On Reading Textbooks

This page was written with calculus students in mind, although much of it applies to anyone taking a college-level math course. I will assume that you are taking a course that requires not only using but understanding a certain amount of material, and that this material is presented to you in a text.

A typical calculus course is built around some kind of text. These texts are not very different from each other, and indeed from other technical mathematics books in general. One of the things you will have to do is to learn how to deal with books like this.

Notice that each section of the text is about ten pages long. If you look at your syllabus, notice that you will be covering about a section per lecture. This means that you will be reading about ten pages every two days. How should you be reading this? Well, not like a novel. When you read Tom Swift and his Jetmarine, you read it very rapidly to find out what happens. If you read War and Peace, you read it slowly to catch the language and literary allusions. A calculus text is like niether of these: unlike an adventure story, it is dense and you have to master it; unlike Serious Literature, it is (supposed to be) written so clearly that you do not notice anything but the content.

There are two general methods for reading a dense work.

  • When you read analytically, you try to get into the book thoroughly, and follow it along where it goes: you try to let the author have his way when you read it.
  • When you read synoptically, you try to keep your own interests paramount, and try to get out of the work what you want to get out of it.
One way to think of the difference is this: when you read a book to find out something that the author has to say, you must read analytically lest your own interests obscure the author's intent; when you are doing a research project, and the needs of the project are paramount, you read synoptically, to try to get what you need for the project.

You may be used to reading mathematics synoptically: reading the text in order to figure out how to do the exercises. This is a practical and frequently successful way to do things: since you are usually tested on your ability to solve problems, this often works in getting decent grades. But you may find yourself not really understanding what is going on, you may find yourself carrying out steps and computations and producing numbers that mean heaven-knows-what if you do it this way. The problem is that in reading synoptically, you are allowing your mind to filter out everything novel that would ultimately help you to understand what is going on.

(Analytic versus synoptic non-technical reading is discussed in detail in How to Read a Book, by Mortimer Adler.)

When reading a book analytically, you have to follow the author. You start with the preface (where he tells you what he hopes to accomplish) and the table of contents (which tells you what he does). Each time you come to a new chapter, you should scan through it to get some idea of what it's about. Each chapter begins with an introductory paragraph to help you with this. This is important because as you are going through section by section, the material might seem pointless unless you know where it fits in.

The same goes through with each section. Scan through it before reading it to try to get a sense of what it's about. Then read it carefully. Notice that the vocabulary is introduced in definitions which you (hopefully) will use so often that memorization will not be a problem. The `facts' that you have to learn are the theorems, which are statements of the form:

hypotheses implies conclusions.

When reading a theorem, you must figure out what the hypotheses are and what the conclusions are.

This may present a problem, for most theorems are rather abstract statements, and you may not be used to this kind of abstraction. One consequence: it may not be entirely clear what the theorem is saying. For example, consider the following theorem (since this is .html, I will use e for epsilon).

Let O be an open interval, and let x be an element of O. Then there exists e > 0 such that for any real number y, if |x - y| < e, then y is an element of O.

How would you make sense of this? The way to analyze long and complicated sentences is to break them into their constituent clauses. That makes it easier to make sense of them. You can try outlining the sentence (try to go through it slowly, to get the logic of it):

1. (Hypothesis) Let O be an open interval, and let x be an element of O.

2. (Conclusion) Then there exists e > 0 such that 2.1 is true.

2.1 For any real number y, 2.1.1 implies 2.1.2 below.

2.1.1 If |x - y| < e, then:

2.1.2 y is an element of O.

This allows you to get a sort of picture of the clauses of the theorem. (This is an example of a hypertext analysis of a statement. For a discussion of this kind of analysis, see Leslie Lamport's How to Write a Proof in the American Mathematical Monthly 102:7 (1995).)

Another way to make sense of theorems is to play with examples. Imagine that O is the open interval (0, 1). Draw a picture of (0, 1) (I really mean this: right now, get out a pencil and paper and draw this interval). Mark some point in the interval with a point for some x. So we have (1): a set (0, 1) and an element x in (0, 1).

Note that if 0 < y < x or if x < y < 1, then y is also in the interval (0,1). So suppose that e is the smaller of x, 1 - x. Then as 0 < x < 1, we have (2): e > 0, provided that we can satisfy (2.1).

But the distance from x to the endpoints 0 and 1 are x and 1 - x respectively, and so any y closer to x than e = min {x, 1 - x} must also be in (0, 1). That is to say: if (2.1.1), y is within distance e of x, then (2.1.2), y is also in (0, 1).

So there's the idea. To understand the statement of a theorem, you should outline it, work out the logic of its language, and toy with a few examples so you can get a feel for it.

It is usually a wise idea to make sure you understand a theorem before continuing. If you find a theorem intractible, you probably should seek help. It is often a waste of time to glide over material, reading words but not understanding what they are saying.

You may have noticed that it takes some time to unravel the statement of a theorem. Again, if you are used to reading fast, this can be difficult, just going over a single statement carefully, as if you were reading a poem. This requires patience, and a willingness to put time into the task. (You are probably beginning to notice that a math course is going to consume serious time.)

And now for some more work. These theorems did not come out of the blue: they were thought up somehow, and justified somehow. The justifications are the proofs. A proof is a kind of convincing (inescapable) argument. In this course, you should learn how to recognize a (valid) proof, and construct (valid) proofs of your own: this boils down to learning how to recognize and develop convincing arguments (it is this aspect of mathematics that convinced pedagogues for centuries that the way to teach people how to think clearly was to teach them mathematics).

And here's the problem: a mathematical argument relies on logic, and logic appears to be alien to human thought processes. (In fact, most people fling phrases like `illogical' or `irrational' about, when they usually mean `you have the effrontery to disagree with me.') ``Logic'' is in fact a precise method for analyzing and/or constructing certain kinds of very rigorous arguments. Most people avoid logic because:

  • It is difficult, and requires going over things slowly and carefully.
  • Using logic can force one to reach conclusions that don't fit one's opinions.
Logic was used in philosophy, but even some philosophers grew tired of finding themselves reaching conclusions that they didn't like. In modern politics, which consists largely of demagoguery and whining, there is very little logic indeed, and advertizers and spin-meisters would like even that little to go away. But if one is not to be the lawful prey of these creatures, one must learn logic. And since logic is indispensible to mathematics, this is as good a place as any.

So here is one use for what you are studying ...
To learn how to detect lies, delusions, and other hooey

Back to mathematics. There are examples of how to solve problems using these theorems. When you come across one of these, try to do it on your own (only peeking at the solution when you have found your own or are stuck): you will get more out of it that way. Remember while reading this: your memory will retain material that you invest emotional capital in, and reject material that you feel indifferent to. So if you read the text with detachment, obsessed with the desire to get it all over with, you will not retain it as well. (It also doesn't help if you are perennially distracted by thoughts of things that you would rather be doing.) This means that you will both (i) not retain individual facts as well, and (ii) you will not get a very good idea of what's going on. Only by trying to connect with the author can you get a clear picture of what this is all about.

After you've done the reading, and come to class to look over the material again, you can do the homework. You will find yourself referring back to the text while doing the homework: this is synoptic reading (now its okay) to try to find precisely what you need for that specific problem. In most texts, the exercises towards the beginning of each exercise section are mechanical problems that can be done by using the examples as templates; notice that the exercises towards the end are quite different --- and more difficult. The first few exercises will consume very little time: the exercises towards the end can consume a lot of time, but that it is these latter exercises that will help you master the subject. In essence, it is impossible to really learn a subject in mathematics by memorizing a handful of templates. One must learn how to deal with problems that you have not seen any templates for: you must learn how to meld templates, vary them, see how far they will bend, and ultimately construct your own. For more on homework, click here. If you don't learn how to do this, you will never be a master of mathematics; fear of mathematics will be a master of you.

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