Reshaping Undergraduate Mathematics Programs

A Cautionary Tale

The following five quotes come from 1.

The Dean at the University of Vermont called in the faculty from the department. His message:

(1)

“Given the crunch on university finances amid the COVID-19 pandemic, he’d made a difficult decision to terminate the geology department.”

Reflections of the faculty:

(2)

“The geology department had a few large, popular courses, but many of our upper-level classes were highly specialized, attracting only a handful of students. Fewer than 10 students a year majored in geology.”

(3)

“Our curriculum hadn’t changed much in decades.”

(4)

“Together, we imagined a wholly new earth science curriculum, one that would prepare students for the challenges of today. We’d rethink all of our courses from scratch, focusing on problems that students—and their potential employers—care about.”

Action:

(5)

“Our new vision looks to the future and leaves the past behind, something we could never bring ourselves to do before the prospect of termination forced us to spring into action.”

No department would want to be called in to the Dean’s office to be given a notice of termination. If we were to replace “geology” with “mathematics” in quotes 1 and 2, how applicable would they be to your mathematics department? Given the nationwide decrease in enrollments, could your mathematics department be slated for termination? Or if not termination, denied the opportunity to offer the mathematics major and relegated to only offering service courses for other departments? Imagine, “The queen of the sciences” relegated to being a servant!

This sort of relegation will be unlikely if mathematics departments take the initiative on their own to look forward and adapt. Consider this paragraph from the Overview of the 2015 Curriculum Guide from the Committee on the Undergraduate Program in Mathematics 2 of the Mathematical Association of America (MAA):

“Major programs in the mathematical sciences should present the beauty, fun, and power of mathematics. They should be designed so that all students come to see mathematics as an engaging field, rich in beauty, with powerful applications to other subjects and contemporary open questions. Each department should create and maintain a community that welcomes and supports all students, including those from groups that have been traditionally underrepresented in mathematics.” 2.

These lofty goals are associated with a set of content recommendations. Keeping the big picture of creating community and supporting students in mind, departments can choose to follow some, most or all of the content recommendations to build majors in many flexible and locally appealing ways. Many departments have taken the initiative to do this, and have been rewarded with increased numbers of majors, increased diversity and even external recognition for their efforts. The AMS Award for Exemplary Achievement in a Mathematics Department 3 recognizes many departments that have developed new pathways for upper division students.

We note that with few exceptions these award-winning and innovative programs are at large public universities, or small colleges with highly selective admissions. In this piece we provide suggestions for resources now available in the hope that more departments, at all types of institutions, will move toward a community that will welcome and encourage all students. We also review the steps taken by some of the exemplary departments who have led the way in this direction.

Transforming Post-Secondary Education (TPSE)

Among the organizations listed as partners for the 2022 Joint Mathematics Meetings was one that may have been new to members of the mathematical sciences community. Transforming Post-Secondary Education (TPSE, see www.tpsemath.org) has a vision statement which reads “Post-secondary education in mathematics will enable all students regardless of their identity, background, or chosen program of study, to develop the modern mathematical knowledge and skills they need for productive engagement in society and in the workplace.”

TPSE began their work by hosting several workshops both nationally and regionally to bring together mathematicians who were eager to recruit and retain students. The founding Board of Directors led by Chair Philip Griffiths and Executive Director Brit Kirwan formed a diverse Math Advisory Group to connect to schools and faculty at all types of post-secondary institutions. Recognizing that transforming organizations takes time and support, TPSE has laid the groundwork to drive systemic change in mathematics education. There are four priority groups (Lower Division Pathways, Upper Division Pathways, Teaching Strategies and Practices, and Graduate Education) that are working to convince mathematicians of the need for change, and also to provide the resources, encouragement, and strategies to implement that change. The authors of this piece are the co-chairs of the Upper Division Pathways (UDP) group, and we hope that by presenting some of our initiatives and future plans we can help spark discussion among our colleagues that can create momentum for change and help keep our fellow mathematicians from that unwanted trip to the Dean’s office.

Making Mathematics Relevant: The Case of Linear Algebra

A mathematics department should play a central role in supporting the educational goals of a university. What changes do we need to make so that we can be a “leadership department” rather than a “service department”? Perhaps instead of emphasizing traditional mathematics courses, a department could instead rename itself to “Mathematical Sciences” and offer a broader curriculum. This would include courses founded on the strong tradition of mathematical thought but expanded to address the needs of society and the educational goals of the university. Tradition plays an overly large role in course offerings, especially at the upper division level. Turning back to quote (2) from 1, we know that mathematics departments risk the fate of the Vermont geology department, “… our upper-level classes were highly specialized, attracting only a handful of students.”

Calculus, linear algebra, and differential equations have been the traditional mainstays of the first two years of the curriculum. However, this is changing as statistics and data analysis now often appear early in the curriculum. To understand the structure of high-dimensional data, a course in Linear Algebra can be valuable. The traditional linear algebra course, however, is available only to students who have passed a calculus sequence even though few topics have a true calculus prerequisite and the structure is meant to ensure students have “mathematical maturity.”

A faculty member or department seeking to make their linear algebra course more attractive to a wider set of students would do well to start with the recommendations of the Linear Algebra Curriculum Study Group 4 that appeared in these Notices early in 2022. This thorough report includes suggestions for timing, topics, and pedagogy to make Linear Algebra more accessible and more applicable. It is especially valuable that the study group worked with industry sources to identify both high-level skills and particular topics of value for students entering the workforce. A recent report on a survey of mathematics instructors in the United States and Canada suggests support for the notion that Linear Algebra is an appropriate first course for college students in the STEM fields, particularly computer science and data science 5. The authors make a persuasive case that making Linear Algebra the introductory college mathematics course could be especially helpful in recruiting and retaining students from traditionally underrepresented groups.

TPSE has supported innovation in teaching Linear Algebra in many ways. A visit to the events and webinars page (https://www.tpsemath.org/events) reveals an excellent collection of resources for changing Linear Algebra in particular, as well as ideas for updating the curriculum more broadly. Some mathematicians will argue that Linear Algebra is the ideal place to teach formal proof techniques. This may be so, but the experience of many of the TPSE contributors shows that the careful writing of proofs can be replaced with a more universal goal practice of effective oral and written communication of mathematical ideas. Employers tell us that students who approach internships and job interviews with experience in giving presentations have a huge advantage in the job market. Note that these goals can be accomplished in traditional mathematics courses without drastically changing the content.

What Will Mathematics Majors Do?

At schools of all sizes, much of the traditional mathematics major was built so that graduates would be well positioned to pursue graduate study in mathematics. However, graduate training in mathematics has been transformed to meet the needs of society, as noted in the previous paragraph. And these transformations now require a different set of undergraduate courses than in the last century. No longer is abstract algebra a requirement for entrance into many graduate programs in the mathematical sciences. Letting go of this topic provides an opportunity to a department to develop a mathematical sciences major that serves its local population. Of course, if the department develops an emphasis in discrete mathematics or cybersecurity issues then abstract algebra remains appropriate.

There are many students arriving at our universities that have mathematical ability and interest but decide not to declare mathematics as a major. Why? Most high school students do not understand that mathematics is now a support for so many academic endeavors. Of course, entering students who choose to major in engineering and physics understand the importance of mathematics. However, all of the sciences, plus economics and some of the social sciences, depend heavily on the analysis of data. If an undergraduate department would add courses in mathematical modeling and data analysis, this would be very attractive to students, much more so than the traditional mathematics major. Providing an undergraduate mathematics major that supports these other academic areas would serve to increase the number of students majoring or minoring in mathematics.

In a previous paragraph, we pointed out that the traditional mathematics major positions students for graduate study in mathematics. It is also the case that this same training positions students for graduate study in any academic area. However, students would more readily choose to add mathematics as another major if they could see more clearly the role that mathematics plays in their other interests. There is no question about it: A student who majors in mathematics and X is very competitive for graduate programs in X.

The claims in this section are supported by a set of reports that TPSE commissioned from the Rutgers Education and Employment Research Center, available at https://www.tpsemath.org/reports. We have heard some mathematicians argue that there is no urgency to change because the mathematics majors who don’t go on to graduate school still get good jobs in business, industry, and government. And indeed, the evidence in the Rutgers reports strongly supports this contention. However this argument neglects the fact that we are only retaining a fraction of the students who could or would major in mathematics if the curriculum was more welcoming and more relevant to their interests.

Some Illustrative Examples

But there is hope! Many schools have increased the number of mathematics majors by changing attitudes, with or without changing names. In formulating a change, as in Quote 4, it would be productive to consult with the administration and the other academic departments as to what their mathematical sciences needs are. With this information, and insight into the career paths of its recent mathematics major graduates, the department could formulate courses for its major that would attract more students to take courses from the department, thereby increasing the relevance of the department in the university. In this section, we review some particular cases that demonstrate how this can be done. TPSE has worked to help departments make these changes, and through many curricular workshops and panels that have promoted these forward-thinking programs.

How does a department redesign an undergraduate program of study for the mathematics major that meets the mathematical/statistical needs of its student body? Together with this question, it is also important to understand the same needs of the nation’s workforce. How do we prepare our students to meet the many challenges that our nation is confronting?

Formulating a program of study that is not overprescribed and recognizes the interests of students serves to increase interest in the major and attract more students. In 6 we see the impact that allowing more variety can have. In particular, we note that in 2016, three universities on the west coast (University of California–Berkeley; University of Washington–Seattle; and University of California–Los Angeles) each graduated over 300 mathematics majors. At UCLA, when you look up the mathematics major (https://ww3.math.ucla.edu/majors-minors-specializations/), there are 6 entries: Applied Mathematics, Atmospheric and Oceanic Sciences/Mathematics, Financial Actuarial Mathematics, Mathematics for Teaching, Mathematics of Computation, and Mathematics/Applied Science. Two more interdepartmental majors are also listed: Mathematics/Economics and Data Theory.

At the University of Arizona (with 125 degrees in 2016), we see the same pattern. There are seven different options (https://www.math.arizona.edu/academics/undergrads/requirements/math) available and also a degree in statistics and data science. In looking over these programs of study we see that it is possible to graduate with an undergraduate degree in mathematics without having to take courses in real analysis or abstract algebra. This is particularly striking when one also views the instructional staff at these universities as steeped in this culture of abstractness, yet this culture is allowing students to take the mathematics that is appropriate to their own interests.

At the University of Arizona, it has been made easier to double major or to graduate with two degrees. For most of these students their main interest is not mathematics, though they appreciate mathematical courses, but rather with a connection to some other subject. Some of these students choose to pursue graduate studies not in mathematics but in their other area of interest. Adding that mathematics major to the other subject matter makes those students much more competitive for graduate studies.

The above-mentioned institutions have the benefit of having large numbers of students and this allows for the creation of different options. Schools with smaller enrollments do not have this luxury. Yet some have managed to graduate a proportionately large number of mathematics majors. Some of these smaller schools have the benefit of students entering the program of study with two semesters of calculus, which certainly adds to the possibility of selecting mathematics as a major. Even so, listing the options for the major and where those options might lead is effective. Luther College (https://www.luther.edu/catalog/curriculum/mathematics/), offers the traditional mathematics majors but also indicates options that do not require real analysis and abstract algebra. At Kean College (https://www.kean.edu/academics/programs/mathematical-sciences) we again see the different options that are available and the programs of study.

The presence of these options is in keeping with the CUPM guidelines. Guideline 9 of CUPM states: “Mathematical sciences major programs should offer their students an orientation to careers in mathematics.” By providing listed options to their students, the department is emphasizing the applicability of mathematical thought.

To quote CUPM, “The times they are a-changin’—rapidly.” The section of the 2015 CUPM that deals with preparing for graduate programs is already somewhat outdated, as it only focuses on preparing for graduate programs in pure mathematics. Graduate programs in the mathematical sciences now include applied mathematics, statistics, and biostatistics, perhaps even some programs in data science. For these latter graduate programs, the options that the universities listed above, and many others, serve as adequate preparation.

Next Steps for TPSE Upper Division Pathways

In the future, our Upper Division pathways group at TPSE will be working to help departments and individuals by organizing and publicizing meetings and workshops around the following topics:

•

Diversity—The importance of having the major reflect the interests and the goals of the student body; and growing a more diverse faculty to make all students feel that they belong.

•

Informed policy choices—Helping departments obtaining data from their university needed to make sound decisions.

•

For smaller institutions: Reimagining the mathematics major and using consortium or post-bac programs to prepare students who do wish to pursue the PhD in mathematics but did not have the appropriate undergraduate preparation.

•

For larger institutions: Providing flexibility in the mathematics major for students to prepare for careers in different fields, as well as having a sustainable traditional major.

•

Providing real world experience for faculty by arranging visiting positions between mathematics departments and employers in business, industry and government.

•

Using the double major to promote the career paths of students, including the increased competitiveness for graduate programs outside of the mathematical sciences.

If you are interested in learning more about TPSE in general and the UDP group in particular, and how we might help your department think broadly about curriculum, please contact either of the co-authors.