Reading Your Textbook

It is essential in this (and all math courses) to read your textbook. The text contains a wealth of knowledge. It is often a work of a master teacher in the subject, who is not only giving examples of how certain skills are performed, but really trying to teach the subject to the reader: the vocabulary and concepts, why these concepts are important, how certain procedures/skills arise from those previously learned, the thought processes when approaching different types of problems, and what aspects of a problem make it different (so that it requires a different approach) from another. What better than to have two teachers, one with whom you can interact and ask questions, and the other who is a recognized leader in the field and whose “lectures” are with you whenever you are free to open the book?

Reading a math textbook is not, however, like casual reading. You must be inquisitive as you read. Each section you read will contain new concepts, new skills, new vocabulary. You should continually ask “why?”: “What questions can be answered using this new material that could not have been answered before? Are these the types of questions that it would be natural to ask? Had they occurred to me? If not, why would someone want to know their answers?” You should also ask “how?”: “How is the new material related to material I have already learned? How is it different?”

At the same time, you must pick up the vocabulary. In some sense it is like learning a foreign language. How can you expect to accurately convey your ideas if you do not speak the language? But it is not quite as simple as when you learn, say, French. Since people in different cultures generally think and converse about the same things, learning a foreign language is usually a matter of word substitution — saying “pain” (French) instead of “bread” (english), “j'ai faim” instead of “I am hungry”. In math sometimes the vocabulary includes words you already know and use in daily speech. Nevertheless, the word will have a very specific meaning in the mathematical setting, often an idea or concept that you have never formulated or thought about before. Learning math vocabulary entails pinning down just what this concept is, and being able to use the word in sentences where this concept makes sense.

Some specific suggestions (along with those mentioned in the last couple of paragraphs) as you read:

If you are not already accustomed to reading mathematics in the fashion described above, be aware that it will definitely increase the amount of time you devote to reading. Nevertheless, it is time well spent, as it generally prepares you much better for doing homework problems (and the kinds of problems you will find on exams as well) than a more cursory reading of the text would. The amount of time required to complete homework should drop as a result of your efforts.

Working in Groups

Though from time to time I make assignments which are meant specifically to be done in a group, you are always encouraged to work with others on out-of-class assignments unless otherwise indicated. In fact, I urge you to find one or two other students and agree to meet at a regularly-scheduled time each week to study together. Here are some reasons for doing this (some of these may apply to you more or less than others, depending upon your innate mathematical ability):

Having said all of this, remember that you are individually accountable for your learning (exams, after all, are not group efforts). The end result must be that you are able to discuss (usually in writing) the concepts of the course. College subjects like mathematics and statistics are not spectator sports! Work on a problem by yourself before seeking help, identifying specifically the place you get stuck. When you get help, ask for the least amount of information necessary to get you going again. Once you've made it to a solution, give that problem a rest and see if you could do it again (without peeking at any notes) the next day. A rule of thumb: Understanding what another has done does not mean that you can generate the same solution on your own nor critique it.

Studying for an Exam

The first most important thing is not to let things go. If there are things you don't understand from class today, do whatever you must — meet with the professor, seek help from a friend, pour over the textbook — to understand it right away. Don't assume it will begin to make sense over time. Treat the day's material as if you need the competence required for exam-readiness by tomorrow.

It's good to first go back over notes and problems you've done before, especially problems on which you got stuck and made it through only with help. It is also good to do problems that weren't assigned. (In fact, do them even if no exam is looming in the near future! It can only help, right?) You may also find it helpful to dream up your own exam questions and solve them. You'll be surprised how much you learn by doing so!

Keep in mind that there are important ways that homework is very unlike an exam. When doing HW you have the luxury of knowing which section the problem comes from, and where to look for examples. Moreover, it's usually the section that was just covered in class, so you're probably already in a mindset of approaching the problem in a manner that will be somewhat productive. All those advantages disappear on an exam. Prepare for it. One possibility would be to write out a few problems (preferably ones that weren't assigned) from each section on note cards. When you've got a pile of such cards, shuffle them up and then work through them. (Think creatively. You can probably improve on this method.)

Besides inventing your own test questions, there are other productive ways to act like the teacher. “Lecture” to the blank wall in your room, telling your imagined audience what you know about various concepts and how they fit together. Imagine questions your students would ask, and dig up the answers when you don't know them. (Yeah, you'll wonder at first if this idea is a little crazy. But, just like Christians, educated people are a little out of step with the rest of society, and there's no call to be ashamed of steps toward excellence.)

This page maintained by: Thomas L. Scofield
Department of Mathematics and Statistics, Calvin College

Last Modified: Wednesday, 11-Aug-2004 17:04:57 EDT