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 (@mathbabe), 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.

Two fundamental challenges of teaching required math courses to students in the first two years of college are (i) engaging students, and (ii) making the mathematics relevant to their future study. Tying the curriculum to the needs of other disciplines can encourage student engagement and also has the potential to rejuvenate the mathematics content of these courses. But where to begin? The SUMMIT-P grant aims to address this question by building on recommendations compiled by the MAA’s Curriculum Foundations Project, a series of disciplinary workshops that began in 1999.

Dr. Bill Haver is Professor Emeritus at Virginia Commonwealth University, and from 2004 – 2010 he served as the chair of the Curriculum Renewal Across the First Two Years (CRAFTY) subcommittee, a subset of the MAA’s Committee on the Undergraduate Program in Mathematics (CUPM). Through his work with the Curriculum Foundations Project, Bill became involved in the SUMMIT-P grant, which he introduces us to in this post.

*How will students and the wider mathematical community benefit from the results of the SUMMIT-P project?*

The overall goal of SUMMIT-P is to have a large segment of the national mathematical community* *respond to the message of the Curriculum Foundations Project, leading to collaborations with faculty* *from partner disciplines and engaging courses that most effectively prepare students to make use* *of the mathematics that they study in course work in partner disciplines and within the workplace.

It is our experience that curriculum materials and teaching approaches that are developed collaboratively at multiple institutions and involving faculty from multiple disciplines has a much greater chance of broad scale adaptation and adoption than materials and approaches developed by one or two faculty members working in isolation.

*Where did the idea for the SUMMIT-P project originate?*

A number of participants in this project were involved in the MAA/CRAFTY Curriculum

Foundations Project that conducted more than 20 weekend conversations with groups of faculty from the various partner disciplines asking these faculty to describe the mathematics and the types of experiences with mathematics that they would like to have provided for majors in their disciplines.

Many of us found the reports from the Curriculum Foundations Project very useful at our own institutions and we wanted this valuable tool to be used across departments nationwide.

*NSF grants are competitive - what factors do you think set the SUMMIT-P proposal apart?*

Our proposal was made from a set of 10 diverse institutions, including small private colleges, large research universities, HBCUs, community colleges, state supported colleges, and colleges with religious affiliations. The institutions shared common commitments: to collaborate with faculty from partner disciplines in learning communities, to improve student learning through active engagement, and to take seriously the recommendations from disciplinary faculty in the Curriculum Foundations Project. All 10 institutions are committed to sharing these experiences with the national community since we believe that this process is as important as the final curriculum and courses that will be developed.

*The SUMMIT-P grant impacts hundreds of faculty and thousands of students. What are some strategies for organizing a project of this scope?*

We have a large number of ongoing opportunities for collaboration and mutual support. These opportunities include periodic Webinars exploring different aspects of our work; course clusters among faculty working on the same mathematics courses; annual in-person meetings of project leaders; regular virtual meetings of both a project Management Team and of the Principal Investigators from each institution. Our Evaluation team solicits information and provides instruments to each college. We also are flexible in realizing that not all faculty from each college will participate in every activity.

Of particular value are the extensive site visits included in our work. Over the lifetime of the project each college will be visited three times, and project leaders from each college will participate in visits to three other colleges. The college will be visited by a member of the Management Team, an evaluator and at least one mathematician and one partner disciplinary faculty member from the visiting team. A detailed protocol for organizing and reporting on the visit has been developed.

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

*Acknowledgements: **Dr. Susan Ganter** is the project lead for SUMMIT-P and provided guidance in the creation of this post. *

**Learn more about NSF DUE 1625244**

Full Project Name: Collaborative Research: A National Consortium for Synergistic Undergraduate Mathematics via Multi-institutional Interdisciplinary Teaching Partnerships (SUMMIT-P)

Abstract: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1625244

Project Website: https://www.summit-p.com/home

Project Principal Investigator: Rhonda Fitzgerald rdfitzgerald@nsu.edu (Principal Investigator); Aprillya Lanz (Former 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.

*Katie Haymaker is a co-editor of DUE Point and an Assistant Professor of Mathematics at Villanova, where her research interests include coding theory and mastery-based testing in undergraduate mathematics courses.*

NAM-MAA Blackwell Lecture, Raegan Higgins, MAA MathFest, Denver 2018

The National Association of Mathematicians (NAM) was founded in 1969. With membership open to all, NAM promotes excellence in the mathematical sciences while serving as a voice for underrepresented Americans in the mathematical sciences community.

The MAA is proud to join NAM to celebrate its 50th Anniversary Year throughout 2019.

In this, Black History Month, it’s useful to take a moment to think about the progress we’ve made, and the work that remains to be done.

In 1933, Dr. Carter G. Woodson, the father of Black History Month and a pivotal figure in the study of black history, wrote The Mis-Education of the Negro. Dr. Woodson argued that the core of classical education at that time served to strengthen the oppression of black people, even for the limited segment of the population who gained access to prestigious schools. It is a biting critique of society at the time.

Of course much has changed since 1933. But power structures in our society remain aligned against many of those who have been historically marginalized. Ibram X. Kendi, the author of “Stamped from the Beginning,” a history of racist ideas, and the founder of the Antiracism Center at American University, argues persuasively that we must move far beyond so-called “neutral” policies that ultimately are bound to establish (majority) norms and values. I agree.

The MAA and NAM have long shared the goal of broadening participation in mathematics, and we look forward to continuing to work with our friends and colleagues in NAM to continue this important work, as well as engaging in the broader discussions needed to move our society towards a fully-inclusive and just society. I note, too, that trends in society around the use of data offer a natural place for our community to engage in discussions around what constitutes appropriate and ethical use of such data.

I’ll finish by sharing the congratulatory letter sent to NAM:

January 10, 2019

Edray Goins, President

National Association of Mathematicians

Pomona College

Claremont, CA 91711

Dear Edray,

On behalf of the Mathematical Association of America, we are writing to congratulate the

National Association of Mathematicians on the occasion of their Golden Anniversary.

The MAA joins with NAM to celebrate 50 years of promoting excellence in the mathematical

sciences and advocacy for inclusion of underrepresented populations in our discipline.

Recognizing that we have made progress but have much more to do, we also affirm our

commitment to continue to work with NAM. The role of mathematics is as critical for our

society now as it has ever been. For both the health of our discipline and our society, we must

continue to expand opportunities for all of our citizens to participate and succeed in

mathematics at all levels.

The MAA has recently adopted a new vision statement: We envision a society that values the

power and beauty of mathematics and fully realizes its potential to promote human flourishing.

We know that working towards this vision will require a shared commitment across our

community to inclusivity, one of MAA's core values, and we are proud to continue to work

with our friends and colleagues at NAM to ensure that mathematics is a key contributor to a

more welcoming and just society.

Sincerely,

Michael Pearson Deanna Haunsperger

Executive Director President

As usual, MAA MathFest will feature the annual David Blackwell Lecture. We’re already planning to co-host additional events to mark NAM’s Golden Anniversary. We hope you’ll join us in Cincinnati July 31–August 3 as we mark this milestone for NAM, celebrate its achievements, and join in discussions of how you can contribute to the critical work that still remains.

]]>In its January 1 Learning Blog, the business networking service LinkedIn published a list of the skills today’s large companies value most in their employees, as obtained from survey data. The report notes that 57% of senior leaders in business value soft (human-centered) skills over hard skills, pointing out that “the rise of AI is only making soft skills increasingly important, as they are precisely the type of skills robots can’t automate.” In the case of mathematics, this aligns with the theme that occupied Devlin’s Angle for most of last year, starting with the January post.

Survey respondents ranked the top five of those soft skills in the following order (most valuable first): creativity, persuasion, collaboration, adaptability, and time management. (For the top 25 hard skills, see the LinkedIn blogpost. I note that mathematical skills do not appear anywhere in that list—at least not under that name, but keep reading.)

As regular *Devlin’s Angle *readers will know, as a lifelong university scholar and educator, I have never viewed K-16 education as being job training; it’s *life preparation*. But as I also always add, jobs and careers are part of life, so it would be irresponsible for educators to ignore the realities that will face the students who graduate from our institutions.

For instance, and to pick up the main theme of last year’s posts, until the late 1960s you had to master numerical calculation (ideally fast and accurate) in order to (1) live a successful, productive, and rewarding life, (2) get many jobs, (3) acquire a mathematics education and use mathematics, and (4) acquire an education in a STEM related field and work in a STEM area. So, it was important that schools taught basic arithmetic. Up until the late 1980s, it was likewise important for colleges and universities to ensure their students master other forms of calculation (most notably, algebraic). But with the arrival of electronic calculators in the 1960s and computer packages like *Mathematica *and *Maple *in the 1980s (and particularly after the appearance of *Wolfram Alpha *in 2009 and *Desmos *in 2011), the need to master any kind of calculation had been eliminated. Since no one in the world (at least the parts of the world with Cloud access) ever needs to do calculation themselves, there is no longer an imperative for schools or colleges to teach it.

At least, there is no need for anyone to teach calculation (of any kind) so their students can execute algorithms by hand. But in place of that now obsolete skill set, there is a new one. In today’s world, we all need to be able to make good use of those new calculation technologies. To achieve that, we need to provide students with a good understanding of the calculations those technologies can perform, and that surely requires that those students achieve some level of mastery. But the goal of calculation instruction today should be understanding, not execution, so the level and nature of the required mastery is different.

With calculation now automated, the creative aspect of mathematics now occupies primary place. But what exactly is the “creative aspect of mathematics”?

Prompted in part by the LinkedIn article, I had a fascinating email exchange about this question recently with a long-time friend in the ed tech industry. A former school teacher (not STEM), he shares my interest in finding ways to make productive use of technology to improve teaching and provide access to quality education in particular to groups currently under-served (for various reasons). Though my friend has spent many years in the tech world, like me he thinks that technologists who approach education simply as another domain in which to find markets for the products they have built are unlikely to create anything of educational value. You need to start with a good understanding of, and some considerable experience in, education and *then* look for ways technology can help—either an existing technology or one that has to be designed and built.

The goal of our exchange was to answer these specific questions: Can digital technologies, in particular digital mathematics learning games, help develop creativity, and can they measure it?

In using the term “mathematical learning game,” I mean a game explicitly designed to support the learning of specified mathematical skills. All games produce learning, and indeed all games can result in the acquisition and development of skills and attitudes useful in doing mathematics. Anyone who does not see that has not played many video games—or does not really understand what it means to do mathematics. But our focus was games developed specifically to provide learning of specified mathematical skills. Hence my choice of term.

Before we could answer those questions, we had to decide what we meant by “creative mathematics.”

Without question, the first step anyone should make when trying to answer a question in today’s world is do a quick Google search. (That is true even if you have some prior knowledge of the topic, which I did—more on that in Part 2 of this post.) In my case, Google instantly brought up some helpful resources, among them:

A youcubed page from my Stanford colleague Prof. Jo Boaler.

A 1997 book called Creative Mathematics aimed at elementary school teachers.

A website called creativemathematics.com providing teaching resources for elementary school teachers.

A 2017 blog post with the wonderfully provocative title Mathematics must be creative, else it ain’t mathematics, written by a former research mathematician now focusing on teaching, called Junaid Mubeen, who I have interacted with productively on social media from time to time.

But pretty well everything that came up near the top of my search assumed we all know what the term “creative mathematics” means. (Actually, I prefer to use the longer term “creative mathematical thinking,” to emphasize that it is the process of doing—or using—mathematics that we are referring to, not the body of knowledge found in textbooks that the word “mathematics” commonly suggests.) The articles I found seemed to be using the adjective “creative” to evoke a word cloud along the lines of “lively, engaging, fun, enjoyable, experiential, multidimensional, open-ended, exploratory, intriguing, satisfying, …”

None of the words in that word cloud are exclusively connected to mathematics, though all can (and to mathematicians do) apply. What was significant, I found, was the absence of two words that definitely apply to mathematics and one that applies more or less exclusively to mathematics. The first two words are “difficult” and “challenging” and the third is “algorithmic” (or “procedural”, which would be equivalent in this context).

Let me start with “algorithmic.” That’s the one that, of necessity, used to be front, center, and most significantly temporally first, in mathematics education, but which has now been relegated to a teaching tool to be used to develop understanding. “Algorithmic” is the elephant in the room that the terms in that word cloud were trying to distance themselves from. In fact, you can simplify the entire cloud with one word: “non-algorithmic.”

The point is, we humans evolved to understand, and we find the act of achieving understanding rewarding (both psycho-chemically and cognitively). We are meaning-seeking agents. The three traits understanding, planning-based-on-understanding, and communicating-our-understanding-and-planning, are *Homo sapiens’ *evolved skills to compensate for our lowly position in the “red in tooth and claw” ranking table.

What we did not evolve to do, and until very, very recently in our history had no need for, is execute mathematical algorithms. We started to do it because, a few thousand years ago, we got to a stage where we had to. But it was difficult for the human brain to do and took time and effort to master. The majority of people disliked it from the start, and many never did succeed. In particular, it required suppressing the very thinking processes our brains found natural and enjoyed doing, as we trained our minds to act like mechanical devices. (Hence the use of derogative colloquial terms such as “number crunching”, “grinding away”, and “turning the handle” to refer to computation.)

Not to put too fine a point on it, “algorithmic thinking” is an oxymoron. The trick to being able to master execution of algorithms was to suppress the brain’s instinct to think and force it to slavishly follow the rules. Few of us were able to do that well.

I think those (relatively few) of us that succeeded were able to do so because we took pleasure in understanding how and why those algorithms and procedures worked, and appreciated the human creativity that went into designing them. That was definitely the case for me. As I have written about often, I was the last kid in my school math class to do well on tests in the lower grades, because I was never able to just learn the rules and apply them; I kept trying to make sense of them. (In later grades, I figured out how to play that game, accepting that to succeed in the education system I had to first master the procedures and get good grades, and *then* try to figure everything out later. I got good at that. So good, in fact, that I was a mathematics graduate student before I really understood the calculus methods I used efficiently to get A’s on tests in high school and as a mathematics undergraduate.)

To get back to my theme: What is the opposite of “machine-like thinking”? “*Creative* thinking” seems to capture the essence. (A somewhat equivalent phrase would be “the kind of thinking the human brain naturally finds pleasurable.”)

And once you are at that point—when you have discarded “algorithmic thinking”—you can safely throw in those other two missing words “difficult” and “challenging” when talking about “creative mathematics.” For the fact is, the pleasure we get from using our minds the way they evolved is all the greater when we succeed in something we found difficult.

So now we have made some progress in answering the question “What is creative mathematical thinking?” It is *non-algorithmic mathematical thinking*. We have defined it in terms of what it is not.

But can we turn that into a positive definition? For that is what my friend and I would need in order to pursue our email exchange about how to develop and use technology to help develop creative mathematical thinking, and even more so if we want to measure it (with or without technology).

*To be continued next month in PART 2*

Read the Devlin’s Angle archive.

]]>The International Congress on Mathematics Education (ICME) meets every four years and produces a series of monographs based on the contributed papers. One important monograph from ICME-13, held in 2016 in Hamburg, Germany, has just been released: *The Legacy of Felix Klein*, published Open Access by Springer. Part II of this volume is comprised of four important articles on one of Klein’s central tenets: the need to place functional thinking at the heart of the entire mathematical curriculum. This month’s column will focus on two of them: Katja Krüger’s “Functional thinking: the history of a didactical principle” and “Teacher’s meanings for function and function notation in South Korea and the United States” by Pat Thompson and Fabio Milner.

**Figure 1**. Cover to *The Legacy of Felix Klein*. Felix Klein (1849–1925)

Klein was frustrated by what he called the “double discontinuity,” first the fact that university mathematics made little or no attempt to connect with the mathematics that students had learned in earlier years, second that those returning to teach in school drew little or nothing from the mathematics they had learned at university. The solution from the German National Teaching Commission for Mathematics that he led was to bring elements of university mathematics into secondary school mathematics, in particular analytic geometry and concepts from differential and integral calculus (in a secondary curriculum that extends to grade 13), and to place “functional thinking” at the center of instruction for grades 5 through 13.

Figure 2. Members of the German National Teaching Commission for Mathematics: Felix Klein, August Gutzmer, Friedrich Pietzker, and Heinrich Schotten.

(Lorey 1938, pp. 18, 20, 26, 41)

Krüger explains the meaning of functional thinking through quotes and examples. It has nothing to do with the formal, static definition employing ordered pairs. Rather, in the 1909 words of Heinrich Shotten, a member of this commission, “It is about making students aware of the variability of quantities in arithmetic or geometric contexts and of their shared dependence and mutual relationship.” In modern language, functional thinking involves understanding co-variational relationships. As Klein would elaborate in 1933, “It [function] should not, of course, be introduced by means of abstract definitions, but should be transmitted to the student as a living possession, by means of elementary examples, such as one finds in large number in Euler.”

Functional thinking lies at the heart of the research and curricular reforms with which Thompson and Milner have engaged. In their article, they explore the *meanings* that high school teachers associate with aspects of function. They come to the depressing conclusion that, rather than Klein’s double discontinuity, too many U.S. teachers exhibit a *continuity* from inappropriate meanings learned in high school that are preserved during their university education and reappear in their understandings as teachers.

**Figure 3**. Question from *Mathematical Meanings for Teaching secondary mathematics*.

© 2016 Arizona Board of Regents. Used with permission

In this article, they consider three aspects of function notation, *f*(*x*). As I reported in my May 2017 column, *Re-imagining the Calculus Curriculum, I*, many students see *f*(*x*)* *as simply a long way of expressing the dependent variable. Figure 3 shows a question given to 253 U.S. high school mathematics teachers. Of those who had taught calculus at least once, only 43.2% correctly inserted *v* into all four spaces, while 33.8% retained the variable names *s* and *t* from the function definitions. Of those high school teachers who had never taught calculus, 29.6% inserted *v*, and 41.3% retained the *s* and *t*. For comparison, this question was also presented to 366 South Korean high school and middle school teachers. Of the high school teachers, 76.9% put in *v*; only 5.3% retained *s* and *t*. Even South Korean middle school teachers did considerably better than teachers in the U.S., with 63.7% inserting *v* and only 5.9% retaining *s* and *t*. The point is that students often view *w*(*t*) and *q*(*s*) as the names of the functions, a misconception that is rampant in the United States and carries forward from one generation of teachers to the next.

There were similarly dispiriting results on a question that explored teacher understanding of the role of the domain in the definition of a function.

The third exploration went directly to the question of whether teachers could use functional thinking. Showed a circle with a dot at the center, teachers were presented with the following problem:

Hari dropped a rock into a pond creating a circular ripple that spread outward. The ripple’s radius increases at a non-constant speed with the number of seconds since Hari dropped the rock. Use function notation to express the area inside the ripple as a function of elapsed time.

What one would hope to see is something like *A*(*t*) = π (*r*(*t*))², with functional notation employed on both sides of the equality. What appeared included *A* = π (*r*(*t*))², *A*(*t*) = π*r*², and *A* = π*r*². The point of the exercise was to see whether teachers would recognize that they could use *r*(*t*) or similar functional notation as a model for this unknown function of time. Answers were categorized by whether teachers employed functional notation on the left, right, both sides, or neither.

.

These results should be a wake-up call to those who prepare our future teachers. Preparing teachers for their role in the mathematical preparation of the next generations of students is about more than filling them with mathematical knowledge. It also must consciously address the misunderstandings with which they enter university and work to correct them.

Read Bressoud’s Launchings archive.

**References**

Krüger, K. (2019). Functional thinking: The history of a didactical principle. Pages 35-53 in *The Legacy of Felix Klein*, Weigand, McCallum, Menghini, Neubrand, and Schubring, editors. Cham, Switzerland: Springer Nature. https://www.springer.com/us/book/9783319993850

Lorey, W. (1938). *Der Deutsche Verein zur Förderung des mathematischen und naturwissenschaftlichen Unterrichts*. Frankfurt: Otto Salle.

Thompson, P. and Milner, F. (2019). Teachers meanings for function and function notation in South Korea and the United States. Pages 54–66 in *The Legacy of Felix Klein*, Weigand, McCallum, Menghini, Neubrand, and Schubring, editors. Cham, Switzerland: Springer Nature. https://www.springer.com/us/book/9783319993850

The landscape of faculty positions is changing, as documented in the CBMS survey report. To get a comparison from 2010 to 2015 you can enjoy a little data dive into table E.5 below and find plenty more online in the full report. Faculty jobs have varying levels of security, benefits, duties and privileges. Job categories such as NTT (non-tenure track) and NTL (non-tenure line) are growing in number. The reality is that most departments depend on these faculty, and I wonder about the impression we give when we describe their jobs as “non-something-we-value.”

A related phenomenon occurs when jobs in industry are referred to as “non-academic.” In forming the MAA BIG SIGMAA, the acronym BIG was adopted to refer to business, industry and government jobs, as a more positive alternative to non-academic. Folks in industry have pulled me aside on many occasions and asked me to help people transition to more positive terminology to describe the work they do. So we adopted this same term for the BIG Math Network and the BIG Jobs Guide. Of course no term is perfect. Another widely accepted alternative is just the term “industry,” which is what we use for MAA PIC Math.

What if we were to create a similarly positive acronym, **VITAL**, for the Visitors, Instructors, TAs, Adjuncts and Lecturers who teach most of the students in many institutions? For many students, these faculty provide their first experience of college mathematics in the critical early years. Their teacher, regardless of title, is the primary facilitator of their learning experience.

The MAA values the many ways all faculty contribute to their institutions and communities. We provide environments where faculty can realize and celebrate the values of inclusivity, teaching and learning, community and communication. MAA Programs such as College Mathematics Instructor Development Source and Progress through Calculus are supporting faculty who serve students in their early college years. The MAA community also can provide continuity when a job location or description changes. I hope to hear ideas from you about additional ways the MAA can honor and support the work of VITAL faculty as their roles and influence continue to grow.

]]>Equity. What does it mean in the context of mathematics instruction? No one argues that all students have the same opportunities to learn. Nevertheless, there is a widespread belief, common among mathematicians as well as the population at large, that there is nothing wrong with what we teach or how we teach it. If students do not learn mathematics, then the fault lies with them. This myth is exploded in an important report from Pamela Burdman of the Just Equations project of the Opportunity Institute, *The Mathematics of Opportunity: Rethinking the Role of Math in Educational Equity*.

Pamela Burdman and the Just Equations report.

The report is structured around three sections. The opening describes the critical role that mathematics plays within the educational progression and the expectation of failure it too often engenders. In and of itself, failure is neutral. Every successful mathematician has failed more often than succeeded. Those who are successful know how to work with failure. But that is not the message of most mathematics instruction. As Burdman writes, “The way mathematics is typically taught and tested, as well as the very requirements students are expected to meet, appear designed to winnow students out, effectively surrendering to the notion that only a few students are ‘math worthy’.” An expectation of failure can be devastating. This is illustrated in the sidebar case of Javier Cabral for whom a bad experience with Algebra I in 7th grade set him up for repeated failures in algebra, blocking his route to a college degree. Burdman quotes Jo Boaler, “Mathematics, more than any other subject, has the power to crush students’ spirits.” (Boaler, 2016, p. x)

The second section of this report lays out three key problem areas. The first describes common misconceptions: that math ability is innate and some people are just not good at math, that there is only one way to correctly solve any math problem with no room left for creativity or expression, and that speed and accuracy are what really matter when doing mathematics. The expectation that there are some students who will never succeed in mathematics is damaging when the student believes it of him- or herself. It can destroy generations of students when a teacher, school, or district embraces it. Mathematicians know the falsity of the second claim. Mathematics does impose structure and rules, but such frameworks can promote creativity. We see this in music, and the strong connection between mathematical and musical ability should not be surprising. The last misconception is particularly pernicious because it is what we test, but it is not what we value once the course has ended and our students need to use their mathematical knowledge.

The second problem revolves around existing inequities, including poorly resourced schools, differential access to strong curricula and good teaching, income inequality, and the nature of the educational support from peers, family, and community. These inequities foster an environment where the common misconceptions flourish.

The third problem area lies in how we use mathematics as a marker of pedigree. As Dan Teague pointed out in the workshop on “The Role of Calculus in the Transition from High School to College Mathematics,” (see my June 2016 column for more on this workshop), a student who wants to go to Duke to study French literature knows that he or she needs to take and do well in AP Calculus. See also my September 2017 column, “Mathematics as Peacock Feathers.” Across the spectrum, we find that—whenever college students are forced to back up and retake a course they thought they had successfully navigated in high school—the result is tremendous harm to self-esteem and motivation to continue.

In the final section of the report, Burdman lays out four areas where work is needed if we are to advance equity. The first is content. We need to seriously rethink what mathematics students really need. Traditionally, we have either directed students into dead end courses or pushed them along the pathway to calculus until they fall off the tracks. This has been directly responsible for the strong reaction from Hacker and others, arguing that we should abolish any requirement for Algebra II (see Bressoud, 2016). These critics are wrong, but what is taught and how it is packaged requires very serious rethinking.

The second area for work lies in how we teach. This is such a huge subject, one on which I have written often, that I will simply point to it.

The third is assessment. Speed and accuracy have roles to play. But if those are all we really care about, then we do our students a serious disservice. Each year, I have put less emphasis on timed tests, more on assignments and projects. They are hard to assess, especially in situations where large numbers of students must be tested. But the AP Calculus exams, written for 450,000 students, do a much better job of this than many math departments (see Tallman et al, 2016). The fact that this task is difficult does not mean we should abandon hope.

And, finally, there are the readiness structures and support. These encompass high school graduation requirements, college demands, and placement procedures. Both Algebra I in eighth grade and Algebra II as a prerequisite for graduation have proven problematic. Algebra is a collection of tools, marvelously refined in the late sixteenth through early seventeenth centuries to replace hundreds of *ad hoc* strategies for solving real problems. It *is* important for students to know how to use these tools, but it is equally important that they are given the opportunity to build and create with them. At the same time, we need to rethink which of these tools are truly essential.

At the college level, I have written about the flaws in many of our expectations and placement programs (see First do No Harm, January 2012). Traditional remediation seldom accomplishes the desired result. There is good news. Pathways programs that direct students toward statistics or quantitative reasoning are gaining wider acceptance. Prerequisites are being replace by co-requisites in a variety of innovative approaches. Universities are re-imagining their support structures for at-risk students.

The need to promote equity is real. This report does a helpful job of laying out the issues and challenges. We are making progress. There is much more to do.

Read Bressoud’s Launchings archive.

**References**

Boaler, J. (2016). *Mathematical mindsets: unleashing students’ potential through creative math, inspiring messages and innovative teaching*. San Francisco, CA: Jossey-Bass

Bressoud, D.M. (2016). Book Review of *The Math Myth and Other STEM Delusions*. *Notices of the AMS*. 63 (10): 1181–1183. www.ams.org/publications/journals/notices/201610/rnoti-p1181.pdf

Burdman, P. (2018). *The Mathematics of opportunity: rethinking the role of math in educational equity*. Berkeley, CA: Just Equations justequations.org/resource/the-mathematics-of-opportunity-report/

Tallman, M.A., Carlson, M.P., Bressoud, D.M., and Pearson, J.M. (2016). A Characterization of Calculus I Final Exams in U.S. Colleges and Universities. *International Journal of Research in Undergraduate Mathematics Education*. 2(1) 105–133.

The primary purpose of K-16 education, we say, is to prepare the next generation for life in society.

CAUTION: The “we” here refers to those of us who are educators, and for an audience comprising only professional educators, my opening statement would stand on its own. But in an open, online forum such as *Devlin’s Angle*, I know from experience that my observation requires elaboration. Specifically, there is no shortage of people who think our job is to prepare our students for work. Of course, work is part of life, so if we prepare young people for life, that should definitely include preparation for work. But that should not be its sole, or even primary, focus. Equally (and arguably more) important, our task is to prepare our young charges to live full and rewarding lives as productively contributing members of society. SECOND CAUTION: Not the society we grew up in; rather, the one they will be part of. That’s a critical consideration.

That’s quite a challenge—particularly during a time of rapidly occurring, major societal changes, like today. Teaching well is hard at any time, doubly or triply so when the world our students will live in is not only different from the one we grew up in, but in all likelihood will have changed dramatically by the time they graduate.

One of the first nations to recognize that was Finland. Back in the 1970s, this tiny nation (population today just over 5.5M) realized that to prosper in the Information Age, they had to ensure maximum benefit from its most valuable natural resource: not the timber or the ships of previous ages, but its people. No, scratch that. Not its people, its society. They did not make the mistake of thinking it was about training people for work; rather, the trick was to create a cohesive, educated society where people can live and work together. They also understood that, as the ones tasked with producing the individuals who would make up that society, teachers were one of the most critical professions, alongside physicians, nurses, scientists, engineers, and business leaders. The result was that, thirty years later, in 2000, Finnish schoolchildren topped the international rankings in the OECD’s PISA (Programme for International Student Assessment) education tests.

To this day, the United States has, by and large, failed to meet the challenge, making up for the huge shortfall in adequately educated school graduates by massive immigration of talent educated elsewhere in the world. To be sure, that solution has many advantages in terms of a culturally more diverse society, but it consigns many native born Americans to less rewarding (and less remunerative) careers, often leading to resentment (and an antipathy to immigrants).

But, as I often do, I digress. My topic for this month’s post was occasioned by reading a remarkable new book from a social scientist at Temple University, who I have interacted with professionally on a few occasions: Jordan Shapiro. His book, The New Childhood: Raising Kids to Thrive in a Connected World, was published at the end of December. Its title suggests a “how to” manual for parenting, and indeed he has structured it that way, with each chapter ending with a summary that provides specific things parents can do to best prepare their children for life in today’s always-on, global society. But, on another level it’s much more than that. It’s a discussion of media that I found highly reminiscent both of Marshall McLuhan’s 1964 classic Understanding Media: The Extensions of Man, in which he coined the famous phrase “The medium is the message”, and of Alvin Toffler’s 1970 bestseller Future Shock. In fact, I’ll go further than that. I think it may well end up being viewed in the same way, as a seminal “taking stock of where we are as a society” study.

Divided into four sections—Self, Home, School, and Society—with three chapters in each (a symmetry sure to please mathematical readers), Shapiro’s book constantly asks the reader to consider the world from the child’s perspective.

As a father of two young sons, Shapiro has an in-house observational laboratory not available to many of us, but the way people (including children) use and react to media is also an area he has studied and written about professionally for many years, which is how our paths first crossed. He was one of the first ed tech commentators to write about my work on educational video games. [His book’s introduction is titled “Plato would have been a gamer”, which is very similar to my own oft-repeated remark “If video games had been around in 350 BCE, Euclid’s *Elements* would have been a video game.” It’s possible my meme inspired his header, but to anyone who really understands the nature of mathematics—particularly classroom geometry—and the nature of video games, as we both do, the sentiment is blindingly obvious.]

Though mathematics and math teaching are referred to throughout the book, they are there merely as examples of subjects that are taught and need to be mastered. The significance of Shapiro’s book for college math educators (or K-12 math teachers for that matter) is his discussion of how today’s globally-connecting technological infrastructure impacts what we need to teach and how best we can teach it.

Regular readers of this column have been exposed to my perspective on that topic for most of last year (beginning with the January post and continuing with just a couple of diversions through to the November post). Everything I wrote in those posts is entirely consistent with what Shapiro says, in large part because for several years we have followed each other’s work and consulted many of the same sources. But there is plenty in his book that was new to me, and my guess is it will be new to you as well. Since he writes superbly, I will for the most part leave it to you to check it out yourself.

I will, though, end by providing two BIG, and reassuring, takeaways, which come from Shapiro’s many years studying media—not just new media but different media stretching back thousands of years.

TAKEWAY 1: Nothing going on today is really new. It seems new to us, because we are in the middle of it. But if you put yourself in the position of people living when writing was introduced, when the printing press came along, when we acquired telephones, radio, film, television, and then all the generations of digital media we did live through, and finally the always-on, global network today’s kids take for granted, you will realize that each of those revolutions must have seemed very much the same to those living through them. Case in point. Already on page 5, Shapiro reminds us that Socrates thought much learning would be lost if ideas were written down. That did not deter his pupil Plato from doing just that, and the scholarly world rapidly adjusted to the radical new idea of learning being based on written texts. (Had video games been available at the time, Plato would have been able to stay closer to his teacher’s insistence that learning should involve active interaction of student and teacher by creating a video game rather than a book. Hence Shapiro’s introduction title. But Socrates would still have been unhappy. That brings us to the second takeaway.)

TAKEAWAY 2: It is an unavoidable consequence of being born and growing up at a certain time that we take our contextual environment as “the way things necessarily are.” That which we grow up with, we take for granted. We have no other choice. Society advances because each new generation eventually finds ways to go beyond what they encountered as children.

For the most part, the advances appear to be gradual and continuous, but every so often there is a kind of phase shift, where an accumulation of small changes has a dramatic effect.

Such was the case in mathematical praxis when, in the late 1980s, we acquired machines that could execute any mathematical procedure, rendering hand calculation unnecessary. (See my provocatively titled January 2017 article in the Huffington Post.) Since then, mathematicians spend their time very differently from that way their predecessors had operated for thousands of years. That major shift in how mathematics is done has been slow to percolate down to how it is taught in schools, but in due course the system will catch up. It has to.

An equally major rift occurred in mathematics education with the invention of the printing press. Prior to that, mathematical texts were written in words and numerals; no diagrams and no symbolic algebraic expressions. Part of learning mathematics back then was sketching diagrams and scribbling symbolic expressions in margins as part of the process of understanding what the prose argumentation meant. After the printing press came along, math textbooks were heavy on symbolic expressions and (in due course) diagrams, and students sketched diagrams and wrote prose comments in the margins as part of the process of understanding the symbolic expressions.

Neither of these specific examples is in Shapiro’s book, but there are a great many different ones of a less disciplinary-specialized nature. You may start out thinking, “Yes, but this particular point in history is different, because … ”, but eventually, Shapiro’s examples will overwhelm you, and you’ll cave. Scholarship does that.

So what is the change that tomorrow’s children will take for granted, but we will think is (a) impossible or (b) a disaster-in-the-making? According to Shapiro it is the abandonment of the fixed-period lesson, be it 30 minutes, 40 minutes, an hour, or whatever. It will, he argues, be replaced by “drip engagement.”

Shapiro introduces that term as “the process of turning one’s attention to small things as they arise… Think of academic content as if it were delivered like raindrops rather than a deluge.” You’ll need to read the book (and by now it should be clear I am urging you to do just that) to see what this amounts to, but the term itself is a good indicator. Alternatively, if you have young children, as Shapiro does, just watch how they study today.

Incidentally, this does not mean replacing the division of learning into one-hour classes, as we do now, by division into smaller chunks of something, say three minute videos—a change that has often been suggested by media technology folk who know virtually nothing about education. As Shapiro points out, the fixed-length class is a model we inherited from the monasteries in thirteenth century Europe. The introduction of devices that measure time accurately, combined with the necessity of bringing students into the same room as a teacher, resulted eventually in the establishment of the credit hour, that is the basic building block around which our entire current systemic educational system is built; from curriculum to finance and budgeting, to educational personnel workload, duties and compensation.

But a fixed unit of time has nothing to do with learning. From the perspective of learning, it is an imposed arbitrary constant around which educators must adjust everything to fit. The ideal unit of learning is not a unit of time, it is a … wait for it … *unit of learning*. Duh! Of course that’s what it should be! If it can be, that is—and in today’s world it can.

Time-to-reach-mastery should be a ** variable**—because it is! It varies from subject to subject, topic to topic, student to student, and it depends of course on the availability of resources and on the degree of mastery required. Drips come in different sizes. (Like all analogies, you have to give the metaphor some latitude to be effective. In this case, ignore physics and think of idealized drips that can vary indefinitely in size.)

This may all seem strange to those of us who grew up in world dictated by the clock, with education delivered in credit-hour units, but to today’s students the only place they encounter that method of learning big-time is at school, and they make it clear they find it arbitrary and they don’t like it.

It’s not that they are not capable of spending long periods of time engrossed in one challenging task. Just watch them playing a difficult video game. In the video-game world, time is flexible; it is all about the challenge at hand. That’s why writer Gregg Toppo titled his excellent book about game-based learning “The Game Believes in You.”

In the case of math learning (and likely many other subjects as well), no one (not least Shapiro, who is both an academic and a parent of young children) is suggesting throwing the educational baby out with the bathwater. But we do need to recognize that the credit hour is part of the bathwater. Bathwater that has now cooled down over time to the point where we need to pull the plug and let it drain away.

I suspect that for math learning the situation will end up being very reminiscent of the way it changed with the printing press. Just as words and symbols swapped roles, so too I expect math learners will view drip engagement as the primary “delivery” medium and “extended periods of focused thought” as a secondary mode to adopt as and when required—the very opposite of the situation today.

[I’ll leave it to those with expertise in teaching math to younger children to determine how things will go there. In the early stages, education is as much as socialization and learning how to learn, as it is about any particular subject matter content. I do know from observation that elementary school classes today look very different from the way they did when I went through the system, so I suspect elementary school teachers are way ahead of the rest of us educators in adapting to today’s kids.]

I am sure this will all seem very strange to us. We may even think it could not possibly work. But the historical record—and Shapiro gives us a lot of it to reflect on—suggests otherwise. We are not in a unique historical moment.

In the meantime, if today’s math educators want to help prepare the way for tomorrow’s learners, we need to start stripping mathematics down to the individual components and reassembling it in a way that permits learning in a drip-engagement model. For, whether ** we** like it or not, that is the future.

This month, it will be exactly 22 years since the MAA first went online. After its initial release in 1994, the web browser *Netscape* had, by 1996, started to acquire users rapidly, in the process turning the new World Wide Web from a scientists' communications platform into a citizens' global network. Like many organizations, the MAA was quick to establish a presence on the new communication medium. In December 1996, the Association launched *MAA Online*. It’s the platform on which you are reading these words, though the word “Online” was eventually dropped, when it no longer made sense to call out its online nature!

At the time, I was the editor of the MAA’s flagship, members (print-) magazine FOCUS, which was sent to all members six times a year. I had become the editor in September 1991, and would continue through until December 1997. As such, I was involved in the process of getting the Association’s new online presence off the ground—or more precisely, into the (ethernet) cable.

With FOCUS being the primary way the Association informed members of its activities, it fell to me to get the word out that there was a new kid on the block. I reprint below the FOCUS editorial that I used to spread the news. If you are under forty, this might provide some insight into how the “world of online” looked back then.

Note in particular that I went to some lengths to reassure members that the new medium would be *an optional addition* to the Association’s existing offerings. There was a general feeling among the MAA officers that not every member would leap to adopt the new technology. Indeed, many of them did not have access to a computer, let alone own one. Note also that I gave assurances that FOCUS would not go away. And indeed, the magazine remains with us to this day. (Though most of the advantages I listed for a print magazine have long been obliterated by technology.)

For the rest of us, it can provide a short trip down memory lane. Enjoy the ride!

* * * *

FOCUS, December 1996 Editorial

**Spreading the Word, at 186,000 miles per second**

Here at FOCUS we put in heroic efforts to ensure that your bimonthly MAA news magazine reaches you as rapidly as possible. But for all our efforts, almost two months elapse between the moment we stop accepting copy and the mailing out of your copy of FOCUS.

Things move much faster for my colleague Fernando Gouvêa, the editor of* MAA Online*. If necessary, he can even beat the New York Times in getting the news out. While FOCUS moves at the speed of overnight delivery during the production stage and the speed of second class U.S. mail for distribution, *MAA Online* travels at the speed of light through optical fiber and electrons through copper wire. Corrections can be made at any time, in an instant.

There is no doubt then that if you want up-to-the-minute information about the MAA, you would be advised to consult *MAA Online*. If you are reluctant to do so because you prefer the professional magazine look of FOCUS that you have become used to, think again. *Online* is no text-only database. It’s a full-color, professionally laid-out, typeset magazine, with masthead, photographs, and illustrations. Just like FOCUS, in fact, only with full colors.

And what’s more, where FOCUS often abbreviates articles or entirely omits important stories, items, and reports, due to limitations of space, *MAA Online* gives you the whole thing—all the MAA news that’s fit to print. Care to look at that long report the Association just put out? You’ll find it in *MAA Online*. Want to know the current members of the Board of Governors? That’s on *Online* as well.

In short, with the arrival of *MAA Online*, the whole news reporting structure of the MAA has changed. Or at least, it is in the process of changing. Aware of the fact that many members do not yet have full access to the World Wide Web, FOCUS is still carrying all the really important news stories—or at least as many of them as it always has. But the writing is on the wall—or more accurately on the computer screen. As far as news and the full reporting of committees are concerned, *MAA Online* is where tomorrow’s MAA member will turn.

What place then for FOCUS?

Well, ultimately that is a question not for me but for the Association as a whole, as represented through its Board of Governors and the appropriate elected committees. But I can give you my thoughts.

I don’t see the growth of *MAA Online* as heralding the end of FOCUS any more than the arrival of radio brought an end to newspapers or the introduction of television brought an end to the cinema. I suspect I share the view of most MAA members that there is something very significant—indeed symbolic—about receiving our copy of FOCUS every two months. Its very physical tangibility makes it a “badge of membership.” Receiving FOCUS, which for many members is the only MAA publication they receive regularly, is a significant part of what it means to be a member of the Association. Apart from renewing your membership once a year, all that is required of you to obtain the latest issue of FOCUS is to empty your mailbox. You don’t have to remember to log on to your computer, launch Netscape, and bookmark into http://www.maa.org. FOCUS may take its time to reach you, but it does so reliably, like an old friend. And what’s more, you can take it with you to read in bed, on the train, bus, or plane, in the coffee room, in the garden, or wherever.

Launched by MAA Executive Director Marcia Sward in 1981, FOCUS is now a part of the very identity of the MAA. Over the years, it has grown and developed in response to the changing needs and expectations of the membership. And that is as it should be. Of course, it will continue to change and evolve, and one of the forces that will guide its change is the newly arrived presence of *MAA Online*. That too is as it should be.

One change you will notice from this issue onward is that FOCUS will carry pointers to articles and reports in *Online*, with just a brief summary or extract appearing in the hard copy magazine you hold in front of you. No doubt further changes will follow.

In the meantime, from the editor of FOCUS, let me say a formal “Welcome” to our new sibling, *MAA Online*.

*The above opinions are those of the FOCUS editor and do not necessarily represent the official view of the MAA.*

Read the Devlin’s Angle archive.

]]>**The National Academies will be holding a Roundtable on Data Science Postsecondary Education: Motivating Data Science Education through Social Good on December 10, 2018. ****Event Website**** **

If I had to choose the most common job title for students who have graduated from Macalester with a degree in Mathematics, it would be *analyst*. Our graduates seldom wind up in jobs where they have to find derivatives or integrals, solve differential equations, or even find eigenvalues. Instead, they are almost always working with and trying to make sense of the data that can inform and shape business decisions. The habits of mind intrinsic to mathematics have generally prepared them for this role. But as the data available to business and industry has exploded in quantity and complexity, there is a growing need for graduates familiar with the increasingly sophisticated tools available for its analysis. The challenge to our colleges and universities is to provide the education that will equip graduates to become the data analysts that we need today and for the future.

Cover of the National Academies’ Report on *Data Science for Undergraduates*.

In response to this need, the National Academies have produced a report, *Data Science for Undergraduates: Opportunities and Options*, that provides a framework for building an undergraduate program in data science. Reflecting the necessarily interdisciplinary nature of such a program, the program is the joint work of the National Research Council’s Computer Science and Telecommunications Board, Board on Mathematical Sciences and Analytics, Committee on Applied and Theoretical Statistics, and Board on Science Education. The official roll out of the report is December 10, 2018 at the round table described at the top of this column.

The need is immense. The report references an estimate that by 2020 the U.S. will have positions for 2.7 million data analysts (p. 1-2). Meeting this need is frustrated by many obstacles, not least of which is the fact that few students understand what data science means or entails. Data analysis is also necessarily highly interdisciplinary, requiring new undergraduate programs that draw on expertise in computer science, mathematics, and statistics. As the report forcefully states, no single one of these fields adequately covers the core concepts of data science. It can *only *be taught as an interdisciplinary program. The breadth that is needed is reflected in this passage from the report:

Building on the work of De Veaux et al. (2017), the committee puts forth the following key concept areas for data science: mathematical foundations, computational foundations, statistical foundations, data management and curation, data description and visualization, data modeling and assessment, workflow and reproducibility, communication and teamwork, domain-specific considerations, and ethical problem solving. (p. 2-7)

The report goes into a detailed exploration of the necessary contributions from each of these concept areas.

It also briefly describes programs for majors in data science at the University of Michigan, Smith College, Virginia Tech, UC San Diego, University of Rochester, MIT, UC Irvine, and the NYU School of Professional Studies, programs that are variously housed within a business school, a department of mathematics or statistics, or a computer science department. The report describes a variety of data science minors and highlights the need to provide a basic understanding of data science for all undergraduates.

Macalester College has its own minor in data science. We are particularly well situated for such a program since we have a single department of Mathematics, Statistics, and Computer Science. This is a department that is strong in all three areas and has a long history of cooperation among these disciplines, including several cross-disciplinary faculty hires.

Our data science program begins with **Introduction to Data Science**, a course on the handling, analysis, and interpretation of big data sets that is intended to be accessible to all students. Students minoring in data science need two computer science courses, which could include our junior-level course in **Database Management Systems**. They also take **Introduction to Statistical Modeling **plus a course in** Machine Learning**, **Survival Analysis**, or **Bayesian Statistics**, and two courses in a single domain such as bioinformatics that provide an opportunity for the application of data science methods. A complete description of Macalester’s data science minor can be found at here.

Most math departments lost their faculty who worked in computer science decades ago. Statistics has long been a separate department at many universities. Far too often applied mathematics has been spun off, leaving a department that is increasingly insular, isolated from some of the most important developments in the mathematical sciences today. Separate departments are not necessarily a bad idea *provided* they are able to work collaboratively and share the work that transcends existing boundaries. If they are to serve their students, today’s departments of mathematics must be engaged in the process of shaping and delivering programs in data science.

Read the Bressoud’s Launchings archive.

**References**

De Veaux, R., M. Agarwal, M. Averett, B.S. Baumer, A. Bray, T.C. Bressoud, L. Bryant, et al. 2017. Curriculum guidelines for undergraduate programs in data science. Annual Review of Statistics and Its Applications 4:2.1-2.6.

National Academies of Sciences, Engineering, and Medicine. 2018. *Data Science for Undergraduates: Opportunities and Options*. Washington, DC: The National Academies Press. doi.org/10.17226/25104.

National Academies of Sciences, Engineering, and Medicine. 2018.** ***Roundtable on Data Science Postsecondary Education: Motivating Data Science Education through Social Good*. www.eventbrite.com/e/motivating-data-science-education-through-social-good-registration-51307330607

The above tweet caught my eye recently. The author is a National Board Certified mathematics teacher in New York City who has an active social media presence. Is his claim correct? Not surprisingly, a number of other mathematics educators responded, and in the course of the exchange, the author modified his claim to include the word “just”, as in “It isn’t just about beauty …” In which case, I think he is absolutely correct.

Like many mathematicians who engage in public outreach, I have frequently discussed the inherent elegance and beauty of mathematics, the wonder of its purity, and the power of its abstraction. And as a body of human knowledge, I maintain (as do pretty well all other mathematicians) that such descriptions of the subject known as pure mathematics are totally justified. (Cue: for the standard quotation, Google “Bertrand Russell mathematical beauty”.) Anyone who is unable to recognize it as such surely has not (yet) understood what (pure) mathematics is truly about.

In contrast, the **activity** of doing mathematics is indeed “messy,” as Pershan claims. That is the case not only for the activity of using mathematics to solve problems in the real world, but also the activity of engaging in pure mathematics research. The former activity is messy because the world is. The latter is messy because the logical elegance and beauty of (many) mathematical theories and proofs are characteristics of the finished product, not the process of development.

And there, surely, we have the motivation for Pershan’s comment. When we teach mathematics to beginners, we don’t do them any service by making claims about beauty and elegance if what they are experiencing is anything but. With good teaching of a well-designed curriculum, we can ensure that they are exposed to the beauty, of course, and perhaps experience the elegance. But it’s surely better to let them know that the messiness, the uncertainty, the repeated stumbles, and the blind allies they are encountering are part of the package of **doing** mathematics that the pros experience all the time, whether the doing is trying to prove a theorem or using mathematics to solve a real-world problem.

By chance, the same day I read that tweet, I came across an excellent online article on *Medium *about the huge demand for mathematical thinking in today’s data-rich and data-driven world. Like me, the author is a pure mathematician who, later in his career, became involved in using mathematics and mathematical thinking in working on complex real-world problems. I strongly recommend it. Not only does it convey the inherent messiness of real-world problems, it convincingly makes the case that without at least one good mathematical thinker on the team, management decisions based on numerical data can go badly astray. As the author states in a final footnote, he takes pleasure in the process of applying the rigor of mathematics to the complex messiness of real-world problems.

To my mind, therein lies another kind of mathematical beauty: the beauty of making productive use of the interplay between the abstract purity of formal rigor and the messy stuff of everyday life.

Read the Devlin’s Angle archive.

]]>MPWR (Mentoring and Partnerships for Women in RUME) is a daylong seminar that began five years ago as a means to support women at all career stages—graduate student, postdoc, faculty and professional—in the RUME (Research in Undergraduate Mathematics Education) community. Each year, the leadership team conducts this seminar preceding the RUME Conference to address the personal and professional needs of women in this community. Dr. Stacy Musgrave of California State Polytechnic University in Pomona describes the MPWR (pronounced empower) seminar and the newly-integrated research component of her work in her answers to the questions below.

**1. What issues did you seek to address with the development of the MPWR seminar?**

As women academics with expertise in RUME, the MPWR leadership team (currently myself, Jess Ellis Hagman, Megan Wawro, and Eva Thanheiser) recognizes a need in our community for increased support and mentorship. Women in RUME are in the position of being part of the gender minority in STEM academic positions and often part of the mathematics education minority in mathematics departments. By providing ongoing support targeted to the unique needs of our members, we aim to foster a community of thriving, successful women researchers who contribute to the RUME knowledge base and instigate changes in the RUME (and broader mathematics/mathematics education) culture.

**2. Tell us about the features and scope of MPWR.**

The annual event has evolved to include:

themed panels (e.g., mentoring + partnerships, gender + equity, individual and collaborative success, how to support each other in making good contributions to science, mental health);

mini-workshops (e.g., writing articles for practitioner journals, conducting research around your own teaching: simultaneously advancing both!, doing well at your job while working only 9:30-5 M-F, equity and social justice in research and teaching);

and formal peer mentoring structures called MPWRment groups.

To date, we have hosted 179 women from the RUME community. To put this in perspective, 325 women attended a RUME Conference during 2014-2017, 44% of whom participated in at least one MPWR. Of the 127 women who have been to more than one RUME, 71% have been to at least one MPWR.

**3. Now that the MPWR infrastructure is established, I understand you’ve added a research component to your work. What questions are you exploring?**

Generous funding from the NSF has enabled us to organize the annual event, and also start investigating several important questions:

What issues do women in RUME struggle with related to career success? What impedes women in RUME from social and academic integration into their research community? How do these issues compare to those faced by other subgroups of the mathematics education research community?

How does MPWR impact the RUME community, specifically attending to women in RUME who have participated in MPWR, women in RUME who have not participated in MPWR, and men in RUME?

What characterizes a successful MPWRment peer-mentoring group?

**4. What surprises have you found during the implementation of your research?**

There has been a lot of validation for the work we’re doing. The vast majority of participants provide positive feedback, affirming that MPWR provides a space to connect with other women as a whole person – researcher, teacher, faculty member, collaborator, mentor, mentee, wife/partner, sister, daughter, mother, dog owner, traveler, etc. Based on how valued this space is, we face the welcome challenges of how to continue to fund and implement MPWR.

**5. What new questions have arisen as your work progresses?**

As participant feedback continues to highlight common themes, we are beginning to question how to increase the impact of MPWR. In an ideal world, we wouldn’t need MPWR anymore – social structures that serve as obstacles to women (and other underrepresented and underprivileged groups) in RUME (and more broadly) would cease to exist, and the treatment of academics as more than paper-writers and grant-seekers would extend beyond the confines of a daylong seminar.

Although we acknowledge the support we provide as a much-needed first step, we are now asking much harder questions about how to create systemic and cultural change within our communities so that a structure like MPWR is no longer needed.

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

**Learn more about NSF DUE 1823571**

Full Project Name: MPWR 2016 and beyond: Fostering sustainable networks for women in RUME

Abstract: https://nsf.gov/awardsearch/showAward?AWD_ID=1823571

Project Website: http://www.mpwr-seminar.com

Project Contact: Stacy Musgrave, PI smmusgrave@cpp.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. *