Understanding Mathematics by Peter Alfeld, Department of Mathematics, University of Utah

Do you find it difficult to prepare for an exam?

Some of the questions most frequently asked in a math class are: Will this be on the test?, or What will be on the test? Those are reasonable questions, but chances are that if you have to ask them you are not spending your time effectively.

Exams are artificial situations. You do not take a class to prepare for a particular contrived and transient situation. Rather you are most likely taking the class so that you can function better in the long term, for example in your career. Ideally, you should be taking the class at least partly because you like the subject. (If not, ask yourself if you set the right goals for yourself.)

So you should focus on the long term aspects of the class, and on the subject, and in the process, as a pleasant side effect, you will become well prepared for the test.

Specifically, I suggest you make a habit of doing the following:

• Before class, read the relevant sections of the textbook. That way the material won't be so new, and you'll understand better what's happening.
• After class go over your notes and relevant sections of the textbook and make sure you understand what you did in class.
• Look at the exercises of the relevant sections of the textbook (or perhaps exercises suggested by your instructor). Think about how you would solve them. Also think about what each exercise teaches you. If you are sure you can do an exercise skip it. If you are not sure, do the exercise. If you don't know at all what to do with the exercise, think about it, and maybe get help. Avoid doing lots of repetitive exercises. You may do easy exercises for fun and the sense of accomplishments, but the learning effect is much higher for exercises that you can barely manage. If there are a bunch of exercises that drill a certain technique do enough so that you understand how the technique works, but don't spend your energy reaching a high level of speed and proficiency. Put emphasis on correctness and accuracy rather than speed. Always question and check your answers.
• Whenever you do an exercise, before you start think about what you expect to happen. There are two possible outcomes: your expectations turn out to be correct, which makes you feel good. Or you find something contrary to your expectations. In this case you may have made a mistake and now that you are alerted to that fact you can find it and correct it. Or you may have misunderstood something and now you have an opportunity to learn something and understand it better. If you don't formulate expectations there is a chance that you don't even notice something strange has happened, and so you may not realize your mistake or miss an opportunity to learn something.
• At the beginning of the course, find a study partner. Get together on a regular basis with your partner and go over the material. It's OK if one of you is having a much easier time with the class than the other. The one having the easier time will benefit from explaining things, and the other from having explained things one more time.
• When you get stuck in your preparations don't spin your wheels. You are trying to take too big a step and need to go back and solve some simpler related problems, perhaps study some more math, talk to your friends, or talk to your teacher. You may be hesitant to do this since you are under time pressure, but continuing to spin your wheels is not getting you anywhere.
• Don't hesitate to seek help. Talk to your instructor. Other help may also be available through tutors or teaching assistants.
• When you actually take the test:
  • First of all, relax! I know that's easy to say and hard to do. But if you've prepared well taking the test should be easy!
  • Before you start actually answering the questions read the entire test. Then answer the easier questions first! Leave the difficult ones till the end. This has several advantages. If you can't answer all questions during the allotted time you answer more questions that way, the questions you do answer are more likely to be correct, and you don't get flustered as easily or immediately.
  • When you get stuck on a particular question, and it's not the last one, put it aside and get back to it only after you answer the other questions.
  • When you are done and have time left, don't leave! You have allotted that time to the test, and the test is important to you. So spend the rest of the available time checking your answers, and correcting them if appropriate.
  • Suppose there is a question you don't understand. Some instructors will let you ask for a clarification during the test. In my experience this only causes disruption and distraction, so I don't answer questions during a test. So suppose you can't ask. Then write on your test that you don't understand the question, describe what part you don't understand, and answer the question as well as you can. Sometimes there are mistakes on the test that make a question non-sensical, or incomprehensible. It shouldn't happen, but it does. In that case say that the question does not makes sense, and if you can describe what the error or the problem is. Any clearheaded instructor will appreciate your insight and your effort, and give you credit.

Different Kinds of Test

Different kinds of tests have different kinds of strengths and weaknesses. Here's a brief summary:

Take Home Tests

One can ask more sophisticated and therefore more instructive questions. The test is more of a teaching tool than an assessment tool. Students can work together and learn in the process. Grading tends to be tedious and more difficult because of the amount of student writing and the larger scope for partial answers and partial credit.

In Class Tests - Closed Books and Notes

This is the standard sort of test. The learning effect comes from preparing for the test, not from taking it. Tests tend to be short and easy to grade. Students tend to dislike this kind of test because it makes their proper preparation more crucial.

In Class Tests - Open Books and Notes

Allowing open books and notes makes it possible to ask more sophisticated, and therefore more instructive, questions. Students tend to like this format since it allows them to think of their books and notes as a crutch. But actually at least some of the questions on the test are likely to be harder than in a closed books and notes test, and students sometimes waste significant amounts of time searching their materials for items that they actually could think of immediately if only they tried. Grading is similar to closed books tests.

Oral Exams

This in my opinion is by far the most effective form of a test. You can adjust your questions according to the student's answers, and find out better than in any other way what the student really knows. Often a side effect of an oral exam is that the student for the first time understands key issues that were previously not understood. The major impediment to using oral exams exclusively is the amount of time it takes if there is a large number of students. Students tend to be terrified of oral exams, but they should look at them as opportunities to prepare in a benign context for similar situations later in their lives and careers. Every interview (or conversation with a co-worker, client, or customer) is a form of oral exam, to various degrees, even though it is not called one.

Fine print, your comments, more links, Peter Alfeld, PA1UM

[25-May-1999]