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:

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?