HOW TO TEACH

by Chester Miracle

Editors’ note: Professor Miracle has earned recognition as one of the best teachers in the department. He has administered for many years our very successful extension program and participated in numerous departmental teaching initiatives as well as being a long-term member of the Curriculum Committee. Recently he played a major role in the development of the K-12 standards, as described in Bert Fristedt’s contribution in this issue. As Director of the extension program, Professor Miracle has supervised and guided many instructors in mathematics. He kindly agreed to share with us
some of his thoughts on the methodology of teaching:

In my view, the most important single thing that a person should do in order to be a good teacher is to prepare the lecture before class. When preparing a lecture, take a realistic view of what the students will be able to understand. Years ago, almost no teacher came to calculus class with written notes. Now, I would guess, at least half of the lecturers come to a calculus class with written notes in hand. Teaching assistants should also always have something prepared.

Lecture to the students you have, not to some ideal student. The University of Minnesota may not be as selective as we might like. But we take the money from these students and so we have a moral obligation to teach a class they understand. Be sure that someone other than yourself is able to follow what you are saying. It is easy for an instructor to find an ‘advanced topic’ to discuss. Mostly, the students do not find this interesting. What they actually think is ‘this lecturer is a jerk and the worst part of it is he is wasting my time.’

Do not spend half a class period going over some ‘interesting idea’ that is not part of the course. Mostly, the students do not pay attention since they feel the instructor is talking about something not in the course. The job of a teacher is not to get up in front of class and just talk, it is to teach students something they feel they want to learn or at least must learn.

Very few students know everything we think they should know. If they need a little review on a topic, give a little review. If you are teaching polar coordinates and you know your students do not remember much trigonometry, then go over the ideas of trigonometry as you teach polar
coordinates. Try to start every lecture at a point where you are sure that the students understand what you are doing.

Spend some time getting to know your students and what they know. One very good way to get to know students is to grade their exams. I have
been teaching IT calculus for 5 years. I always help grade the hour exams and the final exam. This gives me a clearer picture of what the students can and can not do.

I consider attempts to motivate students using jokes or history of mathematics a waste of time. Math is not fun for students. Many students get satisfaction from success in mathematics, but it is not fun. The things like short biographies we once saw in calculus books do nothing to motivate students to want to study math.

Working with students in IT calculus has made me certain that people learn mathematics by doing mathematics. The most important thing about a mathematics course, at any level, is how much mathematics are the students
doing. I have talked with many students who were in lecture, were paying attention and took really great notes. My conclusion is: lecturers are not really as effective as we would like to think they are. Students learn very little during actual lecture. Students learn more by doing other things. These things can make the lecture much more meaningful. With this in mind it is important for the instructor to control what the students are doing with their ‘study time’ outside of class.

I always try to test students on what I teach them. Many times, I have asked a student that I know who is taking another class: “How was the test?” All too often they reply that the test was unfair or a bad test because the instructor asked a bunch of questions that were never discussed in class.

As teachers we often think that our primary job is to give great lectures. However, our primary job is to make students work problems correctly.