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 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 is to use 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, New York City, China, India, Latin America, Germany, or other foreign countries.

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 could 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.   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 be forced into studying on their own after class.   In addition, writing fast allows the professor to cover more material.
  3. Write something while saying something different.   For example, after working out a lengthy problem, tell the class the answer is x + 2y while writing on the board y + 2x.   This forces students to think the problem through on their own.
  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 the topics they missed.   This technique is particularly effective if one uses both hands to write and erase simultaneously.
  5. Stand in front of your work.   By blocking any clear view of the blackboard, the teacher will help improve students' speculative and psychic abilities.   Those instructors who are underweight may find this procedure difficult.

The above "tricks" may be used separately or combined.   It is a good idea to change them occasionally to add some variety to the classroom routine.

It is very important that the professor lecture to the blackboard.   This helps to demonstrate to students how involved the teacher is with the subject.   This enthusiasm will most assuredly rub off on the class.   This has the added benefit of making it easy to ignore questions which forces students to go home and answer the questions themselves.

There is one last point on teaching technique.   It is important that one does not over prepare for a lecture.   Generally, one should arrive at class a few minutes early, open the book, and glance at the topics to be covered that day.   Lectures prepared in this manner have a certain freshness and spontaneity that is often missing in carefully prepared lectures.   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 part of this guide dealt with actual teaching, concentrating on lecturing "tricks," techniques and preparation.   The subject of the last part is general appearances.

Students tend to have more confidence in an instructor if they believe he or she has a thorough understanding of mathematics.   This confidence is enhanced if the instructor appears to be "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 or skirts, baggy pants, very narrow ties, hairy sweaters and dirty sneakers are very effective and even more so in combination.
  2. Don't wash your sweatshirt.   Einstein is remembered for two things -- being a genius and wearing dirty sweatshirts.
  3. Don't comb your hair with anything finer than your fingers.
  4. Walk into the wrong room and begin to lecture to whatever class is in it.
  5. Walk into the correct classroom and begin lecturing on whatever is left on the board from a 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.

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