Wednesday, May 19, 2010

Reading a Math Book

Students really stumble here. They prefer to just look at examples, to help them with the problems. Actually reading a whole section is daunting for most of them. On my syllabus, in a section on preparing for class, I've included this for years:
Read the next section. (Reading a math book is different from other reading. You may have to read a section 3 or 4 times in order to fully understand it.) Take notes, using your own words; I’ll check them a few times during the term. (This gets easier over time, and you’ll become a more independent learner.)
But that's not enough. Last year, I read How to Read Mathematics, and loved it. The level was way too high for most of my students, though, so I made my own handout. I'm still working on it.  Maybe you have suggestions?

Here it is:

How to Read A Math Book

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. I’ve mixed my own ideas together with ideas from an article by Shai Simonson and Fernando Gouvea ( They wrote for higher-level math students, so I’ve just kept the nuggets that seemed approachable for this course. Their words (sometimes modified) are indented.

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 and symbols of those 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 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 have built a framework of understanding. Do this just as you might reread a novel.

2. Be an Active Reader (and Expect to Read Slowly)

A math article 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.


  1. I like the idea of teaching kids how to "read" their math texts. As a high school math teacher, I've found that it is unfortunate that many kids don't have a clue how to decipher the information from their math textbooks and, as a result, they depend only on the information given to them in class.

    Another area in which I see a lot of kids struggle in high school math classes is studying for their tests. Most kids can do alright in their other classes because it is simply a matter of memorization, but in math students actually have to know and understand the process to getting to an answer. While the process remains basically the same, the problem that they must solve can differ in so many ways.

  2. I put this together while I was on sabbatical, and then forgot to use it. I wanted it yesterday, and couldn't find it on my computer. Thank goodness it was here. I'll be using this in Linear Algebra soon.


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