Introduction to the Special Issue: Insights and Lessons Learned from Mathematics Departments in the Process of Change

Abstract

This editorial provides an introduction to the special issue, Infusing active learning into precalculus and calculus courses: Insights and lessons learned from mathematics departments in the process of change. We begin with providing the background for the special issue, including an overview of the current status of active learning in precalculus through calculus, what many of the papers in this issue mean by the term “active learning” (inquiry based mathematics education), and a brief overview of the literature on institutional change. We then offer an overview of each of the three issues, explaining how various papers relate to each other within each issue. We conclude the editorial with some reflections on the institutional changes efforts in light of the COVID-19 pandemic.

KEYWORDS:

• Departmental change
• active learning
• precalculus
• calculus

1. INTRODUCTION

The purpose of this special issue is to highlight early wins and challenges experienced by mathematics departments as they work to improve student success by infusing active learning into their precalculus and calculus courses. The papers collectively respond to a call for stories around the theme of: Infusing active learning into precalculus and calculus courses: Insights and lessons learned from mathematics departments in the process of change. This set of papers is unique in that it represents ongoing stories of mathematics departments in change as they initiate or revise departmental norms and practices for faculty and graduate student professional development centered on active learning strategies, course coordination, the building of a community of practice among instructors who work together to improve learning and teaching as they interact regularly, the role of department leaders and upper administrators in supporting change efforts, and uses of local data to inform and guide innovations.

A recent census survey of all mathematics departments in the United States (U.S.) that offer a graduate degree in mathematics found that 44% of departments report that active learning is very important for student success in precalculus and calculus, but only 15% report being very successful implementing active learning [20]. These data indicate that departments across the U.S.A. are ready to incorporate active learning to improve their programs and would benefit from stories of ongoing change from those in the middle of implementation. One of the most difficult aspects of changing a department’s culture is initiating change efforts; published research about sustained change does not usually describe initial change efforts in sufficient detail for someone to try to replicate such efforts. These papers, therefore, offer stories of change from departments that are in the midst of a multi-year effort to initiate and begin the process of sustaining change efforts.

2. BACKGROUND

Over 90% of the approximately 2.5 million students enrolled in collegiate mathematics courses in the U.S.A. each year are enrolled in Calculus 2 or below [14]. Thus these stories of change focus on first-year mathematics courses: Precalculus through Calculus 2. With DFW rates for these courses often ranging from 25% to 75%, this translates to hundreds of thousands of students failing their first mathematics course; such failures have a significant negative impact on student retention in STEM majors and in college overall.

At the same time, we are in the midst of a marked transformation in the teaching and learning of introductory undergraduate mathematics courses. This transformation is apparent from recent national calls from professional organizations to improve and innovate curriculum and instructional approaches [8, 17, 18, 24, 27]. The National Science Foundation has also been funding projects seeking to understand characteristics of effective calculus programs (CSPCC: Characteristics of Successful Programs of College Calculus; e.g., [5, 10]; see maa.org/cspcc for a full list), how students’ progress through the calculus sequence (PtC: Progress through Calculus, NSF DUE-1430540; e.g., [1]; Apkarian et al., 2017; [20, 28]; see maa.org/ptc for a full list), and how to transform teaching and learning in the calculus sequence (SEMINAL: Student Engagement in Mathematics through an Institutional Network for Active Learning; e.g., [25]; see aplu.org for a full list). Further, mathematics is not alone in higher education in seeking to understand how to positively transform teaching and learning and improve equitable student outcomes (e.g., [14]).

Additionally, there are two recent special issues of PRIMUS that have some commonality to the current special issue. The first is the two-part special issue on inquiry teaching edited by Brian Katz and Elizabeth Thoren [11, 12]. The second is the special issue on improving the teaching and learning of calculus [6] edited by David Bressoud and Paul Zorn. The current special issue complements the inquiry teaching special issue with a focus on active learning and complements the calculus special issue in terms of curricular focus. However, the focus on ongoing, departmental change efforts provides a wider view of how to change departmental cultures to establish new norms for engaging mathematics teaching and learning.

2.1. Inquiry-Based Mathematics Education

Actively engaging students in mathematics can produce better and more equitable student outcomes, but there is a sharp separation between the knowledge of these research-based practices and actual (and effective) implementation (e.g., [26]). Even when people have good intentions, it is not sufficient for them to merely want to reduce lecturing to instead more actively engage their students in learning mathematics: instructors need professional development and other ongoing supports to make sustainable changes to their teaching practices [25].

Most of the papers in this special issue follow a version of inquiry-based mathematics education defined in Laursen and Rasmussen [15] as having four pillars:

• Students engage deeply with coherent and meaningful mathematical tasks

• Students collaboratively process mathematical ideas

• Instructors inquire into student thinking

• Instructors foster equity in their design and facilitation choices. (p. 138)

This definition encompasses both student and instructor behaviors. Whereas it is important for students to collaboratively engage in doing and communicating mathematics, it is just as important for instructors to structure and set norms for student collaborations in ways that foster equity and inclusive practices [10].

The research is clearly pointing to inquiry-based mathematics education as a stance and set of teaching strategies to actively engage in mathematics teaching and learning (Laursen & Rasmussen, 2019). The Mathematical Association of America’s [17] Instructional Practices Guide seeks to break research-based instructional strategies into small chunks for faculty to learn about course design, classroom practices, and assessment. Individual change is needed, on the part of each mathematics instructor. However, for such changes to be possible and sustainable, a broader cultural change is needed.

2.2. Research on Institutional Change

Reinholz et al. [23], in their recent editorial calling for research on mathematics department change efforts, point out the importance of this work:

What is at stake here is core to the mission of RUME [Research in Undergraduate Mathematics Education] – what mathematics is taught, what mathematics is learned, and the mathematical experiences of students. (p. 148)

Understanding how mathematics is taught and learned across classrooms, departments, universities, and countries requires a wide view of research on systemic change and appreciation for the complexities of change. Often, change efforts fail when those attempting to enact the changes fail to appreciate the complexity of the foundational cultural changes required [13]. Effective change processes need a sufficiently broad field of interested stakeholders who develop a common vision, understand the relevant research and evidence-based practices, select change strategies, plan and carry-out an accompanying data collection and analysis process, and then reflect on progress and adjust the plan [9]. Transformation of a culture is not a set of strategies ending in a set destination, but rather is an ongoing and cyclical process that needs to handle turnover in personnel (particularly administrative leaders) and adapt to ongoing and changing contextual factors (such as a global pandemic) [16].

Systemic change needs to “pull” on levers that can effectively initiate, enact, and sustain changes [14]. In mathematics, inquiry-based mathematics education that actively engages students and instructors in teaching and learning mathematics is one such lever [5, 25]. Effective leaders, with both formal and informal roles, within and beyond mathematics departments are another change lever, along with course coordination and professional development for instructors [5, 14, 21, 25].

When thinking particularly about higher education mathematics departments and the calculus sequence, an added consideration is that it seems like a gross misuse of resources for every department to independently reinvent the same wheels to address similar issues. Rather, departments can and should collaborate in networked improvement communities to improve mathematics teaching and learning [7, 16, 25]. Networked improvement communities are centered on cyclical transformation efforts within a professional network. Networked improvement communities assume that change is local but that groups can learn from each other and thus accelerate their own change efforts. Whereas some projects focus on fidelity of implementation, networked improvement communities typically focus on adaptation: assuming that departments learn from each other and seek to adapt strategies that work elsewhere to fit local people and contexts.

With its roots in the field of organizational change, there are relatively few theories of systemic change that have been widely and consistently applied to higher education. Reinholz and Apkarian [22] adapted Bolman and Deal’s [4] popular organizational four frames theory to apply to higher education contexts. Comprising people, power, symbols, and structures, all four frames are both objects of and lenses through which to view change efforts. Clearly, people within and without the department must be involved in transformation efforts. But, who these people are, their relationships with each other, their positions, and the power dynamics involved are all crucial power factors in a change process. Further, the individual and group beliefs, values and stances, particularly about the nature of mathematics, what it means to know and do mathematics, how students learn, and what it means to teach mathematics effectively are all important symbols involved in attempting cultural change. Finally, the people with beliefs and power dynamics operate within a set of university and departmental structures, including course coordination and incentives for instructors to engage in the work of educational innovation. Activities such as hiring a course coordinator can be viewed through the four frames: the person hired needs to be able to work productively with administrators as well as calculus instructors; the beliefs and values that person has about mathematics teaching and learning will influence how they approach course coordination and professional development; the person hired needs to be positioned with sufficient power to enact needed changes; and the department’s coordination structures need to support the coordinator’s efforts (or be changed to align).

Considering the body of research on institutional change in higher education, particularly mathematics departments, one emerging finding is the focus on academic departments as the appropriate unit for change [23]. Departments are not completely independent entities, but transformation efforts that initiate within the department seem to have sustainability potential [25]. Departments often have consistent internal structure, through course design, assignment of instructors to courses, and supports for those instructors. Another finding is that successful change efforts ground themselves in the research and connect with others who are undertaking similar efforts [13, 16, 25]. Finally, approaching transformation efforts as a mechanism for improving equitable outcomes and inclusive teaching practices is a necessary foundation, both to address opportunity and outcome gaps that exist, and to align departmental efforts with institutional priorities.

3. PAPERS IN THIS SPECIAL ISSUE

In order for other departments across the U.S.A. to benefit from ongoing change efforts, papers in this special issue include discussions of local context and departmental culture, implementation plans and expertise of leadership teams, revisions to the initial plan and reasons why, early wins and indicators of student success, local departmental challenges faced and strategies employed to mitigate challenges, and, when applicable, the role that communicating with other departments involved in similar change initiatives has played in their change efforts. Local contexts include norms and practices related to instructional autonomy, the balance between research and teaching, and the involvement of part-time and full-time instructors in teaching precalculus and calculus. We have organized the papers around three themes in the three different issues the papers are published: Student engagement, Coordination, and Stories of transformation.

3.1. Student Engagement in Mathematics Through an Institutional Network for Active Learning

The first of the three special issues focuses on nine papers from institutions participating in the SEMINAL project: Student Engagement in Mathematics through an Institutional Network for Active Learning. This NSF collaborative research project began by studying six departments that had enacted and sustained positive changes [25] and recruited nine additional sites that were seeking to transform calculus sequence teaching and learning. In addition to providing small grants to help initiate local transformation efforts (such as paying for faculty time to redesign courses or initiate coordination structures, hiring undergraduate learning assistants), the two foci of SEMINAL were active learning and creating and leveraging a networked improvement community to help accelerate local change efforts. The SEMINAL research on the first six institutions showed that a commitment to active learning, along with effective leadership and course coordination were among the most important change levers [25]. All nine papers in this special issue highlight these change levers in their own local contexts.

While all of the papers across these three issues focus on active learning through the calculus sequence (with implications for courses before precalculus), four of the papers focus particularly on instructors’ efforts to implement active learning strategies and how the departments changed to support these efforts. Oliver and Olkin discuss how to infuse active learning in the classroom through developing a community of practice among instructors. Such a community seeks to empower its members through providing both resource and implementation support. Soto and Marzocchi’s department also included a focus on the development of a community of practice; Soto and Marzocchi report how they conducted and studied faculty professional development to support the implementation of active learning through developing a community of practice. Lange et al. focus on how they were able to frame their change efforts through collaborative practices to implement active learning strategies in their calculus sequence. The shared ownership of collaborative practices helped to make their efforts part of a larger cultural change, beyond individuals’ improvement efforts. Finally, Ellington et al. focus on how they have sought to infuse culturally responsive teaching in their precalculus courses through initial and ongoing professional development and a focus on meeting their students’ needs.

Most of the papers across all three special issues include dimensions of course coordination. Two articles in the “Student engagement” issue focus particularly on how they were able to successfully increase the calculus sequence coordination in departments spread geographically across multiple campus locations. Villalobos et al. focus on how they implemented coordination at their minority-serving institution and document the large impact on student success resulting even from these initial efforts. Prior to their efforts, there was no coordination. Each instructor could select their own textbook and online homework system; they have made significant progress in coordinating not just materials but also assessments. Vandenbussche et al. also describe transformation efforts focused on increased collaboration; however, their efforts worked through faculty committees called Strand Committees, as their mechanism for orchestrating changes.

Three of the mathematics departments in this special issue are housed in large research-intensive institutions. Coordinating large multi-section courses was already part of their norms, but those structures were now reformed to focus on active learning. Moore-Russo et al. frame their efforts using organizational development and change theory. Their efforts were further centered around the development of a better mathematics pathway for students, including increased coordination and professional development, along with using data to inform decisions. Gruber et al. describe how they centered course redesign on increased coordination to successfully infuse active learning strategies in precalculus. Miller et al. also focus on increased coordination to infuse active learning in large sections Calculus 1. Although the issues of active learning and coordination are constant, implementing new structures and policies in large multi-section courses presents some unique challenges.

3.2. Coordination

The second of the three special issues focuses on nine papers from departments that feature course coordination as an important (but not sole) component to their transformation efforts. Prior research has documented the powerful ways that course coordination can be a lever for change, nudging faculty to increase their use of active learning within a de facto community of practice [3, 21, 19]. The collection of papers in this special issue adds further evidence to these previous findings and illuminates new benefits that course coordination brings to faculty and students.

With course coordination comes many benefits. One such benefit is a more consistent experience for students, regardless of instructor. For example, the paper by Mingus and Koelling and the paper by Dunnigan and Halcrow both highlight the fact that course coordination can decrease the variability in passing rates and hence level the grading playing field across instructors. A related benefit is, as argued by Mingus and Koelling, increased fidelity in addressing requisite course material for subsequent courses in the calculus sequence. As such, students benefit by being better prepared for later courses and faculty in downstream courses have greater confidence that students were learned essential concepts and procedures.

Faculty can also benefit from course coordination as participation often makes their lives easier [21]. All nine papers in this issue relate to this benefit to course coordination. For example, Krause et al. explain that their course coordinators take care of all student complaints and academic integrity issues. Ksir et al. explain how faculty can “save a lot of work” by using the suggested daily homework problems, which are coded into the online, automatically graded homework system affiliated with the textbook. Bazett and Clough explain how instructors’ lives are made easier by being provided with a wide range of resources, including materials for flipped classrooms, unit level learning objectives, several years of past exams and results, in-class worksheets and quizzes, and videos and online quizzes along with additional materials that other instructors share with the group. As demonstrated by these and the other examples in the collection of papers, providing instructional resources is an important and valued component of what course coordinators and participating instructors do. Such resources, especially those that include well-thought out active learning material, are often just the nudge faculty need to increase their use of active learning strategies.

Another major benefit of course coordination for faculty is that it creates a community of practitioners who work together and support each other. This is true for both new instructors and graduate students as well as more seasoned faculty. For example, Faudree found that even seasoned instructors “benefited from discussions with peers about how to present certain mathematical topics and how to handle particular classroom issues.” Bazett and Clough found that course coordination resulted in faculty no longer being “siloed in their individual courses, but instead part of a community that supports and encourages dedication to the craft of teaching.” In addition to the paper by Faudree and the paper by Bazett and Clough, four additional papers in this issue (Mingus and Koelling; Goyer et al.; Ksir et al.; Maciejewski et al.) speak highly of the value of regular instructor meetings where participants provide each other with just in time support, generating a feeling of belonging to a team that is working together to make the course succeed.

Several of the papers in this special issue also revealed site-specific benefits that went beyond individual students or individual instructors. Faudree, for example, found that coordination made “certain ambitious goals possible.” In particular, without coordination it would have been highly unlikely that they would have been able to implement and grade (via mastery grading) proficiency mini-tests for differentiation and integration as a part of the course, which allowed instructors to separate issues of computational facility from conceptual understanding. Faudree writes that, “All instructors felt these were a pedagogical win, but too hard to implement without coordination.” Bazett and Clough explain that course coordination enabled them to ward off a threat from engineering to take over the teaching of calculus. Maciejewski et al. explain that course coordination dramatically changed the nature of course-specific workshops, from homework review sessions where students worked individually to new teaching assistant training for active learning that reinforced and explored material in greater depth.

Although course coordination offers many benefits for students and instructors, it is not immune from shortcomings or pitfalls. In particular, depending on the policies in place regarding participation (mandatory or optional) and the course coordinator’s disposition to their role (mandator-in-chief or community-builder), there is the danger that instructors feel a loss of pedagogical autonomy.

Maintaining a sense of independence is of utmost importance for a healthy and successful coordination system. Rasmussen and Ellis [21] in their study of successful Calculus 1 programs discovered that in the more successful Calculus 1 programs studied, there was indeed pedagogical autonomy within the systems of coordination. They referred to this apparent oxymoron as “coordinated independence.” Successful Calculus 1 programs figured out ways to maintain pedagogical autonomy within a system that had many uniform elements. Three papers in particular in this special issue offer a number of insights on how this can be accomplished. Mingus and Koelling explain how trust in their colleagues to conduct their classes as they see best for their students and open and frank conversations are paramount. They also offer a comprehensive list of best practices for coordination, a list that anyone interested in course coordination would benefit from. Faudree explains how in their approach to coordination there was zero pressure to use active learning strategies, yet with weekly meetings and a supportive environment, instructors felt more comfortable and less intimidated when they were trying out something new as a team. Ksir et al. explained that in their department, autonomy is highly valued. The norm is for each instructor to set their own grading policy and to write and schedule their own midterms. The final is written by the coordinator, but with significant input from other instructors. Within these department norms and expectations, Ksir et al. discovered two connected practices that are supportive of both consistency across instructors and pedagogical autonomy. The first is the separation of the syllabus from the course calendar and the second is a separate course calendar of suggested daily active learning activities and problem sets. More details on these two practices are provided in their paper.

There are three interrelated themes across the nine papers in this issue. The first is the value and usefulness of a wide variety of resources that make instructors’ lives easier. The second is the way in which course coordination can build a community-focused instructional support system that ultimately leads to improved instructional practice. The third is the stance of disposition that the coordinator takes to their role, one that walks a delicate balance between providing resources/logistical support and creating a supportive community of practice.

3.3. Stories of Transformations

The remaining eight articles in this volume all illustrate the intentionality that is frequently needed to initiate desired departmental change. For seven of these institutions that change involves improving instruction in one or more courses in the Precalculus, Calculus 1, Calculus 2 sequence through the introduction of active learning modalities in these courses; at the other institution the desired change was to transform departmental culture around teaching so that it becomes a less solitary and more collaborative enterprise. The sources for these desires were multiple – from some campus initiatives to a departmental leader or a handful of faculty desiring to improve the quality of instruction in the courses under consideration – but in every case the department’s response was thought-out and deliberate.

This deliberateness takes different forms at these eight institutions. At several it involved establishing a faculty committee to think deeply about the perceived problem. For example, McClendon et al. explain the role of the Calculus Working Group and Bleiler-Baxter et al. explain the role of the Teaching Advance Committee in their respective transformation efforts. Other departments hired one or more faculty who were charged with addressing the perceived problem (see the paper by Bennoun and Holm for revamping Calculus 1 and the paper by Diamond et al. from improving Precalculus instruction). Several changes were often accompanied by the department chair or a group of faculty obtaining additional resources, as was the case in the change stories relayed by Bennoun and Holm, McClendon et al., and Pilgrim et al. External funding from the National Science Foundation was an important catalyst in the changes efforts described by Hancock et al. and by Bulancea et al.

The seven institutions that focused their work on bringing active learning to one or more of the Precalculus, Calculus 1, and Calculus 2 courses took similar approaches. Diamond et al. and Carney et al. explain how at their respective departments they flipped their classes, allowing for more collaborative in-class work. Five departments (see papers by Bennoun and Holm; McClendon et al; Bulancea et al.; Pilgrim et al.; and Hancock et al.) prepared group-worthy projects for use in class or in their recitations. As explained by Hancock et al., their work also included developing a one-credit Calculus 1 co-seminar based on group-work active learning.

To support the instructors using the materials they developed, six of these institutions explicitly mention that they developed or improved the professional development given to the faculty or Graduate Teaching Assistants (GTAs). The forms of this professional development vary widely. For example, Hancock et al. explain how their GTA training consists of classroom observations of the graduate students’ teaching. Bulancea et al. detail how they meet bi-weekly with the GTAs for a given course, while Pilgrim et al. explain how their professional development consists of a 3-day, pre-fall orientation and a 2-semester, non-credit bearing pedagogy course. The GTA professional development described by Bennoun and Hold consists of a summer “Instructional Summer Workshop” for faculty, while the professional development efforts described by Carney et al. and by Bulancea et al. built communities of practice to support the faculty teaching these courses.

4. ENGAGING IN INSTITUTIONAL CHANGE

The transformation efforts detailed in these papers were begun prior to 2020, and thus the articles in these special issues do not deal specifically with the impact of the COVID-19 pandemic nor the social upheavals of 2020. However, departments already engaged in cycles of transformation have structures and people in place to make further adaptations to calculus sequence courses as needed. Departments with close coordination were able to quickly arrange to share the work of creating videos and other online materials to support teaching mathematics in remote environments starting in March 2020. Departments with established communities of practice were well-positioned to engage in collective discussion about the intersection of institutional racism and mathematics teaching and learning in higher education.

Departmental change is not something magical that only the elite can successfully enact and sustain. It is true that improvement is not free, but entails significant personnel resources in addition to fiscal and physical resources. However, achieving more equitable outcomes for students requires all of us to do better. Cultural change is hard, but necessary in order to sustain individual efforts at instructional innovation and inclusive teaching practices. We hope the stories and research in these special issues help to influence more mathematics departments to initiate their own transformative cycles to infuse active learning in the calculus sequence and achieve more equitable student outcomes.

DISCLOSURE STATEMENT

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported in part by a grant from the National Science Foundation [grant number DUE-1624643, 1624610, 1624628, and 1624639]. All findings and opinions are those of the authors and not necessarily of the NSF.

Notes on contributors

Wendy M. Smith

Wendy M. Smith is a research professor at the University of Nebraska and focuses her research on PK-20 mathematics, science, and computer science education and change, including institutional change, active learning, teacher change, teacher professional development, teacher leadership, professional networks, and estimating teacher professional learning effects on student achievement. She is the associate director of Nebraska’s Center for Science, Mathematics and Computer Education, and works with teachers and researchers at all levels to improve STEM teaching and learning.

Chris Rasmussen

Chris Rasmussen is a professor of mathematics education and associate chair in the Department of Mathematics and Statistics at San Diego State University. He is currently co-editor-in-chief of the International Journal of Research in Undergraduate Mathematics Education. His research investigates inquiry-oriented approaches to the learning and teaching of undergraduate mathematics, focusing on how mathematical ideas are developed through student exploration and teacher–student classroom discourse. His research program also seeks to better understand departmental practices to improve student success in the introductory mathematics courses required of all STEM majors and the process of departmental and institutional change to improve these courses.

Robert Tubbs

Robert Tubbs is a mathematics professor at the University of Colorado, Boulder. He has held every administrative position within the department (i.e., chair, graduate chair, undergraduate chair, and director of lower division curriculum). For three decades, he has sought to improve instruction at the undergraduate level, especially in the first-year mathematics courses, and has been instrumental in restructuring these courses. His research interests lie in pure mathematics (transcendental number theory and diophantine approximation), mathematics and the humanities (the history of ideas, literature, and art), and in mathematics education (first-year mathematics courses and institutional change).