A Guide for Teaching Mathematics

The first time I saw this guide I laughed so hard I choked. I cam across it in about 1993 at http://bearspace.baylor.edu/Alexander_Pruss/www/teachmath.html. It has been my Robert's Rules, my Strunk and White, my HG2G ever since. The initial introduction is as follows:

I found this text stuck on the door when I moved into my office as a graduate student at the University of British Columbia's Mathematics Department. The text consisted of four pages, typewritten, numbered 449–452, of which you can see pictures here. I do not know the source. The heading and the text at the end are scanned in from the yellowing pages of the original, and the text at the end suggests that what I have must have appeared in some publication, but the editors do not know the original author. Actually, my suspicion is that the original text may have ended with the paragraph starting with the words "There is one last point on teaching technique", since that paragraph would be a natural stopping point. The rest of the text starting with "The first section of this guide has dealt with actual teaching" might well, thus, be by another author.

It is the responsibility of the teacher to actively involve his or her students in the learning process. The most important thing he or she should do is to avoid giving clear, concise, organized lectures. If the presentation of a lesson is too easy to follow, most of the class will not need to learn the new material on their own. They will have a certain degree of confidence in their new knowledge, and this will tend to stifle their intellectual pursuits. If, on the other hand, the lecture is vague, rambling and disorganized, the students will leave with their heads full of questions. In fact, they will be so filled with curiosity that they will try to expand their knowledge on their own. There are many ways to present a thought provoking lecture. One of the easiest techniques to use is a foreign accent. If the accent is thick enough, even a well organized lecture will produce expressions of intellectual wonder among the students. Effective accents can be acquired in Alabama, China, India, Latin America, New York City, Germany, or any foreign country.

For natives of Kansas, that is, for individuals who cannot speak anything but perfect Midwestern English, this technique may offer difficulties. There are two possible solutions: (1) One can teach in a foreign country, or at least in New York or Texas; or (2) One can incorporate a new syllable into one's language. Two very effective syllables to use are 'um' and 'uh'. The chosen syllable should be uttered every second or third word. This reduces the possibility that any coherent concept will be given to the class. For example, one can say, "Um, today, um, we will be, um, discussing, um....um, determinants." After a couple of sentences, most of the class will be staring at their watches or out the windows. Very quickly, they will become very anxious to go out and learn the material on their own.

In addition, to being aware of one's own speech patterns, the teacher should also pay close attention to the written word. Effective use of the blackboard should be considered almost a necessity. Illegible handwriting can stimulate a student's interest in new material almost as effectively as Incoherent lectures. Often students will meet outside of class to exchange interpretations of lecture notes. Thus illegible handwriting encourages students to work together and share ideas. Writing illegibly requires a great deal of practice to be effective. If one does not have satisfactory handwriting (that is to say., if one's handwriting is suitable only for formal invitations and eye charts). certain "tricks" can be learned:

1. Write small. For students in the back rows, this is almost as effective as writing illegibly. The disadvantage is that students in the front rows will probably be able to read the board and may possibly learn something without having to spend hours interpreting their notes. Also, the professor who writes small may find that most of his or her class will try to sit near the front of the room, which may be too close for comfort, especially on hot days during summer sessions.
2. Write fast. The faster the teacher writes, the faster the students will have to take notes. Often the teacher can move on to a new subject while his or her students are still trying to copy what is on the board . Students will be so busy during class that they will wait until after class to try to understand the lesson. In addition to spurring students to learn on their own, writing fast allows the professor to cover more material in a given class period.
3. Write something while saying something different. For example, after working out a lengthy problem the instructor tells the class the answer; is x² + y while writing on the board y² + x. This forces students to rethink the problem in order to decide which alternative is correct. Students are thus actively involved in problem solving even after the problem is finished.
4. Erase quickly. This technique practically forces those members of the class who take notes to pay constant attention to the lectures. Those who doze off for a few moments will awaken to find nothing to record in their notes on the topics they missed. This technique is especially effective if one uses both hands to write and erase simultaneously. If all else fails, stand in front of what has just been written. By blocking any clear view of the blackboard, the teacher will help improve students' speculative and psychic abilities. Those instructors who are short or underweight may find this procedure extremely difficult. The above 'tricks' may be used separately or combined. It is a good idea to change them occasionally in order to add some variety to the classroom routine.

It is very important that the professor lecture to the blackboard when using it. This helps demonstrate to students how involved the teacher is with the subject. This enthusiasm will most assuredly rub off on the class. Also, by facing the blackboard, one cannot face the class. It is therefore easier to ignore students' questions which tend to interrupt the presentation of topics and make the class period seem to last forever.

There is one last point on teaching technique. It is important that one does not overprepare for lectures. Generally, one should arrive at class a few minutes early, open the book, and glance at the topic for that particular day. Lectures prepared in this manner have a certain freshness and spontaneity that is often missing from those which are more carefully organized. In addition, students will gain a greater appreciation for a correct proof if they see how much time can be spent on a wrong approach.

The first section of this guide has dealt with actual teaching, concentrating on lecturing 'tricks', techniques, and preparation. The subject of the last part will be general appearances. Students tend to have more confidence in an instructor if they believe he or she has a thorough understanding of his or her field. To show a class that one has a thorough understanding of mathematics, it is necessary to appear 'spaced-out.' Being 'spaced-out' implies one is so involved with abstract mathematics that one has lost touch with the real world. There are several ways to project such an image.

1. Dress funny. Old suits, baggy pants, narrow ties, and hairy sweaters are all effective and even more so, when worn together.
2. Don't wash your sweatshirts. Albert Einstein is best remembered for two things—being a genius and wearing dirty sweatshirts. Even if you are not a genius, you can still wear the sweatshirts. In a matter of weeks, you will gain such a reputation that no one will come near enough to challenge it.
3. Don't comb your hair with anything finer than your left hand.
4. Walk into the wrong room and begin to lecture to whatever class is in it. (This will help spread your reputation beyond your own students).
5. Walk into the correct classroom and begin lecturing on whatever happens to be left on the blackboard from the previous class.
6. Acquire a facial twitch.
7. Pretend you are deaf if someone asks a question or the bell rings while you are lecturing. Try to keep talking after everyone has left the room.
8. Follow all the guidelines for teaching given above.

By being properly 'spaced-out', one will gain the confidence and respect of one's students. This will make it easier to help inspire the in their study of mathematics. Being properly 'spaced-out' will also help one to acquire tenure at this or any other reputable college or university.