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.

]]>**You can now follow him on Twitter @dbressoud**

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.

]]>**You can now follow him on Twitter @dbressoud**

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

*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. *

I sometimes use this column to float an idea I think deserves attention. Not on a whim, but after considerable thought and discussion with others expert in the relevant domain(s). This is one of those times. I already set the scene with last month’s post. Here is a nuanced, bullet-point summary of what I wrote then:

The heart of learning mathematics is mastering a particular way of thinking – what I (and some others) call “mathematical thinking,” sometimes also described as “thinking like a mathematician.”

You can master mathematical thinking by focusing on any

branch of mathematics – arithmetic, geometry, algebra, trigonometry, calculus, etc. – and going fairly deep.*one*Once you have mastered mathematical thinking, you can fairly quickly acquire an

mastery of*equivalent*branch of mathematics with relatively shallow coverage. (There are limits to this. There is a complexity and abstraction hierarchy of branches of mathematics. But for K-12 mathematics, that is not an issue.)*any*Thus, to learn mathematics effectively, it suffices to (i) master mathematical thinking by the study of one branch of the subject, and (ii) acquire some breadth by branching out to a few other areas.

For the same reason, it meets society’s need for assessment of mathematics learning if we (i) assess mathematical thinking restricted to one branch, and (ii) measure the individual’s

in a number of other branches.*knowledge*

If you buy this picture – and if you keep reading, I am going to try to sell it to you – then it means we can make use of modern technologies to be far more efficient, in terms of teacher and student time and of money spent, for both learning and assessment. I propose calling this approach T-learning and T-assessment, on account of the visualization of the model shown below.

The vertical of the T denotes the topic that is studied ** in depth** to build mathematical thinking capacity. In last month’s post, I discussed a suggestion by mathematics learning expert Liping Ma that school arithmetic is the best subject for doing that, so I have put that down for the vertical. Read last month’s post to see my summary of her argument, and follow the link I gave there if you want to know more. But school arithmetic is not the only choice. Euclidean geometry could also work. In both cases (or with any other choice), it would be important to teach the T-vertical the right way, so as to bring out the general

The horizontal bar of the T represents the collection of topics chosen to provide

In my Introduction to Mathematical Thinking MOOC on Coursera, which has been running regularly since 2013, I used the structure of everyday language as the T-vertical, and some topics in elementary number theory for the bar of the T. I only needed one branch of mathematics on the bar, since the goal was to teach mathematical thinking itself, and for that, one application domain was enough. (I chose number theory since you need nothing more than arithmathetic to get into the early parts of the subject.) If the goal is to cover everything in the Common Core State Standards, you’d need a number of branches of mathematics (though only a handful).

So much for the diagram. Before I launch into my efficiencies sales pitch, let me make a few remarks about the itemized list above.

Dr. Boaler is a former school teacher, education system administrator, and more recently a world-renowned mathematics education scholar of many years standing. She is one of a number of mathematics pedagogy experts I work and/or consult with. I mention that because my primary expertise is in mathematics, a discipline I have worked in for half a century. I do have a fair amount of knowledge of mathematics pedagogy, but purely as a result of studying the subject fairly extensively. I have not and do not engage in original research into mathematics pedagogy. I cannot, therefore, claim to be an expert in that domain. The suggestions I make here are, as always, in my capacity as a mathematician.

Very rarely, in various areas of human endeavors, exceptional individuals come along. That’s simply a feature of distributions with an element of randomization. But for the most part, children classified as “gifted” are simply the offspring of relatively well-off, educated parents who provide their children with excellent early role models and an educationally stimulating start in life. That is their “gift.” And indeed it is a gift; they were

The “math geniuses” question. Unlike some of my colleagues, I don’t mind that term being used, as long as it is understood to refer to an individual who (a) was born with a brain particularly well suited to mathematical thinking, (b) found mathematics totally fascinating (for whatever reason, perhaps a desire to escape a miserable childhood environment by retreating into the mental world of mathematics), and (c) devoted thousands of childhood hours working on mathematics. For those are the three ingredients it takes to produce an individual who could merit being called a “genius.” There are very few such math geniuses in the world. In contrast, I suspect (on numerical grounds) there are a great many children born with a brain suited to mathematical thinking, who never pursue, or show prowess in, mathematics. The term “born genius,” which you sometimes come across, strikes me as idiotic.

The “What was their secret?” question. Those kids in your class who seemed to find math easy were the ones who, for whatever reason, managed to recognize that, for all that math was presented to them as a jumble of tricks and techniques, there was method to the seeming madness. Not just method, but a fairly simple method. Mathematics, they realized, was a theme-and-variations affair. There was no need to

ASIDE: Fortunately, the multiplicative number bonds can be committed to memory by using numbers often enough in meaningful contexts. But to my mind, since they can be mastered by rote (or even better, by playing one of several cheap, first-person-shooter, multiplication video games), you might as well get them out of the way as quickly as possible by a repetitive memorization process. Moreover, there is mathematical thinking mileage to be gained by this approach, when kids discover that there are various patterns that can be used to avoid actively memorizing most of the multiplication facts (x5, x10, and commutativity are three such time-saving patterns), leaving only a handful that have to be actually learned (6 x 7 and 7 x 8 are two such – though to this day I don’t have instant recall of 6 x 7, but rely on commutativity and instant recall of 7 x 6 – don’t ask!).

But I digress. The point is, that kid on the front row who annoyingly seemed to remember everything almost certainly

The point is, the crucial importance of approaching math learning as a process of acquiring a particular way of thinking does not just apply in the elementary grades – where many kids do manage to get by with pure memorization. The same is true all the way up into the more advanced parts of the subject, where memorization becomes impossible. Yet there is

And the really nice thing is, mastering mathematical thinking to an adequate degree is like learning to ride a bike or to swim. Once you have it, you never lose it.

So much for the first two items on my initial list. But in elaborating on those, I’ve essentially covered

In fact, we’ve got so much useful stuff on the table now, it’s pretty straightforward to make that efficiencies sales pitch I promised you.

Current systemic assessments rely to a very high degree on digital technology, where students take a test presented and answered on a computer, which automatically grades their answers. To fit that format, questions are restricted to multiple-choice questions, questions that require a entry of single number as an answer, or some minor variant of one of these question types. (Earlier assessments used multiple-choice tests printed on paper that the student filled in with a pencil, with the completed test-paper then optically scanned into a computer.) This is fine for assessing what a student has learned on the horizontal (breadth) bar of the T. But on its own, the results of such a test are

That is why the better systemic assessment systems on the market also present students with open-ended questions where the student has to solve a problem using paper and pencil, with the solutions for a whole class, school, or district being sent out for grading by trained human evaluators, who follow an evaluation rubric. Though this process does bring in an element of graders’ subjectivity – even with a well-thought-out and clearly expressed rubric, the graders are still faced with an often formidable interpretation task – it works remarkably well. But it is both time-consuming and expensive. It tends to be used only for major, summative assessments at the end of a unit or a school year. The time-delay alone makes it unsuitable for formative assessments intended to provide feedback to students about their progress and to alert teachers to the need for individual-student interventions or changes in the rest of the course.

With the T-model, only the core subject chosen to constitute the T-vertical needs to be assessed this way, of course. Even with existing assessment methods, making that restriction could lead to

To obtain good assessments of mathematical thinking, educators typically present students with what are known as “complex performance tasks” (or “rich performance tasks”), requiring multi-step reasoning. CPTs often (though not always) have more than one “correct” answer, with some answers being better than others. Even when there is a unique answer, there is frequently more than one solution (= sequence of reasoning steps) that gets to that answer. CPTs can range from very basic tasks, perhaps requiring only one or two individual steps (though with a period of reflective thought required in order to start) to the fiendishly difficult.

Some kinds of CPT (particularly in subjects such as arithmetic, geometry, and algebra) can be implemented as digital puzzles, where the student has to manipulate objects or symbols on a computer screen in order to find the solution. When deployed in this format, such CPTs can be used as systemic assessment tools. Not all mathematical subjects or topics lend themselves to this kind of presentation, so it is not a feasible approach for systemic assessment

Of course, the key requirement here is to have a mathematical topic, or set of topics, and a set of CPTs in that area, that is collectively sufficient to demonstrate mathematical thinking ability. For that, remember, is what the T-vertical is all about.

Such digital assessment tools already exist. (Full disclosure: I am a member of one team developing and testing such tools.) So far, they have been subjected to limited testing on a small scale. The results have been encouraging. Conducting large-scale trials is clearly a necessary first step before they can be deployed in the manner I am suggesting. Moreover, to be useful, mathematics education has to be configured according to the T-model, where an in-depth study of one part of mathematics is used to develop the key capacity of mathematical thinking, coupled with much more shallow experiences in a number of other parts of math to achieve breadth.

That’s my suggestion. In putting it out, I might hear back that others have thought about, or advocated, something very much along the same lines. (In fact, Liping Ma essentially did just that in the article I discussed in my last post, albeit not in terms of the use of digital puzzles to provide automated assessments.)

I may also hear from psychometricians who will instantly recognize difficulties that would need to be overcome to put my proposal into practice. In fact, having talked with psychometricians, I am already aware of some issues that would need to be taken into account. Psychometrics is another of those disciplines of which I have some superficial knowledge but in which I have no expertise. But I have not yet encountered any reason why my suggestion cannot be made to work. (If I had, you would not be reading this article.)

To my mind, the really challenging obstacle is for the mathematics education establishment to accept, and then adopt, the T-model. Fifty years experience as a professional mathematician (the first fifteen or so in abstract pure mathematics, the remainder in various applied fields) has left me in no doubt that the T-model is not only perfectly viable, it is far superior to the “broad curriculum” approach we currently use, often referred to (derisively, but justifiably) as “a mile wide and an inch deep” education. But I am not in a position to mandate educational change. Nor, frankly, have I ever wanted to work my way into a position where I could have such influence. I like doing and teaching math too much! Instead, I am using what platform I have to put this suggestion out there in the hope that those who do have influence might take up the idea and run with it.

Of course, I can keep repeating my message. In fact, you can count on me doing that. :)

Read the Devlin’s Angle archives.

]]>Anyone who has taught calculus knows it is rich with real-life applications, especially in the life sciences. While there’s plenty of anecdotal evidence that teaching calculus with an eye towards application has a positive impact on student learning, this project team wants to investigate that claim with firm educational research funded through the NSF-DUE EAGER: Early-Concept Grants for Exploratory Research. Pam Bishop of the National Institute for STEM Evaluation and Research (NISER), a PI on this project explains in the answers below how she and her team are working toward this goal.

**1. Tell us about your project and the people involved.**

There is a common belief that students frequently struggle with mathematical concepts, computations, and applications because they do not see a direct connection between the mathematics they are studying and, for example, their academic discipline interests. Our project team is developing tools for both instructors and educational researchers to assess the impacts on student learning in mathematics from including biological, real-world examples. Our team includes myself (Pam Bishop), an expert in educational project evaluation, and Kelly Sturner, a science education expert. We also have Louis Gross and Suzanne Lenhart, who spent several years developing a text that covered quantitative ideas appropriate for life science students, building around biological examples with data.

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

Our project deals with a critical question in mathematics education about which there has been virtually no research: Does placing the mathematics in a concrete, real-world context help students learn and understand the mathematical ideas and enhance their skills in applying the mathematics? There were several preliminary versions of this project, and we had a lot of discussion with DUE project officers over several years. Eventually, we submitted a full proposal that was funded.

**3. What have you learned so far in this project? What’s the biggest change/adjustment you’ve had to make?**

We have found that it is possible to build a validated assessment instrument, the Biology Calculus Concept Inventory or BCCI, with the capability to address the impacts of concrete biological examples in mathematics courses. We started by looking at many quantitative concepts related to biology, but we quickly realized we needed to tighten our objectives around just calculus concepts, since so many life science students only take a calculus-focused mathematics course.

**4. Tell us about the impact you hope your project will have.**

With this tool, instructors can reinforce the importance of quantitative concepts from calculus in meeting biology learning goals and assess how effective different educational methods are for student comprehension. Educational researchers will also now be able to formally investigate this question of the impact of real-world context on mathematical understanding using our assessment tool. They can compare different modes for teaching calculus to life science students and compare standard calculus courses for life science students to those that include extensive biological motivation.

Already, Robin Taylor, a postdoctoral fellow with expertise in educational evaluation, has carried out surveys using the instrument and plans to publish her results. We hope the information we learn from the use of the BCCI will foster the development of guidelines for how inclusion of concrete examples in mathematics courses may (or may not) enhance learning and lead to more efficient educational models in interdisciplinary science. At the national level, we hope our instrument will serve as a model to assess the impact of interdisciplinary examples on enhancing mathematical and quantitative comprehension and skill development.

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

L**earn more about NSF DUE Award # 1544375**

Full Project Name: Math: EAGER: Assessing Impacts on Student Learning in Mathematics from Inclusion of Biological, Real-World Examples

NSF Abstract Link: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1544375

Project Contact: Dr. Louis Gross, gross@nimbios.org

*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**. *

Welcome to Math Values!

Our goal is to use this platform to, in the words our mission, advance the understanding of mathematics and its impact on our world. Math Values is just one part of our overall strategy to communicate the power and beauty of mathematics to the members of the Mathematical Association of America (MAA), to everyone who shares our passion for mathematics, and to new audiences who are curious about the role of mathematics across our society.

MAA’s core values are reflected in the themes you’ll see on this site: community, teaching and learning, inclusivity, and communication. All of these are meant to support discovery and innovation across the mathematical sciences, as well as developing the mathematical capacity to tackle fundamental challenges in the many fields that depend on our beautiful discipline.

To those of you who are not new to MAA blogs, you’ll find familiar voices, such as Keith Devlin’s “Devlin’s Angle,” and David Bressoud’s “Launchings.” One of our series will be drawn from projects supported by the National Science Foundation. We will also bring you new and diverse voices, views and perspectives that reflect the power and beauty of mathematics.

For example, we’re featuring an introduction to a special issue of *Oceanography* on Mathematical Aspects of Physical Oceanography. I hope you’ll be as pleased as I am to see how our colleagues in oceanography use mathematics to gain new insights. It’s not only diverse fields like oceanography that will be included in Math Values. We’ll hear from experts in data science, quantitative literacy, and the teaching and learning of mathematics about the ways in which math is used to advance their studies and understanding of the world around us.

We have incredibly rich sources that will make Math Values a great source for our community. More than that, we welcome new audiences who will visit to challenge their thinking about the role of mathematics in the world, and (to paraphrase former MAA President Francis Su’s words) the potential for mathematics to promote human flourishing.

]]>Every year the National Science Foundation (NSF) funds hundreds of grants that support undergraduate education, with many of them specifically targeting mathematics or STEM disciplines. The goals of the NSF Division of Undergraduate Education (DUE) are “to provide leadership, support curriculum development, prepare the workforce, and foster connections.” Math-oriented DUE projects range from local scholarship programs for future mathematics teachers to multi-institutional grants aiming to improve the standard introductory undergraduate mathematics curriculum. The projects are diverse in both aim and scope, with funding levels that begin at a few thousand dollars for collaborative research projects to multi-million dollar grants to support wide-ranging educational reform.

The MAA DUE Point blog will feature one DUE project each month, with the goal of sharing the motivation and results of recent NSF-funded projects. The DUE Point blog posts also aim to:

Demystify the NSF funding process through snapshots of successful projects;

Spread the word about program-wide innovation and curriculum development in mathematics;

Share strategies that helped recent projects get funded;

Highlight the work being done to improve undergraduate education by researchers across the country;

Help disseminate products of NSF DUE grants that improve undergrad math education.

**Introducing the Editors**

The DUE Point blog is edited by Audrey Malagon, lead editor, of Virginia Wesleyan University, Erin Moss, co-editor, of Millersville University, and Katie Haymaker, co-editor, of Villanova University.

Audrey is a Batten Associate Professor of Mathematics at Virginia Wesleyan University with research interests in inquiry based and active learning, election security, and Lie algebras. Erin is an Associate Professor of Mathematics Education at Millersville University, where she works with pre-service teachers at the undergraduate level and in-service teachers through the Masters of Education program. Katie Haymaker is an Assistant Professor of Mathematics at Villanova, where her research interests include coding theory and mastery-based testing in undergraduate mathematics courses.

**An Assortment of Acronyms**

Stage 1 of demystifying NSF DUE funding is to sort through some of the common shorthand that the NSF uses to refer to different types of grants. Under the DUE umbrella, we have:

ATE: Advanced Technological Education - this program has an emphasis on two-year colleges and encourages partnerships with other manufacturing and technology programs.

IUSE-EHR: Improving Undergraduate STEM Education: Education and Human Resources - this program supports projects that have the potential for “broad societal impacts” in STEM education.

S-STEM: NSF Scholarships in Science, Technology, Engineering, and Mathematics Program - the S-STEM program funds scholarships and evidence-based curricular activities that encourage student retention and success in STEM fields.

“Noyce Scholarships” (this one is not an acronym, but is sometimes abbreviated in this way): Robert Noyce Teacher Scholarship Program - the aim of this program is to support efforts to recruit and prepare effective K-12 science and mathematics teachers to work in high-need schools and educational programs.

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.

The title of this article is not intended to imply that one cannot use the derivative to find the slope of a tangent line. My point is that we cannot and should not expect students to base their understanding of the derivative on the slope of the tangent.

When I teach either the first or second semester of calculus, I always begin with a short problem set to assess student understanding of a few key ideas. One of the first questions I pose is to give the students a simple cubic polynomial, say *x^3*+ 6*x*, and ask for both the average rate of change over a given interval, say [0,2], and the instantaneous rate of change at a particular value, say *x* = 1. Invariably, almost everyone, even at the start of Calculus I, can calculate the instantaneous rate of change. Almost no one gives me the correct average rate of change.

The difficulty is that finding the instantaneous rate is formulaic. If students remember nothing else from calculus, they know that differentiation turns *x^3*+ 6*x* into 3*x^2 *+ 6. Asking for the average rate of change requires that they know what this means. I am certain that my students all saw average rates of change in their precalculus courses. They probably saw it again when they were introduced to the derivative in high school calculus. But in a calculus class, it is merely a step in the development of the derivative, a case of what the teacher talks about but not what they need to know for the exam.

The belief that average rates of change are not significant is reinforced when, as in Stewart’s calculus, the derivative is introduced as the slope of the tangent line. The problem is that *slope* is a problematic concept for many students. Identifying the derivative with the slope of a tangent line suggests a geometric understanding of derivatives. But too often it does no such thing, instead short-circuiting student development of an understanding of the derivative as describing the multiplicative relationship between changes in two linked variables.

The problematic nature of slope and rates of change was nicely documented in a paper by Cameron Byerley and Pat Thompson that appeared last year in the *Journal of Mathematical Behavior*. In the summers of 2013 and 2014, they administered a diagnostic instrument requiring written responses to 251 high school mathematics teachers.

The following is an example of the kinds of questions that were asked. Part B was asked on a separate page with the answer entered by pen so that teachers could not go back to change the answer to Part A after seeing Part B.

**Part A. Mrs. Samber taught an introductory lesson on slope. In the lesson she divided 8.2 by 2.7 to calculate the slope of a line, getting 3.04. Convey to Mrs. Samber’s students what 3.04 means.**

**Part B. A student explained the meaning of 3.04 by saying, “It means that every time x changes by 1, y changes by 3.04.” Mrs. Samber asked, “What would 3.04 mean if x changes by something other than 1?” What would be a good answer to Mrs. Samber’s question?**

The point that Byerley and Thompson were getting at was whether teachers recognized 3.04 as a multiplicative factor connecting the change in

A chunky explanation of Part A, similar to the student’s response described in Part B, was given by 78% of the teachers. Part B was included to give them a chance to expand to a multiplicative explanation. Only 8% of the teachers who gave a chunky answer to Part A provided a multiplicative response to Part B.

Further teacher difficulties with the concept of slope and rate of change are illustrated in the following two problems (Figures 1 and 2).

**Figure 1**. Item Called *Relative Rates*.

© 2014 Arizona Board of Regents. Used with permission.

Most teachers interpreted the information in Figure 1 as describing a difference, with 54% answering a. Only 28% answered e.

**Figure 2**. Item Called *Slope from Blank Graph*.

© 2014 Arizona Board of Regents. Used with permission.

Only 21% of teachers were able to provide a reasonable approximation to the slope for the problem in Figure 2. Most were unable to give any numerical value.

Given teacher difficulties with the concept of slope, we should expect most of our students to enter calculus with an inadequate understanding of what it tells us about the relationship between the variables. While mathematicians hear “slope” and associate it with the multiplicative relationship between changes in the two variables, most of our students interpret it as nothing more than an arbitrary numerical description of the degree of “slantiness.”

Consequently, when we define the derivative as the slope of the tangent, we fail to convey the meaning that makes the derivative so useful. If we want students to understand this meaning, the derivative must be introduced in terms of a multiplicative relationship between changes in the variables. It must be grounded in a thorough understanding of what average rates of change tell us and what a constant rate of change actually implies.

Read Bressoud’s Launchings archive.

**Reference**

Byerley, C. and Thompson, P. (2017). Secondary mathematics teachers’ meaning for measure, slope, and rate of change. *Journal of Mathematical Behavior*. **48**:168–193.

This post is adapted from the introduction to a special issue on Mathematical Aspects of Physical Oceanography, volume 31, no. 3 of *Oceanography*, the official magazine of the Oceanographic Society, and is published with the permission of The Oceanography Society.

Our knowledge and understanding of ocean dynamics is far from complete, but is expanding thanks in great part to new developments in mathematics. Some of the most important oceanographic discoveries have been made as a result of an integrated, multidisciplinary approach. The deepest understanding and the most interesting results almost always evolve from the interplay between theory and observation. A substantial body of theory to aid in the interpretation of observations has been developed, yet the ocean offers continually new data to challenge existing ideas—modern fieldwork is much more than cataloguing oceanic features, being designed as much to test theoretical hypotheses as it is to detect new phenomena.

The mathematical subject areas that are essential to the description of the changing spatiotemporal processes in the ocean are partial differential equations and dynamical systems. All subfields of physical oceanography rely heavily on these subjects, with analytical and computational aspects often intertwined and mutually reinforcing each other—their combined effect being stronger than the sum of each separate part. With a few notable exceptions, nonlinearity makes it impossible to obtain exact solutions to the governing equations for ocean flows. Consequently, numerical simulations play a prominent role in modern physical oceanography. However, the available technological means cannot cope with the vast range of temporal and spatial scales present in the ocean, and the prediction of fluid flow behavior becomes unrealistic if small disturbances draw energy from the main flow and subsequently grow rapidly until they become large enough to alter fundamentally the overall flow. An ongoing challenge is to reduce the computational problem to a manageable size, a task contingent upon making sensible simplifications that still provide accurate descriptions and predictions. This procedure often not only permits an in-depth study of a known phenomenon but sometimes also uncovers new processes that may not have been apparent or were overlooked. For successful derivation of adequate simplified models, it is necessary to understand the main ongoing mechanisms very well. This allows identification of physical regimes in which certain factors can be neglected, so that the dynamics is captured by a relatively simpler model. Such a model is amenable to in-depth theoretical studies that often reveal unexpected features and close the gap between real-world observations and idealized theoretical flow patterns.

From January 22 to March 23, 2018, the program “Mathematical Aspects of Physical Oceanography” took place at the Erwin Schrödinger Institute for Mathematics and Physics (Vienna, Austria). The presentations at that event inspired a special edition of *Oceanography*, the official (and open-access) magazine of The Oceanography Society, on “Mathematical Aspects of Physical Oceanography.”

In this issue, you’ll find papers that discuss emerging theoretical methodologies and computational approaches, and describe high-precision experimental results. The methods are diverse, and reflect the critical role of deep mathematical tools in advancing our knowledge of the oceans, a critical need especially as we recognize that there are systemic changes occurring that are likely to affect global climate and more.

We invite you to read this special issue, and share with your colleagues and students. We hope that increasing awareness will also lead to increased collaboration between oceanographers and mathematical scientists and, ultimately, to accelerating our understanding of our planet.

**About the authors***Adrian Constantin adrian.constantin@univie.ac.at is Professor of Mathematics at the University of Vienna, Austria.*

*George Haller georgehaller@ethz.ch is Professor of Mechanics, at the Swiss Federal Institute ofTechnology Zürich, Switzerland.*

“In math you have to remember, in other subjects you can think about it.” That statement by a female high-school student, was quoted by my Stanford colleague Prof Jo Boaler in her 2009 book What's Math Got To Do With It? I took it as the title of my June 2010 Devlin’s Angle post, which was in part a review of Boaler’s book. In a discussion peppered with quotations similar to that one, Boaler describes the conception of mathematics expressed by the students in the schools where she conducted her research. To those students, math was a seemingly endless succession of (mostly meaningless) rules to be learned and practiced. Among the remarks the students made, are (the highlighting is mine):*“We're usually set a task first and **we're taught the skills needed to do the task**, and then we get on with the task and we ask the teacher for help.”**“You're just set the task and then you go about it ... you explore the different things, and they help you in doing that ... so **different skills are sort of tailored to different tasks**.”**“In maths, **there's a certain formula to get to, say from A to B, and there's no other way to get it.** Or maybe there is, **but you've got to remember the formula, you've got to remember it**.”*

Given that is their experience of mathematics, there is no surprise that many students that are taught that way give up and bail out at the first opportunity. In fact, a more natural question is, “Why do a few students enjoy math and do well in it, answering questions at the board with seeming ease.”

The answer is, the students who do well in math and enjoy it, are ** doing something very different from the activity described in the above quotes**. Indeed, one of the things that attracts students to math is that it is the subject where

For the few who know the “one big trick” professional mathematicians rely on, math class is an engaging and enjoyable creative experience. How those few get to that point seems to be exposure to an inspiring teacher at some point, hopefully before the rot sets in and the student has been completely turned off math, or perhaps some other fortuitous event. Absent such a stimulus, though, it’s no surprise that when fed a steady diet of math classes focused on mastering one concept, formula, or special technique after another, the majority sooner-or-later give up, and simply endure it (in bored frustration) until they are through with it.

Which brings me to the mathematics Common Core State Standards, rolled out in 2009 to guide developments in education required to meet the changing environment and needs of the 21st Century.

If you go onto the CCSS website, you will find a large database of specific standards items, one such (which I picked at random) being

CCSS.MATH.CONTENT.5.MD.C.5. Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume.

Parsing out the reference code for this particular standard, it relates to Grade 5, Measurement & Data, Geometric measurement: understand concepts of volume, item 5.

It is important to realize that the Common Core is not a curriculum, nor does it stipulate how any topic should be taught. It is exactly what its name indicates: a set of

But those individual CC items are the terminal-nodes on a branching tree that has a regular structure, and it is in that structure that you see not only order but just a handful of basic principles. It is those basic principles that should guide math instruction. There are just eight of them. They are called the Common Core State Standards for Mathematical Practice. Here they are:

**Make sense of problems and persevere in solving them.****Reason abstractly and quantitatively.****Construct viable arguments and critique the reasoning of others.****Model with mathematics.****Use appropriate tools strategically.****Attend to precision.****Look for and make use of structure.****Look for and express regularity in repeated reasoning.**

Those eight principles (the website elaborates on each one) constitute the core of the mathematics Common Core. They encapsulate the key features of mathematics learning essential for anyone living or working in today’s world. Notice that there is nothing about having to learn specific facts, formulas, or techniques. The focus is entirely on ** thinking**.

The same is true of the specific items you will find in the rest of the Common Core website. When you drill down, you will find targets to aim for in order to ** develop thinking**, following those eight principles at each grade level.

When mathematics is taught as a way of thinking, along the lines specified in those eight Common Core principles, then along the way, a student will in fact pick up a whole range of facts, and meet and learn to use a variety of formulas and techniques. But the human brain does that naturally, as an automatic consequence of lived experience. We are hardwired that way!

In contrast, learning becomes hard when presented as a sequence of items to be learned and practiced one by one, each in isolation, based on the false premise that you must first learn the “basics” before you can “put them together” to form the whole. The moment you realize that mathematics is about ** process** rather than content, about doing rather than knowing, the absurdity of the “must master the basics first” philosophy becomes apparent.

Notice I am not saying “the basics” are irrelevant. Rather, they are picked up far more easily, and in a robust fashion that will last a lifetime, by ** using** them as part of living experience. For sure, a good teacher can speed the process up by helping a student recognize the used-all-the-time basics, and maybe provide instruction on how they can be used in other contexts. But the focus at all times should be on the thinking process. Because that’s what mathematics is!

If the above paragraph sounds a bit like learning to ride a bike, then all to the good. A child learning to ride a bike will acquire a good understanding of gravity, friction, mechanical advantage, and a host of other physics basics. An understanding that a physics teacher can use to motivate and exemplify lessons in those notions. But no one would say that you cannot learn to ride a bike until you have mastered those basics! Think of doing math as a mental equivalent to riding a bike. (I wrote about this parallel in Devlin’s Angle before, in my March 2014 post. My final point there was somewhat speculative, but as a mathematician who also rides bikes, I claim that the overall parallel between the two activities is very strong and illuminating.)

Now comes the point where I part company with the CCSS, and indeed much of the focus in present-day American mathematics education and standardized math assessment.

Let me ease myself in by way of my cycling. I learned to ride a bike as a child and used a bike to get around throughout my entire childhood up to graduation from high school. I then hardly ever got on a bike again until I was 55 years old and my knees gave out after a quarter century of serious running, and I bought my first (racing-style) road bike. For the first twenty minutes or so, I felt a bit unsteady on my new recreational toy, but I did not need to seek instruction or help in order to get on it that first time and ride. The basic bike-riding skill I had mastered as a small child was still there, available instinctively, albeit a bit rusty and in need of a bit of adjusting for the first few minutes.

Moreover, when I started riding with a local club, my fellow riders gave me lots of tips and advice that made me able to ride more safely and at higher speeds. Some of what I learned was not obvious, and I had to practice. It was not the same as riding a city bike at low speed.

Likewise, when I bought, first, a mountain bike, and then a gravel bike, I had to take my basic bike-riding ability and transfer it to a different device and different kinds of terrain, and, in each case, once again learn from experts how to make good, safe use of my new machine.

The point is, they were all bicycles and it was all cycling. So too with mathematics. Once someone has mastered — truly mastered — one part of mathematics, it is relatively easy to master another. Yes, you will need to learn some new things, including a new vocabulary, some new techniques, and likely a new ontology, and yes you will almost certainly benefit from (and possibly need) help, guidance, and advice from experts in that new area of math. But you already have the one key, crucial ability: ** you can think like a mathematician**.

In terms of learning mathematics, what this means is that it is enough to devote considerable effort to genuinely mastering ** just one topic **— say elementary arithmetic — and then spending some time going through the process of branching out from that one area to a number of others (perhaps algebra, geometry, trigonometry, and probability theory).

In terms of mathematics assessment, it means that it is enough to test students’ ** mathematical thinking ability** focused on just one topic, and then test to see if they have

The only question that remains is what mathematical topic should we focus on to develop the ability to think mathematically — including, I should add, an understanding of the importance of the precise use of language, the ability to handle abstraction, the need for formal definitions, and the nature and significance of proof.

Well, why not once again take our cue from how most of us learn to ride a bike. What is the equivalent of our first child’s bicycle? Elementary arithmetic.

What’s that you say? “There isn’t enough meat in elementary arithmetic to learn all you need to know about thinking mathematically, with all those bells and whistles I just mentioned.” Think again. Alternatively, check out the article written by mathematics educator Liping Ma in the article she published in the November 2013 issue of the Notices of the American Mathematical Society, titled A Critique of the Structure of U.S. Elementary School Mathematics.

Based on her experience with mathematics education in China, Ma argues forcefully, and effectively, that there is more than enough depth and breadth in “school arithmetic” (as she calls it) to fully develop the ability to think mathematically. True, in the West we don’t teach elementary arithmetic that way; indeed, we present it as a series of basic number facts to be memorized and algorithms to be practiced, as in the Boaler critique. We do so, at some speed I should add, in large part because we are in so much of a hurry to move on to all the other mathematical topics that someone at some time in the past declared were “essential” to learn in school. But as many have pointed out over several decades, the result is that our mathematics curriculum is “a mile wide and an inch deep”, resulting in students leaving school believing that “In math you have to remember, in other subjects you can think about it.”

In the June 2010 Devlin’s Angle post I referred to earlier, where I talked about Boaler’s then-new book, I mentioned Ma, and said I agreed with her argument about using school arithmetic as the topic to develop the ability to think mathematically. I still do.

I also think school arithmetic provides the one topic you need to assess mathematical thinking ability — regardless of whether you are assessing student learning, teacher performance, or district system performance. Given that, assessment of whatever breadth is required can be done relatively easily and cheaply. Because the thinking part is essentially the same, the assessment of the breadth can focus on what is ** known** (rather than what can be done with that knowledge).

And (of course), one really valuable benefit of focusing on school arithmetic is that it provides as level a playing field as you can hope for, with elementary arithmetic the one mathematical topic that everyone is exposed to at an early age.

In a future post, I’ll take this topic further, looking at the implications for teaching, the educational support infrastructure (including textbooks), the effective use of modern technologies, and the educational implications of those technologies.

Also, as the title makes clear, the focus of this article has been ** systemic** mathematics education, the mathematics that states decide is essential for all future citizens to learn in order to survive and prosper and contribute to society. There is a whole other area of mathematics education, where the focus is on the subject as an important part of human culture. That’s actually the area where I have devoted most of my efforts over the years, writing books and articles, giving public talks, and participating in radio and television programs. So I’ll leave that for other times and other places.

Read the Devlin’s Angle archive.

]]>**Figure 1.** Bill McCallum, Brit Kirwan, and Joan Leitzel at the Third CBMS Forum,* Content-Based Professional Development for Teachers of Mathematics,* October 10-12, 2010

I am using this month’s column to announce the next Forum from the Conference Board of the Mathematical Sciences (CBMS), **High School to College Mathematics Pathways: Preparing Students for the Future.** It will be held at the Hyatt Regency in Reston, VA, May 5–7, 2019, run in cooperation with the Charles A. Dana Center at the University of Texas, Austin and Achieve. Details can be found here.

The Forum is designed to develop and support state-based task forces working to bridge the gaps between high school and college mathematics. The ultimate goal is to help states create policies and practices for mathematics instruction that contribute to successful completion without reducing quality. To be truly effective, such a task force will need to be representative of all interests across the state including business and industry as well as those who shape educational policy and those who implement it at both high school and post-secondary levels, including both two- and four-year institutions. The full task force will probably have twenty or more members. The Forum is intended to work with a smaller team of six to eight individuals who will provide the leadership for the task force.

**Figure 2**. One of the breakout sessions from the Third CBMS Forum.

CBMS is the umbrella organization for the professional societies in mathematics, spanning pure and applied mathematics and statistics and including practitioners of the mathematical sciences in education (both PreK-12 and post-secondary), research, business, and industry. Over the past decade, these societies have come to agreement on a series of issues with direct relevance to mathematics education in grades 11–14, the critical transition over which so many students stumble. The Dana Center has many years of experience working with state leadership in formulating effective policies for mathematics instruction, as exemplified by their Mathematics Pathways programs. The Forum is designed to prepare state-based teams to build structures that draw on the expertise of the Dana Center and the professional societies in order to facilitate constructive dialogue among stakeholders.

**Issues**

The Forum will focus on three issues with which the professional societies have wrestled and toward which they can contribute insight:

**Responding to the changing role of mathematics in the economy**. The avalanche of data across all fields is spurring exciting and important work in mathematics. The transition years of grades 11–14 are critical for building the foundations for a workforce that can meet the evolving needs of the new economy.**Ensuring college readiness today and tomorrow.**High school and college mathematics educators are working collaboratively on this issue, recognizing the need for college-ready students, but also student-ready colleges. CBMS societies acknowledge the need for a broader understanding of how mathematics is and will be used, encompassing modeling, statistics, and data science. They also understand the need for active learning approaches that promote problem-solving abilities and higher order thinking.**Articulating the mathematical pathways that will serve all students.**Changes in demographics, economic demands, and the mathematical sciences themselves are forcing reconsideration of the pathways into and through college-level mathematics. It is necessary to evaluate whether the course structures now in place still serve their intended purpose and to understand the alternatives that are available.

**Structure of the ForumFi**The spring 2019 Forum will be built around 20 to 25 state-based teams of six to eight leaders who are committed to the formation of a local task force that will pursue dialogue leading to the creation of structures and policies that address the three issues. Each team should include representatives of the state’s department of education, higher education system, and two-year college system, while also drawing on state leaders who have been engaged in efforts to improve mathematics education at either the high school or college level. In addition to its plenary sessions, the Forum will be offering breakout sessions designed to meet the needs of state leaders at four different stages of development of bridging activities:

**Investigating**. At the introductory level are those state-based leaders who are simply curious about what has been happening in mathematics education focused on grades 11 to 14. The Forum will expose them to a wealth of information and offer suggestions of how they could begin to address the issues of the mathematical bridge.**Initializing**. These are state-based teams that are aware of significant problems at the transition from high school to college mathematics, are ready to start looking at programs and efforts that could improve the situation, and want to learn more about the options that are available and the efforts being undertaken in other states.**Emerging**. These are the states that have begun work on one side of the problem but have not started to coordinate efforts across the gap. The Forum will provide networking opportunities with states that are well down the road of coordinating these efforts.**Implementing**. These are the states that are committed to efforts that regularly bring together leaders from K-12 and higher education and are in the process of developing coordinated programs. We will provide opportunities for them to learn of other efforts and to work with policy experts to deal with obstacles and difficulties that have been encountered.

**Figure 3**. Following lunch discussion at Third CBMS Forum.

The state-based teams will leave the Forum with an agenda for following up on the ideas that they have encountered and with the connections necessary to help them as they flesh out the construction of a task force to address issues at the transition from high school to college mathematics. There will be continuing support from the Dana Center and the opportunity to engage more directly with their expertise in policy formation.

The Forum will be held at the Hyatt Regency, Reston, VA, convenient to both Washington, DC and Dulles airport. It will begin at 5 p.m. on Sunday, May 5 and conclude at 3:30 p.m. on Tuesday, May 7. It will offer a mix of plenary speakers and panelists as well as breakout sessions where participants can receive advice and support from policy experts at the Dana Center and engage with representatives of the CBMS societies around their recent reports and recommendations. Thanks to sponsorship from the Teagle Foundation and expected support from the National Science Foundation and the Carnegie Corporation of New York, CBMS anticipates covering the hotel expenses for up to six team members from up to 25 states.

The day before the Forum, Saturday, May 4, 2019, is the biennial **National Math Festival**, held at the Washington, DC, Convention Center. Those coming to the Forum are strongly encouraged to take in this day of mathematics for all

Diane Briars, Chair of CBMS, also chairs the planning committee. For questions, please contact Kelly Chapman, CBMS Administrative Coordinator, kchapma1@macalester.edu.

Read the Bressoud’s Launchings archive.

]]>I have often cited data from the Sadler and Sonnert FICSMath study (Factors Influencing College Success in Mathematics, sponsored by NSF grant #0813702), a large-scale study of 10,437 students in mainstream Calculus I in the fall of 2009 at a stratified random sample of 134 U.S. colleges and universities. Sadler and Sonnert have just published their insights from this study into the following question: Are the students who will enroll in Calculus I in college well-served by studying it first in high school?

**Figure 1**. Phil Sadler (left) and Gerhard Sonnert.

To allay the suspense, their answer is a qualified “yes.” Sadler and Sonnert demonstrate that, for most students, having taken any kind of calculus in high school raises college calculus performance by about half a grade. However, they also found that the level of mastery of the high school mathematics considered preparatory for calculus varies widely. It is a far more powerful predictor of how well students will do than whether or not they have seen calculus before.

The FICSMath study had a very simple design. Questionnaires were answered in class, exploring a wide range of variables that might influence student performance in Calculus I. These included race and gender, year in which Algebra I was taken, year in college, college precalculus (if taken), career interest, parental education, high school calculus (if taken), preparation for calculus including courses taken, grades received, and SAT or ACT scores. The single dependent variable was the grade received for the course. The authors employed a hierarchical linear model. They found that about 18% of the variation in grades could be explained at the institution or instructor level. Their model enabled them to focus just on the student effect.

**Figure 2.** Relationship between grade earned in college calculus and course grade or SAT/ACT score. The symbol area is proportional to the number of students in each group. The dotted line represents the mean grade (80.7) Source: Sadler and Sonnert, 2018, page 312.

By far the biggest effect at the student level came from preparation for calculus. Figure 2 shows the relationship between grades earned in college calculus and grades earned in high school mathematics courses or on SAT or ACT quantitative exams. The average grade across the entire study was 80.7%, a low B–. We see that less than an A on any high school math course and less than 600 on the SAT or 26 on the ACT suggests a grade of C or less, on average, in college Calculus I. While C is a passing grade, it is a strong signal that there is considerable risk in continuing the pursuit of calculus.

The six variables indicating various aspects of mathematics preparation were combined into a “Calculus Preparation Composite Score” that was very highly correlated with the probability of taking calculus in high school (Figure 3).

**Figure 3**. Relationship between calculus preparation composite and probability of taking high school calculus. Source: Sadler and Sonnert 2018, page 313.

This demonstrates the difficulty of untangling preparation for calculus from whether a student took calculus in high school. With the calculus preparation composite normalized to a mean of 0 and a standard deviation of 1, the authors found that at every level of preparation, taking calculus in high school led to an improvement in the college calculus grade (Figure 4). For students in their first year of college with an average level of preparation, the boost is 5 points, or half a grade. Intriguingly, the benefit is greatest for the students with the weakest preparation. The benefit is less for students who enroll in Calculus I after their first year in college.

In the introduction to their paper, the authors discuss how the debate over the place of calculus in high school echoes a much older and more fundamental disagreement over the extent to which mathematics is hierarchical. Does every mathematical topic have a set of prerequisites that must be mastered before any progress can be made, or can students benefit from a spiraling effect, introducing new concepts while revisiting the mathematics on which they rest?

From my experience, most mathematicians and mathematics educators recognize that spiraling is an essential part of learning. It is commonplace to assert that one never learns a subject until one has moved on to the course that builds upon it. At the same time, they acknowledge that students whose foundational knowledge is too weak will struggle as they move forward. The familiar adage is that a student does not fail calculus because they do not understand the calculus but because they have not mastered precalculus.

**Figure 4**. Relationship between college calculus performance, high school preparation, taking high school calculus, and year taking calculus in college.

To the college instructor who sees students missing exam questions because of mistakes at the level of precalculus or earlier, the rapid expansion of calculus into our high schools seems a misplaced allocation of resources. And yet, requirements of prerequisite knowledge before admission to calculus that are too strict can limit access to mathematically intensive careers, especially for first generation students and those from under-resourced schools. This is compounded by the fact that, generally speaking, we do a miserable job of remediation. I documented this in “First, Do No Harm.” In this paper, Sadler and Sonnert reveal that—with other variables controlled—taking precalculus in college lowered the Calculus I grade by a small but statistically significant amount, an observation described in greater detail in Sonnert & Sadler, 2014.

We must expect that students will enter Calculus I with deficiencies that will need to be recognized and addressed within the context of the new material in this course. The rapid expansion of courses that offer expanded labs, stretched out curricula, or co-curricular offerings designed to address these deficiencies speak to the growing recognition that this is the case. What we can and should expect by way of preparation for college calculus will need to be institutionally specific, dependent on the goals of the course, the implemented curriculum, the nature of the student body, and a continuing data-based appraisal of how well current support structures and curricula are serving our students.

Read the Bressoud’s Launchings archive.

**References**

Sadler, P. & Sonnert, G. (2018). The path to college calculus: the impact of high school mathematics coursework. *Journal for Research in Mathematics Education*. 49(3), 292–329.

Sonnert, G. & Sadler, P.M. (2014). The impact of taking a college pre-calculus course on students’ college calculus performance. *International Journal of Mathematical Education in Science and Technology*, 45(8), 1188–1207.