Teaching How to Read Mathematics

At this time of year many teachers are picking the calculus book for their class to use next year. At the same time you will find teachers complaining, quite correctly, that their students don’t read their math textbooks. Authors, editors and their focus groups try their very best to make books “readable,” to no avail, since students won’t read them anyway.

Maybe this is because students have never learned to read math books, because no one has ever taught them how. Have you? Here are some thoughts and suggestions gathered from several sources that may help.

First, some obvious (to us) comments, which, alas, probably won’t make much of a change in students’ ways:

  • It takes time to read a math book. Unlike a novel or a non-fiction book, a few pages of mathematics will take longer than reading a story or essay.
  • Readers should stop every few lines and make sure they understand what they’ve read
  • Readers should have pencil and paper handy both to take notes, to draw graphs, to work through some of the examples.
  • Math books contain examples to help the reader understand what’s going on; so readers should work (i.e. with paper and pencil) through the examples.
  • Readers should make note of what they don’t understand and ask about it in class.

Here are some things you can do to help your students at all levels learn to read a mathematics textbook. The sooner students learn to read mathematics the better. Work with your pre-Algebra and Algebra 1 teachers (or earlier) to get them started. The sooner the better, but if they have not done it before they get to your calculus class do it then.

  1. Start with short reading assignments and spend some time before and after discussing both what they read and how they read it. Do not do this forever, rather
  2. Don’t reread the text to them or follow the text exactly in you class discussions; make them responsible for understanding what you’ve assigned them to read. Of course, you should answer questions on anything they didn’t understand, but expect them (eventually) to learn from what they read.
  3. A brief but structured reading organizer can be a help. Have them make notes on what they read in a form like this:
    1. In your textbook read section ___.__,  pages ____ to ____
    2. What is this section about? What is the main idea?
    3. There are ____ new definitions (or vocabulary words) in this section. For each, express the definition in your own words, include a drawing if appropriate.
    4. There are ____ new theorems (rules, laws, formulas) in this section. For each, write its hypothesis and conclusion and explain what it means in your own words, include a sketch if appropriate.
    5. Which application or example was most interesting or instructive for you? Why?
    6. Is there anything you find confusing or do not understand in the reading?

The next day in class meet in groups of 3 or 4 and compare answers: Does everyone in the group agree on the new vocabulary? Which paraphrase is better? Which example/application was the most interesting? Why? What questions do you still have?

Hold the students responsible for doing this work by not repeating what they have just read as a formal lesson on the same material. Approach the material from a different way; probe their understanding with questions.

Instead of a lecture on the material they read, just have a discussion on it. Let the students lead the way explaining what they think the text means, why the examples were chosen, and what they are still unsure of.

In my next two posts I intend to discuss definitions and theorems in more detail – their structure and how to help students use the structure to increase their understanding.

The Electronic Discussion Group or EDG run by the College Board for AP Calculus which has now become the AP Calculus Community is an excellent source of help and information. Some of the ideas here are taken from an EDG discussion on helping students read mathematics textbooks. I’ve also used and expanded ideas from Dixie Ross, Stephanie Sains, Jon Stark, and David Wang that appeared on the EDG. Thanks to them all.


3 thoughts on “Teaching How to Read Mathematics

  1. Mr. Macmillan, I have learned so much from you through your blog. Can you help me further by explaining the scoring statistic of 2012 and 2011 exams. I could not understand why there are more than 6 questions. And which is split in two. If split then how to explain which question has what mean score. My students asked for mean score of let’s say # 4, 2011 FRQ. What would be that. 2.29? Or 2.74? Thank you so much.

    Sonal Patel


    • One of the BC questions (2011 BC 3 listed as questions 3 and 4) contributes to the AB sub-score on the BC exam. Part 3A was part of the AB sub-score, 3A, 3B and 3C all counted to the regular BC score. The means for this question are listed in two parts, one for the AB (called question 3) and the other for the BC only (called question 4). The overall mean for this question is the sum of the parts. (listed as 3 to 4). Questions 4 to 6 were renumbered as 5 to 7.
      I hope that makes sense.


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