I modified an article by Shai Simonson and Fernando Gouvea titled How to Read a Math Book, which was shared on a blog. They've since published an improved version here, now titled How to Read Mathematics. Thier new improved version includes mathematical examples. My modified version is below.If you'd like to use these ideas to their fullest, it might help to make your own examples for each idea.
Have you noticed that you read differently when you’re reading different sorts of books? Good fiction takes you away, you can’t put it down. But sometimes just the opposite happens with a good non-fiction book – you have to put it down, so you can think more about the ideas. Math books are extreme that way.
Don’t forget poetry and plays, where reading aloud can be what makes it work, or cookbooks, where every word makes a difference (that’s like math!), or reference manuals, where you wouldn’t want to read straight through, but really need to skip around, and try out the suggestions.
So how do you read a math book? Here are some ideas. (Simonson and Gouvea wrote for higher-level math students, so I’ve edited to make it more approachable for us.)
When we read a novel we become absorbed in the plot and characters. We try to follow the various plot lines and how each affects the development of the characters. We make sure that the characters become real people to us, both those we admire and those we despise. We do not stop at every word, but imagine the words as brushstrokes in a painting. Even if we are not familiar with a particular word, we can still see the whole picture. We rarely stop to think about individual phrases and sentences. Instead, we let the novel sweep us along with its flow and carry us swiftly to the end. The experience is rewarding, relaxing and thought provoking.
Novelists frequently describe characters by involving them in well-chosen anecdotes, rather than by describing them by well-chosen adjectives. They portray one aspect, then another, then the first again in a new light and so on, as the whole picture grows and comes more and more into focus. This is the way to communicate complex thoughts that defy precise definition.
Mathematical ideas are by nature precise and well defined, so that a precise description is possible in a very short space. A mathematical piece and a novel are both telling a story and developing complex ideas, but a mathematical piece does the job with a tiny fraction of the words used in a novel.
1. Look for the Big Picture
“Reading mathematics is not at all a linear experience ... Understanding the text requires cross references, scanning, pausing and revisiting.” (from Emblems of Mind, by Edward Rothstein)
Don’t assume that understanding each phrase will enable you to understand the whole idea. This is like trying to see a portrait painting by staring at each square inch of it from the distance of your nose. You will see the detail, texture and color but miss the portrait completely. A math article (or a section in a good text) tells a story. Try to see what the story is before you delve into the details. You can go in for a closer look once you’ve built a framework of understanding. Do this just as you might reread a novel.
2. Be an Active Reader (and Expect to Read Slowly)
One section of a math text usually tells only a small piece of a much larger and longer story. The way to really understand the idea is to re-create what the author left out. Read between the lines.
Mathematics says a lot with a little. The reader must participate. At every stage, s/he must decide whether or not the idea being presented is clear. Ask yourself these questions:
• Why is this idea true?
• Do I really believe it?
• Could I convince someone else that it is true?
• Why didn't the author use a different argument?
• Do I have a better argument or method of explaining the idea?
• Why didn't the author explain it the way that I understand it?
• Is my way wrong?
• Do I really get the idea?
• Am I missing something?
• Did this author miss something?
• If I can't understand this can I understand a similar but simpler idea?
• Which simpler idea?
• Is it really necessary to understand this idea?
• Can I accept this point without understanding the details of why it is true?
• Will my understanding of the whole story suffer from not understanding why the point is true?
Putting too little effort into this participation is like reading a novel without concentrating. After half an hour, you wake up to realize the pages have turned, but you have been daydreaming and don’t remember a thing you read.
Reading mathematics too quickly results in frustration. A half hour of concentration in a novel might net the average reader 20-60 pages with full comprehension, depending on the novel and the experience of the reader. The same half hour in a math piece buys you a page or two, depending on the topic and how experienced you are at reading mathematics. There is no substitute for work and time.
3. Make the Idea your Own
The best way to understand what you are reading is to make the idea your own. This means following the idea back to its origin, and rediscovering it for yourself. Mathematicians often say that to understand something you must first read it, then write it down in your own words, then teach it to someone else. Everyone has a different set of tools and a different level of “chunking up” complicated ideas. Make the idea fit in with your own perspective and experience.