Tips for Reading Mathematics

Focus on concepts, not exercises.
The most important material in your math text is the stuff between the problem sets at the end of each section! If, in the past, you have opened your math book only to the problem sets, looking at any other part of the book only for an example that might shed some light on the current problem, you may find this strategy less than helpful in this course. Instead, set aside time to read the text, before the material will be covered in class, when you are not working on a homework assignment. This will enable you to focus on the mathematical concepts at hand.

There are an infinite number of types of mathematical problems, so there is no way to memorize the solution method for each. Mathematics is about ideas, and the problems you are assigned are expressions of those ideas, giving you not only practice in problem solving, but alternative ways to look at the ideas the problems embody. If you can learn the key concepts, you will be able to solve any type of mathematics problem (including ones you have never seen before) involving those concepts.

Read the text more than once.
Many of the ideas presented in a typical mathematics course have confounded brilliant minds in centuries past. It is not surprising, therefore, if you have difficulty learning these same ideas, especially if you quickly scan through the reading assignments just once. You should expect to go through each reading assignment several times before you can gain a full understanding of the material.

When reading through the first time, scan for big ideas
The first time you read a chapter of the textbook you should be asking yourself, "what is the main idea of this chapter?" Look for the big picture. The details are important, but you need to be aware of the lay of the land before focusing on the trees on the hill or the field under cultivation.

The second time through, fill in details
After you get the big picture, you should then look at the details. Take some time to think about each of the definitions, formulas, and scenarios you encounter (more on this later).

Read with paper, pen, and calculator
As you are reading through the text, you should be writing notes. Check calculations. Rewrite definitions in your own words. See if you can come up with your own examples.

Read the narrative
There is a story to be told in mathematics. What is the progression of ideas being told? Don't just skip to the formulas and examples!

Study the examples
What points do each of the examples illustrate? Some examples are extreme cases; other examples are supposed to illustrate "typical" situations.

Read the pictures
There are good reasons for the many pictures and graphs in a mathematics text. Ask yourself, "what features of this picture are important to the key concepts?" Focus on how each picture illustrates a particular idea.

Learn the vocabulary and the language
Pay attention to definitions and what they mean. Mathematical language is very precise, and a word may have a different meaning when used in a mathematical context than when used in everyday speech. On the other hand, often the way a word is used in everyday speech can give you an entry into understanding the word in a mathematical context.

Use the index and appendices. Know what every word means
Make sure you understand all of the words and ideas. If there is a particular word which you do not know (or which you want to know better), look it up. Use the table of contents and the index to help you.

Make a note of things you don't understand, and ask for help
Even after using all of the techniques above, you may still find some of the ideas unclear and/or confusing. That's OK! You are studying difficult material! If there is something you don't understand, mark it. Write down any questions you may have. You can then bring up these questions with your instructor or a classmate. Don't be afraid to ask in class! Nine times out of ten, if you had a question about something, many of the other people in the class had the same or a similar question. I cannot emphasize this enough: the most important thing that will happen during your math class is progress. Asking and exploring questions in class is a marvelous way to make progress.