If you search for advice on how to read a math textbook, you’ll find plenty to choose from. I’ll link to some of my favorites at the end of this article. But first, here’s an overview of my experience reading math books, and what techniques I found to be useful.

## Writing Your Own Textbook

An article in The Harvard Crimson warns that to survive Harvard’s infamous Math 55 class, you need to be willing to write your own textbook. That sounds dramatic, but I think it’s the right approach for learning math at any level. That’s because no math textbook is exactly right for you. You can avoid reading books that are much too easy, or that require background you don’t have. But the ones that remain still make assumptions about what you know, and those assumptions aren’t always right.

The solution: write your own textbook. As you read, take notes that spell out the important ideas from the text in exactly the way that makes sense to you. For some ideas, this will mean writing multiple pages in excruciating detail, perhaps drawing on other sources besides the main textbook. For other ideas that you already know well, you might just write a formula that you can use for reference.

If you follow this approach rigorously, you end up with a customized document that provides exactly what you need for future study or application. For every explanation in the original textbook, your customized textbook fills in the details to make it obvious (to you) why each step follows from the previous one. If a section of the original textbook covers a topic you have no use for, your customized textbook can omit it. When you look something up in your customized textbook, you get an answer that’s at exactly the right level of detail. Or if it isn’t, you can keep updating your textbook until it is.

## Reading a Math Textbook

To write a section of your custom textbook, you first have to understand the topic you’re writing about. That’s where the reading process comes in. There are more and less elaborate processes for reading a math textbook. I’ll focus on a simple one involving two reading speeds: moderate speed, and line-by-line speed. When you start a new section in the textbook, begin at moderate speed. The author probably starts with a few paragraphs introducing and motivating the section (though more advanced texts may skip this part). When you reach a definition, theorem, example, or any nontrivial block of mathematics, switch to line-by-line speed.

When reading line by line, it can help to cover up subsequent lines (in a physical book) or adjust the window size (in the electronic version). This helps you focus on the current line. More importantly, it allows you to think about a claim the author is making before you read more details about it. For a theorem, this means you can try to prove it yourself before reading the author’s proof. For an example, you can work it out on your own before looking at how the author solves it.

Reading a textbook section in this way is like getting access to bonus problems that come with detailed solutions. Then when you get to the exercises at the end of a section, you’ll be warmed up to tackle them. And your custom textbook will give you the information you need to get started on the problems without having to refer back to the original textbook.

## For Further Reading

The previous sections provide a general approach to reading math, with some specific advice. But there’s a lot more advice out there. Here are some resources I found useful.

The canonical question for this topic on Mathematics Stack Exchange is How to read a book in mathematics? The answers present a few perspectives, and some links to other resources, including How to Read Mathematics by Shai Simonson and Fernando Gouvea. This essay is worth reading in its entirety, but here is one key concept: To fully understand a mathematical idea, you need to “make the idea your own” by “following the idea back to its origin, and rediscovering it for yourself” and “mak[ing] the idea fit in with your own perspective and experiences.” The authors quote the book *Emblems of Mind* by Edward Rothstein, who makes the same point this way: “Reading mathematics … involves a return to the thinking that went into the writing.” Writing your own textbook formalizes this process by providing the structure for you to explain the topic in the ideal way for your background.

A couple more articles, and key points:

How to Read a Math Textbook by Macalester College

- “Really important mathematical passages are problems that the author has worked out in detail.” Make sure you can solve these example problems without referring to the textbook.
- “You will not be expected to invent a new problem-solving technique on the exam. Your task is to do the techniques already shown to you in class and in the book.” (So make sure you know them).
- “[Y]ou do not merely read a math test – you work through it! The information has to be dug out.”

How to Read a Math Book by Stan Brown

Brown quotes from a 2003 post by Chan-Ho Suh on the Usenet group sci.math. Here’s how the post starts:

Let me make a distinction between two kinds of understanding a math text. The first is in being able to follow, reconstruct, etc. a proof (proofs make up the bulk of most math texts). The second lies in rewiring one’s brain so that it’s obvious why the theorem is true, and indeed, there is no way the theorem could be false in the context of all your knowledge.

Ideally the first type of understanding leads to the second type. As you learn how scholars have proven theorems in an area, you gain a better understanding of that area, until eventually you can rely on your intuition to lead you in the right direction.

*I’m writing about discrete math and competitive programming this year. For an introduction, see A Project for 2019. To read the whole series, see my Discrete Math category page.*