Choosing a good book to learn from can be the difference between a great learning experience and a terrible one. If you are not enrolled in a formal course then finding an appropriate book to learn from can often fall fully on your shoulders. Even if you are fortunate to be in a course and have a recommended/required text assigned to you, you may need to find additional books to support your learning.
I’m here to tell you my methods for quickly determining if a book will work for me or not. For most of these steps I am going to assume that you have the ability to inspect the book in its entirety (as if you are holding it in your hands). If you cannot inspect the book then you should use whatever information you have to make the best guess you can.
Step 0: Decide What Kind of Book You Want
There are many types of math books out there: pop math, undergraduate textbooks, graduate textbooks, pure, applied, introductory, reference, research, expensive, cheap, classic, niche. It will help you out tremendously to spend some time mulling over what your needs are and what kind of book is likely to meet those needs. You do not need to decide exactly what kind of book you want from the get go, but have a least a rough idea will help you to more quickly spot which books might be of interest and which are unlikely to be helpful. You will get better at this part with practice.
Step 1: Choose Where to Search
The most important factor in determining whether you will find a good book or not is where you choose to search. That is why, before you even begin your search, you must consider where to search. If you are looking for a graduate level textbook on algebraic topology then you probably will not find it at your local bookshop. Likewise, if you are looking for a book to help you with basic algebra, going to a research library is probably overkill. The same thinking applies when soliciting book recommendations from others: you should ask people who will understand your needs and have reasonably similar tastes.
There is no universal best place for every search, so your first question should always be: where is the most likely place for me to find the kind of book I’m looking for?
If you are planning to study a broad topic that is covered in a specific class (real analysis, for instance) a good place to start is often to look at what books are being used for that class at reputable universities. This information can often be found in a course syllabus with a simple search online. For example, a search for “real analysis syllabus” returns dozens of results from schools like MIT, Harvard, Stanford, and others. Most of them have one or more real analysis textbooks listed. This gives a starting list of some books to consider.
Some places to consider searching are local bookstores, public libraries, research libraries, and online booksellers. For reviews and recommendations you can generally ask in any online mathematics community or check out some review archives (see the resources page).
Step 2: Look at the Title
Math book titles often have a variety of keywords in them that will help you understand the book’s purpose at a glance. This will help you quickly determine which books to inspect in depth. Books which have the keywords “introduction” or “first course” are generally targeted at the nonspecialist and have lower prerequisites. Books with short, nonspecific titles often broadly focus on a whole field or sub-field, while books with longer and more specific names are likely to cover a smaller set of topics but in greater depth. More specifically titled books may have greater expectations of previous knowledge. Books titled with “computational” or “applied” are, unsurprisingly, more likely to place a greater emphasis on explicit calculation. As you spend more time looking at different books you will gain a better feel of which keywords to look for and what they mean.
Step 3: Look at the Publication Year
One of the wonderful things about results in mathematics is that they do not expire. While ideas in the sciences go through a process of creative destruction, in math new discoveries are added but old ones are not taken away. Many students might be surprised to find out that most of the material they are learning is already hundreds of years old. What matters then is not when the material was discovered but how it is presented. What this means to you, the book seeker, is that newer is not always better and you can worry about the publication year of a textbook much less than a biology student might.
The biggest thing to consider about the age of books is the rapid rise of computing power. If you are dealing with a topic which makes use of computers, such as most areas of applied mathematics, books written before the computer era may have a vastly different flavor than those written after. The other thing to consider is that certain fields have had great leaps forwards in recent decades, and older books may not include some of the more recent results.
My rule of thumb for most math subjects is that books less than twenty years old are ideal and get priority, less than fifty years old are worth looking at, and less than a hundred years old are acceptable if there is nothing else available. I will not generally go for books more than a hundred years old unless I have a specific reason to.
Step 4: Look at the Author
You should not be too quick to judge an author’s skill by their bio, but learning more about who they are can provide clues to the focus and style of the book. What other books have they written? What is their area of specialization, or are they from a different field altogether? What level of education do they have? Are they more focused on applied work or pure theory? Do you recognize them? The best reading experiences are those where there is a synergy between the author’s intentions and the reader’s expectations.
Step 5: Read the Preface/Introduction
Most math books will have some sort of preface or introduction which provides several important bits of information. A good preface should tell the reader what the book will cover, what they need to know to understand it, and how the author expects the reader to engage with the book. Some books are even kind enough to provide one or more rough outlines for how one might proceed through the book.
Step 6: Look at the Table of Contents
Look at the table of contents to see what content the book covers and in what order. You should see if you recognize any of the material and make sure the book will contain any specific material you are looking for. Notice how many pages the author takes to present the different topics. This is an important point to compare between different books.
Step 7: Skim the First Few Sections
Almost all math books start out slow or with a few review sections which lay the foundations for what is to come. Skim over the first few sections to see how comfortable you feel about what is presented there. If I have absolutely no idea how to approach the ‘introductory’ material then that is a warning sign that the book may be beyond my current level. On the other hand, if I am skimming through chapter after chapter and it never seems to become more challenging, that is a clue that I may need to search elsewhere. Note that these are not hard and fast rules. Sometimes the difficulty level of the early sections may be drastically out of step with the rest of the book. (I’m looking at you, Hatcher.)
Step 8: Look for Illustrations
When you are learning new material (especially material with even the slightest bit of geometric representation) a helpful illustration can make a world of difference. Presenting concepts visually can help build intuition which will ease the understanding of the formalities. I normally flip through a book to see if it has illustrations, what their level of quality is (size, color, clarity), and how they are spread through the book.
Step 9: Look at the Exercises
Exercises are crucial if you are going to be using a book to self-study a new topic. There are some books, especially at higher levels of math, which have excellent explanation and presentation of the material but are completely devoid of exercises. Those kinds of book are still great but they need to be supplemented with another book which does have exercises. Look for books with exercises appearing at regular intervals (such as after each section or chapter), some example problems worked out, and solutions or hints for most exercises available. Some authors like to put important material inside the exercises themselves. I’m not a fan of this since it makes it too easy to miss important theorems or concepts and end up wondering why everything suddenly stopped making sense.
Step 10: Compare Multiple Books
You are not just looking for a good book, you are looking for the best book out of a selection. Comparing similar books can also help you better understand the quality of an individual book, even when you are not yet familiar with the topic it covers. Suppose you are looking for a book to learn algebraic topology, but you do not yet know anything about algebraic topology. If you just pick up a single book and look at its table of contents, it may not mean much to you since you will not be familiar with the topics listed. However, if you look at several different books you can start to notice patterns in which topics they all cover, the normal ordering of topics, and which are unique to a particular book.
Conclusion
There is a vast world of math books out there to be explored. The best way to find books which work for you is to go out and search. try some out. You’ll find some duds, but you’ll find some gems too, often in surprising fields. You won’t regret it.