{"id":782,"date":"2019-01-24T09:05:10","date_gmt":"2019-01-24T14:05:10","guid":{"rendered":"http:\/\/blogs.ams.org\/bookends\/?p=782"},"modified":"2020-04-30T08:46:40","modified_gmt":"2020-04-30T12:46:40","slug":"author-interview-john-erdman","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/amathematicalword\/2019\/01\/24\/author-interview-john-erdman\/","title":{"rendered":"Author Interview: John Erdman"},"content":{"rendered":"
\"\"John Erdman<\/a> is an Emeritus Associate Professor of Mathematics at Portland State University.\u00a0 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.\u00a0\u00a0 Read on to learn how this author’s teaching philosophy and methods evolved and developed over time.
\n<\/em><\/h6>\n

Do you have a general philosophy\/approach when it comes to the dissemination of mathematics?<\/strong> <\/em><\/p>\n

I have had over many decades an ongoing disagreement with the great majority of my colleagues over appropriate teaching methods for mathematics.\u00a0 I am not a great admirer of the\"\"<\/a> lecture method.\u00a0 My first question to fellow mathematics instructors is, \u201cIs mathematics primarily an activity or is it a body of knowledge?\u201d\u00a0 The reply, with unfailing unanimity is, \u201cIt is an activity.\u201d<\/p>\n

My second question, then, is, \u201cIf 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?\u00a0 You might make assignments for them to go home and try to play a similar piece on their own.\u00a0 You might even have them record their \u2018lesson\u2019 so you could provide criticism\u2014there were wrong notes in the specific measures, faulty rhythms, incorrect tempi, etc.<\/em>\u00a0 How long would it take for them to develop a reasonable technique under this mode of instruction?\u00a0\u00a0 How would you teach students to play tennis?\u00a0 Have them watch tennis games three times a week?\u00a0 How about ballet?\u201d<\/p>\n

The usual response I get to this second questions is, very roughly paraphrased, \u201cWell, look at what a fine mathematician I am, and I was taught by the lecture method.\u201d\u00a0 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.<\/p>\n

What made you decide to write this particular book?\u00a0\u00a0\u00a0 Was there a gap in the literature you were trying to fill?\u00a0<\/strong> <\/em><\/p>\n

I have never liked the way beginning calculus is taught.\u00a0 In an effort to keep things \u2018simple\u2019 courses usually emphasize routine calculations and abandon any serious attempt at meaningful explanation about \u2018what is going on\u2019. \u00a0One would hope that a course in advanced calculus would fix this imbalance, but, in my experience, it seldom does.<\/p>\n

\"\"Consider the \u2018derivative\u2019.\u00a0 To me \u2018differentiation\u2019 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.<\/p>\n

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 \u2018increments\u2019 and \u2018differentials\u2019 that follow curious computational rules are introduced. (Why, I wondered as an uncomprehending student, if x <\/em>and y <\/em>are just names of two variables, is \u0394x<\/em>= dx <\/em>but \u0394y \u2260 dy<\/em>?)\u00a0\u00a0 After this, the word gets redefined for\u00a0 scalar valued functions of two variables in terms of approximating tangent planes.\u00a0\u00a0Subsequently students are given one of those incomprehensible \u2018and-in-a-similar-fashion\u2019 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 <\/em>dimensions might look like).\u00a0Eventually 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.\u00a0 Perhaps one of\u00a0 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.<\/p>\n

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.<\/p>\n

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:<\/p>\n

Explain why the Fundamental Theorem of Calculus, Green\u2019s Theorem, the Fundamental Theorem for Line Integrals, Stoles\u2019 Theorem, <\/em>and Gauss\u2019s Divergence Theorem<\/em> all say exactly the same thing, but in different dimensions.<\/p>\n

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?<\/p>\n

How did you decide on the format and style of the book? \u00a0Did you consider other formats for this book? Open Source?\u00a0 Online Notes?\u00a0 Self-publication?<\/em><\/strong><\/p>\n

This book was the result of teaching advanced calculus courses over several decades.\u00a0 When I first started teaching, I taught the way I had been taught.\u00a0 I lectured at students and assigned standard texts.<\/p>\n

This, I found, did not work well.\u00a0 The texts, which students seldom read, proved fundamental results, while relegating to exercises peripheral facts.\u00a0 So, most students concentrated on peripherals.\u00a0Those 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.<\/p>\n

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)\u00a0<\/em>should satisfy in order to be useful and a number of assertions, some of which would turn out to be true and others false.\u00a0 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.\u00a0 I would sit in the back of the room and, on occasion, would reluctantly agree with something a student offered.<\/p>\n

This procedure I found to be unsatisfactory in two important ways.\u00a0 Advanced calculus, I think it is generally agreed, should provide an adequate foundation for a subsequent course in real analysis.\u00a0 This involves covering a rather large amount of material.\u00a0 Can students, left to their own devices, come up with, for example, an adequate epsilon-delta characterization of limits and continuity?\u00a0 Yes.\u00a0After 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.\u00a0 They are proud of what they have done, and, I really believe, some of them have learned something important.\u00a0 But this procedure is totally incompatible with covering anything like the great number of topics that are necessary for subsequent courses.<\/p>\n

A second serious problem is that not every <\/em>student invents the epsilon-delta characterization independently.\u00a0 The brightest students come up with the crucial ideas and the rest follow along.\u00a0These \u2018rest\u2019 might just as well have read a text.<\/p>\n

The current book represents a compromise between \u2018learning by doing\u2019 and \u2018covering material\u2019.\u00a0 Students are asked to develop the core material on their own.\u00a0 There are sample proofs, some in the text,\u00a0others available on the internet, that they can learn from and try to imitate.\u00a0 They are not asked to invent definitions or divine the development of the subject material.\u00a0 Initially students received weekly packets of dittoed materials.\u00a0 Later, when Copy Centers opened, they had perfect bound texts.\u00a0For many years now all the material has been available online.\u00a0 And recently the AMS decided to make it available in hard copy.<\/p>\n

I certainly make no claim that the preceding is a perfect pedagogical method.\u00a0 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.\u00a0 Such students get very little from the course and drop out.\u00a0 That, of course, is a great virtue of going to a large urban university\u2014you can always find an easier instructor.<\/p>\n

Was your writing influenced by other books?\u00a0 Which ones?<\/em><\/strong><\/p>\n

\u00a0<\/em>Very definitely.\u00a0 I learned, finally, how differentiation really works by reading Loomis and Sternberg\u2019s Advanced Calculus.<\/em>\u00a0 I took Dieudonn\u00e9 seriously when he says in Chapter VIII of his Foundations of Modern Analysis, <\/em>that it may well be suspected that had it not been for its prestigious name the \u2018Riemann integral\u2019 would have been dropped long ago and that, \u201cOnly the stubborn conservatism of academic tradition could freeze it into a regular part of the curriculum, long after it had outlived its historical importance.\u201d\u00a0 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.<\/p>\n

I have always been deeply grateful to Halmos for his lessons in, and his many examples of, clarity of mathematical exposition.\u00a0And I am indebted also to Stromberg, who, in his An Introduction to Classical Real Analysis,\u00a0<\/em>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.<\/p>\n

Did you find ways to get feedback while writing your book or was it a solitary effort?<\/em><\/strong><\/p>\n

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.\u00a0 Those students who stuck with the course for two academic terms seemed very appreciative of it.\u00a0 Among those who dropped out, several were kind enough to come to my office and offer me some advice.\u00a0 It would have been far better, they explained to me, occasionally in quite elaborate detail, had I <\/em>taken the trouble to write the book, rather than expecting my students \"\"<\/em>to write it for me.<\/p>\n

My very favorite bit of feedback came from a student commenting on one of those teacher-rating websites.\u00a0 He wrote:<\/p>\n

Professor Erdman is the worst instructor I have ever had.\u00a0I learned more in his class than in any other math course I have taken.\u00a0 But it was not his fault!\u00a0 He made me do everything myself.<\/p><\/blockquote>\n

I was absolutely delighted.\u00a0 Hooray!\u00a0Finally a student who gets it.<\/em><\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

John Erdman is an Emeritus Associate Professor of Mathematics at Portland State University.\u00a0 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.\u00a0\u00a0 … Continue reading →<\/span><\/a><\/p>\n

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