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:

• Read at times of the day when you are very alert.
• Keep a writing utensil, paper, a calculator — anything that you would normally have when doing homework — at hand. Use it to verify all calculated answers that the author provides. (In some cases, answers may come from computer output and, while you should consider how these calculations can be verified by hand, you should also exercise good judgement as to the value of carrying all of them out.) You should also make notes about vocabulary, new ideas, etc. as you read — try stating these in your own words. Finally, prepare a list of questions to ask in class on material you do not understand.
• When you come to an example in the text:
• Read the question to understand the problem being posed.
• Without looking at how the author solves the problem, try to work it out yourself. See if you can anticipate what ideas and skills will be useful for getting the answer, and carry out your own calculations.
• If you get stuck, go back and read the author's solution, just to the point where your solution diverges from it. Look for reasons why it might be better to take the author's path instead of yours. Is your method really infeasable? Can your approach be refined so that it does work? (You can expect to come upon similar forks in the road in the future on problems such as this, and you want to have some rationale that guides your choice of direction.) Now that you are on a different path, cover up the author's solution and go back to working out the example yourself. Repeat this process of “getting un-stuck” as often as necessary.
• When you have completely worked through an example and are ready to move on, pause a moment to ask yourself why this example appears where it does in the text. What idea(s) is it meant to convey? Why is it appropriate to present this example here and not someplace else?
• When you have finished a reading assignment, take a few minutes to act like a teacher. See if you can provide (using your notes, of course) your “class” with answers to the overarching “why?” and “how?” questions posed above.

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):

• If you view math as a set of skills to learn, then you are embarking on a very difficult venture indeed when you sit down to learn in a detached way the myriad of skills that go with a specific course. As it happens, these skills are generally based upon a relatively small collection of ideas/concepts that, if mastered, make the skills much easier to apprehend. You will find it easier to master these fundamental ideas through discussions with classmates (always striving to use the terminology/vocabulary appropriate to the course). You should talk about the concepts in each new section, identifying carefully what they are and what they are not. No section is likely to emphasize more than two concepts, and often these are repeats (concepts that have appeared in some form earlier).
• If you spend time demonstrating to others in your group your solutions to various problems, you will find frequently that another has solved a problem in a substantially differently fashion than you. Real learning begins when you grapple together over the merits of different methods, and why they do or do not lead to the same answer.
• You may find that another group member can solve a problem you've tried unsuccessfully to solve.
• Even if you tend to get all of the problems, there is value in explaining things to others. Every teacher you know will tell you that they thought they knew a subject well before teaching it, but gained new insights/understanding as they taught it.
• Studying together provides opportunity for Christian interaction. We can encourage one another, bear one another's burdens/frustrations, and share our gifts with others.
• One day you will seek employment. Companies are looking for people who, while individually competent, have good teamwork experiences and skills.

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