Some Thoughts on Homework Problems

taken from Alex M. McAllister

We really enjoy thinking about why various mathematical statements are true. Most of the time we start by looking at some examples that seem to be related. Sometimes these examples will have some common property and by looking at how that property develops in the examples, we can understand a reason why a statement is true. Such a reason will often be a main idea in an exercise or an argument for why the mathematical statement is true.

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.

As with most things that require great effort, the rewards are tremendous and we are confident that you will see your computational and theoretical abilities in calculus improve each week of this term.

Some Thoughts on Reading Your Mathematics Textbook

by Alex M. McAllister

Mathematics textbooks are often read backwards. Starting with the homework problems at the end of a section, one shuffles back through the book looking for any examples, words, definitions, theorems, and/or vague hints that might be relevant to the problem at hand. This can be a very useful and profitable way of reading a math book. However, this should not be the only direction that you read the text for this course. Rather, you are expected to spend some time reading the text from front to back and, even more, you are expected to do so before coming to the class in which the material is discussed.

Now reading math books is not an easy thing. Once, I had a student tell me: "The book is confusing, but that¹s a given with math books." After some reflection, I identified some reasons that support my student's complaint.

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.

So the moral is: spend time reading your mathematics textbook. Your book is a trusty friend and guide that can provide invaluable assistance in your efforts to understand and learn the ideas of mathematics.

Note:  Alex McAllister was one of Brooke's undergraduate math professors.