Author Interview: John Erdman

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.

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Textbooks for Problems Based Teaching

The traditional approach to teaching rigorous, proof-based mathematics is to provide students with models of excellent mathematical exposition and let students learn by emulation.  Typically students will first absorb by reading the textbook and listening to lectures, and then they work through similar examples and exercises until they have mastered the techniques and thought processes.

This model has been challenged in recent years.  An increasingly favored approach emphasizes learning through independent discovery, with variations like  inquiry-based learning and the  inverted classroom.   In these scenarios students learn by doing.  Starting with minimal pre-guidance, students are given problems to think about individually and in collaboration with others.  Instructor involvement is thus pushed toward the end of each lesson. Though this idea turns the traditional methods of teaching on its head, the number of proponents is growing quickly, and studies suggest that the trend will continue. How do textbooks fit in with this new approach to teaching?    Since class time focuses on group projects and exercises, lectures must be more flexible and adaptable than before.  From this point of view, textbooks may seem inappropriately rigid and only useful for their exercises.  Instead, one could post or hand out lists of notes that can be easily changed on the fly.    There are certain crucial downsides to this approach, especially when accommodating large numbers of students, such as losses in conformity across sections and in continuity within curricula.  Also, in the long run, one of the great values of textbooks is that they help perpetuate a universal language and culture within mathematics globally.   In the old approach standard course textbooks balanced the effects of stylistic differences among instructors.

So what would the ideal textbook for a modern, active-learning oriented classroom look like?   A great book has a sense of narrative — a compelling story that makes you keep turning the page,  and a sense of charm and wit — you can follow “the voice” with confidence knowing that the journey and destination will be full of delights.   These can indeed be incorporated into a problems-based book, as the highlighted example below shows.

Please send more examples in comments!

Featured Book of the Day

A Problems Based Course in Advanced Calculus  by John M. Erdman This textbook is suitable for a course in advanced calculus that promotes active learning through problem solving.  The tone of the book reflects the author’s years of experience balancing the need to give students helpful guidance while maintaining the principle that less teaching leads to more learning.  All this culminates in prose that is conversational and inviting, yet efficient and economical, allowing plenty of room for the reader to discover for themselves.


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Author Interview: Marty Weissman

Martin H. Weissman, Professor of Mathematics at University of California, Santa Cruz, has recently published a book with the AMS called An Illustrated Theory of Numbers. How does one illustrate number theory?   Weissman does it in a visually appealing and pedagogically effective way.  Assuming only a high school background in algebra and geometry, the book takes the reader on a journey through the classical works of Fermat, Euler and Gauss, cutting edge topics including the Riemann hypothesis and the boundedness of prime gaps, and modern applications such as data analysis.  As one reviewer put it: “An Illustrated Theory of Numbers is a textbook like none other I know; and not just a textbook, but a work of practical art”.

What made you decide to write this particular book? Was there a gap in the literature you were trying to fill? Did you use existing notes from teaching?
I had taught “elementary” number theory in a variety of contexts: a course for math majors at UC Santa Cruz, a 2-week program for high-school students, various workshops for K-12 teachers. Then I took the famous one-day course with Edward Tufte, a key figure in the “Visualization of Quantitative Information.” I went on a design kick, read lots of books, picked up Python, and decided to turn my disparate number theory notes into a book.
I understood that the market for introductory number theory books was pretty crowded. There are some beautiful older books, but I thought a newer treatment was needed. Among newer books, I was unhappy with the “textbookification” I saw — bulky expensive books, with clunky layout, Wikipedia-like blurbs posing as history, and a sort of writing-by-committee voice (end-of-rant). So I thought a new book could fill the gap. And, of course, there wasn’t an illustrated number theory book!

What are your thoughts on mathematics publishing in general?
There are so many new modes of publishing, interpreted broadly. Math blogs, MathOverflow, projects like the Stacks project, and the arXiv are part of a flourishing ecosystem of mathematical communication. TeX and the internet have enabled wild openness. At the same time, I worry about the consolidation of publishing houses and neglect of math journals and books. Prices have become absurd, to the point where my library has cancelled journal subscriptions and students can’t afford their textbooks. Moreover, I don’t see the editorial or physical quality I would expect when looking at output from the megapublishers. Since I think that edited and physically printed texts are important, I’m worried. The AMS is a bright spot!

Do you have a general philosophy/approach when it comes to the dissemination of mathematics?
Be clear, concise, and correct. Respect your subject and your audience.

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?
I was very picky about a few issues. One was the physical format of the book, since I designed it with two-page spreads (intentional left and right pages when opened), extensive marginalia, and color illustrations. Another was cost — number theory textbooks in the market cost around \$150, which I think is absurd.
Open source and self-publication would allow the production of a decent physical book at a reasonable cost (around \$60 when I researched it). But publishers like the AMS provide key feedback, editorial guidance, advertising, and a distribution network. The AMS used 4-color offset printing rather than on-demand digital printing, and I think the physical quality is superior to what I would have found through self-publishing. They also offered a reasonable cost, in my mind.
At the risk of going against the open source ethos, I do think that authors should be paid for their creative work. I think my research is supported by salary from my institution, and so it should be (and is) freely available. But this book was completed primarily on nights, weekends, and summers, and I appreciate the royalty checks. I think that nonprofits like the AMS strike a good balance, respecting the needs of the mathematical community and the needs and rights of authors.

How did you choose a publisher? What was important to you when you made the choice?
Since I had specific physical and cost requirements for the book, that immediately eliminated some large textbook publishers. Anyways, I would rather compete with McGraw-Hill, World-Scientific, and Pearson instead of joining them.
That leaves Springer, University presses like Princeton and Cambridge, and the AMS. The AMS seemed most receptive to actively working with me on the book. It was easy to talk to the AMS editors (thanks Sergei!) and production team as I made all sorts of unusual requests. Fundamentally, the AMS is dedicated to the interests of mathematicians, and that played a big role in my choice.

What was the writing process like? Did you write every day on a set schedule, or did you have periods of setting it aside?
I wrote batches of the book while teaching number theory, at UC Santa Cruz, and in Singapore at Yale-NUS College. It mostly came in bursts of days or weeks when time allowed, which is why it took close to 10 years from beginning to end. Sometimes I could set aside a few hours or a day to make an image. But mostly, I needed large blocks of time to get the sort of focus I needed to write chapters of the book. I finished the book on a family writing retreat in Cambodia and Indonesia in the summer before moving back to the U.S..

Was your writing influenced by other books? Which ones?
For layout, I was certainly influenced by Edward Tufte’s books. I used a LaTeX package called tufte-latex, which imitates his layout and fonts. I was also influenced by his principles for “graphical excellence” in the design of illustrations and the integration of graphics and text. Mathematically, I often tried to go back to the original sources and “masters”. For example, I wanted to write a really clear proof of the uniqueness of prime decomposition. I read through a lot of proofs in a lot of books on my shelf; in the end, I thought the proof in Gauss’s Disquisitiones (Art. 16) was best.
Design and mathematics share common goals of elegance under constraint. So it might be the case that learning about visual design helped me to write mathematics.

Did you find ways to get feedback while writing your book or was it a solitary effort?
The book went through some early drafts as a coursepack for students. Since undergraduate students are the target audience, their feedback was most useful. I also showed some early sections to colleagues, friends, and family. They strengthened aspects of the design, treatment of history, and more. My cat tended to sleep on printed drafts, which might qualify as feedback.

Did you have a special place where you liked to write? How did you stay motivated and focused?
I tend to filter out my surroundings, so I can write at my office or at home or a cafe. Coffee and a good Spotify playlist helped too.

What kind of feedback did you get after the book came out?
I’ve gotten lots of emails out of the blue, and I’ve appreciated every one! I’m a bit embarrassed every time someone finds a typo or error, but I track them (with acknowledgment) at the book webpage I really enjoy hearing stories from readers — some are teaching with the book, some are working through the book for enjoyment, some are sharing math problems with their kids.

What advice would you give to new authors?
If you have something to share, create something lasting and beautiful.  Read blogs like this one to understand what you’re getting yourself into.  A practical tip: it’s good to make and track decisions about file directories, layout, indexing, notation, etc., as early as possible. Editing a book-length manuscript is a real headache if you haven’t been consistent along the way.

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Read-together math books for kids

Young kids love books like Goodnight Moon and parents love to read it to them.   Does it matter whether the toddler thinks of the moon, the rhythms of the day, the rhythm of the words, the magic of transitions and change, and so many other beautifully embedded ideas in the same way adults expect them to?  It may be interesting to study what and how a child absorbs through the pictures, the words, the voice, and the reader’s presence during read-along (in fact many people have, including this interesting piece by Anne E. Fernald.), but it is also likely that there are mysteries to a child’s mental process that are far from our understanding.  Whatever the case may be, Goodnight Moon is extremely popular, and one reason could be just the way it helps a child and parent to bond around a single moment of shared experience, both ordinary and familiar yet magical and sublime.

Inexplicable gems like Goodnight Moon are rare but what can publishers learn from them?  Could there be such a thing as read-along math books for kids?  Mathematicians know that appreciating and doing mathematics requires flexibility as well as a structure, imagination as well as logic, but that view of math often does not reach youngsters (or even many adults).  As Paul Lockhart asks in his Lament, would children be inspired by music, if they were forced to first learn musical notation and theory?  Similarly would we require a child to master spelling and grammar before reading them a story?   Aspiring engineers and scientists have examples of useful discoveries and powerful tools to entice them.  How can we present mathematics to kids when math’s highest level practitioners work within a language and form that most people don’t have the tools to perceive?

The contents of math books for children has often been governed by what schools decide is appropriate mathematics for each age group to grasp.   Being good at math is equated with being faster at learning this material, but there is a benefit to exposing all children to mathematics.  For one thing, mathematical talent is not always easy to recognize. There are mathematical leaders whose vast imagination and deep intuition were not  recognized at an early age.  Who knows which child could, after a glimpse of the possibilities, be catapulted onward to the unlimited reaches of mathematical endeavor.  And if people benefit from a broad knowledge of other subjects, and from the arts, they can also be enriched by an exposure to mathematics as a creative and exploratory subject.  Even for the average student, such a view of math could give them something more in this world to contemplate and enjoy.

Mathematical ideas are universal, and there is much for a child of any age to respond to: intriguing patterns, surprising structures that appear as if by magic out of seemingly random chaos, puzzles that sound hard but have simple solutions, and puzzles that sound easy but are very very hard.  Closely identifying mathematics books for children with a an educational agenda (unlike story books, music and art) can severely limit their range.  Mathematically intriguing pictures and ideas, and reading together with an adult or older sibling can lead to pleasant discussions of “why” and “what if”?   Some children may still groan or feel blase about the need to learn multiplication tables, rules of algebra, and geometry proofs, but in the back of their minds they may also recognize that mathematics can be kind of cool, sometimes a bit wacky and unexpected, and sometimes rather entertaining and memorable.

Featured Book of the Day

The Infinite Farm by Richard Evan Schwartz

This is Schwartz’s fourth children’s book published with the AMS, and is written in his recognizable and unique style.    But while there was a significant instructional component of the first three books, this latest is more suggestive and open-ended.  We find ourselves in a world with an unfamiliar geometry that allows infinite objects to live in a bounded space.   Whether you know the rigorous mathematical underpinnings of this world, or just want to explore in it, this book is a fun read.

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The Web and Books of the Future

Given the way the internet has become firmly entrenched in our lives, how do you think books of the future will look and how will these “new books” be read and used?   Is the web making books obsolete, or will a new kind of book emerge from within the web world?

I’m thinking of the possible development of a post web “new book” in analogy to the way our current books evolved from oral tradition.  For example, it seems natural that linear narrative should grow out of oral histories, songs,  poetry, and rote memorization and repetition of speeches.  The text of most books transcribes what could also be narrated.     The major difference is that, being physical objects, books have permanence and can be distributed widely.   Some of the freedom and fluidity is lost when a relatively small group of people decide what to print and disseminate in bulk, but there are established mechanisms to preserve a level of consistency and quality and books can reach beyond the inner circle to unimaginably far away worlds.

With the internet, free-flowing information has re-emerged, and a new establishment (or anti-establishment) has formed. People regularly go to the internet instead of to books for information, enrichment and entertainment.   They “go” not to bookstores and libraries, but to websites that resonate with their beliefs and suit their tastes.  The result is a new sort of “village”.  The modern version of a “village” may be geographically diverse but narrowly focused, and miles apart from other “villages.” In other words, the internet gives us more choices, but  it takes conscious effort not to let it reinforce our prejudices and phobias.

What will be the “new book”?  Multiple screens and interactive features?  Intelligent merging of aural, visual and kinetic elements to optimize absorption?  Will books, music, and film merge into one another?

And in all this, how will the concept of authorship evolve?

As always, your comments are welcome!


Featured Book of the Day

Class Field Theory by Emil Artin and John Tate

This classic book, originally published in 1968, is based on notes of a year-long seminar the authors ran at Princeton University. The primary goal of the book was to give a rather complete presentation of algebraic aspects of global class field theory, and the authors accomplished this goal spectacularly: for more than 40 years since its first publication, the book has served as an ultimate source for many generations of mathematicians.

In this revised edition, two mathematical additions complementing the exposition in the original text are made. The new edition also contains several new footnotes, additional references, and historical comments.

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Solutions to Exercises in Math Textbooks

This is a question for mathematics instructors: How do you feel about having solutions available for the exercises in a math textbook? What if the solutions are available on the internet?

Some colleges and universities have guidelines for how instructors should treat the possibility of cheating on homework (e.g. looking for solutions on the web rather than working problems out). I taught at about five different colleges and universities. Four of them had honor codes, which gave the instructor the luxury of giving students their assignments and instructions and assuming that they complied honestly, whether they did or not. It was up to the administration to sort out egregious problems and the students’ own consciences to deal with mild ones.

My favorite tactic was to simply make homework count for only a small part of the grade, and place more emphasis on in-class tests and quizzes. The homework is useful for studying, students could  work together or work with a solution guide as they pleased.

And there is a third option: to give homework problems whose solutions are not available or very difficult to access online or in books. Is this the high ground approach, or is it simply impractical and too much trouble considering questionable benefits?

What do you think?  Should textbooks contain solutions to problems, or should the problems only be made available to instructors in a separate manual, or online accessible only by password?


Featured Book of the Day

Integers, Fractions and Arithmetic by Judith Sally and Paul Sally.  This book was co-published by the AMS and MSRI as part of a Math Circles Program for K-8 teachers.  The book consists of twelve interactive seminars, and gives a comprehensive and careful study of the fundamental topics of K–8 arithmetic. The guide aims to help teachers understand the mathematical foundations of number theory in order to strengthen and enrich their mathematics classes.

I would describe the approach to exercises in this book as: use very few and explain the solutions carefully and completely.

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Libraries and the art of browsing

Those of you over a certain age may remember when searching for math resources meant going to the library and perusing the subject catalog, spending time in shelving sections devoted to a topic, or leafing through heavy volumes of math reviews. Along the way, you found things you did not expect, leading to further trails and discoveries. You might remember musty smells, occasional hushed sounds, dark lighting, and the heft and texture of the volumes. There was a feeling of timelessness and escape from the real world in a library.

Today libraries seem quite different. Science and math libraries have well-lit, large spaces filled with tables and power outlets, conducive to sitting with a laptop and connecting to web-based resources with hardly a book in sight. Many hard copy books are available only by special order, and browsing is now almost completely digital.

What do you feel are the pros and cons of new library designs? How do today’s libraries affect the way you browse for books?

Your comments are welcome!

Featured Book of the Day

A Mathematical Gallery  by Lisl Gaal

This book started as a picture book by mathematician and artist Lisl Gaal for her children and grandchildren. The illustrations depict whimsical creatures and settings juxtaposed with simple yet far-reaching mathematical ideas that appeal to every age group. Readers are encouraged to explore and understand at their own level and pace. A child of any age could read and re-read this book for years, picking up new insights each time.

(Supplementary text is included for educators and advanced readers.)


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Author Interview: John Roe

John Roe studied with Michael Atiyah at Oxford, and his research has focused on the interaction of index theory and large scale or “coarse” geometry. After teaching at Oxford for twelve years he became Professor of Mathematics at Penn State in 1998. He has written a number of books, starting with Elliptic Operators, Topology and Asymptotic Methods in 1988 and including several published by the AMS. His most recent book, Winding Around, was published by the AMS in 2015, and he has also added several lecture note sets to the Open Math Notes project. Mathematics for Sustainability, a text for a quantitative reasoning course for non-mathematicians, is expected in 2018 from Springer.

What made you decide to write the book “Winding Around”? The spark for Winding Around was lit when I was about nine. My dad drew an incredibly convoluted simple closed curve (something like Figure 4.3 in the book),
made a dot on the paper somewhere in the midst of the convolutions, and asked me, “Is that inside or outside the curve?” I knew about maze puzzles so naturally enough I picked up my pencil and drew a path starting at his
dot, staying between the lines and heading, so I hoped, towards the exterior. After several minutes of wrong turns and entanglements I was finally able to announce, “Outside”. Then he did something I did not expect. He took a straightedge and drew a line directly from his point to the exterior. “But it crosses the curve”, I complained. He didn’t respond directly but just started at the exterior of the curve and marked off O, I, O, I…“outside”, “inside”, “outside”, “inside”, changing at every crossing until he arrived at the original dot. I saw at once what was going on and I have never forgotten that “aha” moment.

Wait, you had really been planning this book since the age of nine? That would be a good story, wouldn’t it? And it’s absolutely true that that experience lit a fire for me. Another source of inspiration was something I learned from Atiyah rather later in my mathematical career: that exciting things happen when different branches of the mathematical family – analysis, topology, geometry, algebra – are made to interact in significant ways. That story is usually told in higher dimensions, as pa
rt of a grad student’s research training, but it can also be told in dimension 2. There’s a beautiful expository paper of Atiyah from the 1960s where he reviews how these different branches of mathematics approach the winding number and then goes on to say, look, if you take the correct higher-dimensional generalization of all this, you will get Bott periodicity. It had been kicking about in the back of my mind for some years that you could build an undergraduate course on that paper and when I had the chance to teach in the MASS program in fall 2013, I decided to give it a try. Winding Around was the result – a book which is centered on the many different definitions of the winding number and the ways they interact.

Tell us more about MASS. This is a unique program that has been held at Penn State for about twenty years. MASS gets a class of twenty to thirty very good students – half from Penn State, half from other institutions across the US and the world – and puts them together in this high-intensity math environment for a full semester. They are focused entirely on mathematics (as Oxford students would be, for instance) and because of that and the strong peer group they learn very fast. It’s a great context for trying the kind of experiment that produced Winding Around.

Was there a gap in the literature that you were trying to fill? To get me energized to write, a necessary condition is the sense that “no-one has ever said these things in exactly this way before, and this is how they need to be said.” Of course that can work out differently in different contexts. For myfirst book it was just, “I wish someone could have put all this together for me when I was starting my thesis”. For Winding Around, it was more “I wish undergraduate students could see all these different kinds of mathematics engaging with each other”. Of course there are plenty of books about complex analysis or plane topology, but I couldn’t find one that gave the sense of deep interconnectedness that I’ve tried to convey.

Did you use existing notes from teaching? For several books but not all of them. As I mentioned, Winding Around comes from a course in the MASS program, for which I prepared detailed notes. Lectures on Coarse Geometry comes from notes of a graduate course. Mathematics for Sustainability is based on a course that I developed for our undergraduate program. In recent years I’ve developed a very specific set of personal procedures for preparing slides and notes (in TeX) for each course I teach. But that doesn’t make it magically easier to produce a book once you have finished!

Was your writing influenced by other books? Which ones? Early in my career I was greatly influenced by Jean Dieudonné’s Foundations of Modern Analysis, which of course is very much in the Bourbaki style – all numbered paragraphs and subparagraphs, and no concessions to “intuition” such as might be suggested by (gasp) a diagram! But later, through reading Milnor I think, and also through listening to Atiyah and his colleagues explain things, I’ve moved away from that style towards something more conversational. In general I would say that exposition has played a vital part in my mathematical life. I am always “explaining” things, even if it is only to myself. I feel that if you really can explain something
clearly, you’re quite likely to discover something new about it. I suppose it is also quite likely that you’ll end up writing a lot of books ☺

How did you decide on the format and style of the book? I wanted Winding Around published in the Student Mathematical Library (as it eventually was) because I had always envisaged it as something to put in the hands of bright final-year undergraduates. But I had to fight for that a bit. Some of the AMS’s reviewers (of the first draft) wanted the book in a graduate series, with one saying something like “the book needs readers who already understand real and complex analysis, measure theory, topology and abstract algebra”. As though all these exciting ingredients have to be carefully synthesized in isolation – in laboratory conditions – before the trainee chef can be allowed to combine them! I’d rather we get cooking, and clean up the mess as we go along.

Anyhow, the compromise that (AMS chief editor) Sergei Gelfand and I arrived at was to leave the main structure of the book as it was but to add a bunch of appendices, A through G, giving capsule developments of these various items just to the extent needed in the main text. I guess this is an example of the influence of Dieudonné, who did something similar with the linear algebra he needed for his book. My ideal reader will more or less ignore the appendices – pushing through the main text, being content perhaps that some things are a little mysterious, and referring forward to an appendix only when mystery has accumulated so much as to impede progress. I wonder if this is how the book is actually read?

What next? Mathematics for Sustainability, out next year, is likely to be my last book. This is quite a departure from my previous works, both in terms of audience and content, but once again feels to me like something that has to be said. I’ve long felt that we mathematicians owe more than we presently offer to the thousands of students who take our pre-calculus courses simply to fulfil a ‘breadth’ or ‘general education’ requirement – and that we should take the opportunity, across our curriculum, to connect what we do with big challenges like climate change. Kaper and Engler’s Mathematics and Climate aims to do this at the graduate level. My co-authors and I are trying for the same connection, assuming nothing but high school algebra. It’s a tall order, but one I am very excited about!

What advice would you give to new authors? Books are magic. Is there a story that only you can tell, or tell right? Do you have the time for a long project and the discipline to add a little more each day, even when the end seems far off? Is this the right point in your career, and is your institution enlightened enough to value your work on a book appropriately? Yes, yes and yes? Go ahead and add to the magic – and good luck!

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Author Interview: Jennifer Schultens

Jennifer Schultens is Professor of Mathematics at University of California, Davis.  Her book Introduction to 3-Manifolds guides beginning graduate students through the foundations of low-dimensional topology to specialized topics such as triangulations of 3-manifolds, normal surface theory and Heegaard splittings.  A fundamental challenge to building up the language of low-dimensional topology is to connect mathematical rigor with geometric and topological visualization.  In her book, Jennifer Schultens does this seamlessly, melding clear exposition with lots of intuition-building examples, illustrations and exercises.

Question:  What made you decide to write the book? Was there a gap in the literature you were trying to fill? Did you use existing notes from teaching?

While in the process of guiding graduate students through the basics of understanding 3-manifolds, I often wished for a reference containing the knowledge that first opened my eyes to the beauty of the subject. The subject had grown and flourished since the publication of the books that I had read as a novice. I especially remember one conversation, probably in the early 2000s, with Aaron Abrams and Saul Schleimer, concerning the need for more current books on the topic of 3-manifolds. The curve complex, rapidly emerging as a central tool in the study of 3-manifolds, needed to be added to the standard repertoire of a 3-manifold topologist. As it happened, I had the opportunity in the Spring of 2003 to teach a course on 3-manifolds to a group of highly motivated graduate students at Emory University. In this course I described the subjects near and dear to my heart. I recorded my lectures in a rather terse set of notes. Over the next 10 years, these notes grew into a book.

Question: How did you decide on the format and style of the book?

I treasure the traditional mathematical style of writing: definitions, theorems, proofs. As a topologist I also find illustrations indispensable. Today’s technology, most notably LaTeX, xfig and AMS style files provide for easy typesetting. Definitions, theorems, proofs and illustrations constitute the skeleton of my book. However, the life of the book derives from the knowledge verbally passed around among low-dimensional topologists that I incorporated into the text, often informally. For instance, I included a proof of the Schoenflies Theorem given in a lecture series by Andrew Casson in China in 2002, but not otherwise in the literature. I did not attend the lectures myself, the lecture notes had been given to me by Yoav Moriah.

Question: What was the writing process like? Did you write everyday on a set schedule, or did you have periods of setting it aside?

I started out with a set of notes recorded during a course I taught in Spring of 2003. In 2006 I spoke with Sergei Gelfand who encouraged me to turn the notes into a book. Most of the writing occurred during four bouts of productivity: Summer and Fall of 2006 at the Max-Planck-Institute for Mathematics in Bonn, Germany; Spring and Summer of 2008 as I grew ever heavier during my pregnancy; Fall of 2010; and Summer and Fall of 2013, again at the Max-Planck-Institute. I found the fallow periods, the months and years when I did not think about the book, indispensable to the maturation of the project.

Question: What did you focus on the most when writing? What was the most challenging aspect? What came easily?

I focused on my vision of the subject and an imaginary reader, either a graduate student or a well known colleague, reading the book. I tried to include all the background material necessary to understand the discussion. Occasionally, I got overwhelmed by the amount of background material still needed.

Question: What were the positives and negatives of the experience? Did anything about the experience surprise you?

Looking up and (re)familiarizing myself with references proved more time consuming than I had imagined. MathSciNet, developed by the AMS, proved indispensable in the process. Through this type of diligence, I learned more about the subjects being exposited and the people involved. The book gained more depth.

Question: How did you choose a publisher? What was important to you when you made the choice?

After speaking with Sergei Gelfand in 2006, I realized that publishing with the AMS meant that the book would be in good hands. Naturally, the AMS looks after the professional interests of mathematicians. In addition, the AMS has an excellent track record regarding publishing at fair prices.

Question: Was your writing influenced by other books? Which ones?

I enjoy reading. Fiction or non-fiction, classical or modern, formal or informal, short or long, I enjoy a well-crafted piece of writing. The first mathematical text that really ‘grabbed’ me was Walter Rudin’s “Principles of Mathematical Analysis” and later his “Real and Complex Analysis.” The topology courses at UC Santa Barbara teemed with good literature: “Topology” by Munkres, “Differential Topology” by Guillemin and Pollack (also “Topology from a Differential Viewpoint” by John Milnor from which Guillemin and Pollack’s book is derived), the books on 3-manifolds by Hempel and Jaco which to me are inseparable from their interpretation by Cooper, Long and Scharlemann. Then there was Rolfsen, not just a book, but an experience. Working one’s way through Rolfsen’s “Knots and Links,” rediscovering so many of the beautiful constructions in knot theory, was a rite of passage for low-dimensional topologists of my generation. Later, I thoroughly enjoyed Silvio Levy’s digest of Thurston’s Lecture Notes and Allen Hatcher’s books.

Question: Did you find ways to get feedback while writing your book or was it a solitary effort?

My writing tends to be introspective. However, my husband, Misha Kapovich, proofread many parts of the book. His feedback helped fill in background information on several subjects, especially the final chapter of the book. This increased the time it took to complete the book, but added depth. It certainly improved the quality of the book.

Question: Did time pressure or other responsibilities help or hurt your writing?

Being an academic involves many types of activities, opportunities and responsibilities. My writing tends to be introspective. Sabbaticals are indispensable to my work on larger projects. However, the busy times, filled with teaching, attending lectures, working on committees, taking care of family, fill my thoughts in a way that informs the quieter periods during which I write.

Question: What kind of feedback did you get after the book came out?

Friends from far and near wrote to tell me that they enjoyed my book. It was nice to (re)connect. Of course, there were also some corrections. I am happy to report that the AMS maintains a web page where corrections are easily posted.

Question: What advice would you give to new authors?

Write about the things you love in a style that suits you.

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Open Math Notes: Free as in beer.

Recently, the phrase “free as in beer vs. free as in speech” caught my attention. It was the first I had heard of this way of distinguishing two English meanings of free, and how it particularly applies to what is accessible on the internet. The explanation I found is here.

The new AMS Open Math Notes site is definitely “free as in beer.” There is no cost to users who browse, upload, or download Notes. It is a place for Notes that are, for a variety of reasons, not published (typically because they are in unpolished form but are at a stage where they can benefit from public view). Notes are submitted by authors and vetted by AMS editors and advisory board members on the basis of completeness, readability, and interest to students, teachers, and researchers in mathematics.

So what about “free as in speech”? Users may only download PDF files for free, but they may not freely change and then redistribute the contents. Authors who post on AMS Open Math Notes retain all rights to their Notes. Of course, they may make their source files freely available on their own webpages or on other free posting sites. (If an author decides to “publish” the Notes commercially, self-publishing or otherwise, then they must withdraw their Notes from the AMS site.)

Let us know what you think of the “free as in beer” vs. “free as in speech” dichotomy for Open Math Notes — examples of both types of posting sites exist on the web. What do you feel are the pros and cons?

Featured Book of the Day

stml-76-covWinding Around: the Winding Number in Topology, Geometry, and Analysis,  by John Roe

This book is a popular item in the AMS Student Math Library Series, a series dedicated to non-standard math topics accessible to advanced and/or extra curious undergraduates. The book takes students on a tour of a range of advanced topics in mathematics as it displays the far-reaching consequences of the concept of winding number.

The author John Roe has also posted several lecture notes on the AMS Open Math Notes site.


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