Here is what I gave out as an extra credit assignment to students who were interested in how to get more out of their textbook. Students mostly got the last problem correct, so the results were that **the people who did this voluntarily really got something out of it**. The handout was 5 pages long.

The last problem was as follows:

- How many pages of the book have we made it through?

(answer: 5)

- How many pages of our own math writing did it take to make it this far?

(answer: 5)

- When you have to read a math book in the future, what is an
important tool you will absolutely need in order to do so
successfully?

*(answer: your own paper and pencil!)

I copy-pasted the activity below although it is out of a google document instead of being properly TeX'd, so you probably won't be able to use it as is (for example, the integral signs didn't make it, nor the exponents or fractions). Still, it's here if you're interested.

The purpose of this exercise is to help you think about how to read a math book. Reading a math book is much different from reading other books!

Step 1: Open your book to section 4.5. Begin reading the “Pattern
Recognition” part. Read page 292 -- everything before the first
example. Don’t worry too much about the details of the “theorem” for
now, but then answer the following question:

Each differentiation rule has an integration rule that tells you how to go backwards. Which differentiation rule goes with “integration
by substitution?”

Step 2: Read Example 1. Then complete this question:

Find the derivative of 13(x2 + 1)3 + C. Why is this answer relevant to example 1?

Step 3. Read Example 2. Then complete this question:

Find the derivative of sin 5x + C. Why is this answer relevant to example 2?

Step 4. Look at the “Exploration” box at the bottom of the page,
after Examples 1 and 2.
4. Complete questions a through e on this page.

a.

b.

c.

d. (multiply by ½ and also by 2, as your first step).

e. (multiply by **__ and also by _** as your first step).

Step 5. Read Example 3.

What does this have to do with the things you did in parts d and e
on the previous page?

There is a sentence after Example 3 that says “After all, if it
were legitimate to move variable quantities outside the integral sign,
you could move the entire integrand out and simplify the whole
process. But the result would be incorrect.”

Use this wrong method (factor everything out of the integral) to find
x2 dx.

What is the actual correct answer for x2 dx?

Step 6. Read the paragraph and formula under the heading “Change of
Variables.”

It says in here that if u = g(x), then du = g’(x) dx. That last
equation looks weird. What happens if you divide both sides of it by
dx? Does that make more sense?

If u = x2, what is du?

Step 7. Read Example 4.
Check their answer to see if it is right (how do we even do that?)

Step 8. Read Example 5.
Check their answer to see if it is right (how do we even do that?)

Following the examples, solve x3x - 1 dx.

Check your answer to see if it is right (how do we even do that)?
Step 9. Read Example 6. It is getting crazy now.

Remember that sin2 x means (sin x)2, and sin3 x means (sin x)3. Check their answer by taking the derivative.

Step 10. How to read a math book.

11a. How many pages of the book have
we made it through?

How many pages of our own math writing did it take to make it
this far?

When you have to read a math book in the future, what is an
important tool you will absolutely need in order to do so
successfully?

"However, they are mostly unable to utilize the expository text in the book -- even the top students." On the one hand, I share your frustration. But from the world of design, there is a different perspective. If the vast majority of students are misusing X, this proves that X is badly designed and needs more user-testing and iteration. The problem is not with the students, but with the design of the book. The solution is to make a book that, by default, students use properly. It will 10x the cost and workload for the author and publisher, but will benefit the students greatly. – WeCanLearnAnything – 2018-02-02T16:32:23.897

@WeCanLearnAnything I agree with most of this, but maybe not the 10x cost; at the freshman/sophomore level, I've had success getting students to learn things from ALEKS explanations. Their style guide for explanations enforces exposition into a small number of consecutive sentences interrupted by examples and pictures, and is delivered on-demand when the student needs it to do a problem. – Chris Cunningham – 2018-02-02T17:39:30.457

@Chris Cunningham - I don't know too much about ALEKS, but my guess is that they user-test and iterate their software with a whole team of employees involved. I would guess that the variable costs alone of their process for developing, say, Pythagorean Theorem materials are AT LEAST 10x as much as a typical textbook publisher would pay to a single author in total. That's probably how they knew how and when to divide their exposition into smaller chunks. – WeCanLearnAnything – 2018-02-04T03:19:24.560

@WeCanLearnAnything I guess I was thinking of the cost from the student perspective, and from that perspective it is not even 1x. That said, I do agree that their review process for exposition is probably a lot more expensive "per page" than most pieces of math exposition. So I think we are in agreement after all. – Chris Cunningham – 2018-02-05T03:54:42.143

1@ Chris Cunningham, yes, I think we are in basic agreement. I'd add, though, that many students have had so many crappy experiences trying to make sense of unintuitive math textbooks (and notes, teachers, etc.) that they perceive almost any sense-making effort to be a waste of time. It's our job to make sure they have better sense-making experiences, but this is tough because so many students in math class are missing so much background knowledge. – WeCanLearnAnything – 2018-02-07T01:44:13.757

12I don't know whether it was "successful", since I was only able to do it for some 60 minutes or so (can't remember) - I was left with some spare time during an introductory course for freshmen. What I did was we read a fragment from a textbook together, discussing it sometimes down to a single word. I'm often wondering whether we mathematicians couldn't use some experience/research on teaching foreign languages - since this seems to have a lot in common with learning to read in a foreign language... – mbork – 2014-05-05T15:13:24.473

2@mbork I think your comment should either be deleted or be converted into an answer! (I prefer an answer! I'll edit the question to allow unsuccessful stories) – Chris Cunningham – 2014-05-05T15:24:43.433

1I'm interested in hearing more about your critical approach. It reminds me of some Engestrom I've read (I think it was "expansive learning" or somesuch, but I'd have to look it up); is it that sort of approach you're taking, allowing students to critique the presentation and representations in the textbook? – JPBurke – 2014-05-05T18:01:51.093

1I'd like to point out that the advice you'd give would depend slightly on the level of the class. If you were teaching topology vs. say calc I, those textbooks can be very different, and they shouldn't necessarily read them the same way. – MHH – 2014-05-06T00:53:52.190

1@MHH I'm aiming at secondary and early-undergraduate courses. – Chris Cunningham – 2014-05-06T01:23:32.507

2

I like Simonson and Gouvea's article on how to read mathematics: http://web.stonehill.edu/compsci/history_math/math-read.htm. However, I haven't used it yet in class beyond pointing out its existence to students. For what it's worth, googling "how to read mathematics" will turn up other pages with handy advice. Perhaps I will try incorporating said advice more actively into my courses in the future.

– J W – 2014-06-23T13:28:35.980