Becoming a better instructor: where to start?

Question

I just finished a PhD in math at a top department, but not one that placed much emphasis on graduate student teaching. Grad students here teach only as TAs, and the training is minimal. I got great student evaluations, but I realize this was due more to holding lots of office hours and making study guides than to any real pedagogical brilliance on my part.

Next year I'll start a postdoc at a similarly research-oriented place, and I'd like to be a more effective instructor. The question is: where do I start? What are some good survey books or articles that would bring me (partially) up to speed on best practices for teaching university-level math? My formal knowledge of good teaching strategies is essentially nil.

I'll have the chance to teach a range of undergraduate courses. I realistically won't have the ability (or longevity, or experience...) to make sweeping changes in e.g. the way the calculus classes are taught, but I'd like to do the best I can within the current framework. I may also have the chance to design a one-off mid-level undergrad course from scratch, so something touching on course development would be helpful.

Answer 1

One thing to watch out for is learning to understand what the student's problems with the subject are. Once you identify that, fixing is typically easy.

Take care when designing exams and homework. Ask yourself what you want to accomplish (what should be measured, what you want them to learn). It is important to think of grading when designing exams, don't ask for overlong derivations (they won't have the time to do more than a very few, and much material won't be examined that way; someone will make the most absurd error off the bat, and force the grader to follow a convoluted path through a completely different jungle).

Collect material from previous terms as a (rough) guide. There are plenty of lecture notes, past exams and homework on a wide range of subjects on the 'web, see if some are to your liking. Even if the policy is that exams (and solutions) aren't published, assume they are available to interested parties. Ask around on what they had before, what follows, where they'll use the material later on. Talk to people who have taught the subject before (or will do so in paralell), ask former students and TAs on the rough (and smooth) spots, request input on them. Yes, that will probably make you look somewhat goofy at the start, but then again, you'll be the new kid on the block.

Look out for trouble (yours, the class', individual students), and make sure it gets handled early. Too often some minor problem grows into unmanageable proportions as term end looms near.

Specific advice: It is often useful to start with a short summary (perhaps list the most important definitions and results from last class for reference). End with a recapitulation of what was done, explaining again what is important and what was just scaffolding or examples.

Answer 2

I recommend the book How to Teach Mathematics by Steven G. Krantz. He covers almost all of what you are asking from low level courses, advanced courses, large and small. The second edition also contains many essays on reflections on teaching that I found enlightening.

Answer 3

After teaching college courses (full disclosure - I'm a biologist) for about 15 years, my main teaching discovery is this:

Teachers overestimate how much students learn from a lecture.

This occurs because a) teachers themselves were part of the select group that learned great from lectures, or they wouldn't have survived the flaming hoops of academia, and because b) teachers learn a BUNCH by preparing a lecture. But math students learn best by doing math problems.

You are entering the field during a time when pre-class video assignments and online quizzes are becoming part of the math curriculum. This presents a great opportunity to lecture less and spend at least 10 minutes of every class time having students work in groups on their first homework problems. I promise that if you clear time in class to tell students to work in pairs on a slightly-difficult problem and walk around and ask, "How's it going? What have you figured out so far?" you will be AMAZED at where students get stuck. And once you learn the sticky parts, you will have a great time trying different techniques to help students understand them.

For some reading on math education, try:

• Robert Talbert's Casting Out Nines blog on Chronicle of Higher Ed
• Dan Meyer's blog on math education (high school focus, but still relevant to college)

(perhaps "real" math educators can recommend additional resources)

For general reading on being a college instructor, try Teaching at its Best by Linda Nilson.

Answer 4

I would like to mention this recent opinion article in the AMS Notices (vol.66, no.7; PDF download) by Colin Adams (author of The Knot Book). His main point is that we should try "to impart a love of mathematics." He ends with this anecdote:

I remember seeing a lecture by a professor who had won a variety of teaching awards. As I always am, I was curious to see what it was he did that made him so successful in the classroom. He gave a relatively standard lecture, but definitely well organized and clear. Okay, that’s fine, but it’s not going to win you prizes. There was only one thing he did differently that really stood out. During the entire lecture, he was grinning from ear-to-ear. It was clear for all to see how much he enjoyed the mathematics. He was in his element and everyone in the room could see and feel that. Of course, this doesn’t mean that you should lecture with a grin plastered across your face. If it is not naturally you, students can tell, and you end up looking like a raving lunatic.

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The "Early Career" section of the AMS Notices (in which the above article appears) is in general an excellent resource.

Answer 5

I read in an article the other day that said rather cynically that a sign of a good classroom is one where students are allowed to say more than one sentence about mathematics without being interrupted by the instructor.

I've been reading a lot of blogs written by university-level mathematicians who are interested in teaching, and the latest buzzword seems to be "inquiry-based learning" - basically, forcing students to develop and prove things on their own rather then merely working rote exercises after someone else has done all the real mathematical work.

IBL comes out of something called the Moore Method, named after a math teacher who would basically give his students a list of things to be proved and the class consisted entirely of students making presentations of their proofs on the chalkboard and writing up those proofs for critique by classmates. Here's a site that gives background information on the Moore Method: http://legacyrlmoore.org/

There's an organization that's based around those principles that offers workshops, grants, and a curriculum library for University-level instructors: http://www.inquirybasedlearning.org/

Basically, if you want students to learn mathematics - maximize the amount of time that students spent doing mathematics in your class rather than WATCHING you do mathematics. The students need to be the ones thinking mathematically.

Finally, a lot of what you might face in teaching mathematics in higher education parallel the issues in teaching mathematics in K-12. So I'd suggest seeking out the education department on your campus and to work collaboratively with them on ideas that you might have.

Answer 6

Study the evidence. People believe all kinds of things about teaching, but in many cases their beliefs are contradicted by evidence. A classic, important paper in physics is:

Hake, "Interactive Engagement Versus Traditional Methods: a Six-Thousand Student Survey of Mechanics Test Data for Introductory Physics Courses," Am. J. of Phys, 66 (1997) 64

These findings appear to hold as well in other fields, including math:

Freeman et al., "Active learning increases student performance in science, engineering, and mathematics"

http://www.pnas.org/content/early/2014/05/08/1319030111