How To Read A Mathematics Textbook

When I was a young and foolish undergraduate, Professor Blyth told us at the beginning of a math course how important it is to read a textbook several times. Once is not enough, he said. I recall very clearly how strongly he emphasized that reading a mathematics textbook is NOT like reading a novel.

Years later, I was struck by the following passage from the article A Writer’s Ten Commandments, by Stephen Vizinczey (his official web site is here), reprinted in Truth and Lies in Literature, pages 4–5 (the quote is in the middle of a commandment “Thou shalt not let a day pass without re-reading something great”):

Don’t commit the common mistake of trying to read everything in order to be well-informed. Being well-informed will allow you to shine at parties but is absolutely no use to you as a writer. Reading a book so you can chat about it is not the same thing as understanding it. It is far more useful to read a few great novels over and over again until you see what makes them work and how the writers constructed them. You have to read a novel about five times before you can perceive its structure, what makes it dramatic, what gives it pace and momentum. Its variations in tempo and time-scale, for instance: the author describes a minute in two pages then covers two years in one sentence—why? When you’ve figured this out you really know something.

Remember, Vizinczey’s comments are directed at aspiring writers. (And in the penultimate sentence, I believe he is referring to one of his great novels, An Innocent Millionaire; see page 35 of the 1983 McClelland and Stewart hardcover edition for, “Thus passed nearly two years.”) So, I suppose that Vizinczey is saying that reading a novel is not like reading a novel either (at least for those who want more than a superficial understanding). But nevertheless, I believe his comments apply more generally to learning mathematics as well.

It’s worth recalling Feynman’s method for reading technical books (I can’t remember in which of his books I read this): He would read the book, and at some point would no longer be able to understand. Nevertheless, he kept reading to the end, not understanding much. Then he started again at the beginning, and he invariably found that he was able to read past the point at which he originally stopped understanding, until he reached a second point beyond which he didn’t understand. He again went right back to the beginning, and started fresh. Eventually, with enough readings, he would understand the whole book.

Repetition is the theme. And it’s comforting, isn’t it, to know that even the Feynmans of the world don’t understand everything instantly. It takes hard work for everyone, even the truly great ones, to understand mathematics and science (and anything of depth and substance; have you ever tried wading through Kant or St. Thomas Aquinas?).

Here are some specific suggestions for students in the early years of university on how to read a mathematics textbook (similar comments apply to physics, and I am guessing that they might also apply to other sciences, too):

1. Balance “reading around” the subject with reading specifics relating to your course textbook. I always liked to find alternative books, which approached the subject from a different perspective from the course textbook. This helped me to get a “big-picture” perspective on the subject, especially if the alternative textbook had lots of words, explained something about the raison d’etre, and had some historical information. It also helps to read general books about mathematics to provide you with some breadth. Despite what Vizinczey says, this sort of reading can be relatively superficial, because it is not meant to make you shine at parties … it is meant to expose you to a broad range of mathematical ideas. Just make sure not all of your reading is superficial!

2. Also balance your reading for conceptual understanding with two other activities, both of which can be assisted by textbooks:
(a) working a large number of exercises. This is essential for developing technical proficiency. It’s important that your technical skill be good, because otherwise you waste a lot of time being stymied by derivations in class or in the textbook. (“How did they get from this line to the next one?”) With poor technical skills, you will also waste a lot of time doing assignments, which will take precious time away from doing a wide range of exercises and problems.
The typical textbook has examples to support technical skill development. So carefully read through the examples, and then do as many exercises as you can. Do the exercises systematically, a little each day, and keep a record of your work. At the start of every study session, review the exercises you completed the day before. Daily work will result in good progress, in the same way that daily work helps you learn a musical instrument, a sport, or a martial art.
(b) Solve actual problems, which are different from exercises. Exercises illustrate techniques that are presented in examples. True problems require deeper thought, and the exercise of some creativity. Training in problem-solving is quite an art, and typical textbooks do it badly. Some specialized books that you might consult for training in problem-solving are The Art and Craft of Problem Solving, by Paul Zeitz, and A First Step to Mathematical Olympiad Problems, by Derek Holton.

3. You’ll have to experiment for yourself when to read and when to attack problems, but I highly recommend that you do both in advance of attending lectures. That is, keep a little ahead of the lecturer. You will be amazed at how helpful this is in speeding your progress, and helping you to understand the subject. Before each lecture, read the relevant sections of the textbook, work through the textbook, and attempt some exercises. Even if you don’t fully understand, the lecture will be so much more helpful to you; seeing the material for the second time will greatly aid your understanding. Then you can follow up by reading the relevant sections again, and working more exercises and problems.

4. Read with pencil and paper by your side. Don’t read aimlessly; if you are not generating questions (“active reading”), then you are probably not getting much out of your time. By the end of each reading session you should have written out a nice list of questions; presumably some of them will be answered in the lecture, but if some have not, then ask the lecturer during the lecture, or see the lecturer during office hours. For best results, try to answer the questions yourself before you finish your reading session; this is another aspect of active reading.

5. Daily work is key. Do your best to budget your time so that you devote some time every day to each of your courses. When I was in second year of undergraduate studies, there was one particular course that I didn’t like, and so neglected it. One Saturday, I resolved to spend the whole day on this course, in order to catch up. And I did spend the whole day on the course. But, I got tied up with deadlines in other courses, and I didn’t look at the troublesome course again until the next Saturday, when I decided to buckle down and have another day-long session. I quickly realized that I had forgotten every single thing that I had done the previous Saturday. What a waste of time! Live and learn … . Learn from my mistake: daily work is the best, most effective way to learn. And because it is effective, it will save you time in the long run, meaning you will be able to do lots of other fun activities besides learning (which also becomes fun when you do it right!).

6. Be systematic. Keep a notebook of questions (might be electronic), and note your answers, too. This will become a wonderful record of the evolution of your understanding. Also keep solutions to your exercises and problems in a neat notebook. Review the notebooks often, ideally daily. Daily work speeds your progress to an amazing degree. And you will only be able to systematically review your exercise and problem solutions if you keep a neat record of them.

(This post first appeared at my other (now deleted) blog, and was transferred to this blog on 22 January 2021.)