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At my institution incoming graduate students must take a semester long course on pedagogy taught by current grad students. I may soon be in the position of having to teach this course and I'm looking for advice for readings to give the students. The problem is that our grad students don't teach till after they pass their quals, so effectively the pedagogy course is teaching people how to teach when they are more focused on their qual courses and know that they won't have to teach for at least a year. Consequently, most students view it as a complete waste of time and gain little from it. When I took this course I really tried to get something out of it and some of the lessons stuck with me. Still, at that time we mostly had readings taken out of guides for high school teachers and I'd rather have more applicable readings.

Here's what I've got so far:

  1. A small booklet the university produces with rules, resources for getting help when you teach, online resources the university uses, and honor code
  2. Old syllabi we can discuss
  3. All students will go observe other grad students teaching and write a report which we can discuss.
  4. Discuss some teaching mechanics, e.g. blackboard use, dealing with student questions, etc. Good resource seems to be the AMS blog and a document from Williams
  5. All students will need to prepare one lecture and deliver it. We can then give them feedback. As recommended by Gerhard, I think I'll try to video tape this and give them the tapes. Below are some further readings...
  6. Some notes by V.I. Arnold, some by John Baez, and some (recommended below) by Bruce Reznick. I'm starting to think a good exercise would be to assign these as readings and ask the students to find 3 things in a given reading which they disagree with. We can then discuss.
  7. Silly, but how to curve grades
  8. Perhaps some discussions of issues of pedagogy, e.g. why teach calculus the way we do, and pros/cons of giving handouts
  9. Discuss evaluations, teaching statements, and teaching letters with an eye towards applying for jobs
  10. Discuss giving talks in general

Does anyone have any other ideas for topics worth discussing or readings worth giving? I'm particularly interested in books on this subject which I could draw readings from.

Also (and this is probably too general), since these students will have at least a year before they actually teach, I'm keeping my eyes open for lessons that will stick with them and come in handy when they finally get in front of a class. If you have any nuggets that fit the bill, I'd be interested to hear them (all mine right now come from John Baez's notes or the notes from Williams)

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This should be community wiki (and, honestly, does not have much to do with research in Mathematics). – Igor Rivin Jun 22 '11 at 20:37
Wow, a meta-pedagogical question! :) – Qfwfq Jun 22 '11 at 22:39
@unknown I suppose it is meta-pedagogical. I think studying higher categories has been rubbing off on me! Everything I do is meta-something – David White Jun 22 '11 at 22:57
I am reminded of a technique George Bergman (UCB) adopted for some mid-level classes. In class, before starting a lecture on a prescribed reading R, he would accept slips of paper (one per student) which contained a question on material from R, to be read out loud in class. If a student already understood the material, they could mark the question "pro forma" to indicate this. It was almost as effective as assigned homework in getting the class to read before lecture, and had other benefits. Gerhard "Email Me About System Design" Paseman, 2011.06.22 – Gerhard Paseman Jun 22 '11 at 23:02
in my history of observing grad students teach, i often noticed most were actually better than average at transmitting information. My charges wrote and spoke clearly, received questions well, were well organized etc... But some fell short in what information to transmit. Some often kept the class as easy as possible, and failed to emphasize the most important concepts. So it was not the basic nuts and bolts that most teaching books are concerned with that was missing. Some just failed to try to convey the main ideas. They were surviving, but they set the bar too low to be effective. – roy smith Jun 23 '11 at 17:01

14 Answers 14

This is a very interesting book on the subject, written by and for mathematicians, and likely to provoke discussions. I am guessing it would be ideal for your course.

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Wow,I didn't know about this book.Looks great,thanks. – The Mathemagician Jun 23 '11 at 6:58
Thanks, this is just the sort of book I was looking for! – David White Jun 23 '11 at 12:07
The link takes you to this book: Teaching Mathematics in Colleges and Universities: Case Studies for Today's Classroom: Graduate Student Edition, by Solomon Friedberg, Boston College, Chestnut Hill, MA – Kevin O'Bryant Sep 15 '11 at 1:34

Among possible sources of readings, I would mention the booklet How to Teach Mathematics by Steven G. Krantz. It is quite practical, and it costs very little.

By the way, one of the other responders mentioned an unpleasant issue. I would like to mention another issue that is potentially much more unpleasant: The TA should keep the door to the office open when a student comes to office hour, most particularly if the student is of the opposite sex. Very often the student closes the door when entering the office, even if the door is open. In such cases, I tell the student to keep the door open "in case someone else comes along." The unpleasant developments that this precaution prevents are extremely rare, but when they occur, they cause quite exceptional amounts of trouble.

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Bruce Reznick wrote a short guide, "Chalking It Up," that I have found helpful.

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Thanks, this is a great reference! It covers a number of things I want to talk about and also covers things I probably would have forgotten to mention. My favorite quote: "You do not need Zorn's Lemma to see that every teacher has been a first-time teacher." – David White Jun 22 '11 at 22:45
At the end of the article he mentions some references, in particular "The Art of Teaching" by Gilbert Highet and "Keys to Improved Instruction", edited by Bettye Anne Case, MAA Notes No. 11. Have you read either of those? – David White Jun 22 '11 at 22:46

Alas, here is something unpleasant you should include: you should explain the basics of how to handle a student you suspect is cheating. Different universities have different policies/procedures. The last thing you want is to have a student get off the hook for a procedural error, or worse yet, be sued for not following procedure.

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my early solution was to monitor my own tests and when a student was suspected of cheating, to ask that student to move to the front, no questions, no explanations. this seemed to work, i.e. prevention is far better than punishment. Later I grew wiser began to pre - empt cheating by offering help to students who seemed puzzled. It made little difference, as they still could do little. But it removed the motivation to cheat when they realized the teacher would help them. The point is to teach, not to punish. – roy smith Jun 23 '11 at 4:18

I don't know if this qualify as "pedagogy" but something i've found useful when I began teaching was the acting lessons I had in high school. I don't know how it is in the US but here in france you can begin teaching as a PhD student with very few experience in talking in front of people.

I find it quite important to know how to use your voice and your body when you give a lecture (or a math talk). Of course acting lessons seems to be a waste of time at this level but a few exercise on breathing and speaking loud enough could be of some use I guess.

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This reminds me of the coaching I got for an appearance on television which later seemed relevant to in class board work. They told us to make the verbal presentation while facing the audience, and only then to turn our backs to write it down, i.e. not to talk over ones shoulder while writing. – roy smith Jun 23 '11 at 16:26
Facing the audience is helpful too for those that need to read lips. Gerhard "Email Me About System Design" Paseman, 2011.06.23 – Gerhard Paseman Jun 23 '11 at 17:19

Here's a phenomenon that any to-be-teacher should be made to experience:

Give teacher #1 a given exercise sheet to grade.
Give teacher #2 the very same exercise sheet to grade.

Their grades of teacher #1 and teacher #2 are likely to be shockingly different
(unless the exercise is either very good or very bad).

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grading is very instructive also for students. at my college orientation they gave us a sample reading, a question, and 2 answers to grade. i gave "A" to the one regurgitating the reading closest and "B" to the one doing it next best. the real grades? the one i gave B to, did get a B for "at least trying to answer the question", but my A got a "C, lucky it wasn't a D", for merely regurgitating the material thoughtlessly. – roy smith Jun 25 '11 at 3:37

Hi All,

I have taught such a course in the UIC MSCS Department for the past 6 years. The class meets once a week for 75 minutes, after the TA's classes on Tuesdays. Each year, this course seems to develop somewhat differently, as the skill levels and students' interests just seem to vary as well. All in all, it is a definite PLUS!

Students especially value the discussions among themselves about teaching. But also, I sweeten the pot by discussing "practical matters". These can be found on the web site for the course, which has some original material, and links to other sites as mentioned elsewhere above. Feel free to borrow whatever might be useful from this site,


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Welcome to Math Overflow, Professor Hurder! – Tom Church Jun 26 '11 at 16:50
The link is now broken, but here's a new one: – David White Jul 24 '12 at 13:51

to teach successfully, first you must decide what is your goal. this may be your innate mission, or your department's imposed outcome, or whatever. Possibilities are: my students will learn to think logically and to pose and solve reasonable problems; as many as possible of my students will choose to major in math, or to take more math courses from our department; my students will pass the departmental exam on calculus skills in as large numbers as possible.

Need I go on? These different goals suggest rather different approaches to teaching. E.g. if you want large numbers of students to go on, your course must be easy and give good grades. If you want the good students to go on, those who have a chance at success, your course should be harder and give grades more accurately,.....

So a teacher should have a goal. It should be a worthy goal, and it should be one he/she has some idea how to implement. But no matter what your philosophy, and no matter how long you have been teaching, you must prepare every night before class. And you must present examples, not just theorems.

Come to think of it I have written an essay on this topic, "On teaching". Perhaps i will link to it or even post it here.

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I didn't see the link to your essay at your web page. – KConrad Jun 23 '11 at 5:51
@KConrad: I also didn't see any link to this essay on Roy's webpage but googling found: – David White Jun 23 '11 at 12:08
Sorry to be slow, but now it's there as #8. – roy smith Jun 23 '11 at 16:08
Rereading it now, I guess that essay is rather wordy, and something of a response to the depressing phenomenon of class evaluations. Still there may be some interesting points/illustrative stories about what matters in a good teacher that is hard to measure at the end of semester. – roy smith Jun 23 '11 at 16:22

How to grade efficiently and fairly. Some things you might cover:

Practical things like "Grade all copies of problem 1, then all copies of 2, etc" and "If you grade out of 5 points, you'll spend less time making and justifying decisions than if you grade out of 10."

Some discussion about the merits of additive vs. subtractive grading. (I'm a big fan of additive, myself, but I know both have defenders.)

How to keep good records of student performance. Discussion of any university policies regarding privacy of these records.

Encouragement to return problem sets as soon as possible.

If you want some more interesting material to discuss, you could check out Noah's blog post.

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I like this idea. Since all the students in the pedagogy course will be doing TA work before leading recitations or teaching their own classes this is the sort of lesson they would view as pertinent and which might stick with them. So thanks! – David White Jun 23 '11 at 12:12
I used to meditate briefly on my finished grades in case anything troubling came to mind. Doing that once, I felt uneasy over an F I had given a student whose performance averaged 40%, surely an F on any scale in US, right? So I read closely the university guidelines at the school where I was visiting. They said an F meant the student had learned "essentially nothing". This surprised me, but it seemed that under that definition probably 40% was a D, not an F, so I changed it, and felt better about it. Written justification for grades also comes in handy if grades are challenged later. – roy smith Jun 24 '11 at 17:15

At the University of Arizona where I received my Ph.D. just like at your institution all incoming graduate students also had to take a semester long course on pedagogy but it was always taught by a postdoc. The mandatory reading was "How to Teach Mathematics" by Steven G. Krantz but in retrospective, I wish, the mandatory reading had been his other book "A Mathematician's Survival Guide". We did lots of exercises like video recording our lectures or grading the same exam (by the whole class) and then comparing the grades.

Besides already mentioned Arnold's article (some of ideas from that article are also eloquently presented in his Ordinary Differential Book 1st one not the Geometric Methods) I really, really like the following writing

by Anatole Katok which is indeed geared towards more advancing courses which you will probably not have opportunity to teach as a TA.

I do not agree with everything Professor Katok is saying but it is a great reading. For example:"After that the picture gets filled in, sometimes straightforwardly using the most elegant or most useful available arguments". I quite on the contrary believe that the picture should be filled by the arguments which the most instrumental in exposing key ideas. I also do not believe in well polished statements of the results as they are usually result of many iterations of the original idea/discovery. I like to state result in its original form when it was discovered (for example Stokes theorem can be stated on the square in $R^2$ in which case its proof become a trivial exercise in applying Fundamental Theorem of Calculus). My advisor Qiudong Wang is a master in delivering such courses (Dynamical Systems, ODE) which are completely revealing.

As of teaching courses taken by unmotivated learners (intro college courses) I have much less to say and in my experience you have to treat crowd on case by case basis. I am not sure that any reading can help you with that (you have to get experience).

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One basic piece of advice I read in a recent book by a master junior high teacher that may apply. To teach well, one must get to know the class. Once you know something about who/where they are you can think of ways to teach them. For this, some grading of ones own homework, not handing it off to a grader, is very helpful. Asking people what their goals are the first day helps too. – roy smith Jun 24 '11 at 21:52
Thanks, I read through that document of Katok's and enjoyed it. It's also nice to hear some of the other ideas that have come up in this thread have worked elsewhere. I like "A Mathematician's Survival Guide" and I'll probably recommend it to the students as a useful reference. – David White Jun 24 '11 at 21:57

I suggest an alteration in procedure, if the department can support such. In addition to video taping and evaluating a performance which the prospective teacher T can review, offer

a) being able to sit in on the first one or two classes given by T to provide feedback to use later in the semester, or

b) better yet, assemble the prospective T's together so they can practice a day or week before they are "thrown over the wall", to borrow a phrase from software design. This can be done at the mutual convenience of the T's and perhaps one or two seasoned graduate students to help organize.

Having a timely practice session or two with copies of the material from the previous year will go much further to keep the lessons learned fresh than anything else I can imagine. It can be arranged so that the T's can do most of the work in 3 days or less. Also having a check list of common errors and common success points to use in the evaluation process would help.

Gerhard "Email Me About System Design" Paseman, 2011.06.22

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Thanks for your response. I especially like your idea of video taping. I thought of having myself taped before teaching but didn't want to get into the logistics with all the students in the classroom. But there's no reason I can't video tape the first-years during their lectures and then give them the tapes with some feedback to help them when the time comes to teach. I don't think we can force them to have people sit in on their teaching if they don't want to, but the idea of a crash course refresher just before teaching seems like it could fly. Thanks again! – David White Jun 22 '11 at 21:54
Nowadays it's a matter of having the right kind of phone/videocam and being able to store the video and playback software long enough to be useful. The review process could start out voluntary; if it receives the success I think it will receive, most T's will want to continue it informally, especially if good teaching points are emphasized along with areas needing work. Gerhard "Also Works In Customer Training" Paseman, 2011.06.22 – Gerhard Paseman Jun 22 '11 at 22:06
video taping can be brutally informative. i did it to myself once early in my career. i gave the clearest explanation i could of an interesting theorem and faced nothing but blank stares the whole period. when i watched the tape played back i saw i had forgotten to state the theorem before proving it. – roy smith Jun 23 '11 at 16:42

It is useful to have a course to go over basic stuff on teaching but there is nothing like standing in front of a classroom with (possibly hostile) live students. Every successful teacher will find his or her own style and having guidance will accelerate the process. What are listed in the other posts are all worthwhile but I think it would be real helpful to have the prospective TAs do some tutoring first. Most beginning graduate students are not used to explaining mathematics, especially to others who may be having problems with very basic material that they had found easy. Also, I think it is more important to have support and feedback when they first start teaching. It would be helpful to have regularly scheduled meetings for the new TAs with more experienced TAs and faculty mentors. There are also other factors to consider. For example, will they be teaching their own classes or recitation sessions? If they are teaching their own classes, then they would need more support and encouragement because it can get pretty depressing for aspiring mathematicians to teach unmotivated students in service courses.

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I agree that it would make much more sense to have a pedagogy course while students are actually teaching (there was a conversation in comments above with Gerhard about this). Sadly, that's not the position I'm in. Still, I can recommend to my students that they seek help from other grad students and professors when the time comes to actually teach. – David White Jun 23 '11 at 12:15

Most answers goes into the practics of teaching. But more philosophical thinking is also important!, and a book that I found extremely helpful (and fun!) was Imre Lakatos: "Proofs and Refutations: the Logic of Mathematical Discovery".

Go read it! It will also help your teaching.

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up vote 1 down vote accepted

Thanks everyone for your advice. The course is over now and it seems to have been a success. I created a webpage which contains all the materials I used as well as the syllabus. Please feel free to make use of this material if you should ever need to

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