by Mark Johnson
Central College, Pella, Iowa

The intelligent reader of a mathematical book desires two things: First, to see that the present step of the argument is correct. Second, to see the purpose of the present step.

It's in the section labelled something like, ``How to use this book: notes for students,'' usually located before the table of contents.

First Reading: look for the Big Picture.
It's easy to get bogged down with details in a math book. No doubt about it. So, it's helpful to let yourself skim through a section your first time through it. Just get a sense for what's there and maybe why. Try to see where the authors are taking you, and worry about the details later.

Second Reading: with paper, pen(cil), and calculator.
Now start worrying about details. Try, as much as you can, to verify everything that is in your text. In other words, make sure YOU think each step is correct.

Many results must be given of which the details are suppressed... These must not be taken on trust by the student, but must be worked out by his [her] own pen, which must never be out of his [her] own hand while engaged in any mathematical process.

Notice that last line: the pen must always be in your hand ``while engaged in any mathematical process.'' To a great extent, people think mathematically through writing. It's hard to do in your head.

Continually ask: What is the Point?
The only way you will learn mathematics is to make it your own. It has to become part of you-not in some weird way, where you suddenly start wearing pocket protectors-but in the way that ``1+1 = 2'' is just a part of what you know. It's not alien.

Asking the question, ``Why is this here?'' or ``What is the point of this?'' can help the process of making the mathematics your own. It is a particularly good question to ask of examples, which are almost always in the text for a specific reason. The reason isn't always stated, but if you look for it, often it isn't too hard to find.

The same advice applies on a smaller scale, and I think that's what Polya had in mind. As you read through an example or proof of a theorem and are checking each step (remember the Second Reading!), try to be thinking of what the point or purpose is of each step. Why do they calculate this here? Another way of saying the same thing is, ``Keep your head up out of the sand.'' Don't get buried too deep in calculations.

Learn the vocabulary.
Mathematics obtains much of its power by constructing a very precise vocabulary. When learning new mathematical vocabulary, it's helpful to distinguish between formal and informal definitions. Informal ones are good for getting a feeling for what the word means. They help to build up your intuition.

Formal definitions are also important for building intuition. What does this mean? Just that working with a formal definition will help you to develop correct mathematical intuition for the concept. It comes from no other place-mathematicians (usually!) aren't born with an intuitive knowledge of, say, continuous functions. They develop it by working with the formal definition.

Because formal definitions are precise, they are one of the few places in mathematics where memorization can be useful. So, during your second reading-with pen or pencil in hand-look for both formal and informal definitions and rewrite them someplace. (Rewriting is one way to memorize. It gets your hands involved.) It may be worth collecting them in one place, to build up a calculus glossary.

Learn the Theorems.
Theorems are another source of power for mathematics. Why? Because a theorem is something we know for sure. Working from precise definitions, mathematicians prove consequences of those definitions, and in doing so, they create theorems. There are few things in life as certain as a good theorem.

Like definitions, theorems are stated precisely and should be rewritten and remembered in the same way.