Reading Textbooks

This is a prescriptive handout: for more on textbooks in general, consult the webpage at

A typical calculus course is built around some kind of text. Notice that each section of the text is about ten pages long, each containg one lecture's worth of material. This means that you will be reading about ten pages every two days. There are two general methods for reading text.

  • When you read analytically, you try to get into the book thoroughly, and follow it along where it goes: you try to let the author have his way when you read it.
  • When you read synoptically, you try to keep your own interests paramount, and try to get out of the work what you want to get out of it.

When reading a book analytically, you have to follow the author. When reading a section, scan through it before reading it to try to get a sense of what it's about. Then read it carefully, not skipping past the hard parts. Notice that the vocabulary is introduced in definitions, while the `facts' that you have to learn are the theorems, which are statements of the form:

hypotheses implies conclusions.

When reading a theorem, you must figure out what the hypotheses are and what the conclusions are.

There are two major approaches for understanding a theorem:

  • Analyze it as you would a sentence in an English course, identifying nouns, verbs, adjectives, and internal logic. Analyze the proof as you would a paragraph in an English course, following the logic of the argument (for that is what a proof is: an argument).
  • Look at examples. Find examples that satisfy the hypotheses and see why the conclusion is also satisfied. Find `counterexamples' where the hypotheses are false, and see if the conclusion still holds.
It often helps if you can figure out how to state the theorem in your own words.

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