Some Thoughts on Homework Problems

Looking at examples, crafting proofs, and solving exercises takes a lot of time. In our fast-paced society, it can be difficult to come to grips with the fact that solving the types of exercises we will work through and that writing articulate and thorough solutions requires focus, attention, perseverance, and effort every day. Students do not successfully cram for an exam in this class! Learning math is a lot like being a part of a sports team -- players must practice every day. In fact, most students say that studying at least ten hours a week is a minimal requirement. You should explicitly think about your schedule and when you can make the study time you need to succeed in this course.

Some math books are confusing because they are garbled and poorly written. Others are confusing because they are written for an audience with more mathematical knowledge. One important responsibility of an instructor is to avoid books that fall into these categories, and, with one exception, I have chosen books for my courses that are written clearly and for the right audience.

Instead, most math books are confusing because the ideas they present are difficult and require a great deal of energy and concentration to accurately understand. In the words of one Isaac Todhunter: "Another great and special excellence of mathematics is that it demands earnest voluntary exertion. It is simply impossible for a person to become a good mathematician by the happy accident of having been sent to a good school." Reading your textbook is one important step in the process of grappling with mathematical ideas and making them your own.

Now we all approach reading different books in different ways; reading a math book is very different than reading a cookbook, or a bible, or a novel (although, a good math book contains traces of all of these). In fact, reading a math book effectively is a different experience requiring a different approach than almost any other type of reading. You need to have a writing utensil in hand and a piece of paper (this could be the book itself; Fermat scribbled his Last Theorem in the margin of his text). As you read, write down important words and ideas, draw pictures, figure out examples, argue with the book. That is, you should actively seek to understand what you are reading. Your book tries to help with boldface words and different colored boxes and pictures, but you need to be able to express the ideas you are studying in your own unique way. Finally, you need to pay careful attention to when you need to back up and reread a passage. If (when) you don't understand some sentence or paragraph you've just read, go back and chew on it some more until the ideas are clear.

With homework assignments, I typically include a list of important terms that will appear in the next section of the book. While reading, find their definitions and write them down in your notes. Also, record an example or picture with the definition which will provide some concrete expression of the abstract ideas with which we are working. And, you should do this before coming to next class. As one Steven Zucker wrote: "That the student must also learn on their own, outside the classroom, is the main feature that distinguishes college from high school." The time we spend together in class should be devoted to clarifying and reinforcing ideas, not considering them for the first time.