John Erdman is an Emeritus Associate Professor of Mathematics at Portland State University. Over several decades, he has devoted himself to developing problems based courses, and one outcome is the recently published book: A Problems Based Course in Advanced Calculus. Read on to learn how this author’s teaching philosophy and methods evolved and developed over time.
Do you have a general philosophy/approach when it comes to the dissemination of mathematics?
I have had over many decades an ongoing disagreement with the great majority of my colleagues over appropriate teaching methods for mathematics. I am not a great admirer of the lecture method. My first question to fellow mathematics instructors is, “Is mathematics primarily an activity or is it a body of knowledge?” The reply, with unfailing unanimity is, “It is an activity.”
My second question, then, is, “If you were going to teach them some other activity, say, playing the piano, would you cram 30 to 300 of them into a hall three or four times a week and have good pianists play for them? You might make assignments for them to go home and try to play a similar piece on their own. You might even have them record their ‘lesson’ so you could provide criticism—there were wrong notes in the specific measures, faulty rhythms, incorrect tempi, etc. How long would it take for them to develop a reasonable technique under this mode of instruction? How would you teach students to play tennis? Have them watch tennis games three times a week? How about ballet?”
The usual response I get to this second questions is, very roughly paraphrased, “Well, look at what a fine mathematician I am, and I was taught by the lecture method.” While it is clear to me that talented and hard-working people can, and often do, succeed despite being subjected to dubious instructional methods, I do not find this response a very convincing argument for the value of lecturing mathematics at students.
What made you decide to write this particular book? Was there a gap in the literature you were trying to fill?
I have never liked the way beginning calculus is taught. In an effort to keep things ‘simple’ courses usually emphasize routine calculations and abandon any serious attempt at meaningful explanation about ‘what is going on’. One would hope that a course in advanced calculus would fix this imbalance, but, in my experience, it seldom does.
Consider the ‘derivative’. To me ‘differentiation’ of a function is a single idea, independent of dimension: it is finding a (continuous) linear map which is tangent to (an appropriate translation of) the function in question.
Most texts indulge in the tortuous process of defining the word first for real valued functions of a single variable in terms of approximation by tangent lines, in the pursuit of which, unsavory, ill-defined creatures called ‘increments’ and ‘differentials’ that follow curious computational rules are introduced. (Why, I wondered as an uncomprehending student, if x and y are just names of two variables, is Δx= dx but Δy ≠ dy?) After this, the word gets redefined for scalar valued functions of two variables in terms of approximating tangent planes. Subsequently students are given one of those incomprehensible ‘and-in-a-similar-fashion’ evasions to explain how to differentiate scalar valued functions of three or more variables (which leaves the typical student desperately trying to imagine what a hyperplane in n dimensions might look like). Eventually the course moves on to differentiation of parametrized curves and surfaces, and, beyond that, finally, engages in a flurry of transfinite arm-waving concerning the differentiability of functions between general finite dimensional spaces, which requires the invocation of an incomprehensible bacchanal of matrices, determinants, and partial derivatives, in which any semblance of any geometric meaning completely disappears. Perhaps one of the worst aspects of this approach is that for students who go on, none of it is of any use whatever in understanding the calculus of infinite dimensional spaces or of differentiable manifolds.
In my opinion it is much better to do differentiation first for a real valued function of a real variable in such a way the almost nothing needs to be changed when one eventually studies differentiation of functions between arbitrary normed linear spaces.
If I were challenged to come up with a single question to ask a student who has taken calculus, or advanced calculus, that would best indicate whether (s)he understands what calculus is really about, I would offer the following:
Explain why the Fundamental Theorem of Calculus, Green’s Theorem, the Fundamental Theorem for Line Integrals, Stoles’ Theorem, and Gauss’s Divergence Theorem all say exactly the same thing, but in different dimensions.
How many students, after perhaps two years of calculus and a year of advanced calculus, can give a reasonable explanation of this truly fundamental fact?
How did you decide on the format and style of the book? Did you consider other formats for this book? Open Source? Online Notes? Self-publication?
This book was the result of teaching advanced calculus courses over several decades. When I first started teaching, I taught the way I had been taught. I lectured at students and assigned standard texts.
This, I found, did not work well. The texts, which students seldom read, proved fundamental results, while relegating to exercises peripheral facts. So, most students concentrated on peripherals. Those few students who did read the text usually did so by simply checking the logic and the computations of the proofs that were offered, but in the end had no idea how to produce similar proofs on their own.
As a result, I switched early on to a rather strict Moore-style format based on dittoed notes that included two things: suggested criteria that a definition of a term (such as continuous) should satisfy in order to be useful and a number of assertions, some of which would turn out to be true and others false. Class time was devoted to student discussion, where they tried to produce decent definitions of terms and determine the truth or falsity of the assertions. I would sit in the back of the room and, on occasion, would reluctantly agree with something a student offered.
This procedure I found to be unsatisfactory in two important ways. Advanced calculus, I think it is generally agreed, should provide an adequate foundation for a subsequent course in real analysis. This involves covering a rather large amount of material. Can students, left to their own devices, come up with, for example, an adequate epsilon-delta characterization of limits and continuity? Yes. After a couple of weeks of failed attempts, acrimonious arguments, frustration, tears, and a dash of covert instructional guidance, they can indeed produce a correct characterization. They are proud of what they have done, and, I really believe, some of them have learned something important. But this procedure is totally incompatible with covering anything like the great number of topics that are necessary for subsequent courses.
A second serious problem is that not every student invents the epsilon-delta characterization independently. The brightest students come up with the crucial ideas and the rest follow along. These ‘rest’ might just as well have read a text.
The current book represents a compromise between ‘learning by doing’ and ‘covering material’. Students are asked to develop the core material on their own. There are sample proofs, some in the text, others available on the internet, that they can learn from and try to imitate. They are not asked to invent definitions or divine the development of the subject material. Initially students received weekly packets of dittoed materials. Later, when Copy Centers opened, they had perfect bound texts. For many years now all the material has been available online. And recently the AMS decided to make it available in hard copy.
I certainly make no claim that the preceding is a perfect pedagogical method. Certain students are highly disinclined to put in a lot of hard work on a subject if they are not provided a detailed algorithm for every step of their assigned work. Such students get very little from the course and drop out. That, of course, is a great virtue of going to a large urban university—you can always find an easier instructor.
Was your writing influenced by other books? Which ones?
Very definitely. I learned, finally, how differentiation really works by reading Loomis and Sternberg’s Advanced Calculus. I took Dieudonné seriously when he says in Chapter VIII of his Foundations of Modern Analysis, that it may well be suspected that had it not been for its prestigious name the ‘Riemann integral’ would have been dropped long ago and that, “Only the stubborn conservatism of academic tradition could freeze it into a regular part of the curriculum, long after it had outlived its historical importance.” I learned from him how to present an elementary form of integration at the advanced calculus level that avoids the quagmires of Lebesgue measure, nets (however artfully disguised), or upper and lower sums.
I have always been deeply grateful to Halmos for his lessons in, and his many examples of, clarity of mathematical exposition. And I am indebted also to Stromberg, who, in his An Introduction to Classical Real Analysis, beautifully demonstrates the art of parsing complex arguments in such a way that students are led to fashion on their own proofs of difficult theorems.
Did you find ways to get feedback while writing your book or was it a solitary effort?
Since I taught from various versions of my text for several decades, I have had the opportunity of receiving a large amount of feedback, primarily from students. Those students who stuck with the course for two academic terms seemed very appreciative of it. Among those who dropped out, several were kind enough to come to my office and offer me some advice. It would have been far better, they explained to me, occasionally in quite elaborate detail, had I taken the trouble to write the book, rather than expecting my students to write it for me.
My very favorite bit of feedback came from a student commenting on one of those teacher-rating websites. He wrote:
Professor Erdman is the worst instructor I have ever had. I learned more in his class than in any other math course I have taken. But it was not his fault! He made me do everything myself.
I was absolutely delighted. Hooray! Finally a student who gets it.