Dr. Eileen Kolman has been the Dean of the First Year of Studies for the previous 15 years or so. She has a Ph.D. in Mathematics Education and has worked very hard with several faculty members in the math department to improve the quality of undergraduate math education here at Notre Dame. Her talk covered a wide swath of topics related to teaching mathematics (and teaching in general).
At the beginning of the meeting, Dr. Kolman gave us an exercise. We were to think through what we would hope a student nominating us for a teaching award might have to say about us and our class. More generally, would makes a teacher good?
After spending a few minutes brainstorming, we were split into groups of two or three and asked to lump our ideas into a few categories. Here are some of the categories that we came up with, and some comments (from both faculty and students present) about them (in no particular order):
Dr. Kolman also suggested that Notre Dame undergrads are pretty much the cream of the crop according to many metrics - we are being spoiled here, and no matter how difficult (or easy) we find teaching here, we are almost certain to find it more difficult later in our careers.
Professor Himonas and Dean Kolman both emphasized that a major problem in society (even at good colleges like ND) is that it is socially acceptable, and perhaps even preferred, to be bad at math. There are quite a few people in the world (even college-educated ones) that believe that people who are good at math are automatically strange or have some sort of mental problem. It is important to show such people that mathematics is a reasonable vocation and that it is very useful and important.
Dean Kolman suggested that, as with any form of talk, it is good to tell students what you will tell them, tell them, and finally tell them what you told them. The stereotypical math class will have the teacher at the front of the room, expounding the virtues of some formula or equation while the students are scratching their head, not grasping the point of it all, and they will walk away not knowing what they saw or why it mattered.
She also reminded us that if we are teaching first year students in the fall, we are not actually teaching college students, but rather very talented high school students. They excelled at jumping through the hoops of high school, but they are not yet in the college frame of mind. If you do not teach them how to think, they will continue to be high school students. Such students can solve equations very well, but if they are not shown how to take a word problem and mathematize it into an equation, they will not be any better off.
Dean Kolman also pointed out that many mathematics professor just lecutre, and that this is not a good thing. She suggested that the best professors work on being interactive. She also said that the line between professor and TA should be blurry, which is not currently the case at Notre Dame. She advocates the problem-solving style of tutorials, not just the regular question and answer sessions as the latter do not result in any actual learning.
A member of the audience suggested that these group-oriented tutorials are nice for classes of 20 students, but that is not the case at ND. Rather, we have 35-40 students in our tutorial sessions. Alex Himonas recommended that we make some noise around the department to have this changed. He suggested that if the grad students are not behind the teaching revolution, it will not work.
Finally, Dean Kolman offered an analogy - teaching a class is like serving as a tour guide on an expedition up Mount Everest. It is important for the guide (teacher) to know the abilities of the tourists (students) and to equip them and guide (teach) them appropriately. The tourists don't care how many times you have climbed the mountain (taught the course/covered the theory) - all they care about is whether you can get them up the mountain safely. Think of Sherpas (the local Tibetans that excel in climbing and often help guide tours) as TAs. Keep in mind that the tourists want to climb and learn for themselves - they do not want you to carry them on your backs. Also, it is interesting to them to point out crevices and crags off the beaten path, without hiking all the way over to see them close-up. Think of multi-section courses as huge expeditions with multiple groups - you don't want one group falling way behind (or one person, for that matter). Think of holes in a student's math training as a lack of some certain skill for rock-climbing that can be taught to them.
Dean Kolman also suggested that mathematicians often work as individuals - that is the nature of some of mathematics. However, teaching is a necessarily communal experience - always seek help and ideas from others, in structured or unstructured settings.
At the end of the period, Alex Himonas suggested that many professors here know very much about research but very little about teaching. Five years ago, when the university asked the department why we teach 100-level courses, we hardly had an answer. This prompted the creation of course goals and justifications, an example of which (Math 108) has been distributed to the seminar participants. It is important to be aware of the goals of the course you are teaching before you start planning and/or teaching. Dean Kolman and Alex Himonas agreed that much work (and arguing) went into the creation of these documents, making them very valuable and important.
(reported by Dan Bates)