Textbook Reading Strategies

Reading mathematics is an active, not a passive endeavour. As you read your textbook, you should be stopping to ask yourself questions to test your understanding. This is a skill that takes time to develop, so here are some beginning strategies:

  1. After reading a definition, stop to come up with an example of something which satisfies the definition. Then come up with an example or two of things that don't satisfy the definition, and ponder which part(s) of the definition they fail.
  2. After reading a theorem, stop and identify the hypotheses and the conclusion. Try to invent examples which don't satisfy the conclusion, and then evaluate which of the hypotheses they fail (if they fail the conclusion, they must fail one of the hypotheses).
  3. After reading a justification, stop to review the reasoning used to reach the conclusion. Sometimes this reasoning is given informally, perhaps in the paragraphs before a theorem. Sometimes this reasoning is a listed afterward in a formal proof. Try to write out a point-form skeleton of the proof, and pick out the places where the hypotheses are used. Practice closing the text and giving a brief overview explanation of the argument.
  4. When you come upon an example problem, read the statement of the problem and then stop and try to work out the solution yourself before you continue reading. If you can't do it -- and that's not a failure -- then, try to outline the ideas that will be used in the solution, based on what you've recently read. Then, as you read the solution, compare your ideas to the solution method to see how or when they are used. When you're done, try to summarize the ideas of the solution, or the key method that was used.
  5. While you read, make a review sheet listing the definitions, theorems and methods you've seen. Don't include everything: try to figure out the most important key ideas and write just those. Commit these to memory and look for them to show up in lecture.