Reading a math book cover to cover is a rite of passage for some, but the process can be pleasant or a nightmare; it can be rewarding or a mindless verification of theorems. This post is a guide to reading mathematics books, and is aimed for someone who is new at the process such as undergraduate or graduate students.
1. Find a book that suits your style of exploration
Some authors will be closer in line with your way of thinking than others. Unfortunately, it's often quite difficult to tell which is which. One way that might help is reading through the first few pages of a book to give you a better impression of the author's style. That, together with looking at the table of contents and introduction ought to be good enough to choose from a list of possible books on a given topic.
For example, I originally starting learning homological algebra through Mac Lane's book 'Homology'. After a while, it seemed as though the author was spending too much time on explicit descriptions of different homological constructions and less time on the big picture, or organising principles, though I didn't know exactly what those were at the time. After about forty pages, I stopped and switched over to Weibel's 'An Introduction to Homological Algebra', and it was much more pleasant and I ended up reading the entire book. This is, by the way, one of the rare times that I've actually finished an entire book.
2. Don't be discouraged by lack of progress, especially with a popular text
An example that frequently comes up is Hartshorne's 'Algebraic Geometry'. Many have read this book and done many exercises from it to learn the theory of schemes, and others have found it frustrating to no end. I gave this book a try a couple of times but never really learned much from it. Early into my attempts I found myself going through proofs and doing exercises without much interest or care in the theorems or goals of the book. The author is very much interested in geometry, but I was interested in the representations of algebraic groups and the fundamental lemma. As a result, the author's interests were so disjoint from mine that the writing style and the theorems presented didn't spark any interest whatsoever.
If you find yourself asking, 'why do I care?' early on, then chances are you should find a new book to read. Don't be discouraged if other people say they were really inspired by the text or that the text has been indispensible throughout their career; that has no bearing on your own progress.
Reading a book because your peers or supervisor have done it is often the worst way to proceed if you are not enjoying it. Serre's 'Galois Cohomology' and Neukirch's 'Algebraic Number Theory' are two other examples of books on topics that I found interesting, but gave up reading after a short while because the style of the book was too different than my own even though they are highly regarded by many.
3. Look for the signs that the book is a good fit
A good fit for you is one that has some or all of the following:
- You are excited to learn the new theorems
- You learn something new about examples you already have in mind and it gives you new understanding of those examples
- You often wonder about questions related to the material and try and solve them or talk about them with others
- Most importantly, you look forward to reading the book and it's fun to do
If I had to condense this entire guide down to one sentence, it would be: keep doing math that you find fun and entertaining, and ignore the rest.
4. Sometimes it takes time to acclimated to a subject
For my PhD oral exam, one of my topics was algebraic groups. At the time, I didn't know too much about them (and perhaps still don't!), so I started reading Humphreys' book 'Linear Algebraic Groups'. At first, I didn't like the subject too much, but after a few weeks I became much more interested in it, primarily because of some interesting examples and theorems I was coming across. This was the only time where I didn't like a subject at first but later warmed to it.
I think in mathematics often there will be topics that are not so interesting to you at first but later become more interesting, maybe due to the discovery of a hidden connection between two different areas, one of which you already like. However, sometimes this process takes a great deal of time and shouldn't be forced. I've noticed that if after a few weeks, I'm still not interested, it's best just to find something else to learn.
5. Don't worry about not finishing a book
Often in books there will be chapters that are not as interesting or optional chapters. Often books have a 'Leitfaden' that describe the logical dependency of chapters and suggest possible reading roots. It's a better idea to read a subset of the book and understand it well than just vaguely having an idea of the whole thing. Finishing a book for bragging rights doesn't help yourself when you've had to force your mind through something unpleasant.
For instance, for Humphreys' book 'Linear Algebraic Groups' that I mentioned above, I didn't read the chapter on representation theory at all, because I found it unenlightening and I simply didn't need it at the time.
6. Look for signs that the book is a bad fit
If you observe the following in yourself, chances are you should put the book down for a while and analyse why you're not making progress:
- You are going through proofs but you are mindlessly verifying them and not taking in the essence of the proof.
- In doing the exercises, you have to refer back constantly even for the easiest ones, and you are just applying theorems without any real understanding of why they are true
- As you go on, you understand less and less until the book becomes just a habit instead of a fun activity
Some of these might be because the book was just badly written. In any case, the best option here is just to try something else.
7. Read actively, not passively
I won't say too much here because this is a piece of advice often repeated on the internet and in books: try and prove some of the theorems, do the exercises, ask your own questions, and come up with counterexamples. I think this process should feel fairly natural when reading, and if it doesn't, it may mean you're just not that interested.
8. Write your own books
If you really would like to learn a topic, a good alternative to reading someone else's book is to write your own. It doesn't have to be a polished book, but rather it could be a set of lecture notes with mathematics gathered from various sources such as papers and books. This will ensure that you're always following your interests instead of someone else's.
9. Create a schedule and track your progress
Use a spreadsheet or other means to keep track of your reading progress and your goals. I've always found it fun to see the number of hours I've spent on various reading projects as a way to keep myself motivated. Some books are quite long so it's nice to see the sum of your efforts on a page.
10. Teach the material to others
Creating a blog and answering questions on stackexchange is a fun way to demonstrate your new knowledge, and gives you another way to synthesise the material besides just doing exercises. Sometimes, if you're lucky, you can find someone else at your university that wants to read the same book as you.
11. Don't read linearly
Feel free to jump around the text, and skip to the later chapters to find results to motivate your progress to understand those results. Also, even if you have understood a chapter or section well, it doesn't hurt to go back and read it differently or do more exercises in that chapter.
12. Don't read too many books
Personally, there have been two books that I have read most or all of and have actually benefited from reading them. Early on as a graduate student, I felt it was very helpful to have a guided tour through certain subjects (homological algebra and commutative ring theory). Later on, my learning style was imbued with more personality and I feel more comfortable in choosing my own, customised learning paths. Be prepared to recognize when this time for comes, and don't be disappointed if reading books actually becomes more tedious and even difficult. Relax! This is merely a sign that you should be paying more attention to the little questions that your mind naturally gravitates to and less attention to the stuff that other people are interested in.
Do you have any more tips? Please write them in the comments below!