I cannot accept that mathematics be taught in a vacuum. Yes, mathematics is beautiful, be it pure or applied. However, in our age of immediacy for students we need to move more of our efforts to teaching mathematics in context, in touch with the real world. We should incorporate more modeling and applications in our mathematics courses to richly support and motivate our students in their attempts to learn mathematics and we should support colleagues who seek to use this approach.

Over the course of time I have moved to this position. At first I used applications of mathematics in course lectures, e.g., error correcting codes in algebra, cryptology in number theory, life sciences in calculus, and engineering in differential equations. Then I assigned students to read articles in other disciplines and share these applications in class. Finally, I incorporated projects in which students could see and practice the application of mathematics. Introducing a modeling scenario makes the mathematics immediate; what do I do right now? Students desire to address the problem at hand, which is real to them, primarily because it intrigues them and piques their curiosity. Thus the mathematics becomes a necessary tool they are ready to learn. I eventually used the application to motivate the learning of the mathematics *before* introducing that mathematics. This is a “flipping” of content.

Some students are a bit shy, even resistant, to this approach. However, in an active and supportive learning environment in which students work in small groups and the teacher works the room by watching, visiting, listening, and assisting the groups, students do amazing things. Sometimes they get off a workable track, but colleagues and teachers bring them along. Students make mistakes, but as we know, learning from mistakes is an important part of learning [BrownEtAl2014]. Indeed, we do it all the time ourselves and call it conjecture and research.

**Practicing What is Preached**

For some time, many colleagues have been calling for using modeling in the mathematics curriculum, be it after the introduction and practice of the mathematical topics or before the mathematics is introduced. An example of the latter is to give elementary school students objects – lots of them – and ask them to describe what they have. Quite often, and quite naturally, they will settle on one attribute, e.g., color, weight, size. The vehicle for description is usually an organized list and quite often an associated visual; something we would recognize as a histogram. We need not formally introduce the notion of a histogram; rather just name it after our students invent it for their immediate purpose.

At a more advanced level, one example [Winkel1997] is to ask students to put one eye on a point on a hillside opposite a mountain across the valley and describe what they can see. In the course of their investigations students invent the notions of partial derivative, tangent plane, and normal to a surface. In another activity [Winkel2008], students invent Fourier series by coming up with the rather natural criteria for measuring best fit of a trigonometric series, motivated by images of spectra from chemistry, voice studies, and seismology. In [LibertiniBliss2016] the authors demonstrate that one can cover traditional topics and techniques in differential equations courses and also introduce rich modeling activities to motivate and consolidate learning. We have found that when students see a modeling situation first, it really motivates the learning of the differential equations material and their grasp of the mathematics is firmer and lasts longer because of the modeling experience. Indeed, Dina Yagodich, of Frederick Community College, says that throughout the semester her students refer to a first day of class activity on death and immigration modeling with simulations using m&m candies [Winkel2014], an indication of the importance and meaning of a modeling first approach to teaching.

**The Big Picture **

In the 2015 CUPM Curriculum Guide to Majors in the Mathematical Sciences, described by Martha Siegel in this AMS Blog [Siegel2015], there is rich support for applications and use of technology in many mathematics courses. From the Course Group on Differential Equations of the CUPM Curriculum Guide material [CUPMODE2015], we note, “There are major applications involving differential equations in all areas of science and engineering, and so many of these should be included in the ODE courses to show students the relevance and importance of this topic.” In the section, “Technology and the Mathematics Curriculum,” of main report [CUPM2015] there is strong encouragement to include technology wherever possible as its use enhances understanding and enables more sophisticated modeling and applications, thereby motivating students.

For years COMAP [COMAP2016] has enriched the repertoire for teachers who seek to motivate mathematics through modeling and application with the production of UMAP Modules, journals, texts, videos, and modeling competitions. I have worked with students who took the Mathematical Contest in Modeling and the Interdisciplinary Contest in Modeling [MCMICM2016] offered by COMAP. This is a four day, team of three, competition in which students apply the mathematics they know and learn lots more mathematics en route to solve a real world problem. Students always say, “This is the best mathematical experience in my life.” Hands down the students tell us that applying mathematics in context and on the spot for the competitions in order to build a model is the most rewarding experience of their undergraduate mathematics.

The Society for Industrial and Applied Mathematics (SIAM) and COMAP have recently released a powerful report, *Guidelines for Assessment and Instruction in Mathematical Modeling Education *[GAIMME2016], in support of modeling throughout the mathematics curricula from K-16. This report is rich in support of why, what, how, and when to both introduce and assess/evaluate modeling efforts in the classroom. The report encourages and supports faculty with little experience in modeling to get into the game and offers practical suggestions and illustrations which should enable more faculty to incorporate modeling in their teaching.

**Taking a Natural Step and Building Community**

We offer an effort to include modeling in one course, differential equations, in the hope that others will join the effort and also do so in other courses. We have created a freely-available community for teachers and students called SIMIODE — Systemic Initiative for Modeling Investigations and Opportunities with Differential Equations. SIMIODE is about teaching differential equations using modeling and technology upfront and throughout the learning process. You can learn more at our dynamic website www.simiode.org [SIMIODE2013] where we offer a community in which colleagues can communicate, contribute, collaborate, publish, teach, explore, etc.

SIMIODE is a teacher repository of materials and references to other useful sources of materials and ideas concerning teaching differential equations using modeling and technology. SIMIODE offers a growing set of Modeling Scenarios. These are key pedagogical components of SIMIODE in which a modeling situation, rich in detail, motivates the study of differential equations. Additionally, there are Technique Narratives which provide techniques and strategies for solving differential equations with motivating examples, activities, and exercises. These materials are double-blind, peer reviewed, and published on-line at SIMIODE. In addition SIMIODE offers videos from which students can collect their own data for modeling with differential equations, both at SIMIODE [SIMIODE2013] and at SIMIODE’s YouTube Channel [SIMIODEYouTube2014].

Examples for learning differential equations with modeling from SIMIODE include such topics as chemical kinetics, sublimation process, Torricelli’s Law, feral cat control, dialysis, word propagation, mixing fish, spread of oil slick, ant tunnel building, pendulum study, machine replacement, pursuit, drug administration, spring mass configurations, shuttlecock fall, design of stadium, hang time, malaria control, electric circuit, whales and krills, and many more. In each case a scenario, often with data, is offered and students are supported in building differential equation models to address the situation. Quite often the model comes before the introduction of the formal differential equation, indeed, the model motivates the mathematics.

SIMIODE is project and inquiry-based learning at its core, for teachers can find (and create and publish their own) activities in which students discover and build differential equation models to address the scenario offered. The use of technology, as appropriate, encourages a rich solution space, addressing technical, graphical, numerical, and symbolic issues in order to demonstrate techniques and address issues for the model under study. Technology permits deeper understanding and richer analyses.

Most importantly, SIMIODE is a community of teachers and students, wherein teachers can collaborate in building modeling opportunities, address issues appropriate to their interests and the interests of their students, and reach out to new colleagues who are interested in teaching differential equations using modeling as the motivation for the subject. Within SIMIODE teachers can build their own course, form groups based on common themes from class rosters to special student teams, and can work on projects with colleagues and students from different campuses.

SIMIODE is sponsoring minicourses at both MathFest in August 2016 and the Joint Mathematics Meetings (JMM2017) in January 2017 as well as conducting a Special AMS Session, “Experiences in teaching differential equations in a modeling first approach,” at JMM 2017. Thus, there is ample opportunity to get first-hand experience in this approach to teaching in addition to collegial support from the on-line community at www.simiode.org.

**Conclusion**

The message is this: Students can learn mathematics in context and we should use mathematical modeling and technology to teach mathematics. The encouragement to use modeling in mathematics coursework and support is growing in the form of resources, professional society encouragement, collegial conversations, and support communities such as SIMIODE.

Just as with radio waves that are everywhere, our receivers tune in and pick up this message about the joy and power of using modeling in mathematics instruction. More importantly, many are engaging due to what they see and hear. We invite you to join us.

**References **

[BrownEtAl2014] Brown, Peter C., Roediger, Henry L., and McDaniel, Mark A. *Make It Stick: The Science of Successful Learning.* Belknap Press, 2014.

[COMAP2016] The Consortium for Mathematics and Its Applications. 2016. www.comap.com. Accessed 17 May 2016.

[CUPM2015] 2015. CUPM. *2015 CUPM Curriculum Guide to Majors in the Mathematical Sciences*. Editor Paul Zorn. http://www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPMguide_print.pdf . Accessed 2 May 2016.

[CUPMODE2015] Devaney, R. 2015. *Ordinary Differential Equations Course Report*. http://www2.kenyon.edu/Depts/Math/schumacherc/public_html/Professional/CUPM/2015Guide/Course%20Groups/OrdDiffeq.pdf . Accessed 5 May 2016.

[GAIMME2016] COMAP and SIAM. 2016. *Guidelines for Assessment and Instruction in Mathematical Modeling Education (GAIMME)*. http://www.comap.com/Free/GAIMME/index.html . Accessed 1 May 2016.

[LibertiniBliss2016] Libertini, J. and K. Bliss. 2016. Using Applications to Motivate the Learning of Differential Equations. To appear in Association for Women in Mathematics publication.

[MCMICM2016] MCM/ICM. 2016. Mathematical Contest in Modeling and Interdisciplinary Contest in Modeling. COMAP. http://www.comap.com/undergraduate/contests/. Accessed 17 May 2016.

[SIMIODE2013] SIMIODE. 2013. Systemic Initiative for Modeling Investigations and Opportunities with Differential Equations. www.simiode.org. Accessed 1 May 2016.

[SIMIODEYoutube2014] 2014. SIMIODE YouTube Channel. https://www.youtube.com/channel/UC14lC-tyBGkDPmUnKMV3f3w. Accessed 1 May 2016.

[Winkel1997] Winkel, B. J. 1997. In Plane View: An Exercise in Visualization*. International Journal of Mathematical Education in Science and Technology*. 28(4): 599-607.

[Winkel2008] Winkel, B. J. 2008. Fourier Series: Optimization Opportunity. *International Journal of Mathematical Education in Science and Technology*. 39(2): 276-284.

[Winkel2014] Winkel, B. J. 2014. 1-1-S-MandMDeathAndImmigration. https://www.simiode.org/resources/132 . Accessed 15 May 2016.

]]>In my Mathematics for Teachers course, students take a fresh look at foundational concepts, such as fractions and place value, from an advanced perspective. For some of them, our work together exposes weaknesses in their backgrounds, and unsettling stories emerge regularly, but B.’s story stands out. B. was a senior Japanese Studies major who offered insightful observations during problem-solving sessions. As the semester progressed, it became clear that there was a gap in his mathematical knowledge. He explained that he moved to the U.S. speaking only Spanish, and missed out on the mathematics being taught while he was learning English. He soon moved to a different city, and never learned how to add fractions. A significant chunk of the college curriculum was inaccessible to him because his middle school had no mechanism for accommodating his language transition. B. has many strengths, and he will do well in the world, but he was shortchanged at a critical phase in his mathematics education.

We have all had students who arrive at college unprepared to do college-level mathematics. While there are many contributing factors at play, it’s clear that inequities in pre-K-12 education systems play an important role. It’s also clear that it is extremely difficult, if not impossible, to make up in four years for disparities experienced over fifteen years. Although we work in higher education, nevertheless we must advocate for greater equity in pre-college education. If we don’t, we’re simply perpetuating injustice.

That injustice is reflected in persistent and significant differences in educational attainment among demographic groups in the United States. It’s not just that students from some groups are less prepared for college. Those college students have too many peers who don’t have access to college at all, for reasons that are well beyond their, or their families’, control.

Consider this chapter title from a recent report of the United States Government Accountability Office (GAO):

**The Percentage of High-Poverty Schools with Mostly Black or Hispanic Students Increased over Time, and Such Schools Tend to Have Fewer Resources.**

The report goes on to describe differences in those resources. For example, 79% of schools described as low-poverty and 0 – 25 percent Black or Hispanic offer Algebra in 7th or 8th grade, compared to 49% for high-poverty, 75 – 100 percent Black or Hispanic schools. (Within that category, the rate is 37% for charter schools.)

Just last week, the U.S. Department of Education released “A First Look” at its Civil Rights Data Collection for 2013 – 2014. From that list of highlights: “Black, Latino, and American Indian or Alaska Native students are more likely to attend schools with higher concentrations of inexperienced teachers … 11% of black students, 9% of Latino students, and 7% of American Indian or Alaska Native students attend schools where more than 20% of teachers are in their first year of teaching, compared to 5% of white students and 4% of Asian students.” Recent research supports the idea that teachers become more effective with experience, and that (contrary to earlier claims) they continue to improve well into their careers.

I focus on high-poverty schools with mostly Black or Hispanic students because students in those schools are getting the least from their education systems by various measures (performance on tests, graduation rates, and college attendance). This is not to say that these are the only students that should concern us. Indeed, here in Vermont, with a predominantly white population, students eligible for free and reduced-price lunch (a common proxy for poverty) are less successful in school than their peers, by several measures. Michael Marder has put together excellent visualizations of data on connections between child poverty and school outcomes. (If you are tired of hearing how poorly U.S. students perform on international assessments of mathematics learning, note from Marder’s slides that if Massachusetts were its own country, it would rank much higher than the U.S. as a whole.)

Remedial college courses have an unimpressive track record overall. An alternative approach offered by the Carnegie Foundation is promising, but it certainly doesn’t absolve us of the responsibility to reduce the need for remediation in the first place.

A look back at recent attempts to reform public education identifies some measures that don’t work to address achievement gaps. Blaming and shaming teachers, for example, is counterproductive. For one thing, many factors influence a student’s learning. Of course teacher training and experience are important, but high-stakes testing that holds teachers accountable for factors beyond their control makes no sense.

So-called “value-added measures” (VAMs), which try to quantify a teacher’s effect on student learning by way of pre- and post-testing, don’t perform as advertised. Indeed, the American Statistical Association (ASA) issued a statement in 2014, which warns that VAMs should be used with care and expertise, because, for example, “VAMs typically measure correlation, not causation. Effects – positive or negative – attributed to a teacher may actually be caused by other factors that are not captured in the model.” For this and other reasons, the ASA states, “(r)anking teachers by their VAM scores can have unintended consequences that reduce quality.”

What are individual mathematicians to do? Given that public schools in the U.S. are largely under local control, we can start by finding out what’s happening in our own towns and states, beginning with the funding disparities among school districts. From the abstract: “…low-salary districts serve students with higher needs, offer poorer working conditions, and hire teachers with significantly lower qualifications, who typically exhibit higher turnover.” This is infuriating, if not surprising. As educators, we should understand the challenges facing our colleagues who teach children and adolescents.

On a more granular level, we might investigate what measures our local districts take to address achievement gaps. Do new teachers have access to effective induction programs designed to reduce teacher turnover? Are high-quality preschool experiences available to all children? Are there programs, like New York City’s Community Schools, that provide services to needy families in order to support learning?

We mathematicians have something to offer to the local and national conversations, given our well-developed attention to detail and our ability to analyze quantitative arguments. One has to be prepared to face, early and often, the irony of poor data analysis and inaccurate terminology being used in the name of improving education. For one example of how to respond, see this excellent piece by John Ewing. For another, we can thank Cathy Kessel.

In our academic departments, we might start by supporting our colleagues who provide appropriate training to future teachers and professional development to current practitioners. We can value mathematics research and mentor future PhD mathematicians while at the same time recognizing the importance, and complex challenges, of bringing substantial mathematics education to all children.

Our professional organizations can provide inspiration and evidence in the form of position statements. The ASA statement on VAMs is one example of a valuable contribution. The National Council of Supervisors of Mathematics and TODOS: Mathematics for All just released a strong joint statement on social justice, while the Principles to Actions document from the National Council of Teachers of Mathematics (NCTM) includes “Access and Equity” as its second principle.

An encouraging development at the AMS is the appointment of Helen Grundman to the newly established position of Director of Education and Diversity. While the focus of that position is on graduate education, this commitment to promoting diversity in mathematics will certainly draw closer attention to conditions at all levels of the pipeline.

In a recent interview, NCTM President Matt Larson reminded us to recognize the power of mathematical understanding:

I think traditionally, especially in the current era, the importance of mathematics education has always been positioned in terms of national defense and economic need and college and career readiness. And all of those issues are absolutely important, but I think we also need to keep in mind that we also teach mathematics to develop democratic citizenship through critical thinking with mathematics and that that is also an important goal for us.

Without quantitative literacy, citizens are unlikely to comprehend, let alone be able to influence, many of the decisions and actions of those in power in political, social, scientific, and economic institutions.

I want to make sure we remember that mathematics teachers in a very real way contribute to a democratic society.

I like to think that all of us who teach mathematics contribute to a democratic society, but we’ll do a better job of it if we pay attention to equity at all levels. In the 1980’s, a consortium of organizations called us to treat Calculus as “a pump, not a filter.” While we search for effective ways to bring under-prepared college students into mathematics, we can also bear witness to the filters experienced by many younger students, and support the construction of pumps to take their place.

]]>One of the iconic messages of the calculus reforms that took place in the 1990s is the “Rule of Four,” emphasizing the use of multiple representations: algebraic, geometric, numeric, and verbal. But what is a numerical representation of the derivative?

In a recent study [1], we asked faculty in mathematics, physics, and engineering to determine a derivative based on experimental data they had to collect themselves, using the apparatus shown in Figure 1. The physicists and engineers had no trouble doing so—but the mathematicians refused to acknowledge a computed *average* rate of change, however accurate, as a derivative. The physicists and engineers knew full well that their computation was an approximation, but they also knew how to ensure that it was a good one.

*Figure 1: The Partial Derivatives Machine, designed by David Roundy at Oregon State University. In this mechanical analog of a thermodynamic system, the variables are the two string positions (the flags) and the tensions in the strings (the weights). However, it is not obvious which variables are independent, nor even how many independent variables there are. For further details, see [1].
*

Context is everything in applications. Ask a physicist how small an infinitesimal distance is, and she will surely ask, “With respect to what?” Furthermore, even when working with quantities \(x\ll L\) for some scale \(L\), she might well add, “but not so small that atomic structure matters!” Physicists know that derivatives do not in fact describe the real world; they are a (very useful) idealization. This awareness of the allowed regime is second nature to scientists and engineers, even if often left unstated.

We recently argued [2] that mathematicians’ “bright line” distinction between average and instantaneous rates of change is therefore misplaced. How does one talk about instantaneous rates of change numerically, anyway? It is not only atomic structure that imposes a lower bound: Roundoff error becomes a problem for “infinitesimal” numerical computations—and experimental error plays the same role when measurements are involved. In both cases, the very notion of numerical derivative *requires* a lower bound on the step size; it is simply not possible to compute actual limits numerically, nor from experimental data.

Should we therefore reject numerical or experimental representations of derivatives? Of course not. Rather, we should move the line; what matters is not whether a rate of change is average or instantaneous, but whether it is “good enough.” Do we need to teach students complicated techniques of data analysis to determine what “good enough” means? Not necessarily, although it wouldn’t hurt to acknowledge that such techniques exist.

Our group has coined the name *thick derivative* for the resulting notion of “good enough approximation to the instantaneous rate of change.”

The lesson here goes well beyond a discussion of how best to teach students what a derivative is. The mathematics community is well aware that negative experiences with calculus are the single biggest factor causing students to switch out of STEM majors [3, 4, 5]. There is clearly a mismatch between what we mathematicians believe such a course should teach and the needs and abilities of our students. Perhaps we are focusing too much on dotting the i’s, and not enough on the underlying concepts.

I once asked my physicist wife whether physicists cared about the difference between the functions \(\frac{x^2-1}{x-1}\) and \(x+1\). Her straight-faced response was, “What difference?” This was not an instance of “sloppy math,” but rather a very deliberate attempt to point out that there are no physical situations where such removable singularities matter. So why do we start our calculus courses with them?

Similarly, mathematicians delight in constructing examples (and, with the advent of 3-d printing, models) of functions with direction-dependent limits, or of critical points that are *not* local extrema. Shouldn’t we be emphasizing the examples that *are* well behaved?

One of my favorite books as a student had the marvelous (and accurate) title, *Counterexamples in Topology* [6]. One thing I learned from this book is that some other mathematician is always going to be smarter than I am. As a successful mathematician, I have learned how to clarify my assumptions. But a calculus student should be learning how calculus works, not the largely unphysical mathematical contexts in which it doesn’t.

Is there a better way? I would argue that calculus is the study of infinitesimal reasoning, not limits. Calculus had been used successfully for 150 years before limits were invented—and the real numbers on which such limits depend were not properly defined until even later. Another 100 years would pass before nonstandard analysis would justify infinitesimal reasoning without limits, but by then it was too late; limits were here to stay.

So what do I suggest? Skip the fine print. Emphasize examples, not counterexamples. Use numerical data, and discuss the implications. Ask students to determine derivatives experimentally. No fancy apparatus is necessary; just measure rise over run! But be sure to include some examples that are not based on graphical data.

Emphasize the need to be fluent with multiple representations, not merely the ability to perform symbolic manipulations.

Much of our own work has emphasized geometric reasoning as the key to conceptual understanding. The dot product is fundamentally a *projection*; the cross product is fundamentally a *directed area*; the divergence is fundamentally about *flux*. In each case, the formulas follow from these conceptual underpinnings, rather than the other way around.

Use and encourage infinitesimal reasoning, the art of working with quantities that are “small enough”. As we have argued in a series of papers [7, 8, 9] and an online multivariable calculus text [10], differentials provide a robust, geometric, conceptual framework for working with such quantities; there are also others, such as power series.

All of these suggestions align well with the recommendations of the Curriculum Foundation Project of the MAA [11], after seeking extensive input from partner disciplines: Emphasize conceptual understanding, problem solving skills, communication skills, and a balance between perspectives.

Small group activities supporting many of these ideas are available through the project websites described below, which include indexes of activities suitable for vector calculus and multivariable calculus.

Each activity is documented separately, in hopes of allowing instructors to use as many or as few activities as they wish. Although our own work has focused on second-year calculus, many of the ideas—and some of the activities—could be easily restricted to single variable calculus. The Partial Derivatives Machine in Figure 1 becomes a *Derivatives Machine *if one string is locked down. Similarly, use just one edge of the surfaces in Figure 2.

Finally, tell a story. After all, there are really only two ideas in calculus: ratios of small quantities, and chopping and adding. Let’s not lose sight of the coherence of that underlying message.

*Figure 2: One of the plastic surface models developed by Aaron Wangberg at Winona State University as part of the Surfaces project. Each of the color-coded surfaces is dry-erasable, as are the matching contour maps, one of which is visible underneath the surface. For further details, see the Surfaces project website.*

**Acknowledgements**

Most of the ideas presented here grew out of more than 20 years of collaboration with my wife, Corinne Manogue, as well as many colleagues too numerous to name. David Roundy deserves the credit for introducing “experiment” as a representation of the derivative, leading directly to the concept of *thick derivatives*. Much of this work was done under the auspices of three overlapping projects.

The Vector Calculus Bridge project seeks to bridge the gap between the way mathematicians teach vector calculus and the way physicists use it.

The Paradigms in Physics project has redesigned the entire upper-division physics curriculum at OSU, incorporating modern pedagogy and deep conceptual connections across traditional disciplinary boundaries; its website documents both the 18 new courses that resulted, and the more than 300 group activities that were developed.

The Raising Calculus to the Surface project uses plastic surfaces and accompanying contour maps, all writable, to convey a geometric understanding of multivariable calculus.

The Bridge and Paradigms projects have been supported by the NSF through grants DUE–9653250, DUE–0088901, DUE–0231032, DUE–0618877, DUE–1023120, and DUE–1323800; the Surfaces project is supported by the NSF through grant DUE–1246094.

Figure 1 first appeared in [1]; Figure 2 is taken from the Surfaces project website, and is used by permission.

**Bibliography**

[1] David Roundy, Eric Weber, Tevian Dray, Rabindra R. Bajaracharya, Allison Dorko, Emily M. Smith, and Corinne A. Manogue, *Experts’ understanding of partial derivatives using the PartialDerivative Machine*, Phys. Rev. ST Phys. Educ. Re.s **11**, 020126 (2015).

[2] David Roundy, Tevian Dray, Corinne A. Manogue, Joseph F. Wagner, and Eric Weber, *An Extended Theoretical Framework for the Concept of Derivative*, in Proceedings of the **18th Annual Conference on Research in Undergraduate Mathematics Education**, (Pittsburgh, 2015), eds. Tim Fukawa-Connelly, Nicole Engelke Infante, Karen Keene, Michelle Zandieh, MAA, pp. 838–843.

[3] *Engage to Excel: Producing One Million Additional College Graduateswith Degrees in Science, Technology, Engineering, and Mathematics*, President’s Council of Advisors on Science and Technology, The White House, Washington, DC, 2012.

[4] Chris Rasmussen and Jessica Ellis, *Who is Switching out of Calculus and Why?*, In:

Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education, Vol. 4, eds. Anke M. Lindmeier and Also Heinze, PME, Kiel, Germany, 2013, pp. 73–80.

[5] David Bressoud and Chris Rasmussen, *Seven Characteristics of Successful Calculus Programs*, Notices of the AMS **62**, 144–146 (2015).

[6] Lynn Arthur Steen and J. Arthur Seebach, Jr., **Counterexamples in Topology**, 2nd edition, Springer Verlag, New York, 1978.

[7] Tevian Dray and Corinne A. Manogue, *Using Differentials to Bridge the Vector Calculus Gap*, College Math. J. **34**, 283–290 (2003).

[8] Tevian Dray and Corinne A. Manogue, *Putting differentials back into calculus*, College Math. J. **41**, 90–100 (2010).

[9] Tevian Dray, *Using differentials to determine the derivatives of trigonometric and exponential functions*, College Math. J. **44**, 17–23 (2013).

[10] Tevian Dray and Corinne A. Manogue, *The Geometry of Vector Calculus*, (online only).

[11] Susan Ganter and William Barker, eds., *Curriculum Foundations Project: Voices of the Partner Disciplines*, MAA, 2004.

In my experience, many students in K-12 and post-secondary mathematics courses believe that:

- all math problems have known answers,
- failure and misunderstanding are absent from successful mathematics,
- their instructor can always find answers to problems, and
- regardless of what instructors say, students will be judged and/or assessed based on whether or not they can obtain correct answers to problems they are given.

As long as students believe in this mythology, it is hard to motivate them to develop quality mathematical practices. In an effort to undercut these misunderstandings and unproductive beliefs about the nature of mathematics, over the past several years I’ve experimented with assignments and activities that purposefully range across the intellectual, behavioral, and emotional psychological domains. In this article, I provide a toolbox of activities for faculty interested in incorporating these or similar interventions in their courses.

**Psychological Domains**

A useful oversimplification frames the human psyche as a three-stranded model:

The intellectual, or *cognitive*, domain regards knowledge and understanding of concepts. The behavioral, or *enactive*, domain regards the practices and actions with which we apply or develop that knowledge. The emotional, or *affective*, domain regards how we feel about our knowledge and our actions. All three of these domains play key roles in student learning. In post-secondary mathematics courses, our classroom activities and assessments often focus primarily on intellectual knowledge and understanding, with emotional and behavioral aspects of learning addressed either implicitly or not at all. A partial antidote to this is found in the many active learning techniques being used in post-secondary mathematics courses, such as think-pair-share, “clicker” systems, one-minute papers, inquiry-based learning, and service learning, among others. A strength of active learning methods is that they challenge students’ unhelpful beliefs and practices through public dialogue and activities. What active learning techniques might not *explicitly* do is frame these discussions and activities within a broader context involving the nature of intelligence and the process of successful learning.

A goal for my courses is to incorporate direct interventions that provide students with three things:

- language that supports articulate reflection and discussion in the context of emotional and behavioral domains,
- an environment in which such reflection and discussion arise naturally and effectively, and
- a contemporary “external source” motivating this language and environment so that our discussion is not driven by the will of the instructor.

The ways in which these interventions are realized in my classes will change over time, and I am willing to follow current educational trends if they are effective tools for my students. Many of the interventions I have used are based on research in psychology regarding mindsets, a topic that I’ve written about previously on this blog. While the literature on mindset research contains contradictory empirical findings, this is not a problem for me since my main goal is to use the language and motivation that this research provides as a tool for engaging students across psychological domains. Mindset research is only one among many possible sources of motivation for meeting the goals above;* what is critical is to make sure that my mathematics courses include activities that explicitly promote student development across all three of these psychological domains.*

**A Toolbox of Interventions**

What follows are student assignments and activities that I’ve used in classes ranging from 20-student upper-level courses for math majors to 150-student Calculus courses for STEM majors. They have a common purpose of promoting student development in one or both of the emotional or behavioral domains, complementing other work that my students do to develop intellectually in mathematics. An important disclaimer: none of these activities are original with me; rather, these are all adaptations of the work of others, to whom I will always be indebted.

*Introductions*. On the first day of class each semester, I begin with students introducing themselves to each other. In a small class with less than 30-50 students, there is time for everyone to take turns sharing with the entire class their name and the reason they are taking the course. In a large-lecture course, I tell students to do the same thing with 4-6 people sitting next to each other. I teach at the University of Kentucky, and many of our STEM majors are primarily enrolled in large lecture courses during their first year. By beginning every course with a 5-minute activity that recognizes the students and promotes discussion, a collaborative tone is set for the remainder of the course, and some of the isolation that students feel (especially as one among many in a large lecture) can be countered.

*Day 1, small classes: reading and autobiography assignment.* During the first week of class, I assign an article regarding mindset research by Carol Dweck along with a one-page autobiographical essay. I have used Dweck’s articles “The Secret to Raising Smart Kids” and “Is Math a Gift? Beliefs that put females at risk” for this with success. I assign a grade to the essay based on completion only, completely ignoring the quality of the writing, editing, or ideas. The goal is to get students to reflect and be honest, not necessarily to train them to write well. If students respond to the prompt in a relevant manner, they get full credit.

*Day 1, large classes: video and small group discussions.* In large classes with 150 or more students, especially in courses that are coordinated across sections, the autobiography assignment is harder to implement. Another way to introduce students to the language of mindsets (or other tools) is to have students students watch a 10-minute video about mindset research during class on the first day. Following the video, have students spend 2-3 minutes free response writing about the video. Following the writing, have students spend 2-3 minutes discussing their response with a neighbor in the class.

*Course policy on supportive language.* I have a course policy on supportive language that I use in all of my classes: *Students are not allowed to make disparaging comments about themselves or their mathematical ability, at any time, for any reason.* I give students a variety of examples of “banned” phrases and suggested replacements that can be found here. The important aspect of this policy is that it must be enforced — if I hear students making negative comments, I say “course policy” and have them create a neutral rephrasing of their negative self-comment. This is tougher to implement in large lectures, but even in this context the policy sets a positive tone for the first month of class. In large lectures with accompanying recitations, it is important that graduate student teaching assistants are aware of this policy and enforce it during their recitation sections. It is also important that students know that the policy applies to faculty and teaching assistants as well. I had a student in a large Calculus II lecture call me out for violating this policy last semester when I was frustrated at making errors during an example, and it was an excellent moment for the class.

*Video regarding effectiveness of science videos.* During class, I have students watch a video about research regarding the effectiveness of science videos. As with the video on the first day of class, students complete a two-minute free writing followed by a two-minute discussion with their neighbors regarding their response to the video. For many students, a common behavioral practice is that if they are stuck on a math problem, they immediately search the internet for videos that explain how to do this type of problem. This is typically an unproductive behavior, and dedicating some class time to confront it directly sets the stage for further discussions regarding the processes students use for completing homework and solving problems.

*Assign an unsolved problem as homework.* As I’ve written before, assigning an unsolved math problem as homework can serve as a gateway to discussions about the nature of high-level mathematical problem solving and the processes, practices, and attitudes that students bring to authentic mathematical challenges. When I assign an unsolved problem, e.g. those given in the article linked to above, I provide students with the following prompt.

This is a famous unsolved problem in mathematics. Work on it for a while — the goal isn’t for you to solve this, but rather to get a feel for the problem. Create an essay by recording your thoughts and attempts as you work. Focus on responding to the following questions: What did you try to do? Why did you try this? What did you discover as a result? Why is this problem challenging? (Seriously, write down everything you’re thinking and every idea you try, even if it doesn’t go anywhere.)

It’s good to grade this problem generously regarding mathematical content, keeping in mind that the goal is for students to be rewarded for demonstrating persistence and good mathematical processes.

*Reflective essay about homework. *In most of my upper-level courses, especially those in which I assign an unsolved problem as homework, I have students write a 2-3-page essay explaining what they found most and least challenging in the homework so far, and what their most and least favorite homework problems have been. The prompt can ask them to directly link to mindset or another external topic, or can be left relatively open-ended to see what connections students make on their own. This can be either graded with a rubric for writing or graded based on completion. The majority of my students have discussed at length their experience working on the unsolved problem, both what they did and how they felt about their work.

*Create-your-own homework assignment.* A recent assignment that I’ve used is to have students write their own homework assignment toward the end of the semester. The specific prompt I used was this:

Create your own homework assignment containing three problems. The homework assignment should be typed. There should be a mix of easy and hard problems that represent a broad spectrum of ideas from the entire course. For each of these problems, type a paragraph explaining why you chose that problem, whether you think it is easy, medium, or hard in difficulty, and what area of the course the problem represents. Once you have created the homework assignment, you should include complete solutions to each of the problems. Your solutions to the problems may be either typed or handwritten, but they should be complete and correct.

It was fascinating to see what the students came up with for their homework. What I found particularly noteworthy was the large number of students who included as one problem a critical analysis essay or short reflective essay similar to what I had assigned in the course to complement mathematical content work. I had honestly expected their assignments to contain a range of standard problems focused on mathematical content, and was pleasantly surprised to see the students incorporating into their homework tasks that addressed behavioral and emotional aspects of doing mathematics.

*End-of-course reflective essay.* In my smaller classes, I assign as the final homework assignment the following short essay prompt. The grade is based only on completion, because I want students to write honestly without fear of being penalized for their opinions.

What were six of the most important discoveries or realizations you made in this class? In other words, what are you taking away from this class that you think might stick with you over time and/or influence you in the future? What have you experienced that might have a long-term effect on you intellectually or personally? These can include things you had not realized about mathematics or society, specific homework problems or theorems from the readings, etc. These can be things that made sense to you, or topics where you were confused, points that you agreed/disagreed with in the readings or class discussions, issues that arose while working on your course project, etc. Explain why these six discoveries or realizations are important to you.

I have found that reading through these essays is a fascinating exercise, because of the wide range of messages that the students perceived as being central to the course. Using this assignment consistently over time has helped me improve my ability to create focused courses with clearly defined and communicated learning outcomes.

**Final Thought**

If you experiment with any of these activities in your own courses, I would love to hear about your experiences!

]]>One common instructional approach during the first two years of undergraduate mathematics in courses such as calculus or differential equations is to teach primarily analytic techniques (procedures) to solve problems and find solutions. In differential equations, for example, this is true whether the course is strictly analytical or focuses on both analytic techniques and qualitative methods for analysis of solutions.

While these analytic techniques play a major part of the early undergraduate mathematics curriculum, there is significant discussion and research about the importance of learning the concepts of mathematics. Many researchers in mathematics education encourage teaching mathematics where students learn the concepts before the procedures and are guided through the process of reinventing traditional procedures themselves (e.g., Heibert, 2013). Additionally, educators who have developed mathematical learning theories often set up a dichotomy between the two kinds of learning (e.g., Skemp, 1975; Haapasalo & Kadijevich, 2000). At the collegiate level, we as professors may agree that these educational ideas hold merit, but also firmly believe that students have a significant amount of content to learn and may not always be able to spend the time necessary to allow students to participate fully in the development of conceptual understanding and the reinvention of the mathematics (including procedures).

However, some researchers, including ourselves, provide evidence that “teaching the procedures to solve problems and find solutions” and “providing ways for teaching concepts first so students will truly understand” can be integrated, and that the notion of learning procedures does not need to be shallow and merely a memorized list (Star, 2005; Hassenbrank & Hodgson, 2007). Our framework to merge these two ways of teaching is titled the *Framework for Relational Understanding of Procedures*. It was developed as part of Rasmussen and colleagues’ work in differential equations teaching and learning (Rasmussen et. al., 2006). Skemp coined the original definition; she defines relational understanding as “knowing both what to do and why” and contrasts it to instrumental understanding as “rules without reason” (1976, p. 21).

Following, we describe the six components of the *Framework for Relational Understanding of Procedures*. The idea is that each category can be used to consider and enhance students’ learning as they study a procedure. For each one, we provide a brief explanation, questions about student thinking, and an example of an exam question related to each component taken from our work in differential equations. Likely, each instructor could add other algorithms in differential equations as well as other courses.

**Components of Relational Understanding of Procedures**

**Student can anticipate the outcome of carrying out the procedure without actually having to do so and they can anticipate the relationship of the expected outcome to outcomes from other procedures. **

This component suggests that a student understands what kind of solution would be expected before solving. A student might need to consider the following: Is the solution going to be a number, or a function? When is the solution one or two functions? Are there different forms to show the answer? How do the answers compare to other answers from similar procedures?

Example:

*Suppose that a differential equation can be solved with either separation of variables or with a general technique for solving first order linear differential equations. Let* \(y_{sep}(t)\)* be the solution for an initial value problem using separation of variables, and let \(y_{lin}(t)\) **be the solution for the same initial value problem using the technique for linear differential equations. Which of the following statements correctly states the relationship between* \(y_{sep}(t)\)* and \(y_{lin}(t)\)**?*

*\(y_{sep}(t)\) is not equivalent to \(y_{lin}(t)\)*

*\(y_{sep}(t)\) is equivalent to \(y_{lin}(t)\) for all t*

*\(y_{sep}(t) = y_{lin}(t)\) only for equilibrium solutions*

*\(y_{sep}(t) = y_{lin}(t)\) only at the initial condition*

**Student can identify when it is appropriate to use a specific procedure.**

Students often can do the procedure when they know that is what is needed. However, they often are unable to decide before they start which procedure is needed. Ultimately, one reason that this is an issue is because of the structure of typical textbooks (e.g., the homework always matches the section). How many of you have had students say, “I could do all the problems in the homework, but then I didn’t know what to do for the exam”?

Example:

*Circle all that apply. A differential equation can be solved with the technique for first order linear ODEs if:*

*it has the form \(\frac{dy}{dx}=ax+by\)**for some constants a and b*

*it has a solution whose graph is linear*

*it has the form \(\frac{dy}{dx}=f(x)y+g(x)\)**for some functions f(x) and g(x)*

*it has the form \(\frac{dy}{dx}=mx+b^2\)*

**Student can correctly carry out the entire procedure or a selected step in the procedure**.

This is what we typically think of as doing a problem, or performing the framework. Can the student do the steps necessary to complete a problem correctly? Can the student analyze where they are in the procedure and know what to do next?

Example:

*A student is solving a first order linear differential equation and at some point in her solution process she correctly gets the expression to \(e^{2y}\left(\frac{dy}{dt}+2y\right)\)**. This expression is equivalent to which of the following?*

*a) \((e^{2t}y)’\)*

*b) \((e^{3t}y)’\)*

*c) \(e^{3t}y’\)*

*d) \(e^{2t}y’\)*

**Student understands the reasons why a procedure works overall. Additionally, student knows the motivation or rationale for key steps in the procedure.**

This step fundamentally involves the conceptual idea behind the procedure. As instructors, we make efforts to teach these ideas in our classes on a regular basis. However, are we concerned about how the students grow to understand the “why” of the procedure? Do the reasons for the steps play a part in the students’ solving? Can the students go back and make modifications because they understand what is really happening?

Example:

*Which of the following would be a justification for one or more of the steps needed to solve a first order linear differential equations? Circle all that apply.*

*Fundamental Theorem of Calculus*

*Mean Value Theorem*

*L’Hopital’s Rule*

*Product Rule*

**Student can symbolically or graphically verify the correctness or reasonableness of a purported outcome to a procedure without repeating the procedure.**

This component is about thinking through the answer in a way that you can decide if it makes sense. Our experience says that if you ask students to check for the reasonableness, they often just repeat the procedure, and this indicates a need to push for the bigger picture of making sense of a solution beyond just doing. Showing competence in this component might involve either checking in terms of seeing if the solution works, or using a graphical or numerical technique to see if the two solutions are compatible. Can the student find a way to check for correctness? Can the student decide if answers are reasonable?

Example:

*Joey is solving an autonomous differential equation of the form \(\frac{dy}{dt}=f(x)\)**, using separation of variables to find the general solution. At one point in his solution process he correctly gets \(e^x=t^2+c \) **. His final answer is then \(x=ln(t^2)+c \)**. We can verify that Joey’s final answer is:*

*a) Correct because \(x=ln(t^2)+c \)** says that graphs of solutions are shifts of each other along the t axis (that is, they are horizontal shifts of each other).*

*b) Correct because \(x=ln(t^2)+c \)** says that graphs of solution are shifts of each other along the x axis (that is, they are vertical shifts of each other).*

*c) * *Incorrect because \(x=ln(t^2)+c \)** says that graphs of solutions are shifts of each other along the t axis (that is, they are horizontal shifts of each other).*

*d) * *Incorrect because \(x=ln(t^2)+c \)** says that graphs of solutions are shifts of each other along the x axis (that is, they are vertical shifts of each other).*

*e) Incorrect because \(e^x\) **is always positive.*

**Student can make connections within and across representations involved in the problem and solution: symbolic, graphical, and numerical.**

Educational literature suggests that one way to demonstrate deep understanding is to make connections among representations. Traditionally, in upper level mathematics, the representations are often symbolic, but in differential equations, linear algebra, and other freshman and sophomore classes, there are several representations, and students who can be flexible and move among them have better understanding.

Example:

*Jung Hee uses a slope field to determine the long term behavior (that is what happens as \(t \to \infty \)**) of the solution to the initial value problem \(\frac{dy}{dt}=0.4y(70-y) \)**. Which of the following methods could be used to corroborate the long term behavior she found by using the slope field? Circle all that apply.*

*The technique to solve separable differential equations.*

*Euler’s numerical method with a small step size.*

*The technique to solve first order linear differential equations.*

*None of the above.*

**Conclusion**

The framework described here and the examples from an assessment developed for relational understanding (Keene, Glass, Kim, 2011) may offer some ways to think about teaching procedures that are the foundation of many of the early undergraduate mathematics class. It may not be a matter of trying to teach the procedures or the concepts (as a dichotomy) but of developing a relational understanding of the procedures so that students can not only find answers, but also understand the underpinnings and development of the procedures. We believe that if students have this relational understanding, not only will they perform better in their classes, they will retain the skills and understandings over periods of time. This will result in students doing better in all their mathematics classes.

We would like to acknowledge Dr. Chris Rasmussen for his contributions to the work.

**References**

Haapasalo, L., & Kadijevich, D. (2000). Two types of mathematical knowledge and their relation. *Journal für Mathematik-Didaktik*, *21*(2), 139-157.

Hassenbrank, J. & Hodgson., T. (2007). A framework for developing algebraic understanding & procedural skill: An initial assessment. In *Proceedings of Research in Undergraduate Mathematics Annual Conference.*

Hiebert, J. (2013). *Conceptual and procedural knowledge: The case of mathematics*. Routledge.

Keene, K. A., Glass, M. & Kim, J. H. (2011). Identifying and assessing relational understanding in ordinary differential equations. In *Proceedings of the 41st Annual Frontiers in Education Conference*, Rapid City, SD.

Rasmussen, C., Kwon, O., Allen, K., Marrongelle, K. & Burtch, M. (2006). Capitalizing on advances in K-12 mathematics education in undergraduate mathematics: An inquiry-oriented approach to differential equations. *Asia Pacific Education Review*, 7, 85-93.

Skemp, R. R. (1976). Relational understanding and instrumental understanding. *Mathematics Teaching, 77, *20-26.

Star, J. R. (2005). Reconceptualizing procedural knowledge. *Journal for Research in Mathematics Education,* 36(5), 404-415.

A good educator must facilitate learning for a classroom full of students with different attitudes, personalities, and backgrounds. But how? This question was the starting point for a new Faculty Teaching Seminar in the math and statistics department at Sam Houston State University. In the conversation that transpired, we looked to identify the most important components of creating a class culture that best enables us to achieve learning outcomes. What are our goals? How do we get the ball rolling each semester? How do we get our students on board? Read on to find out…

**What is a “classroom culture”? What are you after? Why is it important?**

*Taylor*: The environment in my classroom is a necessary component of a successful semester. The rapport that I build with my students, the tone of the class, and the ways that my students interact with each other are just as much a component of learning as the lectures or textbook.

In my experience, setting up a productive class culture can determine the potential for learning for the entire semester. A productive class culture is one where the students feel supported, protected, and valued.

*Ken*: I seek a “learning community” of student-scholars, people who are curious about mathematics and serious about learning. I want calculus students who are *proud* to be taking calculus. I want an upper level mathematics class where the students see themselves as professionals. I want a graduate class where students focus on exploration of mathematics and its mysteries, and where curiosity is the driving reason for study.

I don’t distribute the class syllabus as a hard copy. I collect work every class period and “speed-grade” it to return it the next day. I work with a department secretary to force late registration students to meet with me before adding my classes. I never have office hours before class. When challenged by colleagues about some of these unusual practices, I realized that I desire a certain type of classroom environment. I push, coach, and manipulate my students to achieve that environment.

**What do you do on Day 1 to create a classroom culture?**

*Ken*: My class culture begins with my syllabus, which lays out some “professional” expectations of my students. But I also begin, from Day 1, to set the stage for class expectations. Since much of the material I provide will be online (either via Blackboard or Google Drive), the syllabus is also available there and I do *not* hand out a hard copy. There will not be handouts during the semester; let’s get the students used to this on the first day!

In classes with a prerequisite, I give a quiz the first day. The intended message is, “We are serious about learning and are on the move!” Early in the semester I keep the class at a fairly brisk pace (emphasizing a steady regime of study) and I make sure to model this on day 1. Since many first-year students view office hours before class as an invitation to procrastinate, my office hours are not before class, but afterwards!

I never dismiss class early, not even on the first day.

*Taylor*: The answer to this question depends on what level the class is and what method of teaching I am using in the class, but there are some common themes in all of my classes on Day 1:

- Get the students talking: I always do some form of introductions in my class. Most often, I will have students pair up, introduce themselves to their partners, and then have each students’ partner introduce him/her to the entire class. This takes up a lot of time, but it is worth it! The students quickly learn that they are expected to participate. They must contribute to class, and this exercise makes them more comfortable speaking up. This also helps me to start learning their names and eliminates the need for me to do a roll call, inevitably stumbling awkwardly through hard-to-pronounce names.
- Be a cheerleader: I use some type of unconventional or atypical pedagogy in all of my classes. I always start with the assumption that my students will be new to this teaching method. I must begin to sell my teaching style on Day 1! I achieve this by explaining to students what they can expect from a typical class day and why we do things the way that we do. I also make sure to tell them what my expectations are.
- Include some content. I want to make sure that my students take my class seriously. Hard work begins on Day 1; like Ken, I never dismiss class early.

**Does classroom culture vary by class level?**

*Taylor*: Yes, absolutely. I usually focus on one or two aspects of a successful class culture and hone in on developing those aspects. In a Calculus class, for example, I most want the students to learn to justify their thought processes. To achieve this, I will ask them to buddy up every day – literally push their desk next to someone else’s. I tell them, “Turn to you partner and ask, `Why is it true that…?’” I’ll then solicit feedback in a way that supports their collaboration by asking a student, “What justification did you and your partner come up with?”

In an Inquiry Based Learning class, I most want students to value productive failure as an integral part of the learning process.

I will then carefully praise mistakes and encourage participation from students who know they are wrong. In the photo, you see my IBL Algebra students writing proofs on the board; I have them visit each other’s work and circle anything they don’t agree with. Since we have a safe space where it’s ok to be wrong, my students are professional but thorough when it comes to correcting mathematical errors.

*Ken*: Yes, certainly this varies by level. At the lower level my expectations are typically overly optimistic. I don’t abandon them, but I recognize that students have been trained to focus on grades and testing. At the graduate level a classroom culture can be relatively easy to create, particularly if the students are already in a cohort and beginning to form a community.

The emphasis on a classroom environment is even important at the grade school level – see this article by Yackel and Cobb on creating a productive classroom environment in second and third grade!

**What about students who don’t buy in? How do you create/enforce “buy in” of your culture?**

*Ken*: Some students, in first- or second-year classes, don’t buy in to the steady stream of new material, and the necessary consistent study discipline. I routinely remind everyone of the expectations, and I attempt to motivate these expectations, in the same way that the coach of an athletic team might create team pride. For those students clearly not keeping up, I eventually chat with them briefly about the fact that this class is probably not for them. I encourage these students to either catch up quickly or find a more constructive use of their time. (There is an art to this. I often write an email to a poorly-performing student in which I express concerns about the progress and suggest some constructive alternatives that include starting fresh in the course next semester. I write these emails with a view to Mom reading over the student’s shoulder!)

At every level there is a fair amount of coaching. “Here is where we are going! Here is what we are trying to achieve! Look how far you’ve come!” I’ve coached competitive youth soccer teams and the speeches are similar. “You are working hard to reach this level! Keep it up! Here is our game plan for today…”

*Taylor*: I want all my students to take charge of their own education, so I will let a challenging student make his or her own decisions on how to participate in class, as long as the behavior isn’t disruptive. I may gently remind that student that I would prefer her or him to be fully engaged. In general, though, I think that if your class culture is based on a genuine desire to facilitate learning, students recognize and value the effort.

**What are pitfalls, mistakes, disasters?**

*Taylor*: A few semesters ago, I had a mutinous Calculus class. Somehow, I encouraged so much communication and collaboration among my students outside of class that a vocal minority opposition sprung up from within the class. I later discovered that there were students campaigning for the class to give me bad course evaluations (which happened). My feelings were hurt for a bit, but I learned valuable lessons that semester. I had been uncompromising in my desire for them to ask themselves “Why?” and this group wasn’t academically ready to do that. I now pay more attention to differentiating instruction, for example when a student asks a question in class.

*Ken*: My goal is a community of students all going in the same direction. If just one or two students are not swimming with the rest, the general flow of students will often pull them in to the current. But if a significant minority resist the direction of the class then things can go bad quickly. I must keep up with class morale and make sure that the program is flowing (somewhat).

Long ago, in an abstract algebra class where students were supposed to do small projects without discussing their work with others, I uncovered a collaborative ring that included a majority of the class. The students had ignored my published restrictions on collaboration. Rather than punish over half the class for this “plagiarism”, I backed up and restarted the process, admitting that I had not been sufficiently aware of the stress my problems generated. (The memory of that class is still a bit painful.)

**In summary, **effective learning occurs in a class environment in which curiosity, exploration and even mistakes are part of the norm. We seek to create that culture even before the first class day!

*What do other teachers do to facilitate this? We would like to know!*

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While one important component of successful teaching and learning is what happens inside the classroom, an equally important component involves decisions made at the administrative level that impact our classroom environments. A challenge that mathematics departments face is to make successful arguments for resources that support high-quality programs and courses for our students. Such arguments are often bolstered when the activities of a department are placed within the context of recommendations from professional societies.

In this article we survey a selection of recent reports and recommendations related to courses in the first two years of college study, with the goal of providing an overview of these reports for faculty and department leaders. It is worth noting that most of these were created with grant support from the National Science Foundation (NSF). There are at least seventeen professional societies involved in mathematics education efforts, of which six are represented in these reports: American Mathematical Society (AMS), Mathematical Association of America (MAA), American Statistical Association (ASA), Society for Industrial and Applied Mathematics (SIAM), American Mathematical Association of Two-Year Colleges (AMATYC), and National Council of Teachers of Mathematics (NCTM).

**A Common Vision for Undergraduate Mathematical Science Programs in 2025 (AMS, MAA, ASA, SIAM, AMATYC)**

The MAA *Common Vision* project brought together leaders from the AMS, MAA, ASA, SIAM, and AMATYC to collectively reconsider undergraduate curricula and ways to improve education in the mathematical sciences. This was the first time that these five professional societies had engaged in a joint project regarding postsecondary mathematics education, reflecting the current emphasis in the mathematical community on developing coherent responses to the challenges we face across all types of institutions. Project participants represented not only these mathematical sciences associations, but also partner STEM disciplines, higher education advocacy organizations, and industry. The resulting report includes an in-depth examination of seven curricular guides published by these five associations, with the primary goal of identifying common themes in the guides. The report reflects a synthesis of these themes with other research and input from project participants and other thought leaders in the mathematical sciences community.

Some of the prominent common themes from these seven curricular guides identified by the report are:

- the role of careful curricular development, including both content issues and consideration of pathways into and through coursework needed for majors in STEM fields and other partner disciplines;
- the crucial role played by modeling and computation in mathematics education;
- the need for students to develop communication skills in mathematical contexts;
- the need to diversify pedagogical methods in mathematics courses, e.g. incorporating a blend of traditional lecture and active learning techniques;
- using technology appropriately to enhance the student learning experience;
- developing meaningful partnerships with faculty in other disciplines; and
- the strong institutional support for resources and faculty development necessary to establish and maintain these qualities of effective mathematics education.

From two-year colleges to research-focused universities, from the context of teaching STEM-focused students to engaging students struggling with basic quantitative literacy, it is important to have in mind the common challenges of teaching and learning mathematics. While there are certainly unique challenges for different teaching environments and student populations, the Common Vision report helps identify the ways in which we can provide a coherent response to all these challenges, given a solid level of support and adequate resources. The full 2015 Common Vision report is available at http://www.maa.org/programs/faculty-and-departments/common-vision.

**Partner Discipline Recommendations for Introductory College Mathematics and the Implications for College Algebra (MAA)**

College Algebra is one of the courses that plays a role across the mathematical sciences. Between 1999 and 2011, the MAA Committee for the Undergraduate Program in Mathematics conducted a series of NSF-funded activities as part of their CRAFTY project, i.e., Curriculum Renewal across the First Two Years. In the first phase of this project, a series of workshops involving mathematicians and faculty from partner disciplines were organized to identify desirable student learning outcomes for mathematics courses. In the second phase of this project, the focus narrowed to developing guidelines for College Algebra based on these workshops. Of particular note is the coherence between these guidelines and the main themes from the Common Vision report, especially with regard to:

- the importance of modeling;
- the emphasis on development of student communication skills; and
- the importance of careful curricular development including both conceptual understanding and mastery of procedural algebraic techniques.

The reports from CRAFTY are of interest to faculty involved in revising their college algebra courses, as well as to faculty who are searching for a starting point for discussions with faculty in partner disciplines. Further, the CRAFTY reports contain chapters detailing the experiences of various institutions through modeling-based revisions to their college algebra courses, including successes and failures. These reports, including guidelines for College Algebra courses, can be found at the CRAFTY website.

**Characteristics of Successful Programs in College Calculus (MAA) and the MAA/NCTM Joint Position Statement on Calculus**

While College Algebra is important across all partner disciplines, in STEM disciplines it is Calculus courses that play a central role. Beginning in 2009, with support from the NSF, the MAA has undertaken studies regarding the state of college Calculus. The first phase of this project consisted of a national survey of Calculus students and instructors followed by site visits to programs identified as effective based on the survey results. The second phase of this project began in 2015 and will broaden its scope to include Precalculus through Calculus II. The primary result of the first phase of the Calculus study was the identification of the following seven characteristics of successful calculus programs:

- Regular use of local data to guide curricular and structural modifications;
- Attention to the effectiveness of placement procedures;
- Coordination of instruction, including the building of communities of practice;
- Construction of challenging and engaging courses;
- Use of student-centered pedagogies and active-learning strategies;
- Effective training of graduate teaching assistants; and
- Proactive student support services, including the fostering of student academic and social integration.

The motivation for an ongoing study of Calculus in the United States is that despite its centrality in the postsecondary mathematics curriculum, Calculus instruction is in a state of crisis. In 2012 the MAA and NCTM released a joint position statement on Calculus, including a background document motivating the position statement in which the authors conclude:

The United States has fallen into a seriously dysfunctional system for preparing students for careers in science and engineering, guaranteeing that all but the very best [students] rush through essential parts of the mathematics curriculum [in high school] and then are forced to sit and spin their wheels while they try to compensate for what was missed.

University Calculus courses are taught in a complicated broader context involving high school Calculus courses, the AP exam system, the rapid increase of dual and concurrent enrollment programs, and other factors, significantly complicating postsecondary Calculus instruction. For departments that are interested in rethinking their Calculus courses, these resources can help clarify our conversations and provide refined focal points for improvement.

Articles and reports from the MAA Calculus Study can be found at http://www.maa.org/programs/faculty-and-departments/curriculum-development-resources/national-studies-college-calculus. The MAA/NCTM joint position statement on Calculus, including a background document with more information, can be found at http://www.nctm.org/Standards-and-Positions/Position-Statements/Calculus/.

**Other Resources**

There are several other reports that are worthy of attention from faculty and department leadership, a few of which are briefly discussed here. The NRC Mathematical Sciences in 2025 report is a comprehensive review of the mathematical sciences, including a vision for the future over the next decade. The NRC report was completed at the request of the NSF, and includes information about a wide range of topics in the mathematical sciences, including education and diversity. The MAA CUPM Curriculum Guides from 2004 and 2015 are discussed in the Common Vision report, and together form a substantial set of recommendations for courses, departments, and programs. The other curricular guidelines discussed in the Common Vision report are each worth serious consideration beyond the summaries given in Common Vision. For mathematics departments that teach large numbers of preservice K-12 teachers, the CBMS Mathematical Education of Teachers II from 2012 and the NCTM Principles to Actions report are important and informative.

Regardless of the specific focus of an individual department or institution, framing our activities in a broader context and making use of resources produced by the professional societies can significantly strengthen our arguments in favor of increased support and resources for our mission of teaching and learning. These resources also serve to inform and inspire us as we revise our courses and programs.

]]>Many college and university students do volunteer work in local communities, and can learn valuable lessons in the process. The term “service learning” refers more specifically to service activities that are integral parts of academic courses. It can sometimes be difficult for mathematicians to envision how such projects could be included in their courses, especially courses focused on “pure” topics; for example, I have difficulty imagining how one would include such activities in Abstract Algebra. I have found myself, however, teaching courses in which service learning made sense, and I’ve implemented some service-learning projects with varying outcomes. Below I share some lessons I’ve learned in the process.

First, though, I offer some context. Here at Middlebury College, every entering student takes a writing-intensive first-year seminar (FYS), and every department contributes to the FYS program. For my most recent seminars, I’ve taught “Mathematics for All,” which explores questions of equity in K-12 mathematics education. Students develop their writing and reasoning skills by, for example, comparing contemporary critiques of mathematics education, and examining what is meant by “high-stakes” testing. Each version of the course has focused on a different age group, and the students did projects at local schools. These seminars are capped at fifteen students, and the instructor is the students’ academic advisors until they choose their majors, which might not happen until sophomore year.

The other relevant course is “Mathematics for Teachers,” a math content course for aspiring educators. I offer it jointly with the Education Studies department; the aims are for students to strengthen their understanding of fundamental mathematics concepts, grow as mathematical thinkers, and gain appreciation for the complexities of teaching math to children. That last aim was the motivation for getting students into classrooms. This course has had a more varied audience, including first-years through seniors, only some of whom are firmly committed to teaching after graduation, and from 15 through 30 students.

Here are my notes to self about service learning:

**Include a service-learning project only if it supports the learning goals for your course. ** This may seem obvious, but well-meaning administrators with the worthy goal of community engagement might conflate “life lessons” with the intellectual development for which you are responsible. Be sure that your own goals for your students are primary.

In the first of my FYS projects, students learned about the statewide school assessment program then in place, and wrote a brochure about it for parents of elementary-school children. A more recent cohort learned about the nature and importance of math acquisition for preschool children, and then designed and played math games with children in the local Head Start classroom. Next, they wrote a report about their activities for a college committee that was reviewing community engagement efforts. In both cases, I felt that the writing-to-learn objective of the FYS program had been met, along with the content goals concerning testing and early-childhood learning, respectively.

**Get help from people on campus.** Those enthusiastic administrators in your Office of Community Engagement (or Campus Compact liaison) may have lists of potential community partners, and may even have some funds to cover expenses, from van rentals for site visits to cards for thank-you notes to partners. They can also connect you with colleagues in other departments who have tried projects.

**Make sure you have clear communication with community partners ahead of time about what you and they expect**. Remember the brochure my students assembled? We ceremoniously delivered a couple of boxes to the school principal at a nice dinner on campus, which also included the teachers whose classes we’d visited. Only later did I learn that the principal never distributed the brochures. What we thought would be helpful – an accurate Q. and A. list, in plain language, about the testing program – didn’t serve the principal’s needs. In the future, I will make sure that at a minimum, expectations and needs for both my students and our community partners are laid out in writing. Bringing the students into that conversation is particularly helpful.

**Invite your community partner to visit your class ahead of the project.** Before the visit, have your students prepare some questions. Having learned from my first experience, I had the principal of the second school come to my “Math for Teachers” class. She emphasized, in a way that I couldn’t, the importance of maintaining confidentiality, especially given that some of her students were children of college faculty and staff. She also was explicit about the kinds of conversations she wanted my students to have with hers: “Ask them to explain their strategies.” A view into a fourth-grader’s mathematical thinking was exactly what I hoped my students would get.

When the Head Start teachers came to my latest FYS class, they too offered valuable background. One had been a Head Start parent before she went back to school in early childhood education, a life quite different from most of my students’. They emphasized the importance of play in their room, and exuded a love for their charges that was contagious. Most notable to my students was the advice to get down on the kids’ level rather than stand around, and I saw that happen as soon as we entered their classroom. The children engaged with my students immediately, to everyone’s delight.

**Be prepared to invest a lot of time in logistics. **Scheduling site visits and making sure students get to them can be a challenge. It didn’t make sense for me to send more than three students at a time to a classroom, and schools have frequent alterations to their schedules, for example, so my plans went through many drafts. Consultation with my colleagues in Education Studies was invaluable here, but even so, there were a lot of moving parts to monitor.

**Invite your community partner to come to class once the project is complete.** This doesn’t have to last for a full class period, but it can be valuable. For example, one group of my students had engaged the preschoolers in a game in the gym, and were a bit discouraged with how it had gone. The teachers, however, said, “You got them to stand on that line – that’s an accomplishment!” My students got a little more insight into the world of three- to five-year-olds, and a bonus lesson in setting realistic expectations.

** Have your students reflect on their experiences.** Allow time for the class to debrief, and, where appropriate, assign writing that requires them to integrate what they’ve learned in the field with the rest of the course. Cognitive science has identified the importance of metacognition to the learning process (see, for example, *How People Learn)*, and it’s also useful to read students’ reflections to inform your next project. This was my not-so-secret agenda in having students write about the Head Start project for the committee; their report included a brief description of the readings and discussions that preceded the project, and explained how working with the children had built on those readings. It also made it clear to me that should I do the project again, I will need to help my students fine-tune their games, given the developmental range between an immature three-year-old and an almost-six-year-old.

**Read about other mathematicians and statisticians who have done service projects.** For much more information and advice, I highly recommend *Mathematics in Service to the Community*, edited by Charles Hadlock (also the author of the lovely *Field Theory and its Classical Problems*). There you’ll find a chapter on service-learning in mathematical modeling courses, with case studies such as “The Baltimore City Fire Department Staffing Problem,” as well as chapters for instructors of statistics and education courses, and a detailed “How-To” chapter. There is also a special *Primus* issue on the topic.

As a self-described skeptic, would I include a service-learning component in a future course? At this point, I would say yes for the first-year seminar, but no for the Mathematics for Teachers course. In the latter, the semester is already too short to meet my mathematics learning objectives, and students can go into local school classrooms as part of Education Studies “methods” courses. In the seminar, though, if I want my students to hone their analytical skills by asking what an equitable mathematical educational system would look like, the benefits of experience at the children’s eye level are well worth the trouble. The added benefits to my own learning have been a welcome bonus.

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At the 2016 Joint Mathematics Meetings in Seattle this past January, an unusual mix of mathematicians and mathematics educators gathered for an AMS special session on Essential Mathematical Structures and Practices in K-12 Mathematics. This was the fourth consecutive special session at JMM organized by Bill McCallum and other folks at Illustrative Mathematics that focused on work in mathematics of mutual concern to mathematicians, mathematics educators, and K-12 teachers. The theme this year was inspired by a conversation between Dick Stanley and Kristin Umland about ratios and proportional relationships, and the talks were selected and ordered to highlight the development of mathematical ideas that are both upstream and downstream of this terrain.

Academic mathematicians are able to describe mathematical ideas in an efficient way. Across specialties, they share tools of language and habits of communication that have been shaped in order to facilitate the exchange of abstract knowledge. One purpose of the special session was to apply this cultural skill to selected topics in K-12 mathematics. The participants sought to create clearly expressed and easily understood descriptions of topics that are rarely developed clearly in the K-12 curriculum, such as measurement, number systems, proportional relationships, and linear and exponential functions. Although many people have been working in this area in recent years, much more needs to be done.

The mathematical community can—and should—contribute to the mathematics of the K-12 curriculum by applying to it the same principles of logical clarity that are used in the best expositions at the college level and beyond, so that teachers can have the best possible curricular materials and support. The coming of the Common Core State Standards in Mathematics has created a new opportunity for positive change in all subjects. Still, old habits change very slowly, and the path in this new direction is going to need many more signposts. This means that it is more important than ever for mathematicians to partner with teachers and mathematics educators to lay out a mathematically coherent path that fills in the outline of the standards.

To consider an example for which this is especially true, let’s take proportionality, the subject of two of the talks (Madden and Umland) at the special session. Mathematicians who look in detail at sources on this subject written over the past, say, 30 years will find a disconcerting jumble. Much of what is written is a throwback to procedures that were practiced in the middle-ages—neither wrong, nor useless, but out of sync with contemporary mathematical practice. Some things are needlessly obscure, ambiguous, confusing. It is easy to identify the difficulty here. Traditional treatments of “ratio and proportion” in school mathematics are rooted in a tradition that goes back to Euclid and entered the school curriculum in the Middle Ages. They are based on the concept of two equivalent ratios relating four fixed quantities.

There is no pathway through the traditional curriculum formulated on the basic idea of the modern understanding of proportionality: One (changing) quantity is proportional to another if their ratio is always the same. To understand why this is so, one must understand the different definitions for a ratio implicitly used by different people. In the traditional curriculum, people talk of a ratio \(a/b\) as a comparison between quantities \(a\) and \(b\), by which they mean that when making a multiplicative comparison between \(a\) and \(b\), the scale factor is \(a/b\) (or \(b/a\) depending on which direction one wishes to compare). So “the same ratio” in this context means “the same scale factor in a multiplicative comparison of two quantities of the same type,” and when people speak of a ratio, they sometimes mean the whole notion of a multiplicative comparison, and they sometimes mean just the scale factor.

In modern times, we have expanded and abstracted the definition of a ratio. We talk of a recipe as a ratio and we are not limited to two ingredients, and ratio equivalence is characterized by multiplying all quantities by the same scale factor rather than taking their pairwise quotients. In addition, we have extended our conception of a ratio to include quantities of different types, like distance and time. Formally, the traditional definition of a ratio \(a \mathbin{:} b\) is limited to quantities of the same type (like length and length) and by definition the ratios \(a \mathbin{:} b\) and \(c \mathbin{:} d\) are equivalent if and only if \(a/b = c/d\). The modern definition does not restrict us to quantities of the same type, and equivalence classes are characterized as \[\{sa \mathbin{:} sb \mid \text{for all}\ s > 0 \in {\mathbb R}\}\] it follows from this definition that ratios of two quantities are equivalent if and only if their related quotients are equal. This extension of the idea of a ratio is what allows us to define proportionality as we do in modern times: two variable quantities (i.e. quantities that take values from a specified subset of the real numbers) \(x\) and \(y\) that are related to one another in such a way that the ratios \(x \mathbin{:} y\) (for all values of \(x\) and \(y\)) are always equivalent, from which it follows that \(y/x\) is constant. This may be expressed in the familiar form \(y = kx\).

Unfortunately, the classical/medieval treatments of proportionality do not lead to this view. In fact, in many traditional treatments, the phrase “proportional to” does not appear at all. (Just as, in many traditional treatments of “ratio”, the idea that equivalent ratios have equal quotients is not mentioned.) What is unfortunate is that many textbooks do not go beyond the classical/medieval paradigm, and many teachers are unaware of the need to do so. An earlier post showed in a striking way that many teachers and others in the mathematics education community are not able to bridge that gap and demonstrate a conceptual understanding of proportional relationships. This is true even though they could readily solve procedural problems involving four quantities that were proportional.

What is surprising (and this was the point of the earlier post) is the “learning curve” that separates the classical/medieval paradigm from the modern formulation. Teaching experience shows that going from one perspective to the other is not a natural transition, but a discontinuous conceptual change. Many people simply do not think in terms of variables. In the meantime, those who have acquired the ability to do so regard it as easy and natural—so much so that they appear to have as much difficulty understanding how anyone could fail to grasp it, as those who do not grasp it have in acquiring it.

There is another approach to proportionality that aims to fix this problem, outlined in the Common Core State Standards. Still, the key idea related to variable quantities is very brief and easy to miss. (Four lines in 7.RP 2c, page 48.) In fact, this is the only place where the phrase “proportional to” occurs in the treatment of proportionality in the standards. So it is easy for people to fall back on the old habit of looking at the subject in a static way in terms of four numbers that form two equivalent ratios.

As a result, the old habits of the traditional ratio and proportion curriculum are still firmly entrenched. It will take a coordinated effort by mathematicians and mathematics educators working together to bring the middle school treatment of proportionality into the modern era. We need not jettison the “Rule of Three” (the old manner of reasoning with two four quantities in two equivalent ratios), but the curriculum needs a clear pathway joining this to the \(y=kx\) paradigm. This work is not so different from something mathematicians do all the time: Find the right definitions that lead to the best explanations and descriptions of mathematical phenomena.

We should make sure that ideas are accompanied by grade-level appropriate definitions, so students and teachers have something reliable to refer to when they have questions or doubts. We should provide grade-level appropriate explanations of where formulas and other results come from. We should develop ideas in a way that leads smoothly to subsequent topics and supports them. No one wants to lead students into computational dead ends. In short, this means considering what we often call “grade school mathematics” as a legitimate part of the field of mathematics and treating it as such.

This will not be easy. The culture and practices of school mathematics are not familiar to most mathematicians, but understanding them is essential for success. Moreover, for the most effective work to be done, mathematicians need to get together and talk to each other and to mathematics educators in detail about how we can most fruitfully participate in efforts to improve this situation. Although there has been much work by educational researchers on ratios and proportions (see here and here for instance), we have not yet come across any that works with the modern view of proportional relationships of varying quantities, instead of solving equations about equivalent ratios.

The AMS Special Session at the Joint Meetings is one example of the work that mathematicians are doing related to school mathematics. Several of the speakers are also involved in curriculum development efforts and teacher professional development, as are mathematicians at many institutions across the US. MSRI sponsors Math Circles, and AIM sponsors Math Teachers’ Circles and both have online resources to help mathematicians who are interested to get started. Jason Zimba’s recent article in the Notices about the Common Core also has many suggestions, including especially the final section, “What Mathematicians Can Do”.

To summarize, there is a lot more work to be done. Will you join us?

]]>As statisticians in mathematics departments, we have both spent many department meetings, departmental reviews, and water-cooler conversations discussing the merits of various different curricular decisions with respect to the calculus sequence (“Why not take linear algebra before calculus III??”), upper division electives (“But those classes are needed for graduate school!”), and number and order of courses required for the mathematics major/minor. Recently, more of those discussions have related to critical components of the statistics curriculum, and how courses from mathematics ensure that statistics students have a solid quantitative foundation. These kinds of conversations reinforce the fact that there are strong connections between mathematics and statistics, and these connections can and do affect decisions about undergraduate curricula.

More generally, this is an exciting time to be in a quantitative field. The amount of data available is staggering and there is no end to the need for models that harness the deluge of information presented to us every day. Mathematicians, Statisticians, Data Scientists, and Computer Scientists will all play substantial roles in moving quantitative ideas forward in a new data driven age. To be clear, there are challenges as well as opportunities in what lies ahead, and how we move forward – particularly with respect to training the next generation of mathematical, statistical, and computational scientists – requires deep and careful thought.

The goal of this blog post is to share some of the recent pedagogical ideas in statistics with our mathematician colleagues with whom we – as statisticians – are intimately engaged in building curricula. We hope that the description of the recent developments will open up larger conversations about modernizing both statistics and mathematics curricula. Many of the ideas below on engaging students in and out of the classroom, connecting courses in sequence or in parallel, and assessing new programs are relevant to all of us as we work to better our own classrooms.

We spent 18 months as part of a committee whose purpose was to revise the American Statistical Association’s Undergraduate Curriculum Guidelines (posted here). These new recommendations provide a flexible structure to ensure that students receive the necessary background and critical and problem solving skills to thrive in our increasingly data-centric world. Through our work on the undergraduate statistics guidelines committee, we were excited by many interesting and innovative ideas our colleagues were implementing in and about their own classrooms. This led us to co-edit a special issue of The American Statistician on Statistics and the Undergraduate Curriculum (December, 2015, including a guest editorial). In this blog post, we briefly describe several of the articles of this special issue, with the goal of familiarizing the readers with some of the issues and innovations that statisticians have implemented in their undergraduate classrooms. Many mathematicians teach introductory and advanced statistics courses, and we believe they have a vested interest in what statisticians can and should know mathematically. Additionally, they are likely interested in additional reflections on and ideas for their own classes. Our hope is that the special issue of TAS can be a valuable resource for those interested in statistics and the mathematical sciences at the undergraduate level.

The issue brought together a set of articles designed to help undergraduate statistics curricula be forward thinking. We begin with an editorial that includes a list of key papers discussing statistics at the undergraduate level. George Cobb provides a particularly provocative article encouraging all of us to “tear down” current curricula and start over. His piece is accompanied by 19 responses from renowned statisticians and education experts across the world. A link to these responses and George’s spirited rejoinder can be found here.

A number of the articles in the issue work to answer the question: “Do our bachelor’s graduates have the needed skills to compute with data?” Chamandy et al. describe problems they have encountered at Google that required sophisticated understanding of theoretical statistics. They describe excellent case studies for advanced undergraduate statistics students, by demonstrating problem solving in context. In another paper, Nolan and Temple Lang report on their summer program, “Explorations in Statistics Research”, which exposed six summer cohorts of students to the process of posing statistical questions and solving real world industry problems. In his paper, Grimshaw provides a framework for adding real data into a course with metrics arranged on two axes: data source and data management. On each axis the data is considered to be good/better/best for developing student skills of computing with data. Each of these responses provides practical examples of pedagogical innovations that statisticians have developed to help their students become more adept at working with real data in more realistic settings.

On the curricular side, many of the articles provide structure for a specific class designed to modernize the curriculum. For example, Hardin et al. and Baumer discuss different approaches to implementing a data science / computational statistics course in the modern era. Their articles provide templates for structuring the course to integrate data science with statistics while simultaneously allowing students opportunities to practice communicating results to a larger audience. Green and Blankenship provide an updated data-focused approach to the traditional Statistical Theory course. Their paper provides multiple great examples for making both Probability and Statistical Theory courses more modern and more interactive. Blades et al. and Khachatryan describe second courses (after introductory statistics), which are well suited to the goal of building students’ skills at making decisions based on data. Blades et al. discuss using design of experiments as a second course in statistics, which can follow introductory statistics combined with any level of mathematics. Khachatryan brings case studies into a time series course to help students engage with the real world connections.

The issue closes with a pair of articles (Chance & Peck and Moore & Kaplan) that address the curriculum from the assessment perspective, providing a framework for basing programmatic decisions. The set of articles will be of particular interest to readers who seek ideas for program evaluation, new approaches to teaching statistics, and to incorporate some new ideas into their existing courses and programs.

We hope that The American Statistician special issue and the extended bibliography of papers in the guest editorial as well as the online discussion provide useful fodder for further review, assessment, and continuous improvement of the undergraduate statistics curriculum. As mathematicians and statisticians we will work together to ensure that the next generation of students is able to take a leadership role by making decisions using data in the increasingly complex world that they will inhabit.

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