On June 10^{th}, the AMS closed for “business as usual” in order to participate in #ShutDownSTEM. AMS Governance spent much of the day building an action plan to account for our own Society’s history of racist behavior and to address inequities in our mathematics community, while much of the staff used the day for self-education and self-reflection. There were several “office hours” with executives, as well, which I, relatively new to both the AMS and the discipline of mathematics, found very useful and informative as I learned a great deal about the history of the AMS that the organization continues to reckon with.

This blog is maintained by the Publications team here at the AMS: the marketers, editors, graphic designers, and a host of others who bring you great content and help to make it easy to find. We are content people; we love books, movies, podcasts, journals, and magazines. This is why many of us work in publishing.

Our team found June 10^{th} to be a valuable opportunity to engage with content and reflect on ways we can tackle systemic racism. For some of us, it sounded like a luxurious opportunity: “I will be paid to read all day? To watch that documentary I have wanted to see? To engage with my children about this content and these issues?” And while it was generous of the AMS to give us the time and space to do those things, we all found it energizing, distressing, challenging, gut-wrenching, but not definitely not luxurious.

We wanted to share the resources we found useful. We hope you will, as well.

This tribute by Stephen Kennedy (Carleton College), AMS/MAA Press Acquisitions, originally appeared in the most recent issue of MAA Focus and is shared with permission.

The news of Richard Guy’s passing was a blow. Not only because he was a dear friend, but also because I knew that the appearance of his last book, The Unity of Combinatorics, was imminent and that he would never see it. When I first met Richard decades ago I was too much in awe of him to actually talk, we had a nod-and-smile relationship for a long time. That changed about 15 years ago. I was sitting at an airport gate leaving JMM to come home and Richard in his familiar brown tweed jacket with his ever-present Peace is a Disarming Concept lapel button sat down next to me and asked about the math on the pad of paper in my lap. At the time I had just discovered Geometer’s Sketchpad and was using its capability to combine Euclidean geometry and motion to generate undergraduate research problems, questions like: What’s the locus of centroids of all the triangles that share a circumcircle? With Geometer’s Sketchpad you could make a little movie and observe that locus being generated in real time. It was thrilling to watch.

I don’t remember exactly what problem I was struggling with at that airport gate but it was something close to the above and Richard listened thoughtfully and we spent an hour swapping ideas and pictures. It was clear that he knew about a thousand times as much about geometry as I did and also clear that his brain worked at about twice the speed mine did. But my awe melted away in the face of his kindness and modesty. He was genuinely interested in my ideas and in working together on the problem. He also had a razor-sharp wit and after one of his jokes would flash his disarming, but devilish, grin. It was great fun to do mathematics with him. Eventually he started telling me about the lighthouse problem [2]: What is the locus of the point of intersection of two rotating lighthouse beams? The cited paper is a great place to go to understand Richard’s approach to mathematics and to experience his sense of humor. For another quick taste of the latter, check out the MAA Review of The Inquisitive Problem Solver by Richard’s alter ego, Dick Fellow.

When I got home I had an e-mail waiting from Richard with some more ideas about my problem. We continued that e-mail correspondence for a while. He always did me the kindness of pretending that I was knowledgeable about geometry; I think it was enough for him that I clearly loved it. A few years later I was in Calgary visiting Richard to talk about a possible book on combinatorial games. I spent a week with him, every morning we’d go to his office at the University of Calgary. He taught me about Sprague-Grundy theory and we analyzed dozens of games together. Every evening we’d go back to his home and eat one of the dreadful frozen pot pies he favored for dinner, then get back to work. For a time I thought I could understand three-car Dodgerydoo, Richard did me the courtesy of taking seriously the possibility that I did. (Of course, I didn’t. I think he probably suspected as much all along but was too polite to say so.) We never got the book put together. In spite of that, it was one of the best mathematical weeks in my life.

The Unity of Combinatorics is the latest volume in the MAA Carus series and its genesis was a paper by that name that Richard published in 1995. Richard was reacting to the perception that combinatorics was nothing more than a bag of disconnected clever tricks for toy problems. It is clear today that combinatorics is a mature mathematical discipline with deep problems, subtle results, and intriguing connections to other areas of mathematics. Twenty-five years ago that was not clear and combinatorics’s connection to recreational mathematics made it seem slightly disreputable and frivolous. This book was first imagined by Don Albers who encouraged Richard to expand his article and recruited Bud Brown as a co-author. The result reflects both authors’ personalities, their mathematical interests and their beguiling expository skills. It’s a pure pleasure to read; the perfect mixture of Richard’s gentle wit, Bud’s down-home, welcoming enthusiasm, and both authors’ deep knowledge of, and absolute joy in, the combinatorial landscape.

Let me give you a taste. Suppose you want to find a collection of five-element subsets of the eleven-element set $\{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, X \}$ with the property that each pair of elements occurs together exactly twice. It’s not obvious, at least to me, that such a collection is even possible. A quick count— each of 55 pairs occurring twice is 110 pairs, a five-element set contains ten pairs—will tell you that any such collection will contain 11 sets. But that’s no help in finding it, or even proving it’s possible. It just reassures you that it is not obviously impossible. The Brown-Guy example is given in Table 1, but you’re encouraged to try and construct your own example before peeking.

It is also not obvious why you want to do this. The respectable answer is that it is an example of an $(11,5,2)$ symmetric block design, objects that arose in the design of statistical experiments in agriculture. (The parameters correspond to the bolded numbers in the previous paragraph.) The frivolous answer points to the obvious analogy with Kirkman’s Schoolgirl Problem. You are of course wondering which values of $(v, k, \lambda)$ actually correspond to achievable symmetric designs. You should read Chapter 7. I’m more interested in following up on $(11,5,2)$ right now. Brown and Guy call this gadget a biplane. It is worthwhile to understand why.

Suppose that instead of requiring each pair of elements to occur twice we will be satisfied with a single appearance. As noted above there are 55 pairs and each five-tuple contains ten, so $(11,5,1)$ fails the obvious divisibility test and no such object exists. But, to take one example, $(91, 10, 1)$ does not fail and, so, is not obviously impossible. (Those numbers are a clue to what’s happening, but you might not recognize that.) If we think of the elements as points, we are looking for ten-point subsets such that each pair of points is in exactly one subset. Replace “subset” by “line” and you recognize the description of a finite projective plane of order nine. Thus, the $(11, 5, 2)$ biplane. More saliently, perhaps you begin to see Brown-Guy’s “Unity.”

Brown in [1] asked himself how he might draw a useful picture of the $(11, 5, 2)$ biplane. He wanted the picture to reflect some of the symmetries of the design. For example, note that the product of the two five-cycles $(1, 3, 9, 5, 4)(2, 6, 7, X, 8)$ is a permutation of our original set of order five. Note that it preserves the block structure, e.g., 2456X goes to 61478. This corresponds to the five-fold rotational symmetry in the figure. In fact, as Brown and Guy show, the symmetry group of the biplane actually has order 660 and can be shown to be $PSL(2, 11)$. Many of these symmetries can be seen directly in Figure 1.

One final unification observation. It’s interesting to notice that $2^{11}=1+\binom{23}{1}+\binom{23}{2}+\binom{23}{3}$. It is known that this equality is exactly what is required for the existence of a perfect three-error-correcting binary code. The alphabet has 23 symbols and the codewords are length 12. Similarly, the fact that $1+2\cdot\binom{11}{1}+2^2\cdot \binom{11}{2}=3^5$ means that there exists a perfect two-error-correcting ternary code. In this case the alphabet has 11 letters and the codewords have length six. Each of these codes can be realized as the row space of a particular matrix. Suppose one were to construct the $11\times 11$ incidence matrix for the $(11,5,2)$ biplane by putting a 1 in the $(i,j)$ entry if element $i$ is in subset $j$ of the biplane and a 0 if not. (NB: The subsets are indexed by their first listed element in Table 1.) This incidence matrix lives inside the code matrix as a submatrix in the case of each of those codes. You are invited to explore why.

All of the above was taken from just one chapter of Brown-Guy, and we have already run into statistics, group theory, linear algebra, coding theory, recreational mathematics, and projective geometry. Perhaps The Ubiquity of Combinatorics would have been a better title. Whatever we call it, it is full of wonders and breathes with Richard’s spirit. It is a fitting memorial to a mathematical giant whom we were lucky to have for 103 years.

******************

[1] Ezra Brown, The Fabulous (11, 5, 2) Biplane, Math. Mag., 77:2,
87–100, 2004, DOI: 10.1080/0025570x.2004.11953234.
[2] Richard K. Guy, The Lighthouse Theorem, Morley & Malfatti—
A Budget of Paradoxes, Amer. Math. Monthly, 114:2,
97–141, 2007, DOI: 10.1080/00029890.2007.11920398.

This post by Robert Harington originally appeared on The Scholarly Kitchen blog; access the original post here. Robert Harington is Associate Executive Director, Publishing at the American Mathematical Society (AMS). Robert has the overall responsibility for publishing at the AMS, including books, journals and electronic products.

Back in 2017, I penned a post for The Scholarly Kitchen entitled “The Value of Copyright: A Publisher’s Perspective“. We are now in 2020, hunkering down in isolation, working remotely. As we ride out these difficult times, we can’t help but look ahead and consider what a post-COVID-19 publishing landscape will look like. In this article, I want to revisit the history of copyright, steering into Creative Commons Licensing, and weigh the value of protection and reuse in light of an inexorable push towards global openness. There is value in publishing in an open setting, but do we fully understand how openness will stimulate or hinder creation and expression of ideas? Publishers, and indeed all players in the publishing ecosystem, have not moved far in helping our communities understand the rights and licensing landscape. On the one hand, authors are mainly concerned with disseminating their research and doing so in a way that maximizes use and citation. On the other hand, authors be they authors of journal articles or books, may find their content repurposed in ways they did not expect by publishers they did not sign on with. I am deliberately not stepping into the treacherous waters of whether publishers pay royalties for differing kinds of content to authors. The issue I address is how to equip authors to be able to ask the right questions, and sign up to be published knowing how their content will be treated. An author’s bandwidth to consider complexities of licensing and rights associated with their publishing output is limited. However, it is important that authors grapple with such complexities, as their ability to create may rest on being able to navigate the right path for publicizing their research and communicating their ideas.

Copyright law is complex and varies greatly across countries – one of the main reasons that authors do not grapple with its complexities. Here I am referring specifically to American copyright law, though of course such law was established when the printing press was introduced to England in the late fifteenth century. Printing presses were firmly in control, and the Licensing Act of 1662 cemented what was effectively their ability to censor publications. By 1710, England’s parliament enacted the Statute of Anne, which established principles of how authors may own rights to their work – copyright. It set a fixed term of 14 years for protection of an author’s work, which could be renewed if the author was still alive when the first 14-year period expired. Copyright law continued to evolve, although you will no doubt be grateful that I will spare you the details of all of its extensions.

However, some key innovations are worth examining. For example, the Berne Convention, which came into being in 1886 and was signed by the US in 1989. The notion was to place the US approach to copyright in context of a broader international approach. Effectively, it recognized that there is a myriad of approaches to copyright laws across the world, which to this day confuses authors and publishers alike, given the global nature of research.

Copyright case law is convoluted: hundreds of important cases up to the present day further refine copyright law. One significant American law is the US Copyright Act of 1976, which substantially revised the Copyright Act of 1909 and essentially provided protection for all authors’ works between 1978 and the present day. Why does this matter? Well, it was the first real recognition that an individual’s work is worth something and needs to be protected, steering away from bookseller and printer monopolies.

A question we have not addressed is, why develop copyright at all? The notion behind copyright law is that authors are more able to express their creative ideas in the arts and sciences if they are protected through ownership of their work, by establishing rights that prevent unauthorized use of content. Too few people recognize that academic product – that is, ideas and knowledge – is the result of considerable hard work, work that should be recognized through attribution at least. An author can look to copyright law to help prevent others from re-purposing their work inappropriately, or altering it to say something different and republishing it under their name. Copyright protects the integrity of the research. This is a primary concern for humanities authors, where the argument is the result. They don’t want anyone else changing their carefully chosen words.

This, in turn, allows authors to benefit financially from publishing their work. I do find this notion appealing, and I sense that in the rush to demonize copyright law in the publishing industry, it is often easy to forget that copyright is indeed in force to protect authors themselves, not so much the publishers.

Let’s fast forward to the present day and take a look at how we have evolved into a world where Creative Commons Licensing is the new normal. Creative Commons Licenses were a successful venture into allowing authors to retain copyright, and allow for publication of their work through licenses that allow for reuse. These licenses come in a variety of flavors and courtesy of the Creative Commons organization, I list them here in their full complex glory, with Creative Commons’ short summary descriptions:

ATTRIBUTION: CC BY

This license lets others distribute, remix, adapt, and build upon your work, even commercially, as long as they credit you for the original creation. This is the most accommodating of licenses offered. Recommended for maximum dissemination and use of licensed materials.

ATTRIBUTION-SHAREALIKE: CC BY-SA

This license lets others remix, adapt, and build upon your work even for commercial purposes, as long as they credit you and license their new creations under the identical terms. This license is often compared to “copyleft” free and open source software licenses. All new works based on yours will carry the same license, so any derivatives will also allow commercial use. This is the license used by Wikipedia, and is recommended for materials that would benefit from incorporating content from Wikipedia and similarly licensed projects.

ATTRIBUTION-NODERIVS: CC BY-ND

This license lets others reuse the work for any purpose, including commercially; however, it cannot be shared with others in adapted form, and credit must be provided to you.

ATTRIBUTION-NONCOMMERCIAL: CC BY-NC

This license lets others remix, adapt, and build upon your work non-commercially, and although their new works must also acknowledge you and be non-commercial, they don’t have to license their derivative works on the same terms.

ATTRIBUTION-NONCOMMERCIAL-SHAREALIKE: CC BY-NC-SA

This license lets others remix, adapt, and build upon your work non-commercially, as long as they credit you and license their new creations under the identical terms.

ATTRIBUTION-NONCOMMERCIAL-NODERIVS: CC BY-NC-ND

This license is the most restrictive of our six main licenses, only allowing others to download your works and share them with others as long as they credit you, but they can’t change them in any way or use them commercially.

What are the pros and cons for authors thinking about using Creative Commons Licenses? Advantages to Creative Commons Licenses lie mainly in the open nature of an author’s content, with the license allowing for reuse under conditions set by the license used. The con to using these licenses is in the lack of control an author may have over their content. A reuse of content can’t be revoked under a Creative Commons License once granted – unlike removing copyright permissions. Also, authors need to be very careful about which license they use. If an author is adamant that they do not want their work used by another party to make money, then they need to know to use CC BY-NC. If an author does not want their work to be the basis of a derivative work then ND comes into play and so on. Even with CC BY, where all that is asked is correct attribution in reuse, who amongst our authors knows how to follow up on abuses – and abuses are common. Even if attribution is given, an author may not initially realize to what use their content may be put. For examples of this you can turn to Rick Anderson’s excellent article of 2014 entitled CC-BY, Copyright and Stolen Advocacy, For example:

Last year we saw a troubling (if far less repugnant) example of how something like this can happen in the academic realm. Apple Academics Press published a book titled Epigenetics, Environment, and Genes. The book was comprised almost entirely of articles taken, without their authors’ permission, from OA journals in which they had been published under CC-BY licenses. It is now being sold on Amazon for just over $100. Although members of the scholarly community have responded with outrage, Apple Academic Press has done nothing illegal or even unethical. As long as the authors of the articles are given due credit, this kind of reuse is one of the many that are explicitly allowed under CC-BY terms. If the authors feel mistreated by Apple Academic, it’s because they failed to read (or understand) the agreements they signed when they submitted their articles for publication in OA outlets.

Joe Esposito pointed out in his article of 2019, entitled “Internal Contradictions with Open Access Books“, that publishers themselves do not always understand, or compute the implications of a CC BY license, and can get quite upset when content they have published is reused quite legally.

An author essentially has to ask themselves how important control is to them when publishing their content. As it stands, publishers, and in fact all stakeholders in the publishing ecosystem, do a poor job of explaining how to navigate these questions.

The issue for many authors here is that publishers, institutions, and funders are not making it clear what their licensing and copyright expectations are of authors. A journal may require copyright transfer, but an author’s institution may require use of a Creative Commons License. How does an author resolve such a quandary? (If, indeed, they are even aware of it?) Publishers perhaps are not helping here, as we do very little to explain how rights issues may affect authors as they publish their content. An author deserves to be able to make an informed choice to publish based on the rights they want associated with their content, and to do that they need help understanding their rights.

Some of you have likely been readers of the BookEnds blog by AMS Consulting Editor, Eriko Hironaka. We have made the decision to broaden the scope a bit to provide a wider look at the publishing activity of the AMS. BookEnds will remain a key part of this endeavor in this new “home” (as a content category) and we will also occasionally “borrow” from Beyond Reviews, the MathSciNet blog by Edward Dunne, Executive Editor of Mathematical Reviews. We also plan to highlight content, authors, and editors from our journals program, and to look at issues and changes in the world of mathematics publishing.

Who will be blogging?

Eriko Hironaka will continue to contribute the occasional BookEnds post and she will also be joined by her Editorial colleagues from time-to-time. Other contributors will include Nicola Poser, Director of Marketing and Sales, Eric Maki, Senior Marketing Manager, and Robert Harington, Associate Executive Director for Publishing. We also hope to have guest contributions from our authors and Editorial Board members, and from our readers – please feel free to be in touch if you have an idea for a guest post! You can reach Nicola at nsp@ams.org.

Why a blog about AMS Publications? Why now?

In this time of the COVID-19 pandemic, communication is more important than ever. At AMS Publications, we hope to maintain open communication with all our stakeholders: authors, readers, teachers, students, librarians. Our hope is that this blog will become a place you find useful information. Read on: the next post gives an overview of some important information from AMS Publications related to access to our content.

In response to current challenges that colleges and universities face as a result of the spread of COVID-19, the American Mathematical Society is offering our community additional support in line with recommendations in the International Coalition of Library Consortia (ICOLC) Statement on the Global COVID-19 Pandemic and Its Impact on Library Services and Resources.

For libraries:

We are extending grace access for content hosted on our platforms (including MathSciNet) through the end of May for our existing customers. We will re-evaluate this timing as needed.

In normal circumstances, remote access and mobile pairing for access to AMS content can be set up while on campus or while connected via institution VPN (in order to validate IP-based access). We realize many students, faculty, and researchers did not have an opportunity to initiate this access before leaving campus and we reached out recently with a unique link and instructions for our library partners on how patrons can connect to our content. If you are setting up online access for the first time or have not received instructions to share with your patrons, please email us at cust-serv@ams.org.

We are aware that many institutions are closed globally. If you subscribe to our journals in print and wish to suspend delivery of print journals, please contact cust-serv@ams.org.

All of our print journal subscriptions include complimentary online access. If your library has not yet activated online access and you would like to do so, please complete the license agreement and send it to cust-serv@ams.org.

For teachers, students, and researchers:

As courses transition to online, we can provide instructors with complimentary electronic “reserve” copies of our textbooks for cases in which students do not have access to their print copies. Please visit the book page for your textbook on bookstore.ams.org and use the “request desk copy” link to request an e-copy that can be posted to a course website, course management system, or place on e-reserve.

E-books purchased through the perpetual access model on the AMS platform are always available DRM-free with unlimited simultaneous use. In addition, we are partnering with ProQuest to allow multi-user access through mid-June to all e-books purchased on their platforms. Read ProQuest’s statement.

We have also made relevant mathematical modeling content available free-of-charge; please visit this page for more information.

We also offer freely available content for teaching at Open Math Notesa repository of freely downloadable mathematical works in progress. These draft works include course notes, textbooks, and research expositions in progress. They have not been published elsewhere, and, as works in progress, are subject to significant revision. Visitors are encouraged to download and use these materials as teaching and research aids, and to send constructive comments and suggestions to the authors.

We are providing remote access and mobile pairing for access to all our content online, including MathSciNet. In normal circumstances, this remote access can be set up while on campus or while connected via institution VPN (in order to validate IP-based access). We realize many students, faculty, and researchers did not have an opportunity to initiate this access before leaving campus, so we have given instructions to our library partners on how patrons can connect to our content. Please contact your librarian for assistance.

Finally, our colleague, Abbe Herzig, Director of Education at the AMS, has compiled useful resources and practical strategies for transitioning to online teach, which you can find here.

We hope these resources and policy updates will be helpful. Please do not hesitate to reach out to us if you need something as we all adjust together to this “new normal.”

Do we really need textbooks? In this age of swelling enrolments in undergraduate math classes, students with diverse interests and backgrounds, new modes of teaching, and alternative media, are textbooks too rigid? Are they too expensive? Would it be better for department faculty to write-up specially tailored notes for their students to download for free?

As a teacher, I see two important reasons for textbooks in academia. The first, mundanely, is time. Faculty members are busy, and it doesn’t make sense to reinvent the wheel for each new course or move to a new institution. The second deeper reason is orientation. Though there should always be room in teaching for variation and individualization, it is also handy to have a few universally recognized reference points from which to measure knowledge in a subject. For students, books help to give structure to their study, and a way to reference the material in later years. Every now and then a textbook will be so good at capturing how the mathematical world sees a subject that it becomes “the canonical textbook”, a sign-post.

In these days with so many sources of information, maybe the role of textbooks is less clear. Maybe we are preparing for a jump in the evolution, similar to the jumps from oral tradition, to scribing, and on to mass publications. In the current system, a professionally produced textbook has a panel of reviewers to decide whether a book meets high standards of academic rigor and language, and has the necessary scope for its purpose. A variety of specialists put care in copy-editing, lay out, packaging and marketing the book to its intended audience. All these additions to the value of the book incur costs. Even for a non-profit publisher like the AMS, the expenses entailed lead to prices that can seem high when so much information is available to the public for free.

If the jump in the evolutionary process is leading to a brand new form of “book”, we have not yet seen a consensus on what it should look like. What will be the new landmarks in mathematical history? What are your thoughts?

This collection of 39 short stories gives the reader a unique opportunity to take a look at the scientific philosophy of Vladimir Arnold, one of the most original contemporary researchers. Topics of the stories included range from astronomy, to mirages, to motion of glaciers, to geometry of mirrors and beyond. In each case Arnold’s explanation is both deep and simple, which makes the book interesting and accessible to an extremely broad readership. Original illustrations hand drawn by the author help the reader to further understand and appreciate Arnold’s view on the relationship between mathematics and science.

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.

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.

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 illustratedtheoryofnumbers.com. 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.

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.

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.

The opinions expressed on this blog are the views of the writer(s) and do not necessarily reflect the views and opinions of the American Mathematical Society.

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