Greetings all,

There's a never-ending story that many of us have sunk our teeth into. How do we go about teaching subjects like calculus and analysis "well?" Most universities that I'm familiar with do a significant amount of "service" work where the majority of our students have their focus on other subjects, be it engineering of physics or computer science or economics or whatever. So our curriculum frequently has to strike awkward balances between issues such as:

  • How we view mathematics.

  • What the students are ready to learn.

  • What our students would like to learn.

  • What other departments expect us to teach our (their) students.


In universities where resources are not limitless (I want to exclude examples like Harvard, MIT, Stanford, Oxford, etc -- not to say they don't have these problems, but the focus is different) this leads to endless compromises and fussing about with strained resources. Sometimes the compromises are extremely far from ideal.

I'm curious to find out what some "joe average" math departments have been doing. Are there some interesting success stories out there? Some novel approaches to teaching things like calculus and/or analysis to a broad audience, on a tight budget?

Have any departments out there got away from the expensive "phone book" style textbooks? Into on-line material? Interactive software? Has anyplace started seriously using things like Wikipedia as a resource for elements of their courses? Are any departments having success using "muscular" calculus books like Hubbard's "Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach" ?

I've seen some examples of on-line homework management, like "WebWork". I believe there are a few others similar platforms out there. We use content-management software here, things like "Moodle" and "Blackboard".

How about interesting ways of merging (or separating) highly-motivated math students into/away from the service curriculum? Does your department have honours courses starting in the n-th year where students would learn axioms for the real numbers? Set-theoretic constructions of the real numbers? Do you ease them into foundational issues slowly (axioms for real numbers before a definition, etc?), or do you whip it out right away? Do you avoid the issue completely?

What kind of background do your students have before learning things like basic point-set topology? Modules over rings? Manifold theory? Lie groups? Representations of finite groups? Basic differential geometry? The uniformization theorem for Riemann surfaces? -- if they have chances to learn anything of the sort. ie: what are the "high points" of your curriculum?

This is a massive sprawling question but I'm curious to hear your insights. In case there is any confusion I do want to keep to specifics as much as possible, things like: we tried A, it was a problem because of X, then we tried B and it worked well with Y. What I'd love to see more than anything is a response like: here at the University of Z we just started trying C and it has doubled the enrollment in our analysis classes!! That'd be candy.



4 Answers 4


Our department has had a lot of discussions about resources. I rather like the following ideas and good practices that have been implemented or proposed here:

  1. The Calculus Room. Instead of having each calculus TA sit in his or her office and wait for students, they are assigned times in one big help room called "The Calculus Room". This is a much more efficient system that helps more students per hour of labor.

  2. We have a grade distribution system called "MyUCDavis" that I use a lot. Students can see all recorded grades quickly, throughout the entire quarter. They can also see their rank on each test (and HW) and a histogram of test scores. I imagine that Moodle can do something like this too. I like it because the students can know where they stand, and because I don't like face-to-face questions about grades. Also, I of course recommend posting homework and solution sets on the web, but these days that should go without saying.

  3. WebWork. We just started using WebWork after a negative experience with another system. It is not perfect, but it is a genuinely helpful educational tool. Enough human attention to homework is better in principle, but grading homework in calculus can easily deteriorate to the point that WebWork is better.

  4. Cheap/free textbooks. This is more about saving the students' money than ours, but in the face of a 30% fee hike in one year, we are eager to create goodwill. When a good choice is available, I like the model of using a book that is both sold in print and has a free or nearly free PDF. (Or maybe we can arrange to print and bind such a book at the copy shop.) The first really good book in this model that we used was Hatcher, Algebraic Topology, but more recently there are others. I respect ideas such as wiki-books and teaching with Wikipedia as experiments and supplements, but they are not presently a good substitute for a tried-and-true, structured textbook.

  5. Slowly, incrementally try to raise standards. For instance, we recently shifted our 3-quarter intro analysis so that the first quarter is lower division. The first quarter is taught in the style of Spivak's classic, Calculus. (But Spivak is expensive. As of this year, we use Thomson, Bruckner, Bruckner, because it's a very nice textbook, and the PDF is only one dollar.) More commonly, we just revise the syllabus of this or that course to make it more interesting. We do not have a two-track system for good students vs bad students and I suspect that I wouldn't want it. The students are free to take harder or easier courses within a certain range.

  • $\begingroup$ Point 5 "... the first quarter is lower division" does that mean students see analysis earlier in their programs? Sorry, this is curriculum jargon I don't know. $\endgroup$ Dec 19, 2009 at 17:50
  • $\begingroup$ That's right. Roughly speaking, lower-division courses are large freshman/sophomore classes for non-majors, majors and pre-majors. The main such courses are calculus, linear algebra, and intro diff equations. Upper division are junior/senior classes such as intro topology, algebra, intro PDEs, etc. We use course numbers $n < 100$ for lower div, $100 \le n < 200$ for upper division, and $200 \le n < 300$ for graduate. We renumbered analysis from Math 127abc to Math 25 + 125ab. $\endgroup$ Dec 19, 2009 at 18:22
  • $\begingroup$ Regarding your "I suspect that I wouldn't want it" comment, I'm curious where in your undergraduate program are your "hooks"? I'm thinking of an undergraduate program as like a pop song -- you need something to grab students attention to get them interested in mathematics. Maybe you're targeting students that have always liked mathematics? I find that potentially dangerous since many students haven't had the opportunity to see mathematics in a positive light. $\endgroup$ Dec 19, 2009 at 18:48
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    $\begingroup$ I am all in favor of having a range of courses, from routine to advanced or exotic, and from easy to very difficult. But I don't like to segregate students by ability, when instead they can make their own choices. Even for the common question of prerequisites, I often say: "You're a grownup now. I'm willing to waive the prerequisite if you think you know best." $\endgroup$ Dec 19, 2009 at 19:11
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    $\begingroup$ +1 on the Calculus Room. Berkeley started doing "collective office hours" for upper level courses a while ago. In retrospect, it seems absurd that they haven't done this for the main first year calculus course. Office hours are very unevenly utilised -- and perhaps some of the TAs would learn something from seeing the more effective ones in action. $\endgroup$ Dec 21, 2009 at 15:25

Here at University of Michigan, we have a completely separate introductory sequence for highly motivated freshmen. It's modeled after the ones at Harvard and Chicago, from what I've been told (although having not taken them, I can't honestly say for sure). We happened to use Spivak, but that's really the only book that is really 'phonebook-sized'.

Essentially, within the first semester, we're fairly familiar with point-set topology (to the point that taking the introductory point-set topology course was a waste of time since I'd already seen it all), the basics of group theory, and a decent amount of analysis (last year, we finished the first semester with the proofs of taylor's theorem and the different error estimate formulas for series expansions.)

The second semester was finishing up covering some things like uniform and pointwise convergence that were left out of the first semester for time constraints, then linear algebra for pretty much the rest of the semester. We covered topics in a similar order to Hoffman and Kunze (i.e. abstract vector spaces before real/complex inner product spaces.). Then we returned at the end of the year to calculus on real and complex inner product spaces (frechet derivative, inverse function theorem, et cetera). During this semester, we were assigned mostly point-set topology and algebra problems unrelated to linear algebra. These problems usually led to the final step on a later homework being a proof of something like the fundamental theorem of Algebra or Sylow's first theorem (which are two I can remember explicitly.) (No textbook necessary for this course, although the official textbook was Hoffman and Kunze.)

The third semester was spent pretty much entirely spent on measure theory and integration, although a little bit of time was spent on complex analysis, but only enough to prove the statements about the fourier transform. (No textbook was assigned for this course. Anyone who actually purchased the course text indicated on the internet was urged to return the book.)

From what I've been told, next semester, we're doing differential geometry, but as of yet, I don't know how our professor intends to teach it (I asked him if he'd talk about the subject from a category-theoretic point of view (i.e., http://ncatlab.org/nlab/show/nPOV, but he laughed as though I were joking).

The upshot on this sequence is that by the end of the first three semesters, we also receive 'credit (toward completion of a math major, not actual credits)' for linear algeba and real analysis in addition to completion of the intro sequence.

Were I forced to take one of the other introductory sequences, I probably would not have become a math major. The rest of the math curriculum is that awful.

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    $\begingroup$ I'm very sorry to be the one to tell you this, Harry, but here goes: University of Michigan is nowhere close to an "'average' research university". $\endgroup$ Dec 19, 2009 at 11:10
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    $\begingroup$ No, but the question seemed to emphasize resources: the top tier of state universities in the US have many features distinguishing us from "joe average" universities, but "limitless resources" is not one of them. $\endgroup$
    – JSE
    Dec 19, 2009 at 14:58
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    $\begingroup$ I would prefer to think that all research universities are above average. (As in the Lake Wobegon News.) $\endgroup$ Dec 19, 2009 at 15:56
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    $\begingroup$ The resources that are scarce is the university's willingness to indulge its faculty in teaching courses that have small numbers of students, especially when those students form a tiny segregated minority. Some universities do a good job of getting a diverse collection of students taking honours type courses, but that takes time to build up the good-will among departments, and indulgence on the part of the administration. $\endgroup$ Dec 19, 2009 at 23:32
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    $\begingroup$ Harry, often the scarce resource is students motivated and prepared to take such a course. I think this is one of the key points where Michigan could be very different from a lot of less well-known schools. $\endgroup$
    – Ben Webster
    Dec 20, 2009 at 21:34

When an undergraduate mathematics department designs (or redesigns) its curriculum one variable that I believe has not been given enough attention is the nature of the student body the department serves. Small branches of a large public university system serve a different niche of students from the flagship school of the system or the niche served by small or large highly selective private schools. Furthermore, even schools that do not attract an "elite" group of students often find themselves with a few students who could have gotten into an "elite" college or university.

This issue was addressed in the United States in the last CUPM (Committee for the Undergraduate Program in Mathematics) report issued on the auspices of MAA. One recommendation was the mathematics departments get feedback about the goals and aspirations of the students it serves, and monitor to the extent they can what mathematics majors do with their degrees after they graduate. Departments can certainly take pride in math graduates who go on to get a doctorate in Mathematics, CS, or some allied discipline. However, they can also take pride in the dedicated high school or middle school teachers they train and inspire, the students who get jobs that put their mathematical skills to work, or students who do not directly use any mathematics in the careers they ultimately pursue.

In prior CUPM reports often "sample" syllabi for various courses were produced. In the last report it was decided not to provide sample syllabi. I believe this to have been a mistake. In fact, I would favor having samples of several different syllabi for the same course which would show how schools with different niches of students might try to serve their students as well as possible. When departments are aware of the needs of their students even when they primarily serve students who do not intend to get a doctorate degree in the future, they can still try to meet the needs of the students who do plan to get a doctorate, or might consider doing so if circumstances allow. Similarly, departments which serve very strong mathematics students can find ways to meet the needs of their "weaker" students as well.

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    $\begingroup$ I'm surprised someone didn't like your comment. Perhaps it is a medicine some people don't want to swallow? Thanks for your comment. It's something I've been considering but I also haven't been here at Victoria long enough to really know how to fully consider these thoughts. $\endgroup$ Dec 19, 2009 at 17:00
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    $\begingroup$ I feel like the distinction between "strong" and "weak" math students is also very much a matter of motivation (although natural ability surely does have a lot to do with it). There's this idea that 'real' math is "too hard" for the average person, so they're instead taught calculus from something like Stewart. Math should be taught like any other subject: it should be taught the way that 'real mathematicians' use it. Accommodations should be made for students who are unable to do well, rather than those who have no interest in doing well. $\endgroup$ Dec 19, 2009 at 23:15
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    $\begingroup$ Harry: The "world of mathematics" is very complicated. There are certainly issues of ability, motivation, and passion. One can be passionate about mathematics but perhaps not that good at it. One can have great talent and not work hard. Some people who find calculus very hard (or perhaps not interesting) are capable of doing original work in discrete mathematics while some people who are very good at analysis don't seem to have much creativity in discrete areas of mathematics. The way mathematics grows as a subject reflects all of these complexities. $\endgroup$ Dec 23, 2009 at 19:40

Let me just share a paper on Integrative Education that I and my favorite co-author presented at a conference concerned with the future of the university.

The conference version:

Awbrey, S.M., and Awbrey, J.L. (September 1999), “Organizations of Learning or Learning Organizations: The Challenge of Creating Integrative Universities for the Next Century”, Second International Conference of the Journal ‘Organization’, Re-Organizing Knowledge, Trans-Forming Institutions: Knowing, Knowledge, and the University in the 21st Century, University of Massachusetts, Amherst, MA. Online.

The published version:

Awbrey, S.M., and Awbrey, J.L. (May 2001), “Conceptual Barriers to Creating Integrative Universities”, Organization: The Interdisciplinary Journal of Organization, Theory, and Society 8(2), Sage Publications, London, UK, pp. 269–284. Abstract.


Speaking of "resource constrained", the comment boxes are a bit short on amenities, so I'll add further comments below.

On a more concrete level, my sources tell me that Virginia Tech has been running a very successful "Math Lab" or "Math Emporium" program for several years now. A number of other universities have taken this approach as a model and begun to adapt it to their particular requirements. Here are a couple of links:

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    $\begingroup$ Your essay doesn't appear to be directly relevant to my question. I'm asking about specific steps people have tried and been successful with, not so much personal opinion on the way things should be done. $\endgroup$ Dec 20, 2009 at 5:31
  • $\begingroup$ "This is a massive sprawling question but I'm curious to hear your insights." Sorry, I took you at your words — all of them. $\endgroup$
    – Jon Awbrey
    Dec 20, 2009 at 16:24
  • $\begingroup$ I hope that revision clarifies things. $\endgroup$ Dec 20, 2009 at 20:23
  • $\begingroup$ Ask a massive sprawling question → you'll get a massive sprawling answer. But seriously, generalizing the problem is not only second nature in mathematics, it's an official heuristic from How To Solve It. Many of the issues that arise in reform are peculiar to mathematics, but many more are universals. And if you read your own text carefully again, you'll see that you mentioned contextual and interdisciplinary factors very prominently, even "pragmatic" and "rhetorical" factors like the affective inclinations of the students and their individual objectives in learning the material at hand. $\endgroup$
    – Jon Awbrey
    Dec 21, 2009 at 3:42

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