Students often struggle to visualize concepts in multivariable calculus. While technology can help, virtual surfaces can still seem out of reach. Aaron Wangberg of Winona State University describes how this team created dry erase surfaces and supporting materials to enhance students’ interpretation of the mathematical objects they were studying.

**NSF grants are competitive. What do you think it is that set your proposal apart and got the project funded?**

We created contextualized activities with concrete manipulatives to deepen students’ understanding of key concepts in multivariable calculus. There was a clear end goal with a concrete plan to create materials that could be widely disseminated for a broad impact. We had prototypes of the products and promised to disseminate the final materials to 30 institutions. Our proposal included letters of support and names of participants from 20 institutions.

**Where did you get the idea for this project?**

Teaching multivariable calculus, I realized students could use the formula for the gradient, but couldn’t tell me what was important about the resulting vector.

I spent hours taping together bowls and using paper-mache or plaster-of-paris to make surfaces my students could manipulate while exploring partial derivatives and the gradient. When I used these surfaces in class and asked students where the gradient pointed, several groups listed their vector components until one group said “It points uphill!” This spurred a lot of activity as other groups double checked their results on different surfaces. When I asked students to re-measure partial derivatives and re-form the vector after they’d rotated the surface, there was another ‘aha!’ moment when they realized the gradient vector stayed the same relative to the surface even though the numbers had all changed.

I realized then the importance of hands-on manipulatives in helping students identify the key features of a gradient vector. I wanted other students to have the same experience and other instructors to have available the resources I had painstakingly created by hand.

**How did the project evolve from taped bowls and paper-mache to the sophisticated models you have now? **

Ben Johnson, a practicing mold-maker and my former student, advised me how to produce smoother surfaces. Over the course of a year, we developed surfaces that were carved with a CNC machine out of blocks of wood and then finished with white surface that acted like a dry-erase finish. Ben and I also developed several activities exploring multivariable calculus ideas: level curves, partial derivatives, and constrained optimization (Lagrange multipliers). He presented the project at the Joint Mathematics Meetings.

I tried the materials in my multivariable calculus course in order to introduce new concepts and found students were exploring connections between ideas that we wouldn’t typically cover until weeks later.

**What inspired you to apply to this grant program?**

I knew the materials were in high demand. Every time I spoke at a conference, instructors would ask how to get the materials. I had worked with three composite engineering students at Winona State University in order to find plastic materials and a process for turning the wood models into clear plastic dry-erasable surfaces, and then the TUES program provided funding equipment to turn that theoretical process into a concrete, cost-effective application.

Since my graduate advisors had used NSF funding to develop instructional materials for vector calculus and physics, I decided to apply here for funding to expand this project’s impact to where it is today.

**Tell us about the impact this project is having. **

The materials and activities have been created to help students develop productive understandings of multivariable functions, derivatives, and integrals. In the first year of implementation, the materials have been used in over 40 courses with over 1000 students.

Because students don’t have the formulas for the surfaces, they instead have to reason using context and geometric relationships. Outside of the activities, instructors are noting how students are participating in their small groups in class when they use the materials and also noting how students are contributing ideas to the classroom.

One instructor, who used the materials in one of two multivariable calculus courses, noted how students in the course that used the materials were much more inquisitive and asked questions during non-surface days. The materials are impacting teaching styles as well, encouraging discussion and inquiry.

*Are you interested in using these surfaces? Check out their project website **https://raisingcalculus.winona.edu/** for more information and access to resources. *

Editor’s note: Q&A responses have been edited for length and clarity.

**Learn more about NSF DUE #1246094**

Full Project Name: Raising Calculus to the Surface

NSF Abstract Link: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1246094

Project Website: https://raisingcalculus.winona.edu/

Project Contact: Aaron Wangberg, awangberg@winona.edu, Principal Investigator

*For more information on any of these programs, follow the links, and follow these** blog posts**. This blog is a project of the Mathematical Association of America, produced with financial support of NSF DUE Grant #1626337.*

*Audrey Malagon is lead editor of DUE Point and a Batten Associate Professor of Mathematics at Virginia Wesleyan University with research interests in inquiry based and active learning, election security, and Lie algebras. Find her on Twitter** @malagonmath**. *

Examining a first edition copy of Pacioli’s book *Summa *(1494)

What’s your reaction when you see the term “double-entry book-keeping”? Do you associate it with cool, societal-changing innovations like the Internet, Google, social media, laptops, and smartphones? Probably not. Neither did I—until I was asked to write a brief article about the fifteenth century Italian mathematician Luca Pacioli, to go into the sale catalog for the upcoming (June) Christie’s auction of an original first edition of his famous book *Summa de arithmetica, geometria, proportioni et proportionalita *(“Summary of arithmetic, geometry, proportions and proportionality”), published in 1494, which I referred to in last month’s column. (I also gave a talk at a public showing Christie’s organized in San Francisco on April 24, which gave me an opportunity to examine the book myself.)

Sure, I knew what double-entry book-keeping was. Indeed, I spent several days in March going through my QuickBook records as I prepared my annual tax filing. Though by no stretch of the imagination am I in the big-income category, my tax-filing situation is not simple. I have several sources of income from around the world—from my university position and my ed tech startup company BrainQuake; fees from writing, speaking, and consulting; royalties from books; and more recently income from a number of pensions and annuities. As a result, I long ago started keeping meticulous records, using Excel spreadsheets to keep track of individual activities and QuickBooks to bring it all together. Excel is a digital implementation of ninth century commercial arithmetic and algebra, as laid out by al-Khwārizmī in his two famous books on the subjects; QuickBooks an implementation of the book-keeping methods described by Pacioli in Chapter 9 of *Summa*.

Having spent time studying both al-Khwārizmī and Leonardo of Pisa (a.k.a.* *Fibonacci) while I was researching my three books on Leonardo and his hugely influential mathematics text *Liber abbaci* (the project began with a single book in mind), I had long ago come to appreciate the magnitude of the mathematical developments those two authors had catalogued—and thereby contributed to—in terms both of human intellectual progress and the impact of those methods on the way people around the world go about their daily lives. Double-entry book-keeping, on the other hand, the one topic Pacioli covered that those previous authors had not, simply never caught my attention. It’s just keeping records, right? What’s the big deal?

Given my experience with al-Khwārizmī and Leonardo, I should have known better. But there is a reason why people hardly ever give any thought to just how revolutionary, in their time, were numbers (and the associated innovation of money), arithmetic, the Hindu-Arabic representation, the classical arithmetic algorithms, and algebra. Each of those innovations changed human life in such fundamental ways that, once humanity had them we incorporated them (and the products and activities they brought in) into our daily lives to such an extent that we no longer gave them any more thought. Their fundamental role became no more remarkable than the presence of air and water.

The same is true of more recent innovations, such as radio, the telephone, computers, the Internet, laptops, and mobile phones. To those of us who lived through at least some of those innovations, we still think of them as life-changing developments. But ask anyone of school age and it is clear that to them, all of those technologies are just part of the everyday environment. Nothing remarkable. It is a measure of the greatness of any innovation that completely transforms the way we live, that before long we no longer recognize how profound and remarkable it is.

Time, then, to take a fresh look at double-entry bookkeeping.

The benefit of keeping detailed records of financial transactions was recognized back in ancient times. For example, in ancient Rome the first emperor, Augustus, created imperial account books and established a tradition of publishing data from them. While Augustus’ primary purpose may have been propaganda—to publicize his personal spending—he made use of the accounts to plan projects and think about how the empire was managed. According to historian Jacob Soll in his excellent book *The Reckoning*, Augustus’ attention to the accounts enabled Rome to flourish.

But the beginnings of modern bookkeeping came much later, in the emerging city-states of northern Italy in the eleventh century, where the Crusades sparked a massive growth in commercial activity. As trade flourished, merchants in Florence and Venice, in particular, developed a method of accounting that became known as bookkeeping *alla veneziana *(“the Venetian method”).

In their ledgers, the Venetian merchants listed debits and credits in two separate columns. As Pacioli subsequently explained in *Summa*, this was the key to the new form of bookkeeping: “All the creditors must appear in the ledger at the right-hand side, and all the debtors at the left. All entries made in the ledger have to be double entries—that is, if you make one creditor, you must make someone debtor.” Today, we call this “double-entry bookkeeping.”

One important benefit of this system is it provides a built-in error detection tool; if at any moment in time the sum of debits for all accounts does not equal the corresponding sum of credits for all accounts, you know an error has occurred.

In Florence, in the fifteenth century, the bank run by the Medici family adopted double-entry accounting to keep track of the many complex transactions moving through accounts. This enabled the Medici Bank to expand beyond traditional banking activities of the time. It started opening branches in different locations, offered investment opportunities, and made it easy to transfer money across Europe using exchange notes that could be bought in one country and redeemed in another. This growth allowed them to dominate the financial world at a time when Florence was the center of the world for trade and education.

This then was the environment in which Pacioli grew up and lived. As a result, when he set out to write an account of all the commercial mathematics known at the time, his list of contents included one topic that could not be found in Leonardo Pisano’s *Liber abbaci *: book-keeping.

**Pacioli**

Portrait of Luca Pacioli, generally attributed to Jacopo de’ Barbari, ca.1500

Luca Bartolomeo de Pacioli was born between 1446 and 1448 in the Tuscan town of Sansepolcro, where he received an *abbaco *education, the package of commercially-oriented Hindu-Arabic arithmetic, practical geometry, and trigonometry that had been common in Italy since Leonardo published *Liber abbaci*, on which the schooling was based. (I tell that story in my 2011 book The Man of Numbers.) With texts written in the vernacular rather than the Latin used by scholars, *abbaco *focused on the skills required by merchants.

Around 1464, Pacioli moved to Venice, where he worked as a tutor to the three sons of a merchant. It was during this period that he wrote his first book, a short text on arithmetic for the boys he was tutoring.

In 1475, he started teaching in Perugia, first as a private teacher, then, in 1477, becoming the holder of the first chair in mathematics at the university. In 1494, he published his book *Summa*, which made him famous. In 1497, he accepted an invitation from Duke Ludovico Sforza to work in Milan, where he met Leonardo da Vinci, with whom he worked and taught mathematics to until their paths diverged around 1506. Pacioli died at about the age of 70 on 19 June 1517, most likely in Sansepolcro where it is thought he spent his final years.

In addition to *Summa*, published in Venice in 1494, Pacioli wrote a number of other mathematics books:

*Tractatus mathematicus ad discipulos perusinos * (Ms. Vatican Library, Lat. 3129) is a nearly 600-page textbook dedicated to his students at the University of Perugia, where Pacioli taught from 1477 to 1480. It covers merchant arithmetic (barter, exchange, profit, mixing metals, etc.) and algebra.

*De viribus quantitatis* (Ms. Università degli Studi di Bologna, 1496–1508), a treatise on mathematics and magic.

*Geometry * (1509), a Latin translation of Euclid's *Elements*.

*Divina proportione * (written in Milan in 1496–98, published in Venice in 1509). Two versions of the original manuscript are extant, one in the Biblioteca Ambrosiana in Milan, the other in the Bibliothèque Publique et Universitaire in Geneva. The subject was mathematical and artistic proportion, especially the mathematics of the golden ratio and its supposed potential application in architecture. Leonardo da Vinci drew the illustrations of the regular solids in *Divina proportione*, while he lived with and took mathematics lessons from Pacioli. Leonardo's drawings are probably the first illustrations of skeletal solids, which allowed an easy distinction between front and back. The work also discusses the use of perspective by painters.

Two of Pacioli’s books were to have a lasting impact. One was his book on the golden ratio, which gave the initial impetus to a cottage industry of writings that continues to this day, claiming to have identified the number Euclid referred to as the “extreme and mean ratio” in any manner of worldly objects and human artistic creations. While a few of those claims have substance (mostly the ones about the botanical world), the vast majority are entirely spurious, originally no doubt inspired in part by the use of the adjective “divine,” plus the proximity (to Pacioli) of Leonardo da Vinci, and thereafter driven by a dangerous intellectual mix of wishful thinking and mathematical naiveté.

Pacioli’s other influential book was *Summa*. Yet, in many respects, *Summa *is little more than an updated, vernacular version of *Liber abbaci*, which itself was an updated Latin translation of al-Khwārizmī’s Arabic books on arithmetic and algebra. But two factors resulted in *Summa *having a degree of impact that greatly exceeded those two earlier works.

First, thanks to the recent invention of the printing press, *Summa *was the first major *printed *mathematics text, a format that could be duplicated and sold on a wide scale. In the days when manuscripts were hand-written, authors of mathematics texts avoided any use of the abstract symbols they used to do calculations—other than the basic numerals—because they could not rely on accurate copying of formulas and equations by the scribes who made copies. But with print, there was nothing that prevented them having entire pages consist of little else than formulas and equations. (The reason people today associate mathematics with symbols is a result of the printing press. Before then, mathematics was a subject presented in prose.)

Indeed, as I recounted in *The Man of Numbers*, we would today not know about Leonardo’s work and the major role it played in the development of the modern world, were it not for Pacioli’s acknowledgement that *Summa *was based largely on Leonardo’s teachings. The thirteen whole or fragmentary handwritten *Liber abbaci *manuscripts that are now treasured items in the libraries lucky enough to have them would likely still be gathering dust in archives, unseen by modern eyes.

That they are not is due to Pietro Cossali, an Italian mathematician in the late eighteenth century, who came across the reference to Leonardo while studying *Summa *in the course of researching a mathematical history book he was writing. Intrigued by Pacioli’s brief reference to an unknown “Leonardo Pisano” as having been the source for most of the contents of *Summa*, Cossali began to look for the Pisan’s manuscripts, and in due course learned from them of Leonardo’s important contribution. (A French historian subsequently invented a surname for the newly re-discovered Leonardo: “Fibonacci,” and thereby helped give rise to a modern-day mathematics legend.)

To return to my main theme: because it was a print book, *Summa *achieved a far wider readership than *Liber abbaci*, or any of the other handwritten manuscripts that were based on Leonardo’s work. And so its impact was far greater. For that, Pacioli was simply lucky that he wrote his book after the printing press became available.

On the other hand, we can definitely credit Pacioli for the other factor that made *Summa *unique: his inclusion of a chapter on accounting.

As with *Liber abbaci*, *Summa *was more than a business person’s “how to” manual. Both were scholarly mathematical texts, written in the rigorous logical fashion of Euclid’s *Elements*.

*Summa *consists of ten chapters covering essentially all of Renaissance mathematics. The first seven chapters form a summary of arithmetic; chapter 8 explains contemporary algebra (initiating the use of logical argumentation and theorems in studies of the subject); chapter 9 covers various topics relevant to business and trade (including barter, bills of exchange, weights and measures, and bookkeeping); and chapter 10 describes practical geometry and trigonometry. As I noted earlier, none of the methods described are due to Pacioli himself; his contribution, which was significant, was the comprehensive, comprehensible exposition.

Significantly, *Summa *was also the first printed book to codify and give a comprehensive explanation of modern, double-entry bookkeeping, a system of accounting with a long history going back to Jewish bankers in Cairo in the eleventh century (maybe earlier), and used by Italian merchants and bankers, including the Medicis in Florence, throughout the fourteenth and fifteenth centuries.

Pacioli clearly viewed chapter 9 as significant, devoting 150 pages to its coverage of mathematical techniques for business. It is in the section titled *Particularis de computis et scripturis *(“Details of calculation and recording”) that he describes the accounting methods then in use among northern-Italian merchants, including double-entry bookkeeping, trial balances, balance sheets and various other tools still employed by professional accountants today. (The chapter also introduces the “Rule of 72” for predicting an investment’s future value, a technique that anticipated the developmentof the logarithm over a century later.)

In deciding to include a substantial chapter on business mathematics, Pacioli was simply reflecting the local needs of the time, just as al-Khwārizmī wrote his algebra book in response to the changing practices of the merchants around him in ninth century Baghdad, who were developing ways to “scale up” arithmetic to handle multiple trades where the same calculation was being repeated often with different numbers.

In Florence, the Medicis were using double-entry accounting to keep track of the many complex transactions moving through accounts. As a result, the Medici Bank was able to expand beyond traditional banking activities of the time, setting up branches elsewhere and offering customers investment opportunities, as well as making it easy to transfer money across Europe using exchange notes that could be bought in one country and redeemed elsewhere. The Medicis’ mathematically-driven financial expertise enabled them to dominate the financial world at a time when Florence was the center of world trade.

Pacioli’s *Summa *showed others how it was done. He was surely writing a book for which he knew there was a great need. In short, *Summa *did for accounting what *Liber abbaci *had done for Hindu-Arabic arithmetic: it made it go mainstream, presenting it in a way that enabled ordinary people (at least those with some facility with numbers) to master the mathematical techniques required for finance and commerce. That’s the reason Pacioli is sometimes referred to as “the father of accounting.”

When we look back at the development of human society, we tend to see major leaps forward, initiated by a single individual or a small group, with longer periods of more steady progress. The bold initiators who launch society on a new path seem like superhuman geniuses, made of different mental stuff than the rest of us. But, in fact, each major leap forward is always a cumulative effect resulting from many individuals, each making small steps over years, decades, and even centuries. The intellectual giants we see absolutely deserve credit for what they did, but they are still mortals. As one of the greatest “giants” of all, Isaac Newton, famously and revealingly said, “If I have seen further than others, it is by standing upon the shoulders of giants.”

What those giants did that resulted in their names being prominent in the history books is bring together many accumulated small advances, interpret and synthesize them into a whole, and then *package *that whole in a fashion that is readily accessible to others less immersed in the details and the history. In some cases, among them Archimedes (around 250 BCE), Newton (seventeenth century), and Einstein (twentieth century), the influencers brought their own originality into the synthesis.

With others, although their own original work was in some cases significant, it was solely their synthesis and packaging of the work of others that they are known for. Such was the case for Euclid (whose mammoth text *Elements*, ca 350 BCE, established the modern canons of geometry and number theory), al-Khwārizmī (the author of the ninth century text that established algebra as a widely used tool in commerce and then later engineering and science, who we met earlier), Leonardo of Pisa (Fibonacci—also encountered earlier—whose 1202 book *Liber abbaci* brought Hindu-Arabic arithmetic and algebra to the West), and Pacioli with his *Summa*. Though each of these authors produced other books where they presented their own work, it was the breadth and accessible quality of their expository works that changed the course of human history.

The same is true today, with technology. Two giants who changed the world in the 1980s are Steve Jobs and Bill Gates. But neither made any breakthroughs in the design of computers or the creation of software systems. Rather, they took the best of what was available, and packaged it in a way that millions of others could use. In the worlds of science, mathematics, and engineering, the professional cultures direct their admiration towards the innovators who come up with new ideas, and tend to downplay or even dismiss the individuals who package those new ideas into an accessible form that others can use. But both invention and packaging/marketing are required in order to change the world.

It is then, as a “packager” that we must view Pacioli in order to recognize the major impact he had on the course of history.

Because of the power of the recently invented printing press to spread multiple copies of identical texts relatively cheaply and quickly, Pacioli’s book-keeping treatise, as the first printed synthesis of the method, led to a rapid adoption of Venetian book-keeping, and by 1800, use of the system was standard across Europe. But that was not the end of the revolution.

Not long afterwards, the business world found another, far-reaching use for “bookkeeping alla veneziana.” It came about as a result of the desperate efforts of an English potter to prevent his company going bankrupt.

**Josiah Wedgwood **

Today, the name Wedgwood is synonymous with fine pottery, sold all around the world. Less well known, is the major influence this eighteenth century English potter had on mass-market manufacturing in the early days of the industrial revolution.

Born in Staffordshire, England in 1730, Josiah Wedgwood was a highly talented potter and, it turned out, a skillful entrepreneur. Having learned the basic skills of pottery from his father, also a potter, he founded his own company while still very young. That company (the Wedgwood Company) was one of the first to adopt an industrial, mass-production approach to manufacture (and the first to do so for the manufacture of pottery).

By the late 1760s, his traditionally produced, expensive classical designs had found a ready market among the nobility, among them Queen Charlotte (the wife of George III), who he persuaded to grant him permission to refer to his crockery sets as the “Queen’s Ware”. (A smart marketing move.) But Wedgwood wanted more.

In order to grow his company beyond that limited market, he looked for ways to manufacture cheaper sets to sell to the rest of society. This involved both experimenting with different materials and developing ways to produce and sell at scale.

By staying abreast of scientific advances, he was able to adopt materials and methods to both revolutionize the production and improve the quality of his pottery. In particular, his unique glazes began to distinguish his mass-produced wares from anything else on the market.

He also proved to have a flare for marketing, and today he is credited as the inventor of modern marketing techniques such as illustrated catalogues distributed by direct mailings, money-back guarantees, traveling salesmen carrying samples, self-service, and free delivery.

In 1764, he received his first order from abroad. Just three years later, he was able to write of his pottery, “It is amazing how rapidly the use of it has spread all most [*sic*] over the whole Globe.”

Unfortunately, however, that rapid growth brought problems of finance, and by late 1769, Wedgwood and his partner, Thomas Bentley, had serious cash-flow problems and an accumulation of stock. Like many entrepreneurs, too much early success brought him to the edge of bankruptcy.

In response, in 1772 Wedgwood decided to use double-entry book-keeping to examine his firm’s accounts and business practices to see if there was a way for his company to survive. The results proved enlightening and, for the business world, far reaching.

He found that the firm’s pricing was haphazard, its production runs too short to be economical, and that it was spending unexpectedly large amounts on raw materials, labor and other costs, without collecting its bills fast enough to finance expanding production.

Statue of Josiah Wedgwood at the Wedgwood factory in Staffordshire, UK

He also made an important discovery: the distinction between fixed and variable costs. He immediately understood the implications of their difference for the management of his business.

He told Bentley that their greatest costs—modelling and molds, rent, fuel and wages—were fixed: “Consider that these expenses move like clockwork, and are much the same, whether the quantity of goods made be large or small.”

He realized that the more their factory produced, the cheaper these fixed costs would be per unit of production.

In other words, by scrutinizing his books using double entry, Wedgwood had uncovered the commercial benefits of mass production.

To take advantage of his observation, Wesgwood had to take Pacioli’s book-keeping system and apply it beyond its mercantile origins in an exchange economy, to the world of manufacturing, where the emphasis is on the production of goods. That was a major shift, with enormous consequences, both for his company and for the world.

The need to incorporate new elements—labor and materials per unit of production—into an enterprise’s accounting system so that managers could calculate the cost of each unit of production posed significant conceptual difficulties. (That today we don’t give this a moment’s thought, is further indication of how fundamentally Wedgwood’s revolution changed the world.)

The challenge was that the transactions needed to incorporate the manufacturing of products into the existing double-entry system were not financial; they did not involve the exchange of goods, rather activities such as adding the cost of labor acquired or of materials bought. These “non-financial” transactions were new, and to fit them into the 300-year-old accounting system was not easy. Only after a century of factory production had such accounting problems become better understood.

Meanwhile, and not altogether unconnected, the rise of the joint-stock company brought double-entry bookkeeping center stage, giving birth to a new profession: accounting.

The huge amounts of capital expenditure required to build railways—raised from private investors on stock exchanges and managed by joint stock companies—also generated new issues of accounting and accountability.

As a result of all these advances, by the 1860s, accountants in Britain were legally required at every phase of a company’s life. Financial statements had gone from being an incidental product of an enterprise’s book-keeping system in 1800, to being bookkeeping’s raison d’être a century later.

Looking back, we see that Venetian bookkeeping proved to be an ideal system for generating the financial statements that were required for the modern industrialized world. It could accurately record capital and income (as required by law and investors), it could distinguish between private expenses and corporate costs, and it could produce data that helped to evaluate past investment decisions.

It doesn’t get more relevant and important to today’s world than that.

For additional photos from the Christie’s event where I spoke about Pacioli, see the April 27 blogpost on the Stanford Mathematics Outreach Project site.

Read the Devlin’s Angle archive.

The Education Committee of the U.S. House of Representatives is currently preparing for the reauthorization of the Higher Education Act. In anticipation of the committee hearings, they have asked for input from interested parties. A group of us put together a joint statement that is being submitted on behalf of MAA and CBMS. It is available here. The “ask” in this statement focuses on what is increasingly seen as an important catalyst for facilitating educational change at the department level, the presence of discipline-based education specialists (DBESs). As we state in our request to the committee:

The colleges and universities that have been at the forefront of these improvements have usually had discipline-based education specialists within the departments who led, studied, and adapted local interventions. Training and embedding specialists responsible for: i) implementing, studying, and adapting local innovations; ii) adapting assessments and curriculum to be more conceptually focused, and iii) supporting faculty to implement more engaging instruction constitute one of the most significant steps that could be undertaken with additional resources.

We have a very small number of education specialists who are trained both in the mathematical sciences and in the research in undergraduate mathematics education. Expanding this corps and encouraging all departments to embrace their expertise would greatly facilitate the improvement of the mathematical experience for all students.

To me, one of the most striking observations from multiple studies of change that has led to effective undergraduate programs is the presence within the mathematics department of faculty with training in undergraduate mathematics research. It is not a necessary presence, Macalester’s small department is innovative and progressive without need for a faculty member with formal preparation in mathematics education. But large departments usually need someone who is familiar with the literature that supports a variety of interventions and is accepted by members of the department as someone who can help guide the process of change.

The physics education literature has demonstrated that the greatest obstacle to sustained change is neither awareness of what can be done nor willingness to try something different (Henderson et al, 2012). The greatest problems arise when faculty try something new and discover that it is both harder to pull off than anticipated and not as successful as experienced at its pilot sites. This is where assistance from someone embedded in the department can be of critical importance: to assess what is really happening, to provide support that makes it easier for those attempting change, and to help tweak these efforts to improve the outcome.

Warren Code and Stephanie Chasteen and the cover of The Science Education Initiative Handbook.

This past fall, the Science Education Initiative, the product of Carl Wieman’s work at the University of Colorado-Boulder and the University of British Columbia, produced a handbook written by Stephanie Chasteen and Warren Code that serves as “A practical guide to fostering change in university courses and faculty by embedding discipline-based education specialists within departments” (Chasteen & Code, 2018), available free here.

Their guide is structured in three parts

1. **For those seeking to bring about change**: What is the role of a DBES? What should be the nature of the position? What does it take to make this person successful? What is the organizational structure that is required?

2. **For department leaders**: How do you set up an organizational structure that will support a DBES? How do you set the stage for successful course transformation?

3. **For those who would be a DBES**: How do you get started on transforming a course? What are the keys to partnering with the faculty? How do you handle your multiple roles within the department and build your skills?

The guide concludes with an appendix of case studies of different institutions using DBESs, illustrating the variety of approaches for this model.

The emphasis in this handbook is on the use of postdocs or instructors for these positions. It is not enough to simply hire a postdoc, contract instructor, or permanent lecturer who will focus on issues of teaching. I know universities where highly capable, informed, and motivated instructors find themselves frustrated by their lack of influence. This represents a tremendous waste of potential and underscores the importance of the organizational structures that must be built by the departmental leadership.

Another line of attack in supporting change is to embed Science Faculty with Education Specialties (SFESs). Analysis of this approach can be found in Kimberly Tanner’s work. Admittedly, re-envisioning faculty in mathematics to include those engaged in research in undergraduate mathematics education (RUME) requires a broadening of the view of who belongs in a math department. It also requires an expansion of the vision of RUME from its traditional focus on student understanding of specific concepts to a broader study of departmental change

This can be done. Arizona State, Oklahoma State, San Diego State, and Colorado State Universities are prime examples of what is possible. Moreover, because of the nature of their work—which requires funding for team efforts—bringing researchers in undergraduate mathematics education into the department inevitably creates funding for graduate students and postdocs who can carry much of the load of monitoring and assessing ongoing efforts.

Most department chairs and their deans recognize the need to improve student outcomes. The SEI Handbook provides direct and practical advice for facilitating and supporting the requisite changes.

**References**

** **Chasteen, S.V. and Code, W.J. (2018). *The Science Education Initiative Handbook*. Accessed at https://pressbooks.bccampus.ca/seihandbook.

Henderson, C, Dancy, M. and Niewiadomska-Bugaj, M. (2012). Use of research-based instructional strategies in introductory physics: Where do faculty leave the innovation-decision process? *Phys. Rev. ST Phys. Educ. Res.* **8**, 020104, available at https://journals.aps.org/prper/abstract/10.1103/PhysRevSTPER.8.020104

** **

The federal government has long played a central role in collecting and analyzing data, for all sorts of reasons. Our founders recognized that this should be so; the U.S. Constitution calls for a census (Article 1, Section 2) in order to appropriately allocate votes. Of course the scale of the data gathered by the federal government in our own time was unimaginable even 50 years ago, as technology has advanced to provide mechanisms for collection and storage of data of all kinds.

I was reminded of this most recently by several efforts to which we were alerted by our colleagues at the American Statistical Association (ASA). One of which is an effort to oppose a reorganization of the Bureau of Transportation Statistics. Essentially, the proposed reorganization would move the Bureau into a policy-making office, risking the perception of objectivity for which the Bureau has long stood.

A second effort, which MAA joined, was to oppose the relocation and reorganization of the USDA Economic Research Service, again moving the office from a strictly research arm to a policy-making arm. The ASA sent a letter to appropriate congressional leaders, and MAA joined a similar letter sponsored by the Friends of Agricultural Statistics and Analysis.

Another reminder of the value of the large-scale efforts that only the federal government can undertake, and the risk that comes with substantial reorganization driven by policy objectives, came through the recent publication of The Fifth Risk, by Michael Lewis, and reviewed in the NY Times.

Discussions that Lewis had with DJ Patil, who served as the first Chief Data Scientist of the United States Office of Science and Technology Policy from 2015 to 2017, under that Obama Administration, were featured in Lewis’s book, and help support Lewis’s larger claims that current efforts by various groups hostile to the role of the federal government in collecting and managing data to support effective policy-making do constitute the “risk” of the book’s title.

DJ, in particular, had done substantial work while in graduate school at the University of Maryland, using data from the National Weather Service (NWS, a division of the Department of Commerce), to provide measures of uncertainty to improve our ability to forecast weather. DJ’s methods have in fact helped improve forecasting. And it’s worth noting that the data used by all major weather forecasting companies are drawn from the NWS. Without careful collection, management, and sharing of this data, the U.S. economy would stand to suffer billions more in weather-related losses than we already do.

I note in passing that DJ was also an advocate for mathematics, and worked to allow the MAA-sponsored U.S. team to the International Mathematical Olympiad to visit the White House on Pi Day 2016.

The notion that federal support is essential for the long-term health of our society is not new. At the end of World War II, Vannevar Bush laid out the case in Science: The Endless Frontier, delivered to President Truman in July 1945. Bush was the Director of the Office of Scientific Research and Development, and had been asked by President Roosevelt to produce the report the previous year. In the summary, he wrote:

*Science can be effective in the national welfare only as a member of a team, whether the conditions be peace or war. But without scientific progress no amount of achievement in other directions can insure our health, prosperity, and security as a nation in the modern world.*

Vannevar Bush is credited with laying the groundwork for the founding of the National Science Foundation.

The MAA cannot single-handedly effect large-scale efforts to persuade policy makers to undertake meaningful efforts to preserve the enormously valuable and successful ways that the federal government supports the scientific enterprise, and ultimately uses the results to effect policies. However, we can, and do, engage in myriad ways with our colleagues and allies to support math and science, and the appropriate inclusion of data and evidence to inform policy decisions that will improve the quality of life for all our citizens.

It’s an exciting time to be part of the mathematical community, and I’m happy that MAA can represent our members to, in the words of our mission, advance the understanding of mathematics and its impact on our world.

Jeanette Shakalli, Executive Assistant of the National Secretariat of Science, Technology and Innovation of the Republic of Panama

Have you heard about the fantastic mathematics outreach program in Panama run by Dr. Jeanette Shakalli?

Jeanette is a mathematician who currently serves as Executive Assistant of the National Secretariat of Science, Technology and Innovation of the Republic of Panama. You might recognize her from "Meet A Member" in the August/September 2018 issue of *MAA FOCUS *magazine.

I had heard about Jeanette from several mathematicians lucky enough to travel to Panama and present mathematical wonders as part of her program to bring mathematics to a general audience. You might recognize some MAA Math Ambassadors in the collection of posters below (click on the thumbnail to see a larger image)!

Poster featuring MAA Math Ambassadors

Jeanette’s role is to help connect people to mathematics and share the power and beauty of the subject with Panamanians. After Jeanette and I met at the Mathematics Section Meeting of the AAAS Annual Meeting 2019, we started an email correspondence. I was pleased to learn Jeanette is an enthusiastic MAA member. When I asked her why she joined MAA, Jeanette wrote:

“Actually, I became a member of the MAA because the AMS President of the time, David Vogan, recommended I attend MAA MathFest because he thought I would really enjoy it, so I did. I followed his advice and joined the MAA, and it has been the best decision I ever made. I believe I am a self-proclaimed math ambassador for Panama. In 2017 I became IMU's CWM Ambassador for Panama so maybe I am not self-proclaimed anymore =).”

“Whoa!” I responded. “Best decision you ever made?!?!? Tell me more…”

Jeanette responded: “It's true!!! If it wasn't for all the great connections that I have made through the MAA, I would not have been able to come up with my Program on Math Outreach at the National Secretariat of Science, Technology and Innovation of Panama (SENACYT). If you notice, almost all (or maybe all?) the mathematicians who have visited Panama so far are MAA members. Moreover, they have somehow become my mentors. For example, today Robert Vallin sent me a message on another activity that involves pi that I could do next time I organize a Pi Day Celebration. (I have to mention that yesterday was my first time organizing a Pi Day Celebration and it was SO much fun!!!) Arthur Benjamin is always introducing me to remarkable mathematicians who he thinks are wonderful public speakers and would love to come visit Panama. Michael Dorff has helped me think about what I want to do with my life (I know, it sounds intense!) and has pushed me to consider creating my own math presentation for Panamanians and spreading the love for math in my own country (in other words, besides bringing talented international mathematicians to Panama, he says that I could also do it myself!). Deanna Haunsperger invited me to be one of the Twenty Faces of the MAA. In a way, the MAA has become my family and I am very happy that I followed David Vogan's advice.”

Jeanette’s comments made me start thinking about what the last “A” in MAA really stands for. Before I was the Deputy Executive Director of the MAA I was the Vice President for Education for SIAM, appointed under the Presidency of Dr. Irene Fonseca (Carnegie-Mellon University). Irene liked to say that the “I” in SIAM not only stood for industrial but also for international. The mission of the MAA is "to advance the understanding of mathematics, and its impact on our world.” Sometimes I like to think of us as the Mathematics Association of Ambassadors!

This map of our sections includes the US, Canada and four US Territories. But our community extends beyond these boundaries through our mathematics competition program, our members who come from all over the world, our global mathematics teaching, learning and research networks, and the global society and ecosystem that we serve.

However, Jeanette shared this perspective: “I do not consider the MAA to be an international organization since whenever I go to the conferences, I rarely see people from outside the US, and all the MAA meetings are in the US.” She encouraged us to consider not how many MAA members live outside of the US but instead how many MAA members are international (not US citizens or residents). She suggested many ways the MAA could choose to be more international, such as making new alliances with international organizations, organizing MAA meetings/events outside of the US, promoting MAA membership in other countries, and preparing flyers about math in general and about the MAA in other languages.

This is a timely suggestion! Just last week the staff leadership of MAA, AMS, ASA and SIAM were invited to the National Academies of Sciences to discuss research trends with leadership from the British Research Council (EPSRC) and the UK Research and Innovation USA. Our new MAA President Michael Dorff is well known for traveling internationally so I expect we will hear more from him about global mathematical perspectives.

Next time you are in Panama, let Jeanette know, and if you are lucky, she will show you around her beautiful country and connect you to the mathematics community there!

]]>Students participating in DAoM

Mathematics for Liberal Arts (MLA) is one of the largest undergraduate mathematics cohorts. Unfortunately, MLA coursework is largely disconnected from students’ academic interests and frequently fails to engage them in research-recommended active learning approaches. Discovering the Art of Mathematics (DAoM) curriculum materials provide unique and inexpensive alternatives to traditional methods and texts. Read on to find out more about this important project, described by co-PI Philip Hotchkiss of Westfield State University.

**Q: What issues did you seek to address at the outset of your project? **

Many liberal arts and humanities students are served by single-semester courses generally known as Mathematics for Liberal Arts (MLA) courses. Because many MLA students’ negative prior experiences in mathematics have left them with unhealthy perceptions of the subject and their own abilities, it is incumbent on faculty to provide our students with positive experiences that challenge these perceptions. This can seem like an enormous task to many faculty. Our project was designed to address these challenges by creating an active learning environment that supported MLA students in connecting mathematics to their disciplinary interests.

**Q: Briefly describe your project. **

The guided-discovery curriculum materials that make up DAoM meet the need for more active involvement of mathematics students in the learning process. We provide a library of 10 inquiry-based learning (IBL) guides that involve students in authentic mathematical experiences that are both challenging and intellectually stimulating. These experiences nurture healthy and informed perceptions of mathematics, mathematical ways of thinking, and the ongoing impact of mathematics not only on STEM fields but also on the liberal arts and humanities.

We also have extensive teacher resources, as well as many professional development opportunities for mathematics faculty who wish to transform their classrooms in response to current research on learning and meet the needs and interests of MLA students.

As this project has progressed, we have come to understand how fundamental this vision is for many other audiences beyond MLA students. Central aspects of this project provide for teaching and learning in courses at all levels, as well as for homeschoolers, for Math Circles, and for mathematical enrichment programs.

**Q: How do you hope your work will positively influence students and the mathematics community?**

After graduation, we hope that our students will help change the negative stereotypes of mathematics in their communities and families. We also hope that our materials will inspire and support faculty to change the way they teach and interact with their mathematics students.

**Q: What unanticipated challenges did you face during the implementation of your project?**

One of the unanticipated challenges we encountered was the need for workshops on using IBL and support for our books. We assumed faculty would download our materials and then easily use them in their classrooms. This misconception required us to provide additional workshops and resources on our web page, such as the e-book, *Discovering the Art of Teaching and Learning Mathematics Using Inquiry*.

**Q: Tell us about someone impacted by the project. **

After Steven Strogatz saw our materials at JMM in 2014, he invited us to Cornell to do a workshop and then started using our materials. In a blog post on our website about our workshop, he wrote, “This experience gave me powerful insight into what it must be like for students in an IBL classroom. It made me realize the importance of providing a safe and nurturing space for the math explorers I was about to start working with in just a few days.”

In an interview with Jessica Lahey that appeared in *The Atlantic*, Strogatz said, “I have to say that teaching this class has been a joyful experience in a way that no other class I've ever taught has been. I love teaching, and I certainly love teaching students who already enjoy math – don't get me wrong. But there's something remarkable about working with a group of students who think they hate math or find it boring, and then turning them around, even just a little bit … I was so proud of them. They were having a true mathematical moment. That is, they were deeply engaged with a puzzle that made sense to them, and they were enjoying the struggle, and no, they did not want a hint! They were feeling what anyone who loves math feels, the pleasure of thinking. The pleasure of wrestling with a problem that fascinates you. No one in the class was asking, ‘What is this good for?’ Or ‘Where will I ever use this?’ Those are questions that students ask only when they are not engaged.”

*Editor’s note: Q&A responses have been edited for length and clarity. *

**Learn more about NSF DUE 1225915**

Links to the guest blogs from Steven Strogatz can be found at http://www.artofmathematics.org/blogs/cvonrenesse/steven-strogatz-reflection-part-1, and a link to Jessica Lahey’s entire interview with him can be found at: http://www.artofmathematics.org/news/steven-strogatz-and-daom-in-the-atlantic

Full Project Name: Discovering the Art of Mathematics: Mathematical Inquiry in the Liberal Arts

Abstract: https://nsf.gov/awardsearch/showAward?AWD_ID=1225915

Project Website: http://artofmathematics.org/

Project Contact: Julian Fleron, PI jfleron@wsc.ma.edu

For more information on any of these programs, follow the links, and follow these blog posts! This blog is a project of the Mathematical Association of America, produced with financial support of NSF DUE Grant #1626337.

*Erin Moss is a co-editor of DUE Point and an Associate Professor of Mathematics Education at Millersville University, where she works with undergraduates from all majors as well as graduate students in the M.Ed. in Mathematics program. *

At the beginning of my first semester here at Bates, I received some jarring news. My high school math teacher, Marty Badoian, had passed away.

I think everyone has that professor or teacher. You know - “the one” that all paths lead back to in your life. That was Marty Badoian. There are many more figures in my life who have made a difference, who encouraged me to pursue math, who guided me, who provided me with opportunities. There were many moments that were pivotal, where I can almost see the fork in the road and see myself deliberately choosing a path that led to the outcomes of today. But Marty was a part of my life for four years formative to earning my PhD in math. Outside of my family, he is the one I credit with my ability to achieve what I have achieved today. Marty was one of the crucially necessary, though not sufficient, conditions.

While in his memorial Facebook group, many of Marty’s students posted of his kindness and mentoring. I had a bit of a different relationship with him. Marty Badoian was the one that made me struggle with math, that put the bar so high I never thought I was good enough. And yet, that early experience of failing, of working so hard, of being given the chance to compete at high levels, and of being supported regardless - well he was my Jaime Escalante.

Let me back up a bit and put the story into perspective. Marty was a Brown University graduate who eventually settled as a math teacher at Canton High School - a public high school in the suburbs of Boston. Anyone who has been a part of the math competition scene in New England knows of Canton High School and likely also of Marty. In 1966, he founded the Mathematics Massachusetts League and the Greater Boston Mathematics League (see his obituary for more information and a longer timeline of his life and accomplishments). He coached his mathletes like he coached basketball - practice every day and during the summer - and at least during my time we were Canton’s most winning team. In 1984, he was awarded the Presidential Award for Excellence in Mathematics and Science Teaching. By the time I moved to the Canton School system in 1994, just in time for high school, Canton had won 14 out of the last 15 New England Mathematics League [NEML] Championships.

When I was in 8th grade, my dad got a new job in Boston. We had to move and he was keeping in mind the reputation of the math team as a factor since I had shown an interest in math. After we chose Canton, my meeting with Mr. Badoian that spring wasn’t quite what I expected - I left crying! He quizzed me and took my measure as a burgeoning mathematician. Some of the mathematical operations had different names that I didn’t recognize (factor by grouping was called a “2-3” or a “3-2,” for example). For once in my mathematical life, I was worried I wasn’t good enough. But Marty told my dad to get a senior to tutor me that summer to catch me up on these differences, and in the fall I took both Marty’s 9th and 10th grade math classes.

In these classes, I always felt like the one that struggled the most. It was intimidating to be with colleagues a grade ahead. But there was something about the culture that Marty created that made this reach something that the students admired. You weren’t a weird nerd if you were on the math team, you were on top of the world. You weren’t stupid if you went after school for help every day on your math class, it was expected and normal. While his self-made curriculum of worksheets was unorthodox, it introduced our junior math class to vectors and distance between planes, proof by induction, and introductory number theory. This combination of high expectations, innovative approaches to curriculum, and culture around work ethic and expected mathematical growth made Canton’s math students a powerhouse. I might have struggled in this powerhouse, but I was quite unaware that even a struggling person in Marty’s room was quite a good mathematician. The first year I made the American Regions Math League [ARML] Massachusetts team, I turned it down because I had a family trip that week. I truly had no idea it was an honor.

I started competing in math competitions, like most Canton students, in my sophomore year. We shared a bus, we did a lot of math, and we traded wins with our rivals at Lexington. We also always had dinner after competing, and it always involved a discussion of the answers to the problems on paper napkins and paper placemats and how to do better next time. Many weekends were spent volunteering at the high school football concession stand and the Foxboro (now Gillette) Stadium concession stand raising money for the team. I had no idea at the time, but this didn’t just pay for our bus rides and dinners - this gave me a scholarship that paid for what my University tuition scholarship didn’t cover in my first year. Marty also recommended me as a tutor, which I loved, and which in retrospect, really helped me practice my mathematics. And thinking back now, I realize also that Marty also had two women and two men as senior team captains - reflecting a diversity that I don’t see in many of today’s top math teams. So sure, it was the math - but it was more… it was a package of opportunity, a team, and a growth mindset culture.

So, when did I realize what I had? It is funny how even now when I look back at my scores as I try to reconstruct my experience to write this post, I didn’t remember making the [tied-for] highest individual score for my team on the NEML in ‘97 and second highest in ‘98. I only remember our new theme song from Queen coming over the radio “We are the Champions” on the way to dinner that night. I think the understanding didn’t set in until I was on the ARML team that won the international competition at PennState. We had reporters from all over, including the Boston Globe come and interview us, and I was awarded a State Senate official citation! But by then I had also decided that I wasn’t a mathematician. You see, I liked doing mathematics and I loved teaching mathematics, but I didn’t love investigating mathematics for the beauty of math. It was a few years later (and probably a future separate blog post worth sharing), when I finally realized I could study biology questions with mathematics!

In 2004, Marty was thrown a huge party to celebrate his 50th year of teaching. I told him I was getting my PhD in Mathematics. He seemed surprised. At the time I thought he didn’t think I was capable. But as I reflect now, I think he saw me as a bit stubborn in pursuing my dreams of zoology, and never thought I would ever give mathematics serious consideration. We didn’t have the close mentoring relationship some had because I feared him - or rather feared disappointing him. And, sadly, I missed the opportunity to tell him thank you and show him that I did indeed make it! However, now that he has passed, I think finally it is possible for him to see all of me from where he is, all of my challenges, all of the forks in the road, and can be proud of where I am today. RIP Marty Badoian. You created the environment of opportunity, teamwork, and mathematics learning that has helped many of your students accomplish what they have done today.

**Note**: A big thank you to former Canton High Math teammate, Dave Archibald, who shared these photos of newspaper clippings from *The Patriot Ledger*, *The Canton Journal*, and *The Boston Sunday Globe*.

Janet Ray, a dear friend who was one of MAA’s representatives on the project team that launched the AMATYC ACCCESS program, recently sent me a copy of the seventh edition of a 19th century text, Elements of Algebra, advertised by its author James Thomson, as an “abridgment of Day’s Algebra adapted to the capacities of the young.”

The edition Jan sent along was printed in 1846, and is inscribed as “Papa’s Algebra that he went to school with, Townsend Vermont Academy.” Since the book comes from the library of Jan’s great-aunt, Bertha Ray, who retired in the 1930’s after serving as the principal of a school in Brooklyn, Jan reckons that the book was used by her great-grandfather.

The Preface of “Elements” opens with the statement that “Public opinion has pronounced the study of Algebra to be a desirable and important branch of popular education.” Mr. Thomson concludes his lengthy preface by noting that “it remains to his fellow teachers and an impartial public to decide” if he met his goal “to divest the study of algebra, once so formidable, of all its intricacy and repulsiveness; to illustrate its elementary principles so clearly, that any school-boy of ordinary capacity may understand and apply them; and thus to render this interesting and useful science more attractive to the young.”

The roughly 175 years since the first edition of Mr. Thomson’s work was published (1843) bear witness to the less than enthusiastic verdict of the “impartial public” to whom he appealed, which as far as I can tell remains largely indifferent if not hostile to the study of algebra at any level.

Among teachers, most of us continue to agree that the study of mathematics is worthwhile, and goes beyond utilitarian purposes. Further, most of us agree that development of conceptual understanding, as well as procedural fluency, go hand in hand, that both are essential elements in teaching and learning of mathematics.

There’s also recent evidence that the inherent value of knowing at least some mathematics -- baseline numeracy -- contributes to more effective decision making in multiple domains. I recently interviewed Ellen Peters, director of the Decision Sciences Collaborative at The Ohio State University, on this theme (see “Learn Math and Live Longer,” in the Aug/Sept issue of MAA FOCUS). It’s great to see evidence for improved critical thinking that those of us in the mathematical sciences have felt we bring to the table all along.

We also have an increasing body of evidence that supports the use of active learning strategies in classrooms at all levels. The recent publication of MAA’s Instructional Practices Guide provides information designed to improve outcomes for all students. As the authors note, “It is our responsibility to help our colleagues improve and to collectively succeed at teaching mathematics to all students so that our discipline realizes its full potential as a subject of beauty, of truth, and of empowerment for all.”

Just as Jan spent years honing her skills as a teacher of mathematics, and giving back to colleagues through the MAA and collaborative programs such as Project ACCCESS, the MAA community continues to develop programs and opportunities to build capacity “to advance the understanding of mathematics and its impact on our world.” It’s a great place to be!

]]>Okay, with an expected sale price at auction of $1M to $1.5M, this is not your typical, used, math textbook. In fact, you can get a sense of the anticipated buyer by the title of the sale: the subtitle of Luca Pacioli’s book is “Summary of arithmetic, geometry, proportions and proportionality”*, *but for the sale announcement Christie’s (who are experts in handling sales of this kind of work) replaced that phrase by “the birth of modern business”.

There is no shortage of mathematicians who have made fortunes as quants in the financial markets over the past few decades, who have the monetary resources to indulge their love of mathematics to the extent of acquiring a first print of one of the most influential mathematical works of all time. (Only 120 or so first edition copies are known to exist. The copy on sale appears to be from the second of three printings, dated around 1502.) I have to admit, I would love to hold it in my hands for a while, as I did two early (hand-written) copies of Leonardo Fibonacci’s *Liber abbaci*, when I was researching my books The Man of Numbers, Finding Fibonacci, and Leonardo and Steve. Owning it, not so much—but if I were in a position where a million dollars were small change, maybe I’d feel differently.

Be that as it may, Christie’s sale-oriented title is absolutely not inaccurate, any more than my subtitle for *Finding Fibonacci*: “The Quest to Rediscover the Forgotten Mathematical Genius Who *Changed the World *”, or the publisher’s catalogue summary of *The Man of Numbers*, which I approved: “The untold story of Leonardo of Pisa, the medieval mathematician who introduced Arabic numbers to the West and *helped launch the modern era*.” (Emphasis added in both cases.) Those works by Leonardo and Pacioli really did change the course of history.

Pacioli’s book *Summa de Arithmetica* was in fact the next major mathematical publication landmark after Leonardo Pisano in the development of the modern commercial and financial world. In fact, as I described in both those Fibonacci books, were it not for Pacioli acknowledging in his introduction that his treatment of Hindu-Arithmetic followed that of Leonardo, we would likely never have known of the role the thirteenth century Pisan and his book *Liber abbaci *played in that major historical development.

Both *Liber abbaci* and S*umma de Arithmetica* were as much business books as they were mathematics texts. In *Finding Fibonacci*, most of Chapter 15 is devoted to the work of William Goetzmann, Professor of Finance in the Yale School of Management, who wrote:

“The five-hundred-year period following the appearance of

Liber abbacisaw the development in Europe of virtually all the tools of financial capitalism that we know today: share ownership of limited-liability corporations, long-term government and corporate loans, liquid and active international financial markets, life insurance, life-annuities, mutual funds, derivative securities, and deposit banking. Many of these developments have their roots in contracts that were based on the mathematical analyses Leonardo introduced to Western Europe throughLiber abbaci.”

As I noted already, Pacioli was a major part of that commercial thread. As the first printed book on algebra in a vernacular language, and being written by a respected, established mathematician, *Summa* reached a far wider and more influential audience than any of the earlier, handwritten manuscripts of Leonardo and others.

It was also the first printed book to codify and give a comprehensive explanation of modern, double-entry bookkeeping, a system of accounting with a long history going back to Jewish bankers in Cairo in the eleventh century (maybe earlier), and used by Italian merchants and bankers, including the Medicis in Florence, throughout the fourteenth and fifteenth centuries.

*Summa* did for accounting what *Liber abbaci* did for Hindu-Arabic arithmetic: it made it go mainstream, presenting it in a way that enabled ordinary people (at least those with some facility with numbers) to master the mathematical techniques required for finance and commerce. For that reason, Pacioli is sometimes referred to as “the father of accounting.”

But as with *Liber abbaci*, *Summa* was more than a business person’s “how to” manual. Both were scholarly mathematical texts, written in the rigorous logical fashion of Euclid’s *Elements*.

*Summa* consists of ten chapters covering essentially all of Renaissance mathematics. The first seven chapters form a summary of arithmetic; chapter 8 explains contemporary algebra (initiating the use of logical argumentation and theorems in studies of the subject); chapter 9 covers various topics relevant to business and trade (including barter, bills of exchange, weights and measures, and bookkeeping); and chapter 10 describes practical geometry and trigonometry.

None of the methods described are due to Pacioli himself; his contribution, which was significant, was the comprehensive, comprehensible exposition. Something I have spent a large part of my mathematical career focusing on.

The significance of good expositions, and more recently well-designed technologies, in ensuring the new ideas and methods “change the world” is the focus of my short book *Leonardo and Steve*, referred to earlier. The “Steve” is Steve Jobs. Pacioli was a talented polymath, and a collaborator of Leonardo Da Vinci, who did many things. But it was his *Summa* that had a major influence in the development of modern society. That’s why his book is a major historical artifact.

For some wonderful images of pages from Pacioli’s *Summa*, see this MAA webpage from 2011.

For details of Pacioli and his other works, his Wikipedia entry provides an excellent summary.

ADDENDUM: For those of you in the San Francisco Bay Area, I will be giving a short talk about Pacioli’s *Summa* at a showing of the manuscript on sale, being organized by Christie’s in San Francisco on April 24 from 6:00 to 8:00pm

Read the Devlin’s Angle archive.

]]>*Notice: I have considered myself part of the IBL community in mathematics since my first IBL conference in 2008. I currently serve on the Board of Directors of The Initiative for Mathematics Learning by Inquiry (MLI), a public nonprofit organization that supports this community of practitioners. The opinions expressed here are not necessarily those of MLI.*

Inquiry Based Learning (IBL) is a term that took on a specific meaning within the mathematics community in the 1990s when it was adopted by those attracted to certain aspects of the Moore Method. Two recent papers in the *International Journal of Research in Undergraduate Mathematics Education* help to illuminate the special role and the potential of IBL within our community.

The first of these, “What’s in a name? Framing struggles of a mathematical education reform community” (Haberler et al, 2018), examines the sociological phenomenon of the IBL movement. The origin of this movement within mathematics was the effort by the mathematical descendants of R.L. Moore, led by Harry Lucas through his Educational Advancement Foundation, to preserve the legacy of Moore’s approach to mathematics instruction. Moore’s method was idiosyncratic. Best exemplified in graduate courses in topology, instruction began by presenting students with a set of definitions and a list of theorems to be proved. Students were responsible for working individually to prove these theorems without recourse to any supporting resources, including classmates. Class time was restricted to the presentation and critique of these proofs.

Moore handpicked the students for his classes, and there were many notable successes, such as Mary Ellen Rudin, who would build on this experience to become leading research mathematicians. But pure Moore had many drawbacks, leading his successors to introduce variations that came to be known as modified Moore method. Those attracted to the inquiry aspects of the Moore method but wishing to distance themselves from his elitist approach and promote collaborative learning latched onto the term IBL, which had recently been popularized in the Boyer Report (1998).

Sandra Laursen at the 14th Annual R.L. Moore Conference, June 2011, Washington, DC

The Haberler article describes the process that has led to today’s vision of IBL in mathematics. It clarifies that, while it did not originate from educational research, its practices are generally well aligned with research and there are good pedagogical foundations for an inquiry-based approach to learning. What makes it special is its development as a social phenomenon, bringing together those who have discovered the power of student-centered and active learning approaches to undergraduate mathematics instruction. It now embraces a “big tent” that is characterized by a willingness to cede large chunks of time from lecture so that students have structured opportunities within class to explore concepts, investigate key ideas, and build understanding through inquiry.

The second paper, “I on the prize: inquiry approaches in undergraduate mathematics” (Laursen and Rasmussen, 2019) iterates the observation from the first paper that IBL did not arise from within the research community for mathematics education. The authors contrast it with Inquiry-Oriented Instruction (IOI), which did. While IBL has become a social movement for a wide variety of practitioners of student-centered learning, IOI is strongly rooted in the discipline of research in mathematics education, drawing directly on educational theory to explore student thinking and build activities that seek to address common difficulties.

Today, the authors recognize IBL and IOI as more similar than different. Their goals in this research commentary were to offer ways to see complementarity in current IBL and IOI approaches (as opposed to focusing on historical differences) and to present a research agenda for moving forward. They propose labeling this merged agenda Inquiry-Based Mathematics Education (IBME) and identify four aspirational characteristics. The first has been the core requirement of the Moore Method, IBL, and IOI:

**Students engage deeply with coherent and meaningful mathematical tasks**.

This falls directly in line with the definition of active learning adopted and promoted by the professional societies in the mathematical sciences, “classroom practices that engage students in activities such as reading, writing, discussion, or problem-solving, that promote higher-order thinking” (CBMS, 2016).

The second requirement diverges from the Moore Method but is firmly rooted in educational research into big tent IBL:

**Students collaboratively process mathematical ideas**.

The third requirement is the piece that has not been an explicit pillar of IBL research but has been a practice taught in the IBL workshops for a long time. It is central to IOI:

**Instructors inquire into student thinking**.

IBL as practiced today goes a great way toward equalizing opportunities for students from underrepresented groups without harming the performance or prospects of students who have benefited in the past from traditional modes of instruction (Laursen et al, 2014 and 2016). Both IBL and IOI researchers and practitioners are taking up the national charge for equalizing mathematical opportunity. Laursen and Rasmussen see this as more than a happy coincidence. It should be one of the driving rationales for the adoption of IBME:

**Instructors foster equity in their design and facilitation choices**.

IBME has enormous potential to improve mathematics instruction. It seeks to bring together the community of practitioners who have discovered student-centered and active learning as a means of improving the outcomes of their students with the community of researchers who seek to further our understanding of which approaches should be most successful for which students. For a sense of what this community looks like today, there is no better illustration than the page of IBL workshop leaders at http://www.inquirybasedlearning.org/workshopleaders.

Read the Bressoud’s Launchings archive.

**References**

Boyer, E. L. (1998). *Reinventing undergraduate education: A blueprint for America’s research universities.* Stony Brook: The Boyer Commission on educating undergraduates in the research university. https://files.eric.ed.gov/fulltext/ED424840.pdf

Conference Board of the Mathematical Sciences (CBMS). (2016). Active learning in post-secondary mathematics education. https://www.cbmsweb.org/2016/07/active-learning-in-post-secondary-mathematics-education/

Haberle, Z., Laursen, S. L., and Hayward, C. N. (2018). What’s in a name? Framing struggles of a mathematics education reform community. *International Journal for Research in Undergraduate Mathematics Education. *4:415–441. https://doi.org/10.1007/s40753-018-0079-4

Laursen, S. L., Hassi, M. L., Kogan, M., & Weston, T. J. (2014). Benefits for women and men of inquiry-based learning in college mathematics: A multi-institution study. *Journal for Research in Mathematics Education*, 45, 406–418. https://doi.org/10.5951/jresematheduc.45.4.0406.

Laursen, S. L., Hassi, M. L., & Hough, S. (2016). Implementation and outcomes of inquiry-based learning in mathematics content courses for pre-service teachers. *International Journal of Mathematical Education in Science and Technology*, 47(2), 256–275. https://www.tandfonline.com/doi/abs/10.1080/0020739X.2015.1068390

Laursen, S. L. & Rasmussen, C. (2019). I on the prize: Inquiry approaches in undergraduate mathematics. *International Journal for Research in Undergraduate Mathematics Education. *https://doi.org/10.1007/s40753-019-00085-6

**Using Popular Books in Teaching and Learning**

Last semester I organized a book club for my program. I was teaching a Discrete Modeling and Structures course, which attempted combine mathematical modeling with the study of structures for programming. One of the elements of the course is the discussion of the design of programming, and because our program is rooted in Equity as a value, I wanted to give the students the option to think about Equity issues in modeling and programming. I chose Dr. Cathy O’Neil’s book, *Weapons of Math Destruction* as a class reading (also discussed in my last post).

I have often chosen a popular reading book to accompany my class. I am an advocate for using reflection as a tool to promote metacognition around mathematics. In calculus, I would have students read *The Practicing Mind* by Thomas Sturner, to prompt students to reflect on “talent” as being something that is practiced often. Where as some books like *The Math Gene * by Keith Devlin emphasize the intrinsic capability or “nature” within all of us to do math, *The Practicing Mind*, offers a companion tale about deliberate practice as a mechanism to “nurture” it. I credit that idea to Dr. Erin Bodine, who uses the book *Talent is Overrated,* by Geoff Colvin in her calculus course and to my graduate Statistics professor, Charles Cweick, who assigned books like *Freakonomics* by Levitt and Dubner and *Calculated Risks: How to Know When Numbers Deceive You *by Gigerenzer to make the statistics I was learning relevant to my life. When I let the bookstore know that I would be using *Weapons of Math Destruction* as a companion text to my course, they helpfully let me know I was not the only professor to have recommended it for reading - and so the idea of a community book club was born.

The students really took to the book. I think no one was quite sure the first time we met - and we met once a month to tackle a third of the book each time. But by the second meeting, the enthusiasm was readily apparent and by the last meeting students were grappling with how they might fight for the kind of social justice that Dr. O’Neil describes. We also had visitors from all over campus - from not just our program, but from the neuroscience program, mathematics department, and our academic support centers in writing and mathematics. My class was a bottom up sampling of discrete modeling and structure, which only took students as far as 2 dimensional nonlinear difference equations and basic network analysis. This book was a top down approach, showcasing the broader impact of programming design and modeling with data on real lives. It provided the kind of real life connections that motivated students to do delve deeper.

So - if I could have interviewed Dr. O’Neil, what would this community ask her? What would I have asked her? What would my students ask? Imagine my excitement when I realized that this year’s Mathematic-Con SIAM-AMS-MAA Gerald and Judith Porter Public Lecture at the Joint Mathematics Meetings would be delivered by none other than Dr. O’Neil herself! Imagine my shock when the MAA connected me to her so that I could ask her those questions that had been burning in my mind towards the end of the semester. Instead of transcribing the entirety of the interview, I have chosen some highlights to share with you. Passages in ** are my own reflections to her words.

Cathy O’Neil and Carrie Diaz Eaton at the Joint Mathematics Meeting in Baltimore 2019

**MAA Math Values: Community, Inclusivity, Communication, and Teaching & Learning**

To me, I saw the book club as an exploration of all of the principles of Math Values, developing community among students and faculty in a new program, examining issues of equity, inclusion and justice, and encouraging communication in the teaching and learning environment. I thought readers of this blog might be interested in her perception of these same math values.

**Diaz Eaton: **When I read your book, I felt that it really resonated with the Math Values: Community, Inclusion, Teaching and Learning, and Communication. Do you feel like those values resonate with you?

**O’Neil: **I really don’t distinguish between “communication” and “teaching and learning...” those are the way you communication. If you value communication… you have to learn to teach, so that’s the same idea for me. Community and Inclusion also mean the same thing to me. To have a vibrant, growing, modern community, is to be inclusive, is to find ways to include - which of course means always to be on the lookout for better ways of communicating, better ways of learning, better ways of teaching. The training I got when I became a mathematician was really good for all of these things... In my education, I learned how to not be defensive about being wrong. Mathematicians ideally get to that point - where if I try to teach you something, and you find a mistake, I appreciate it. I am grateful, because it has taught be quickly, it has skipped over a bunch of time I would have wasted, it has moved me faster. And that is what a good community does. It doesn’t simply include, a good community progresses - progresses even through things that are hard and things that are uncomfortable. The idea is I like to just go through that discomfort rather than try to around it, because there is no way around it.

Community is the only thing that mathematics has. Mathematics is not intrinsically a thing. Mathematics is defined by what the community of mathematicians think is interesting and worthwhile. We do a relatively good job, not perfect job, of doing that as mathematicians. If we focus on a little more on that, on curating explicitly and openly why we think this thing is interesting, why we think this unsolved math problem would be good to know, without relying on fake explanations - because the truth is at the end of the day it’s because we think it is beautiful, this is why. If we could focus on that - acknowledging that we don’t really have any argument beyond we think it is beautiful, but spending a little more time on explaining why we think it is beautiful and why we think this is interesting and worthwhile - then our community would be better off and we would be more inclusive. And we would have spent more time educating and learning, so I think community is the first, but it is only thing that matters.

*My own commentary and reflection, I’ll bracket by an asterisk from now on: From my perspective, I think what I hear is “if you are doing Community right, you are doing the others,” which I wholeheartedly agree with. I was surprised that she centered mathematics itself as a human enterprise and therefore completely centered in community. But this is a useful framing, because it might be easy for some to strip the human element from mathematics, but to remind us of its human-constructed origin, reminds us that the two cannot so easily be untwined.*

**Diaz Eaton: **This is interesting because I hear some similarities in something I was trying to explain to someone else, which is being inclusive in terms of mathematics versus being inclusive in terms of mathematicians. What are we really talking about here - are we talking about the mathematics or are we using it as a proxy for talking about the inclusivity of mathematicians?

**O’Neil:** The fact is different parts of the community will consider different things beautiful and they should have their say, so that is a form of inclusivity. Just like, I’m a classical musician fan, but I’m going to listened to Cardi B - which I did yesterday, and I actually didn’t like it - but I want to be the kind of person that actually listens. I want to try to like new things. And mathematicians should be like that too - and this is my opinion - but we shouldn’t be snobs about things because that is self-defeating and it narrows and restricts us.

**Diaz Eaton: **Especially as fields like data science emerge?

**O’Neil:** There’s no proof in data science - there is evidence… It is not a thought experiment, data science is empirical, which is good and bad. The good thing is it has real applications and power. And the bad thing is, it has lots of power, and people are often measuring the wrong thing to see what is the consequence of that power. They are measuring their profits rather than ensuring social justice, so it is an unconstrained potential science - not a science yet, but it’s not mathematics. When I do data science, I do thought experiments. Mathematical thinking makes headway in data science, but what I do as a data scientist is not math, but the way I decide what to do is math.

*I think this is good dialogue for us all to have, because we should be expansive in the ways we approach the inclusivity of mathematics, but then it is healthy to consider both sides of the argument when it comes to considering whether what we do is mathematics or not. I see Dr. O’Neil engage in both sides of that debate. I personally tend to be as inclusive as possible about my definition of mathematics, and as long as a domain involves mathematics, I am happy to say it is part of the mathematics family. Perhaps this is a cultural perspective too - all my relatives are just “cousin” and it does not matter whether 2nd or twice removed, and all are family.*

**On her book, Weapons of Math Destruction**

**Diaz Eaton:** In your book, you use a lot of examples of the everyday things, motherhood, the everyday tasks, the shopping. So who were thinking about as your audience - who were you writing this to, and how did that mold and shape the book?

**O’Neil:** I wasn’t trying to go for only nerds, I was trying to go for people who aren’t nerds and consider themselves unqualified. We are cowed and overly trusting of algorithms because we are not experts in math and science, but we are definitely smart enough. We need people, humans, to basically understand what they are all about and ask probing, tough questions. Because at the end of the day, we find out they are working in our best interests, because we need to educate ourselves - not about the math, it’s not a math test. I consider it much more of a political fight. So it is a question about what is fair. And I don’t think anyone is like: sorry I’m not qualified to talk about what is fairness. Plenty of people say they are not qualified to talk about the math. The real damage is being wrought by the marketing and arrogance that allows us to consider this the realm of technical knowledge rather than the realm of questions of fairness. When it is affecting people like it does, and I’ve demonstrated in the book how much it does, it can no longer be an expert realm. It’s just not okay.

**Diaz Eaton: **What is the bridge that gets the students that don’t have the technical knowledge but want to make contributions to this field?

**O’Neil: **There is no bridge, unless they want to get a job in building algorithms. This is ultimately a question of solidarity and activism. But it is really really tricky. Teacher unions can do something about the teacher value-added model. And the Chicago teacher union strike was an example of that. I suggested they could hire me as an expert witness that might testify in court. That is what it looks like to fight this. It doesn’t happen at the individual level, it happens at the union level or the organizing level. But how do you organize everyone that got rejected from a job unfairly? They don’t even know they got a rejection unfairly and they don’t know everyone else who got rejected unfairly either. How do you organize? How do you unionize? So you can’t. So you have to talk at the society level about how data is used against us. The good news is that politicians are starting to think about this. There was a Bloomberg article just last weekend, where he talked about my book and that we need to start thinking about how the law should be employed when algorithms are involved. There are laws that are being bypassed by algorithms right now. If regulators got on that and figured out how to enforce their own laws - they should hire me as a consultant, by the way (*laughs*). That would be great, and it would also pave the way to see what other kinds of things we can demand.

*At this point, I am pretty excited that she brought up collective action as a key part of the social justice and equity challenges mathematicians, statisticians, computer scientists, and data scientists are now embroiled in today - whether we have yet accepted that ethical responsibility or not. It was important to me that she validated this type of work.”

**Diaz Eaton: **This is great because the idea of collective action is part of another course I am teaching, because we are helping to co-organize this conference called “Bringing the Conversation of Inclusivity and Data Science to the Ecology and Environmental Science Community.” And I was just at the Data 4 Black Lives conference this past weekend (link prior post here).

**O’Neil: **Yes, I went to that last year.

**Diaz Eaton: **The side by side work of community organizers and people who had the power of data at their disposal was incredible. We are not getting as close to that as what that could look like. Do you think that is the model for moving forward?

**O’Neil: **Yes, absolutely. I am writing a new book. It’s about shame as a social mechanism. It is not a math book, but I talk about *Weapons of Math Destruction*. I talk about math shaming - the power that is wielded when people say it is math, you won’t understand it - and people give up their rights, because they are like “Oh it is math, I won’t understand it.” That’s shame. And it is a powerful and potent weapon. And the only way to disrupt that weapon is by organizing. I have this framework of punching up shame versus punching down shame. What we need more is punching up shame, which is essentially solidarity and activism. Putting a spotlight on bad uses of data and preventing punching down shame, whether it is in the form of Jim Crow Laws, in the form of food stamp work requirements, or if it is in the form of automatic cessation of welfare checks, because an algorithm decided you were a leech. Those are all examples of punching down shame, and the only way you can punch up against them is through action by broad-based grassroots solidarity and activism.

**Diaz Eaton: **Hey did you see the AWM shirt? (I show off the shirt I am wearing that I bought the previous day at the AWM booth and read it to her) Change, Group Action, Unity, and Equality.

**O’Neil: **Love it.

**Diaz Eaton: **Isn’t it great - I bought two

*And then we selfied the occasion to immortalize it on Twitter forever.*

Data Science is taking the world by storm. Are we preparing our students for this rapidly growing industry? MAA’s DUE Point blog sat down with StatPREP principal investigator Doug Ensley to discuss this five-year project to help faculty incorporate data-centered components in introductory stats courses.

**Q: What are the main goals of the StatPREP project?**

First, we want to convince instructors to use modern software tools instead of a graphing calculator for data analysis in order to facilitate use of larger and more realistic datasets in assignments and on exams. Second, we want to help mathematics instructors without statistics training to understand the importance of teaching the introductory course using statistical habits of mind.

Our project creates local professional development programs for (especially community college) faculty at 10 different sites around the country, with the goal of creating regional hubs for innovative teaching of introductory statistics.

**Q: Who is involved in the StatPREP project?**

StatPREP partners the Mathematical Association of America with the American Statistical Association and the American Mathematical Association of Two-Year Colleges. Danny Kaplan (ASA) and Kathryn Kozak (AMATYC), along with Jenna Carpenter and Mike Brilleslyper from the past MAAPREP program lead the project.

**Q: What is your hope for the impact of this project at the end of the five years?**

The recent flood of jobs in data science creates a demand for mathematics programs to modernize statistics instruction. Five years from now data science programs will be common, and the first statistics course will need to be aligned with such programs. Intro stats is already a large part of faculty load at four-year and two-year colleges, especially among part-time instructors. We hope professional development in data science methods will be commonplace and StatPREP can provide an evidence-based model for these programs.

**Q: What unanticipated challenges have arisen since you wrote the proposal?**

The greatest surprise has been the demand for this sort of training. The first workshops had barely finished when people starting inquiring about how the resources can be accessed online. Our proposal focused almost exclusively on the dynamics of regional community hubs, so we will have to think about how to meet the demand beyond the planned sites. It is hard to complain when the success of the initial idea creates the biggest challenge!

**Q: NSF grants seem very competitive, so what do you think it was about your project that was unique or innovative enough to get funding?**

I believe there were several key reasons. First, the project has formal endorsement from AMATYC and ASA, and it is run by the MAA. These endorsements give reviewers confidence that the project teams will have the resources (like access to top presenters and a robust network for recruiting participants) to execute the program. In this case, we also had the benefit that the MAA ran the NSF-funded PREP programs for ten years, so there is a long history of executing the logistics of distributed professional development workshops. Finally, and most importantly, the project addresses a clear need for math curriculum to adapt to the growing field of data science.

Editor’s note: Q&A responses have been edited for length and clarity.

**Learn more about NSF DUE 1626337**

Full Project Name: Professional Development Emphasizing Data-Centered Resources and Pedagogies for Instructors of Undergraduate Introductory Statistics (StatPREP)

Abstract: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1626337

Project Website: http://www.statprep.org

Project Contact: Jenna Carpenter, carpenter@campbell.edu

For more information on any of these programs, follow the links, and follow these blog posts! This blog is a project of the Mathematical Association of America, produced with financial support of NSF DUE Grant #1626337.

*Audrey Malagon is lead editor of DUE Point and a Batten Associate Professor of Mathematics at Virginia Wesleyan University with research interests in inquiry based and active learning, election security, and Lie algebras. Find her on Twitter **@malagonmath**. *

This week’s indictments of prominent, wealthy individuals for illegal actions as they sought admission for their children to elite colleges and universities brings national attention to inequity in access to higher education. While the extreme (and illegal) acts at the center of these stories are outliers, the underlying narratives, and the ways in which these particular acts occurred at the fringes of the (legal) college admissions counseling industry, serve as reminders that wealth, power, and privilege are still key drivers of the makeup of freshman classes across the U.S.

It’s a fortuitous circumstance that the current (Feb/March 2019) issue of *MAA FOCUS** *is devoted to issues of equity and inclusion in the mathematical sciences community. I hope that you will read the issue carefully, and consider how the questions our colleagues wrestle with there are related to the admissions scandal.

The theme that came to me across numerous articles is the need to challenge norms and biases within the mathematical profession that serve to maintain power structures across society more broadly. Mathematical knowledge and the ability to deploy quantitative skills in varied settings privileges those who have that knowledge and skills. How do we, then, carry out our professional responsibilities in ways that serve to facilitate growth in knowledge and skills? Mathematics courses are often viewed as barriers to students succeeding in higher education, but what about our responsibility to encourage students to develop mathematical capacity to succeed in their lives?

The recent completion of the MAA Instructional Practices Guide makes a strong case for our responsibilities. In their introduction, the authors write:

Inequity exists in many facets of our society, including within the teaching and learning of mathematics. Because access to success in mathematics is not distributed fairly, the opportunities that accompany success in mathematics are also not distributed fairly. We in the mathematical sciences community should not affirm this inequitable situation as an acceptable status quo. We owe it to our discipline, to ourselves, and to society to disseminate mathematical knowledge in ways that increase individuals’ access to the opportunities that come with mathematical understanding.

I’m going to go out on a limb and extend this call for change. To be sure, our educational system is built on a framework of assessment that has many influences. But it’s not hard to make a case that the eugenics movement, and subsequent efforts to measure human intelligence, have had profound influences on the way our society thinks about competence, how we reward performance, and ultimately the values we place on different populations. Mathematics has been used as a tool to implement, and even to justify, these efforts.

We’re now living in a world where artificial intelligence, which depends entirely on mathematics for its implementation, looms as a key driver of change in the daily lives of our citizens. Perhaps it’s time we reflect on not only how we teach, but what we teach, and how we assess our students, to ensure that as we move towards an uncertain future, we can more effectively contribute towards reaching our vision of a society that values the power and beauty of mathematics and fully realizes its potential to promote human flourishing.

]]>International Mathematical Olympiad (IMO) is the World Championship Mathematics Competition for high school students. The first IMO was held in 1959 in Romania and only 7 countries participated that year. Now, the competition has expanded to over 100 countries spanning major regions of the world. The team from the United States won the first place title in the most recent IMO, the 2018 International Mathematical Olympiad and also won the title in both 2015 and 2016.

Dr. Viorel Barbu, a participant in the first IMO, who has become President of the Mathematics Department at the Romanian Academy brilliantly wrote that “Mathematics has always been a fresh and dynamical field of human creativity and a fundamental science to the benefit of scientific knowledge and technical achievements. It is the role and duty of young mathematicians to bring and develop new ideas and to construct new bridges between mathematics and other scientific fields.”

I have always wondered about the contribution of IMO participants to the field of mathematics and science overall. I came across this fascinating research from Dr. Agarwal and Dr. Patrick Gaule. These researchers analyzed data examining the career and scientific output of participants who competed and performed well in IMO over a 20 years period. This research points to a very positive correlation between the points scored at the IMO and the mathematical knowledge produced, which was measured by the number of mathematical publications and mathematics citations. It also proved that students who performed well on IMO are more likely to become professional mathematicians, measured by getting a Ph.D. in mathematics.

I found some really interesting observations in the research, listed below:

–*Strong performers at the IMO have a disproportionate ability to produce frontier mathematical knowledge compared to PhD graduates and even PhD graduates from elite schools.*

*-The conditional probability that an IMO gold medalist will become a Fields medalist is two order of magnitudes larger than the corresponding probability for of a PhD graduate from a top 10 mathematics program.*

*-Dr. Maryam Mirzakhani, who passed away at a very young age, was an IMO gold medalist with a perfect score, and the first woman to win the Fields medal, the most prestigious award in mathematics. Terence Tao received a gold medal at the 29th IMO and went on to win the Fields medal and is one of the most productive mathematicians in the world.*

*-Around 22% of IMO participants have a PhD in mathematics; of those, around a third have a PhD in mathematics from a top 10 school (7% of the total IMO participants). 1% of IMO participants became IMC speakers, and 0.2% became Fields medalists.*

This research paper clearly articulates the contributions of IMO participants to the field of mathematics. This paper gives strong reason to encourage everyone to participate in math competitions beginning in elementary school, and through college, as problem-solving skills acquired through participating in math competitions have long lasting positive effects that helps you whether you pursue a professional or academic career.

The last time a female qualified for the IMO from the United States was in 2007 and 3 female US students have scored medals at IMO. Their mathematics career and contributions validate the research findings. Sherry Gong represented the United States in 2005 and 2007, winning a Gold Medal in 2007. She famously scored over a 100 in Harvard’s problem solving course, Math 55, and went on to get her Ph.D. at MIT in mathematics. Alison Miller represented the United States in 2004 and also won the Gold Medal. Alison Miller studied mathematics at Harvard and finished her Ph.D. in mathematics at Princeton University. Melanie Wood represented the United States in the 1998 and 1999 IMO and won Silver Medals in both years. She was the first female to qualify for the IMO from United States. She completed her Ph.D. in 2009 at Princeton University and is currently a Vilas Distinguished Achievement Professor of Mathematics at the University of Wisconsin.

Rachel Levy, MAA Deputy Executive Director and Francis Su, past MAA President

I have been thinking lately about reports of microaggressions at math conferences and in the classroom. These can take the form of thoughtless comments that might make someone feel unwelcome. Thoughtless because maybe the person didn’t think what they said would be a problem or perhaps they didn’t recognize possible negative impacts before speaking. Micro because certainly if the person intended to cause harm, it would be a plain old aggression.

We all make mistakes. We all sometimes cause pain with our communication. We need ways to work through these situations without causing undue burden, especially to those feeling the injury.

We also want people to feel free to interact, to ask questions, to learn and grow, to make mistakes. To repair relationships that have gone awry. We want a variety of opinions to be welcome. We want to promote respectful discourse among people who disagree.

As a mathematical modeler, I engage in respectful discourse by working to understand people’s assumptions, objective functions and logic. In her MathFest talk, Eugenia Cheng used category theory to analyze human perspectives. Megan Squire used data mining in her 2016 paper “Differentiating Communication Styles of Leaders on the Linux Kernel Mailing List.” Notably, in September 2018, Linus Torvald, identified in the study as one of the most offensive communicators, publicly recognized the issue and stated that he would take time away from online communication to work on his issues. Mathematics can help us understand each other and ourselves.

Mathematical modeling provides me insight about other people’s points of view because I can recognize what assumptions are being made. This helps identify why we disagree and how we each developed our beliefs, values, conclusions or opinions. When I warmly and respectfully listen to people who have fairly different political or religious views, they usually reciprocate and ask for more conversation, even when the conversation is uncomfortable. This includes conversations about mathematics and pedagogy. I value these conversations. I want to know where people are coming from. I generally learn more when our perspectives differ.

This leads me to humor, because humor (like politics, religion and sometimes research) has a way of bringing people together or dividing them. It provides a mechanism for raising uncomfortable issues. It can build and release tension among a whole crowd in a remarkably short time. It can help us see ourselves in a new light.

And yet, much humor has been based on laughing at someone’s expense. Think about late-night humor that picks on celebrities and politicians. Think about slapstick humor that might involve a pie in the face, a slip on a banana peel, a coyote falling off a cliff, or a crack in the head with a rotating piece of lumber. Think about self-deprecation of a comedian who makes fun of their own culture, relationships or misfortune. We laugh at other people’s pain, while we flinch or cringe as we recognize our own. As a child watching TV, I remember wondering, is this cruelty and violence the main way to make people laugh?

We don’t want to lose humor at our conferences, but when we make jokes in talks or in writing, we must be aware of whether the humor functions at someone’s expense. With this in mind, I found it very challenging to create cartoons for the BIG Jobs Guide.

The way (kind not mean) humor was valued attracted me to work at Harvey Mudd College. I learned humor was part of HMC from the start via founder Joe Platt, who was quite the prankster and wrote silly nerdy songs. More than 50 years later, pranking is still part of the tradition of the college, with rules about how to opt out of being pranked and the requirement to self-report if a prankster doesn’t follow the rules. These include leaving info about who did the prank, and how the pranksters can be contacted to do any necessary clean up.

A shared laugh about who should be predator (chickens) and who should be prey (velociraptors) in my differential equations teaching talk was probably an important factor in getting my job in the Mathematics Department. The funny included chalk drawings of the VLCs and VSVs (very large chickens and very small velociraptors) to show off their relative size and my marginal art skills, along with comments about the 13 chickens that I had at the time. For example, I had a rooster that one of my daughters named "sweet pie brownie" (who was not at all sweet). The fact that humor helped land a job is amusing because in general, I am pretty sure I am not funny, although puppeteer Paul Zaloom says I should develop a growth mindset about that.

I encourage you to thank and appreciate anyone who brings kind humor into your work and recreation. I feel lucky that I still get to cross paths with punster Francis Su in my new gig at the Mathematical Association of America! In the pic above we are giving talks on the same day in Feb 2019 at the Louisiana / Mississippi section meeting. He is a model for me of someone always ready for a smile and a laugh without causing pain or joking at someone’s expense. I am looking forward to the release of his book on human flourishing, and its potential to help our mathematics community take a look at our microaggressions and affirmations, our use of humor, and our strong desire to build an inclusive community.

At the section meeting, in a Section NExT Instructional Practices Guide led by Gulden Karakok the participants shared kind humor, engaged in respectful disagreement, and developed strategies to welcome more students to mathematics. These themes echoed again in a panel (really a town hall meeting) with section members and Mathematics Magazine Editor Michael Jones, MAA Executive Director Michael Pearson and myself. We heard longtime members talk about the history of desegregating the section meeting on the Louisiana shores even before it was legal to do so. We heard section leadership concerned that we increase participation in general, and in particular by members from underrepresented groups. I was honored to be present for that conversation, and by the warm welcome in my first experience as Section visitor as Deputy Executive Director. Thank you in particular to Jana Talley, who oriented me to the meeting and did heroic late night and early morning airport runs and Judith Covington, who kicked off the new NSF-funded Get the Facts Out project!)

]]>Last year, I was checking what’s trending on Twitter (@mathprofcarrie) and started seeing a lot of really cool posts from individuals I respect, such as Dr. Piper Harron (@pwr2dppl) and Dr. Cathy O’Neil (@mathbabedotorg), being retweeted by my community. They all had the same hashtag: #data4blacklives.

Let me back-up: Twitter and other social media platforms have the power to amplify conversations (in good ways or bad). I use it as a personal learning network (PLN) to become aware of opportunities, to keep connected in two-way conversations with people from afar, and to spread the news about what I find in my circles. Learning about this conference from my network is exactly the reason why I use Twitter as my PLN. What was this conference, and what was going on?

Data 4 Black Lives is a social movement, the product of MIT’s Yeshimabeit Milner’s grand vision for interrupting the impact that data and algorithms are having on systematically marginalized people. Cases were investigated by Dr. Safiya Noble’s book, *Algorithms of Oppression*, which documents the ways in which Google’s search, ranking, and monetization algorithms shape narratives about black girls. Consequences were also well explored by Dr. Cathy O’Neil’s book, *Weapons of Math Destruction*, in which algorithms designed to determine credit rates have used zipcodes of applicants as part of a riskiness algorithm - a system in which the most impoverished are hit with the highest loan rates and fees. It was established by MIT’s Joy Buolamwini, who found that the facial recognition system of top companies, a technology now employed by US police departments, worked extremely well at identifying and distinguishing white men, but systematically misclassified people of color and women, and particularly women of color.

So I signed up for the mailing list and registered as soon as registration was open for this year’s conference, and went with my new Digital Humanities colleague, Dr. Anelise Hanson Shrout, to MIT. Why? Well the first reason is that I am co-organizing an NSF INCLUDES-funded conference this April called “Bringing the Conversation of Inclusion and Data Science to the Ecology and Environmental Science Community,” so I felt like I needed to be learning from the leaders in the field. Second, I am a new faculty at Bates College this year. I was previously a mathematics professor at an environmental college and was recruited as an expert in interdisciplinary curriculum design to help build a new program in Digital and Computational Studies, in a way that embraces community and inclusivity as a cornerstone (just like the MAA Math Values!). Third, I suffer from a serious case of #FOMO (fear of missing out), and let me tell you - seeing all of the really cool Twitter #data4blacklives chatter made me what to be there to experience it for myself.

It would take me a hundred posts to tell you everything about this experience, so somehow I’ll try to hold myself back, and instead share with you the highlights from my Twitter feed. Disclaimer - Tweets sometimes reflect my own thoughts, often in response to the amazing speakers I was hearing, so I want to give credit also to everyone I heard from even if it was not an explicit quote.

In addition to being inspired and tweeting out what I was learning, I want to share one of the most significant realizations for me. This conference was not just academics coming together and trying to figure out how to improve some “othered” community. This conference invited community organizers alongside the data scientists - and not just to be data scientists or be more literate, but to collaborate with them, doing the community organizing that was also equally important.

As a network scientist and community organizer myself (of professional academic communities), I admit this was a major blindspot, even though this is not the first time I’ve heard this message. I have just never experienced how that could change everything, and ashamedly needed to have that experience to really understand. We often invite academicians with “lots of important papers” into our conversations. When was the last time we invited half our audience or more to just be the people in our local community to do the community organizing work for advocacy that they are best at? To ask them what they need to know from the data so that they can take action. To let them tell us where we are failing in terms of policy? And to trust them to take that information and create change.

As I write that out, I am reminded of the same tension that has occurred between mathematics and biologists - with mathematicians using unmeasurable parameters or making unrealistic assumptions that could lead to misleading outcomes. On the heels of trying to figure out what it looked like to be a boundary researcher - a true interdisciplinary mathematician, we were introduced to the idea of “team science” and “sustainability science” which suggested that complex problems need to involve all stakeholders and the lenses of multiple disciplines. That is a philosophy that seems to embrace the MAA Math Values: Community, Inclusion, Teaching and Learning, and Communication. To what extent do we practice these values and to what extent do we all have blind spots?

]]>AAAS gets my vote for the best ribbons to attach to attendee badges!

This year’s annual meeting of the AAAS was held in Washington, DC, in mid February. Since it was only a few blocks from MAA’s offices, I decided it was a great opportunity to listen to colleagues from outside the mathematics community and, in particular, look for ways that mathematics is relevant to current issues that affect all of us.

Not surprisingly, there were a number of sessions that, broadly speaking, fall into the “big data” category. The release by the White House of the American AI Initiative on the same week as the AAAS meeting makes this topic even more important for us to address. The AI Initiative calls in a broad sense for the U.S. to maintain leadership in the development and deployment of AI across all sectors of business, industry, and government. Relevant for those of us concerned about post secondary mathematics education, the report calls for us to “train current and future generations of American workers with the skills to develop and apply AI technologies to prepare them for today’s economy and jobs of the future.”

Of course that is extremely broad, and most MAA members will intuit that undergraduate mathematics must be central to the training of future AI professionals (even if that designation is not well-defined). However, at the AAAS meeting, I was struck by two basic ideas that relate active and deep research to undergraduate mathematics. Now I’d have to do a lot of hand-waving in any discussion of the underlying mathematics, but I’ll leave it to others to do the necessary work to make deeper sense of my observations.

First, there were sessions on deep learning and neural networks. As I understand it, a neural network is a realization of the Universal Approximation Theorem. Basically any function (think “input-output” in the broadest sense) can be approximated arbitrarily closely as a (finite) sum of just about any non-constant function, with appropriately chosen parameters as coefficients in the sum, and both shift and scaling factors.

Approximation of data is something we routinely study in undergraduate courses, from polynomial approximations (think, e.g., of Taylor polynomials), to more complex approximation using trigonometric series and orthogonal polynomials, and moving on to interpolation and splines on data sets, to least squares approximations of noisy data.

In fact, the basic study of polynomials that we usually begin in high school algebra classes seems to me to be justified (if it is -- another topic) almost solely because they serve as such a convenient class of basic approximators. I wish someone had explained this to me when I was in high school. I had to intuit this in much later studies of PDEs and harmonic analysis (though yes, I should have made the connection the first time I saw Taylor series and the results around convergence of the same!).

A second topic that really intrigued me is differential privacy. The session I attended first told the unfortunate tale of the 2010 census data being reconstructed at the record level by running the publicly-released data against other publicly available databases (e.g., of names and addresses). This is definitely something the Census Bureau wants to avoid repeating, but at the same time it’s critically important that these large public datasets are made available for appropriate analysis and research. This work drives public policy decisions that affect all of us.

In undergraduate statistics courses, it’s fairly common to deal with messy datasets, and, assuming one knows something about the kind of noise that has corrupted the data (e.g., gaussian or biased because of the underlying mechanism used to record the data), there are methods for analyzing the data and measuring the confidence in the conclusions drawn from that analysis.

Differential privacy is in some sense the reverse of this. Starting with the clean (census) data, noise with known properties is used to perturb the dataset in such a way that large-scale analysis can effectively be carried out, and conclusions confidently drawn, while also safeguarding against reconstruction of record-level data.

It’s an exciting time to be in mathematics. Through MAA’s Preparing for Industrial Careers in Mathematics project and our participation in the National Academies Roundtable on Data Science Postsecondary Education, as well as other initiatives and partnerships, I expect MAA to continue to serve as a source for our community to find effective ways to engage in these important issues.

]]>“What is creative mathematical thinking?” That’s the question I set out to answer last month. The discussion got this far: Creative mathematical thinking is non-algorithmic mathematical thinking.

The question arose when a long-time friend (and former teacher) from the ed tech world and I had an email exchange, prompted in part by the publication of a LinkedIn survey of industry leaders which ranked creativity as the number one skill they look for in employees.

The online magazine EdSurge picked up on the LinkedIn survey results to conduct its own (informal) survey of various thought leaders in different domains (film, writing, teaching, museums, and technology companies of different sizes), asking, “Is creativity a skill (that can be developed through practice and repetition)?” They published the results in the January 21 issue.

The answers given ranged all over. An associated Twitter poll EdSurge came down slightly in favor of “yes.” None of this is scientific, of course. The relevant takeaway is that professionals in different areas for whom creativity is a relevant notion do not agree as to what it is. (Nor did my ed tech friend and I.)

Moreover, the EdSurge survey was by no means specific to mathematics. Indeed, the only responses that came close having particular relevance to mathematics or mathematics learning were acclaimed teacher and *Moonshots *author Esther Wojcicki’s view that creativity is not a skill but a mindset, and Google Education Evangelist Jaine Casap’s observation that:

“[Creativity is] embedded in problem-solving, for example. You must use creativity to think of new ways to define and solve problems. Creativity also separates us from machines or robots. For example, an algorithm is a prescribed process, a pattern of commands a machine (or technology) follows. A human can look at issues from a variety of angles—in a nonlinear way! Creativity can be the ‘how’ part of problem-solving.”

None of those asked gave a definitive answer to the question as to whether creativity could be objectively measured. For my ed tech friend and I, however, leaving the question unanswered was not a viable option. We wanted to know if it were possible, in principle, to develop digital tools that developed creative mathematical thinking and measured it. We needed a definition. It did not have to be “the correct definition.” That seems out of reach given where we all are today, if indeed there is a definitive, clean, concise answer. But is there a notion of “mathematical creativity” that (1) makes a reasonable claim on being referred to by that name, (2) can be implemented in a digital math learning tool, (3) is developed by engaging with the tool, and (4) permits automated assessment by the tool? As long as the notion is easy to understand and clearly specified, such tools could be built. Everyone would know exactly what skill or ability (or mindset, etc.) is being developed and measured, and researchers could take on the task of determining how the defined notion and its implementation compare with other learning outcomes and metrics.

As it turns out, there is such a notion, which had been doing the rounds since the early 1990s. Before I say what it is, it’s probably a good idea to watch (or, re-watch) two excellent TED talk videos on creativity by Sir Kenneth Robinson: His talk Do schools kill creativity? given in Monterey, CA, in 2006 [SPOILER: The answer is “yes”] and the sequel Bring on the learning revolution!, given at the same venue in 2015.

Most people I have talked to about creativity have already seen those videos, and agree that Robinson is absolutely right in saying that creative thought comes naturally to humans, with young children exhibiting seemingly endless creativity in all manner of domains. Anyone who has spent any time with young children, as parents, teachers, or whatever, has surely observed that. But as Robinson correctly, and eloquently, observes, systemic education tends to drive the creativity out of them.

In the case of mathematics education, creativity is suppressed by the adoption of an excessive focus on the mastery of basic algorithmic skills. To be sure, mathematics educators could, until recently, defend that emphasis by pointing to the crucial need to master calculation—a need that lasted throughout the three millennia period up until the 1990s, when calculation was a crucial life skill but there were no machines to do it for us.

ASIDE: While that defense has some merit, I find it hard to accept that the need for calculation “drill” meant the almost total suppression of creative mathematics. “Drill of skill” turned into “drill and kill”—the precious commodity killed being any interest in mathematics as a pleasurable mental activity. There was never an either-or choice; time could have been devoted to engagement with creative mathematical thinking.

Be that as it may, with Robinson’s talks fresh in my mind from an N’th re-watch, I went back and looked at the one notion that, by and large, mathematicians had agreed was a reasonable first definition of mathematical creativity. (At least, the relatively few mathematicians who had spent some time trying to come to grips with the elusive concept so agreed.)

That notion has a history going back to the 1940s, which seems to be when mathematicians, mathematics educators, and philosophers first started to reflect on the issue, of particular note among them being Henri Poincaré (1948), Jacques Hadamard (1945), and George Pólya (1962).

**Mathematical creativity – a definition**

The definition mathematicians and mathematics educators settled on is very much along the lines of the

*mathematical creativity is non-algorithmic decision making*

we eventually arrived at in Part 1 of this post.

Taking that general idea as a starting point, Gontran Ervynck, an educator in the Faculty of Science at the Katholieke Universiteit Leuven, in Belgium, came up with a definition (Ervynck 1991) of mathematical creativity that I personally find productive (as do many others).

I’ll elaborate a bit about the background to Ervynck’s contribution later, but first let me cut to the chase and present his definition. I should, however, preface it by noting that he was trying to define creativity in advanced mathematical thinking. What I find attractive, however, is that his definition distills mathematical creativity to an essence that works equally well for learners of all ability levels, both for learning and assessment. Moreover, that notion could be implemented in digital learning tools.

Ervynck approached mathematical creativity in terms of three stages of mathematical competence (Ervynck 1991, pp.42-43):

The first stage (Stage 0) is referred to as the *preliminary technical stage*, which consists of “some kind of technical or practical application of mathematical rules and procedures, without the user having any awareness of the theoretical foundation.”

The second stage (Stage 1) is that of *algorithmic activity*, which consists primarily of performing mathematical techniques, such as explicitly applying an algorithm repeatedly.

The third stage (Stage 2) is referred to as *creative *(*conceptual, constructive*) *activity*. This is the stage in which true mathematical creativity occurs, and consists of non-algorithmic decision making. Ervynck comments that “The decisions that have to be taken may be of a widely divergent nature and always involve a choice.”

Although Ervynck describes the process by which a mathematician arrives at the creative thinking stage after going through two earlier stages, his description of mathematical creativity nevertheless ends up very similar to those of others who have considered the topic of mathematical creativity, such as Poincaré and Hadamard.

I should point out that, in accepting Ervynck’s concept as a working definition of mathematical creativity, mathematicians and mathematics educators are really taking the word “creativity” and giving it a specific meaning within mathematics. (Mathematicians do this with everyday words all the time.) In this case, the result is a notion that (1) makes sense within mathematics, (2) makes sense within mathematics education, (3) can be applied to all mathematics learners, regardless of experience or ability, and (4) can be applied to mathematics learners in a graded fashion, based on the nature of the choices they make. In addition, it accords very well with the kind of creativity Ken Robinson talked about in his talks. That’s why I like it so much.

What the definition does not capture, however—at least not directly—is the notion of mathematical creativity that is tacitly assumed when we talk about highly creative people. That kind of population was the focus of Einav Aizikovitsh-Udi’s 2014 study The Extent of Mathematical Creativity and Aesthetics in Solving Problems among Students Attending the Mathematically Talented Youth Program.

While Ervynck’s three-stages concept still applies to exceptional individuals, the essence of creativity that Aizikovitsh-Udi studied involves making *highly unusual choices *that lead to *unusual results *that stand out from most others. The mathematics community as a whole has very little difficulty recognizing that kind of creativity when we see it, just as is the case for exceptional creativity in all other domains. But do we understand it? Do we know how to develop it? Do we know how to measure it?

Regardless of any progress we may one day obtain on those questions, the Aizikovitsh-Udi is interesting as it stands as a study of exceptional mathematical creativity as it exists. Certainly, the goal of the study was not to figure out if that kind of creativity could be effectively assessed algorithmically, by technology or by hand. To do so would presumably require analyzing the sequences of choices that lead to the desired result, but such an approach seems highly unlikely to be successful. Algorithms can identify unusual sequences of steps, but as any research mathematician knows from long and frustrating experience, the vast majority of those unusual sequences don’t work—even if they seem like wise choices at the time.

In contrast, the thought experiment my ed tech friend and I were having was the degree to which technology could develop and measure the (mathematical) creativity *in regular children* that Ken Robinson was talking about. Such a technology, it one were possible, would clearly be a significant benefit to the mathematics education community. I don’t think that is necessarily out of reach. In fact, starting with the Ervynck notion of mathematical creativity, I see real potential to make useful progress. But time alone will tell.

Finally, I promised I’d say something about the history of studies of mathematical creativity that led to the Ervynck definition.

The earliest attempt I am aware of to study mathematical creativity was a fairly extensive questionnaire published in the French periodical *L’Enseigement Mathematique* in 1902. This questionnaire, and a lecture on creativity by Henri Poincaré to the Societé de Psychologie, inspired his colleague Jacques Hadamard to investigate the psychology of mathematical creativity (Hadamard, 1945). Hadamard based his study on informal inquiries among prominent mathematicians and scientists in America, including George Birkhoff, George Pólya, and Albert Einstein, about the mental images they used in doing mathematics.

Hadamard’s study was influenced by the Gestalt psychology popular at the time. He hypothesized that mathematicians’ creative process followed the four-stage Gestalt model of preparation–incubation–illumination–verification (Wallas, 1926). That model provides a characterization of the mathematician’s creative process, but it does not define creativity *per se*.

Many years later, in 1976, a number of scholars interested in the notion of mathematical creativity came together to form the International Group for the Psychology of Mathematics (PME), which began to meet annually at different venues around the world to share research ideas. In 1985, a Working Group of PME was formed to look at creativity in advanced mathematical thinking. The volume *Advanced Mathematical Thinking*, edited by mathematics educator David Tall at the University of Warwick in the UK (Tall 1991), resulted from the work of that group. In Chapter 3 of that book, Ervynck presents his analysis of mathematical creativity.

The PME volume is a mammoth, comprehensive work, full of powerful insights, that I have done no more than delve into from time to time. From what I’ve read (and from what Tall says in his Preface), at the end of the day, we really don’t know how the logically-sequenced solutions and proofs mathematicians write out relate to the mental processes by which they arrive at those arguments. Tall writes (p.xiv):

“[T]here is a huge gulf between the way in which ideas are built cognitively and the way in which they are arranged and presented in deductive order. This warns us that simply presenting a mathematical theory as a sequence of definitions, theorems and proofs (as happens in a typical university course) may show the logical structure of the mathematics, but it fails to allow for the psychological growth of the developing human mind.”

Salutary advice for teachers and students alike.

**Final thoughts**

My take-home conclusions from my discussion with my ed tech friend? With today’s technologies having eliminated the need for humans to master computation (of any kind), learning and assessment have to focus on creative mathematics (as defined above).

Teaching computational skills was relatively easy—albeit too often done in a way that turned people off the subject—and assessment could be done with automation. In contrast, developing and assessing creative mathematics are much more problematic.

Technology may help for the early school grades, say through to middle school, but even then it is likely to be a challenging task to develop systems that work really well, and in my view it’s highly likely that if they do work well it will as supplementary tools dispensed as and when appropriate by an experienced teacher.

As to higher grade levels, I’d look to the considered opinions of experienced mathematics educators and developmental clinical psychologists. They, perhaps informed by conclusions generated by machine-learning algorithms, can certainly have (some) value in terms of identifying creative mathematical talent. Such an approach could be useful in deciding who should be given the benefit of a focused mathematical education and when to conduct an educational intervention for a particular student. Decisions about resources allocation have to be made, and it’s always better to make them with as much information as possible. And from society’s perspective, technology can surely help develop creativity and provide useful measurements of an individual’s creative potential. But at the end of the day, each *individual* decision is at best an educated bet.

In particular, the most dramatic forms of creativity are often missed as such at the time. Georg Cantor’s theory of infinite sets was initially regarded as the wild mental ramblings of a deranged mind; only later was it recognized as a work of creative genius. In earth science, it took fifty years before the scientific community recognized that Alfred Wegener’s theory that the surface of the earth consisted of separate plates, whose drifting led to the formation of today’s continents and were the cause of earthquakes, was a creative explanation having scientific validity–supported by evidence not available in Wegener’s time. And in music, Stravinsky’s *Right of Spring* met a similar fate. Etc.

Leaving creative genius aside, however, I should conclude by acknowledging that these Final Thoughts about the potential for ed tech in the development and assessment of creative mathematical ability, are at present no more than a considered (and somewhat informed) opinion from an experienced mathematics educator. Pass the salt.

**References**

Aizikovitsh-Udi, E. (2014). The Extent of Mathematical Creativity and Aesthetics in Solving Problems among Students Attending the Mathematically Talented Youth Program. In *Creative Education 5*, pp.228-241

Ervynck, G. (1991). Mathematical creativity. In D. Tall (Ed.), *Advanced mathematical thinking *(pp. 42-53). Dordrecht: Kluwer.

Hadamard, J. (1945). *Essay on the psychology of invention in the mathematical field*. Princeton, NJ: Princeton University Press.

Poincaré, H. (1948). *Science and method*. New York: Dover.

Pólya, G. (1962) *Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving*.New York: Wiley

D. Tall (Ed.) (1991). *Advanced mathematical thinking*. Dordrecht: Kluwer (2002 edition available on Google Books)

Wallas, G. (1926). *The art of thought*. New York: Harcourt, Brace & Jovanovich.

Online homework systems have become ubiquitous in introductory mathematics courses. The 2009 AMS survey found that 65% of PhD-granting departments of mathematics, 70% of Master’s degree-granting departments, and 43% of Bachelor’s degree-granting departments were using online homework (Kehoe, 2010). For Calculus I, the 2010 MAA survey undertaken as part of *Characteristics of Successful Programs in College Calculus* (CSPCC, NSF #0910240) showed that of the 429 instructors who answered the questions about their homework assignments, 14% collected no homework. Of those who did grade homework, 46% collected written homework only, 26% collected online homework only, and 28% collected a combination of written and online homework (Sonnert & Ellis, 2015). Incidentally, this last has been my personal policy. Using online homework for procedural knowledge frees me to ask a few penetrating questions on the written homework.

Online homework systems have several advantages over traditional, hand-graded homework. They give the student immediate feedback, they usually provide an opportunity to retry this or a similar problem, and they are able to provide a degree of individualization so that different students see different problems. In addition, they are either inexpensive (WeBWorK) or bundled into the cost of the textbook (WebAssign and MyMathLab). But there have been concerns, chief of which is that students do not need to show their work, eliminating the opportunity to provide feedback that goes beyond whether the answer is right or wrong.

There have been several studies of online homework. Most (Halcrow & Dunnigan, 2012; LaRose, 2010; Zerr, 2007) have compared graded online homework to ungraded assignments. These have generally shown some benefit from online homework, though often that benefit has not been statistically significant. A 2001 study at Rutgers (Hirsch & Weibel, 2003) found that students in general calculus who had some of the written homework problems replaced by WeBWorK assignments did show a small but statistically significant improvement in their test scores.

Last year, Larry Smolinsky and Gestur Olafsson at Louisiana State University (Figure 1) published their results of a controlled comparison of hand-graded versus online-graded homework in Calculus II (Smolinsky et al, 2018). These two lead authors taught four sections of mainstream Calculus II in Fall 2016, two with hand-graded assignments, two using an online system (WebAssign). To control for the effects of class size, there were two small sections (40 students) and two large sections (90–150 students), one each for each type of homework. To adjust for possible instructor effects, each lesson was taught in all four sections by one of the two instructors. The study also controlled for gender, student ACT/SAT scores, and whether or not a student was on a Pell grant.

**Figure 1.** Larry Smolinsky (left) and Gestur Olafsson

Outcome was measured by a composite score based on midterm and final exams, which had both open-ended and multiple choice questions. The authors also tracked student performance on the open-ended and multiple choice questions separately. In addition, they looked to see if hand-graded homework had an effect on student performance on three of the exam questions that dealt with graphing, since only hand-written assignments provide an opportunity for students to draw graphs.

**Figure 2.** Composite scores from exams. Hand-graded versus online homework.

There was no evidence that online grading is detrimental (Figure 2). Looking at class size, grading type, and gender, the only interaction that was even moderately significant (*p* < 0.056) was for hand-graded homework in large sections, where women performed about 0.1 points above men. The authors concluded that “It does not seem necessary in this era to assign homework that does not provide feedback to students.”

It is also worth mentioning that Smolinsky and Olafsson found that with large and small classes taught with exactly the same lectures, there was no significant difference in composite score performance. This is line with the data collected in CSPCC where class size was not correlated with our outcome variables of confidence, enjoyment of mathematics, or desire to continue the study of calculus. Other variables totally swamped any effect from class size.

The 2015 MAA survey in *Progress through Calculus* (NSF #1430540) revealed that only 45% of PhD-granting departments and 17% of Master’s degree-granting departments have a uniform policy across multiple sections of mainstream Calculus I on the use of online homework. In most cases, this is left to the discretion of the instructor. This is unfortunate because one of the findings of CSPCC was that the most successful calculus programs have a high degree of coordination among the different sections of each course, including policies on how homework is collected and graded (Rasmussen & Ellis, 2015). Today, with easy access to online homework for single variable calculus, there is no excuse for not having a uniform policy that requires the use of this tool as part of the assessment mix.

Read the Bressoud’s Launchings archive.

**References**

Halcrow, C. and Dunnigan, G. (2012). Online homework in Calculus I: Friend or foe? *PRIMUS*, 22(8), 664–682.

Hirsch, L. and Weibel, C. (2003). Statistical evidence that web-based homework helps. *FOCUS*, 23(2), 14.

Kehoe, E. (2010). AMS homework software survey. *Notices of the American Mathematical Society*, 57, 753–757.

LaRose, P.G. (2010). The impact of implementing web homework in second-semester calculus. *PRIMUS*, 20(8), 664–683.

Rasmussen, C. and Ellis, J. (2015). Calculus coordination at PhD-granting universities: more than just using the same syllabus, textbook, and final exam. In D. M. Bressoud, V. Mesa & C. L. Rasmussen (Eds.), *Insights and recommendations from the MAA national study of college calculus* (pp. 107–116). Washington, DC: Mathematical Association of America.

Smolinsky, L., Olafsson, G., Marx, B.D., and Wang, G. (2018). Online and handwritten homework in Calculus for STEM majors. *Journal of Educational Computing Research*. doi.org/10.1177/0735633118800808

Sonnert, G. and Ellis, J. (2015). Survey questions and codebook. In D. M. Bressoud, V. Mesa &C. L. Rasmussen (Eds.), *Insights and recommendations from the MAA national study of college calculus* (pp. 139–169). Washington, DC: Mathematical Association of America. Data available at https://www.maa.org/CSPCC.

Zerr, R.J. (2007). A quantitative and qualitative analysis of the effectiveness of online homework in first-semester calculus. *Journal of Computers in Mathematics and Science* *Teaching*, 26(1), 55–73.

**Teachers at the NCSSM 2019 Teaching Contemporary Mathematics Conference discuss broadening participation in the MAA AMC and field test materials from MAA Mathematician-at-large James Tanton**

When I began my work as MAA Deputy Executive Director in Denver at MAA MathFest 2018, I knew that part of my job would include the MAA American Mathematics Competitions (AMC): the AMC 8, AMC 10, AMC 12, AIME, USAJMO, USAMO and Putnam. I have to admit that I wondered if I was the right person for the job because I come from the world of mathematical modeling challenges, which seemed like a very different venture. I wondered how might experience with math modeling challenges inform our competitions program?

I was delighted to see that at MAA MathFest 2018, Jo Boaler and Sol Garfunkel would be presenting their research, which is also featured in the Feb/March 2019 issue of *MAA FOCUS**.* They looked at why modeling challenges attract about half female-identified particip=ants, when some mathy sports do not. We met for breakfast to talk about their research and how it might inform our work at the MAA. We have continued the conversation since.

In her 2018 MathFest invited lecture, Eugenia Cheng contrasted the experience of competitions with other ways of engaging in math as a creative endeavor that are more analogous to doing crafts at a table with friends. Important takeaways from conversations with Jo and Sol have been that the emphasis on individual participation, short time limits, and experiences with little or no writing component might all be aspects of the MAA AMC program that could benefit from further investigation.

Some women enjoy competition. As a child, that usually showed up in my kickball game more than my math habits, but that could be because there was no math team in sight. I had no kid-focused and welcoming math community. Much later, when I did become aware of sports like Olympics of the Mind, I felt out of place watching experienced students prepare with their team for those events. I probably would have enjoyed it and there were general announcements welcoming new members, but I did not feel personally invited, and did not ask to join.

When Dr. Boaler and I were talking, she expressed how interesting she found some of the mathematics in the competition problems, and how students would benefit from more exposure to these kinds of challenges. Several faculty have let me know that they wish they had known about the competitions, and would have appreciated an invitation as a young person. Personal invitations are powerful. We want to think about how to extend more of them, and support the young people and their teachers who say YES.

This work has started already under the direction of the MAA AMC Executive team, which includes Jenn Barton, MAA AMC Director of Competitions Operations; Bela Bajnok, MAA AMC Director; Paul Zeitz, Chair of the Competitions Committee and myself. Here are five of the ways we are working on broadening participation and building community:

This Fall we created new editorial boards for our competitions programs, which used to be composed by committees. These boards are composed of over one hundred people from a variety of careers, and they include historic leadership participation by women (they comprise half of the co-editors in chiefs). The new editors met at JMM 2019 and they are already engaged in conversations about what features make the competitions fun, meaningful, beautiful, and challenging.

We want to connect with more teachers and students, and have several approaches. James Tanton’s work as MAA Mathematician at Large provides direct support and encouragement. We are also partnering with other organizations to conduct research on recruitment and retention of teachers and students as well as competition development. We also are working on support materials for teachers, to help them engage students in tackling problems that are not directly found in their curricula.

The MAA SIGMAA-MCST (Math Circles) is planning to share at MAA MathFest 2019 some group-oriented tasks developed for the Julia Robinson Festival. These tasks have embedded qualities of choice, do not emphasize speed, and are conducive to teamwork. The MAA AMC is exploring additional kinds of MAA AMC events for the future that could build in some of these qualities to broaden participation in competitions.

If we want to broaden participation, we need input from the students and teachers that we want to reach. Jo connected me with our new Math Values blogger Meera Desai. As a high school student Meera has shared her love of math competitions through her blog awesomemathgirls.org and has organized events to get more girls involved in the MAA AMC. Look forward to Meera’s posts soon!

I led two workshops focused on broadening participation in the MAA AMC at the Teaching Contemporary Mathematics Conference at the NC School of Science and Mathematics this Spring. These teachers had great ideas about broadening participation and also shared their local challenges and successes in engaging students in mathematics.

Please share your ideas with us. If you are at a college or university, let us know if you are interested in working with a teacher in your area to form a team. If you are a teacher, let us know what we can do to help you get your students involved. We are excited about the future of the MAA AMC - join us in making the vision a reality.