mathematics that one doesn’t understand seems just like meaningless gibberish, it’s always considered polite to ask a question during a mathematical lecture.
If a person at the blackboard keeps writing x where you think y should be, ask about this right away. A person writing on a black-board needs help from the audience to get the details right. And even if your intended correction is wrong, you need to know that it’s wrong in order to understand the rest of the argument.
Often the students who have the least confidence are actually the best students in verbal mathematics. So take heart when you feel that everyone else knows more than you do. It probably isn’t true.
7. When you do ask a question in class, you may be disconcerted to find that the professor does not understand what you are asking. This can be frustrating, but it is a perfectly natural occurrence. What we are studying in this class, even more than in verbal mathematics in general, lies right at the threshold of intelligibility. Until your question can be translated into formal mathematical language, its meaning is genuinely unclear. Pronouns such as “it” seem innocuous to the questioner but confuse the hearer. Be patient.
8. A professor who seems irritated by a question is not angry at the questioner. The professor does not understand the question, and is annoyed at not being able to figure it out. Be patient. Professional mathematicians have difficulties akin to those of beginners. Mathematics is not personal, and no one will ever be personally angry with you for asking a mathematical question. The gruffer-seeming mathematicians are not angry; they are just trying to think under pressure, concentrating on the mathematics and not on smiling reassuringly at the student. Gruff professors are much kinder than they seem, and often turn out to be good company.
9. Sometimes the professor makes mistakes, and sometimes mathematical texts contain mistakes. What’s great about mathematics is that it makes sense. If you and I make opposing mathematical claims, we can use mathematics to settle which of us (if either) is correct. Don’t believe a mathematical statement just because a book or a professor says that it’s true and claims to prove it. Is the proof correct? Is the statement true? Can you disprove the statement?
When a professor writing on the blackboard seems to be making a mistake, be sure to ask about it. Whether you’re wrong or right, you want to know the truth. One of the most refreshing features of mathematics is that mathematical disagreements are not personal. Any disagreements should be resolved as soon as possible.
10. No methods of thinking, figuring, or computation are beneath the professional mathematician. If we can add better by counting on our fingers, we will do so without shame. If we feel that a picture will help us think about a problem, we draw a picture. (If you would like a professor to draw a picture to illustrate a proof, just ask.)
Like a magician, a mathematician does plenty of work behind the scenes. When you write a proof for homework, you are performing a magic trick. Your first draft can be messy, with all sorts of false starts and side figuring. Your final draft contains nothing but the finished proof. Looking at your final draft, a reader would think that you had written the proof easily and smoothly, without effort, getting everything right the first time. But you, the magician, know better.
11. Finally, mathematics is one of the great pleasures of the human mind. Even calculating is fun for those who can do it well. The real fun in verbal mathematics comes in seeing something familiar in a new way. When you suddenly understand how an argument works, when you get the joke, that’s where the real fun lies.
Here in the U.S.A. at the start of the twenty-first century, most people are completely unacquainted with verbal mathematics. Thus the fun of verbal mathematics has become an unintentional secret. By learning the language of mathematics, you are putting
