Some thoughts on teaching Michigan calculus

I just finished teaching two sections of first semester calculus at the University of Michigan. Michigan calculus is somewhat famous — it is very focused on conceptual and graphical understanding, spends a lot of time on “real world” data, and achieves very high scores in national measures of teaching effectiveness. Moreover, while the course coordinators are highly experienced professionals, almost all of the day-to-day instruction is done by a small army of grad students and postdocs; I was one of the very few tenured or tenure track people teaching calculus this term. I was very curious to see how this was made to work.

My intended audience here is others who are about to teach Calculus at Michigan, or people who are wondering what it would be like to set up a Michigan-style program in their own departments. A word to any of my students who find this post. Feel free to read, and feel free to call me David rather than Professor Speyer. But please don’t blast this all over your Facebook pages. I’ve said nothing that I would be unwilling to defend in public, but I’d rather that all my students not find this next time I teach the course. Let’s just keep this little moment of sharing between me and those of you who stumble upon it.

Self congratulation

My students’ median grade was a B-, almost exactly the same as the course median. Of course, I am a highly competitive person, and I wanted my sections to blow the others out of the water. But I’m going to repeat the same thing I told many of my students: Michigan is a very competitive school and everyone who gets in here is excellent. A median grade at Michigan should be a point of pride. In my case, I was competing with grad students who had taken calculus far more recently than I had; had taught it several times before; and who were often extraordinary competitors with a string of Olympiad medals and Putnam victories. Landing in the middle of that pack is an accomplishment, and I have decided to be proud of it.

I also had a lot of fun. It was more work than I expected, and it didn’t give me the same sort of stimulation that teaching an advanced research course does, but I found it really interesting thinking about how to present some of the oldest accomplishments of mathematical thought to my students. If you are at Michigan and are worried that you will be bored stiff teaching this course, I can tell you that I wasn’t.


I was assigned two sections of 32 students each, which dwindled to 30 each over the course of the term. I did the day-to-day teaching (3 times a week, 80 minutes per meeting). I also wrote and graded weekly quizzes, and graded the team homework assignments. The rest of the students’ homework, the basic decisions as to what I should cover, and the exams, were written by a small team of experienced instructors, known as the course coordinators. The coordinators also held weekly meetings to brief us on where the course was going.

The course coordinators were really hardworking and insightful people. I applaud Michigan for recognizing that it is worth paying full time staff to do this job, and finding ones who did it so well.

The course website (including a secret section), was full of resources to help me plan my lessons. In Spring 2011 I often was unable to sleep for fear that I had misplanned my Hodge Theory lectures, so it was a major relief to be so well taken care of. I drifted from the given schedule by a day at times, but I basically followed it and found that it worked well.

I don’t think that this course could possibly work as well without the extensive guidance that we instructors received. If you are thinking of importing MI calculus into your school, you should think about how to take care of your teachers.

The applied and conceptual focus of the class

Calculus at Michigan focuses very heavily on working with data and on understanding what computation to do, rather than how to do it. It also focuses on getting students to be able to explain what they are doing to people with even less mathematical background than they have. I think these are very appropriate goals in theory. I feel that the former was achieved fairly well in practice, though I have some complaints, but the latter was not.

I enjoyed the real world examples. My students seemed to really get excited when I brought in data about a subject which interested them. The athletes really got a kick out of table 1 from this paper, showing how world class runners accelerate second by second; a lot of students enjoyed figuring out how long a runway a Ford Mustang needs to accelerate to 120 mph using the data here.

What was difficult about this was that it made grading very difficult to predict. We would ask questions asking people to give a “practical explanation” of such and such, or to read some data off a graph with no grid lines, and then had very specific grading policies as to what we would accept. It was hard to prepare the students for this, and the best way to do so was to have them memorize certain formulations of “practical explanations”, which were very far from any actual understanding.

The goal of teaching students to write well is a great one, but I don’t think I had the resources to actually do so, and I certainly didn’t succeed. Mathematical writing is a specialized skill. I don’t see how I could have taught it without spending far more class time on it than I had. I wound up grading the writing on the team homework much more leniently than I was told to, because I didn’t feel that I could take off points for expository errors I didn’t have time to explain.

The exams

You can see the exams here: Midterm 1 (PDF), Midterm 2 (PDF), Final. The exams are the major determiner of the course grade, and one feels a very strong pressure to focus one’s teaching on what the exams will cover. I have no problem with that, because I think the exams cover the right topics. If you don’t like the idea of “teaching to a test”, you might dislike teaching at Michigan.

The exams, especially the last two, were almost all challenging conceptual problems. It is strange for someone like me to say this, but I don’t think I approve of this. I would be happier if the exams had one or two more basic questions, with the rest staying as they are.

Consider a student who has learned the basic mechanical skills of numerically approximating a derivative from data, or of differentiating some complex expression. But he doesn’t understand what this really means or when to do it. This student has not learned nearly as much as we want him to. He should do poorly on the exam — a C minus or a D.

But he also has mastered a nontrivial skill. I think we should give him a question on the exam where he can display this mastery. The course’s approach is, instead, to give all difficult questions and then set a curve which will bring this student’s stammerings up to the C minus or D range.

Partly, I think this is important for maintaining student motivation. There are very few people who will enjoy an activity at which they regularly have no small successes. (I am curious — the median score on the Putnam exam is a zero. How many of those competitors come back the next year? If some of my readers were among those who did, can you tell me how you stayed motivated?) I feel that, psychologically, there is a big difference between having one or two questions that my hypothetical student can solidly get right, and having him pick up a smattering of partial credit on each of nine or ten questions that are too hard for him.

Also, I think that it is worth being able to detect the difference between the student I describe here, and one who has not even mastered those skills.

Team homework

My students were organized into groups of four who met weekly to work together on more challenging problems, which they wrote up as a group and received a single collective grade for. I was very concerned about this going into the course. It worked better than I expected, but I still have some concerns.

First of all, a note to any students in Michigan calculus. I highly encourage you to work hard on your team homework, and to talk in depth with your teammates about the problems. Over and over, I saw students who really committed to their teams get great benefits out of it. This happened even if none of the students on a team was strong — I had one team made up of four good friends, all of whom were in the bottom quarter of the class. Once they started working together, their performance on quizzes and in class improved dramatically.

So why the misgivings? Because, if I were a student in this class, I would have highly resented being forced to work with students not at my level. I spent a lot of time in high school and college helping my fellow students, and I learned a lot by doing so. But I wouldn’t have appreciated being forced to do it. I remember the first time — sometime in my Freshman spring — when I told my classmates “Look, it’s 3 AM and I finished this problem set yesterday. You’re welcome to keep looking at my notes, but I’m going to bed.” That was really hard. I felt guilty and awkward about it. If my grade had depended on staying awake until they understood the material, I can’t imagine how much harder that would have been.

I saw some hints of this kind of unpleasant dynamic in the teams, and I suspect there was a lot more trouble that I didn’t see. It wasn’t just the strong students who were stressed, either: I had two students who, when I asked them for teammate preferences, specifically asked not to be put with students in the top half of the class.

Working with people, no matter what, will be a source of tension. Working with people whom you don’t choose to work with, and being dependent on them for your grade — that may be how the real world works, and I see the benefits of it, but I don’t like it in my classroom.

The mathlab

Michigan has a large room called the mathlab (it could probably hold 200 people or so) filled with round tables and staffed 8 hours a day by tutors who can provide help with mathematics courses. It is open without appointment to anyone who wants to come ask a question. If you are taking an intro calculus course, you’ll find lots of help. If you are taking a complex analysis course, then helpers might be rarer — but sometimes you’ll find someone like me at the table, glad to see what I can do.

I think this is a great idea, and every university should set this up.

I had a lot of fun going to the mathlab. I expect that next semester, when I’m not officially teaching, I might drop in anyway to take a break. The mathlab is like a live-action version of math.SE and, while the questions are less exciting, the challenge of being face to face with the questioner makes up for it.

Some things I wish I’d known/would do differently

\bullet My more experienced colleagues told me I didn’t need to spend class time teaching mechanical differentiation skills at all. Just emphasize how crucial it was to pass the Gateway (a computer administered exam testing this skill), and the students would teach themselves. I didn’t quite follow this advice; on one Friday, I dismissed class early for those who had passed the Gateway already and spent 40 minutes drilling differentiation for the remainder. But 2/3‘s of the class had already passed the Gateway by themselves before I did this, and all but one of my students eventually did. So this advice really did turn out to work, despite my skepticism.

\bullet One of my fellow instructors offered to cook breakfast for his class if all of them passed the Gateway a week before the deadline. He wound up paying off and, not only did this make him hugely popular, but it probably relieved him of a lot of the stress I felt in the final days of the Gateway period. When I teach this class again, I will make a similar deal.

\bullet It was really hard to remember how bad my students are at systematic computation. For example, when counting boxes under a curve, they pointed at the squares in a random order, rather than sweeping from left to right, top to bottom. Every time they copied an expression from one part of the page to another, there is a high probability that a plus sign will change to a minus. If they multiplied (a+b+c)(d+e), the terms will not appear in lexicographic order.

This has two consequences. (1) Except when I want to test/improve this skill, I should not assign problems that involve more than a few lines of computation. I missed this over and over. (2) I should think about how to directly teach this skill.

\bullet The course maintains a problem bank of suggested questions to use on quizzes and in class. There are a lot of trick questions in there! For example, one question asked students to numerically approximate \frac{d}{dx} \sin(\sqrt{x}) at x=0. Another geometric optimization problem asked students to construct a solid shaped like a cone on top of a cylinder to optimize some quantity, and the optimum was the degenerate case where the cone had height zero! Since there are no solutions provided, you have to really be on guard not to assign one of these without catching it.

\bullet The webhomework system assumes that (1) any graph which looks piecewise linear actually is piecewise linear and (2) the students will use that fact to make any computations with that graph precise. In my own nitpicky way, I don’t think that’s fair. I very much doubt you can visually perceive the difference between a straight line and two lines making an angle of 1^{\circ} with each other and this can make a substantial difference in an integral. This paradox demonstrates the point very vividly.

From a more practical perspective, almost none of my students naturally grasped this idea. I should have taught it explicitly.

\bullet There are a lot of problems in the webhomework (mainly sections 5.2-5.4) which can only be reasonably done by typing integrals into a calculator. (The intended focus of the problem is on how to set up the correct integral in the first place.) I didn’t realize this was the intended method at first and spent a lot of time teaching students how to approximate them by Riemann sums or how to compute them exactly, while thinking to myself that this was way too hard for these students.

\bullet My students really got settled into their teams and didn’t like spending class time working with other students. (I guess this shouldn’t have surprised me.) However, when I did force them to, they often got a lot out of it. Next time, I should make sure that they spend a lot more time interacting outside their teams from the start of class.

\bullet The course coordinators really stressed how much the course focused on groupwork. I think I took these messages too strongly to heart. Especially in the first half of the term, I tried to spend almost the entire class period working with small groups. I think this is a reasonable interpretation of what I was being told, but it apparently wasn’t what they meant.

11 thoughts on “Some thoughts on teaching Michigan calculus

  1. @Jason fixed, thanks! Turns out that URL’s within are case sensitive (which I knew) and that capitalization was not used consistently with these files (which I did not).

  2. Dear David,

    Do you have any thoughts on what benefits your students might have had from having you teach the class rather than a proficient graduate student? If not for your students specifically, are there benefits for the calculus teaching program as a whole from having you teach the class? After all, you cost the department significantly more than a graduate student!

  3. Before answering, I want to point out that university departments are not idealized firms, who aim to produce a service as efficiently as possible. I’m sure Alex knows this, but the framing of his final sentence tends that way. If the department saved money by getting rid of me and using lecturers (or, given the constraints of tenure, by not replacing the next mathematician who retires), that money would not go to the department budget; it would go to some other department or to a decrease in our rate of tuition increase. Even if the math dept got to keep the money, I suspect there is nothing we would like to do with it more than hire a top researcher!

    I don’t think that I was overall better than a good grad student or lecturer. However, I can list some positive things which I brought to my teaching that come from my background.

    * I think I did a pretty realistic job talking about working with numerical data. I pointed out that, when numerically differentiating a signal, if you take your time points too close together, the noise will overwhelm the derivative. I talked about how the average of a left hand and right hand Riemann sum will converge much faster than either side alone. I talked about how the error rate in a left hand or right hand Riemann sum is controlled by the steepness of the function, while the error in the average is controlled by the curvy-ness. Of course, I didn’t focus on these things, but I hope I said some useful things.

    * It helped sell students on the course to show that I was someone who really did think and care about these things.

    * A number of my students asked me for help with other mathematics from their other courses, such as physics and economics. I think I did a good job explaining it and fitting it into the course material.

    I want to take the opportunity to brag about the most fun example of this: I had a CS student who, for a final project, was making a winamp-like program and was trying to figure out how you take an audio signal and extract a sequence of musical pitches, so I got to tell him about Fourier windowing, and he really did get it to work!

    * I have a huge mental repertoire of functions. If I wanted something with slowly decreasing oscilations, or with a very narrow tall spike, I knew exactly which example to bring up. Similarly, I knew how I should expect numerical computations to behave. I could look at a table of values which one of my students had computed and guess pretty quickly whether they had entered them into the calculator correctly.

    * I am very fast at computation. I could start new problems on the board in response to student’s questions and count on doing them well. I could jump into a group of students who had been working together, glance over their shoulders and start immediately start talking with them about how far they’d gotten.

    None of this comes directly from anything that I learned in grad school or in research. But I think a lot of it comes from continuously thinking about a wide range of mathematics over the last twelve years. Even though I knew almost all this material when I arrived in grad school, I don’t think I could have summoned it up with the necessary speed, perspective and confidence.

    I think I really did stand out from other instructors in these regards. My students commented (in person and on evaluations) at how willing and able I was to discuss anything mathematical, and the course coordinator who observed my teaching commented on how effectively I could enter in the middle of a mathematical conversation.

  4. Alex didn’t ask me, obviously, but having been through similar teaching experiences and having dealt with that question before, I’ll submit my views. Among the benefits:

    It builds personal connections within a department. Grad students graduate; postdocs leave. If a department’s answer to “How long have you known the applicant?” on a letter of recommendation is always “two years or less”, students are shortchanged in ways that go beyond the classroom.

    Most grad students are new to teaching and not initially as good as they become later. Inexperienced teachers often do best with advanced material, or at least advanced students— contexts where something has filtered out the biggest teaching challenges (and the students resemble less-trained versions of the instructor). Introductory material is best taught by people with more experience than grad students generally have.

    At public schools in times of budget cutbacks, taxpayers ought to feel that in funding a math department they are not simply subsidizing an academic lifestyle that has nothing to do with them. Our non-teaching activities are increasingly public: preprints go on the arxiv, talks go on conference websites, some of us blog, or ask/answer questions on MathOverflow. So students/parents/alumni who Google us see us interacting, daily, with people we have no actual obligations to. One can make indirect arguments for academic value like “students learn more and better from instructors who interact a lot with the community”, blah blah. But the simplest argument for math professor value is low-level teaching. It’s an easy-to-appreciate demonstration of a commitment to serving a chunk humanity that is wider than simply “math people.”

    Conversely, when a department never allocates faculty to teach such classes, it conveys a strong message that these classes (and maybe math itself) serve only a “weeding” role and have no intrinsic value.

    People who teach introductory classes generally do better at teaching advanced classes. Every department has three distinct curricula: the things listed in the descriptions in the course catalog, the things the students actually see when they take those courses, and the things that students who have taken those courses can realistically be expected to remember. People who don’t teach introductory classes lose touch with these second two curricula, and their advanced teaching suffers. (I fondly recall an analysis 1 instructor who assumed that calculus 1 included a detailed treatment of proof by induction, a linear algebra instructor who assumed that calculus 2 was a differential equations class, and a differential geometry instructor who assumed that calculus 3 included a detailed treatment of the implicit and inverse function theorems. In each case, for many of my classmates, that was the beginning of the end.)

    Similarly, grad students generally don’t know what parts of an introductory class are used elsewhere. So they don’t know what is absolutely essential, and what can be rushed/omitted if necessary. (New instructors can get the gist by talking to older faculty, but without close supervision, they still have to guess a lot.) With service classes this is especially important— grad students, even when proficient in the underlying subject matter, generally have *no idea* of what goes into hammering out a curriculum that touches multiple departments. And they often only know the “math major version” of introductory material. So when advanced faculty cease to be involved in an introductory class, the class can kind of “drift away” from its actual purpose in a curriculum.

  5. I have just one item to add to your lists: I can help my students put their struggles in perspective by authentically saying that there are math problems that have no procedure for their solution, take months to solve, and result in dense, barely readable, 50 page papers.

    As for the comments on departments not being idealized firms, I will just point out that most public institutions other than flagship universities, as well as lower tier liberal arts colleges, are under such budget pressures that they have no choice but to have most of their teaching done by graduate students and adjuncts. In these situations, departments may only have enough professor lines to staff the upper level (i.e. analysis/algebra and above) undergraduate (and graduate if any) courses with full-time, doctorate holding, faculty.

    I don’t imagine David’s department would on its own decide to replace a professor teaching calculus with a lecturer (for the very good reasons David mentions), but I could easily imagine David’s provost looking at the course schedule and telling the math department it doesn’t need to hire a professor to replace the one that just retired because it still has professors teaching calculus. Even worse, I can see David’s state legislature using the “wastefulness of using him to teach calculus” as an excuse to cut the already meagre funding it provides to his university.

  6. Alex’s additional point is a good one. I also can, and did, tell them that clear writing is essential to a successful mathematical career, though that one didn’t have much effect. :)

    I am going to say something which is obviously against my professional interest. If a university (or its funding source) does not value research, it is not clear to me that it should hire researchers as opposed to lecturers.

    I do think there are advantages to having instructors with more mathematical experience then a grad student can obtain, and I talked about some of them above. In a better economy, perhaps, such people would be unwilling to work as instructors because they would prefer positions in industry/finance, so we would need to offer appointments with a research component in order to attract them. But, at least in this economy, I don’t think that’s true. There are plenty of people who love math so much that they will continually learn and study new mathematics. Some subset of those people will also have the personality and drive to be teachers. I think they could teach low level math courses as well as I do.

    Now, I think pure mathematical research is a crucial investment in the future of our society. Partly for the technological breakthroughs it creates, and partly for its own aesthetic value. And I think that, to teach college math well, you need to be aware of a great deal of mathematics, well beyond the official level of your course. But I don’t see why you need to do that research yourself.

    Obviously, I would be very glad to hear counter-arguments!

  7. One point I should make, though, before the legislators start grinning too broadly. You still need to give those teachers many hours a week of reading mathematics and thinking about mathematics, to keep their minds fresh and to keep them up to date with applications. I don’t think you need to insist that they produce original research, but there isn’t that much money savings in having them thinking without writing.

    And, while skilled instructors with an excellent command of the mathematical field may be more common than skilled researchers, they are going to be rare as well.

    Right now, instructors are much cheaper than professors. I don’t think that’s because comparably skilled instructors are so much more common than professors. I think its because instructors enter into the grad school pathway drawn by the promise of academia, and then feel that they don’t have a good way to work outside that world. If the promise of academia were withdrawn, would people with a passion and skills for mathematics still become instructors? That argument won’t appeal to a truly selfish legislator, as one state could always free-ride on the others. But I hope that many politicians would see the point.

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