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This article reports on the evolution of prospective secondary teachers' knowledge (meanings for dy/dt) and beliefs (about teaching and learning mathematics) in a semester-long inquiry-oriented differential equations class. Students entered the course with limited, primarily procedural, meanings for dy/dt. Throughout the semester, they engaged in collaborative mathematical inquiry using a research-based curriculum. As viewed through the emergent perspective, students' meanings for dy/dt and their beliefs about teaching and learning mathematics co-evolved with community norms and practices through the classroom discourse. Students' end-of-term portfolios and portfolio presentations were analyzed for evidence of meanings for dy/dt and beliefs about students' roles, instructors' roles, and the general nature of learning mathematics. In these, students expressed rich, multifaceted meanings for dy/dt and beliefs about learning as an active process of meaning-making. While many prospective teachers do not see the relevance of advanced mathematics coursework to their career, these students reflected on their experiences in the course and volunteered ways in which their emerging knowledge and beliefs would influence their future practice. We emphasize that the classroom norms supported students in pursuing connections themselves, and conjecture that similar norms would support the development of reflective practitioners in other contexts.

Mounting evidence of the effectiveness of active learning strategies has prompted many mathematics departments to start engaging in transformational change efforts. Change, however, especially change in instructional practice, is a challenging endeavor. Some departments are using coordination of multi-section courses as a vehicle to enact changes and have designated or hired course coordinators to oversee efforts to transform instruction via active learning. This study explores the role of coordinators as instructional change agents for active learning using data collected from five university mathematics departments, all of which are successfully sustaining such efforts. We use Shadle et al.'s drivers for change as a framework for examining data from five retrospective case studies of change. Specifically, we investigate how these speculative drivers connect to coordinators' roles as change agents in each story. We argue that coordinators are positioned to leverage three key drivers for change: providing materials and tools, encouraging collaboration and communication, and encouraging (and providing) professional development. Coordinators must also understand the local contexts and culture in order to engage in effective processes for supporting departmental change. To conclude, we discuss implications of this research for departments seeking to transform instruction.

One way that university mathematics departments across the United States are making efforts to improve their introductory mathematics courses is by implementing or increasing the level of course coordination. This can entail creating uniform course elements for multi-section courses and efforts to build a community of practice among the instructors teaching these courses. In this paper we explore coordinator orientations that implicitly (and in some cases explicitly) guide their work. In particular, we identified two orientations toward coordination, a Humanistic-Growth orientation and a Resource-Managerial orientation. Data analyzed in this manuscript consists of interviews with 19 course coordinators and 53 instructors from seven different higher education institutions across the United States. We introduce a framework that aligns two approaches to professional development with these two coordinator orientations and conclude with a discussion on how department chairs and others looking to implement or improve coordination may leverage this framework provided in this paper.

The field of biology education--and other science, technology, engineering, and mathematics (STEM) disciplines more broadly--has witnessed two major shifts in the past decade: (i) the increased awareness of research-based instructional strategies (RBIS) that are beneficial for student learning, and (ii) a recognition of the importance of computing in industry and academia. However, uptake of RBIS remains low and opportunities for students to learn computing skills are limited. In this article, we present a combination of novel bioinformatics computing apps and associated anticipatory activities embedded in context. This case study examines students' experiences with these apps and activities as enacted in a student-centered classroom that embraced a variety of RBIS.

This study explores how the goals and enactment of three tailored calculus courses, referred to as course variations (standard calculus, life sciences calculus, and physics calculus), at the same university impact students' mathematical identity. In this mixed-methods study, we draw on the framework of figured worlds to interpret how students' mathematical identities can be refigured by the enactment of three calculus course variations. Results demonstrate that students in life sciences calculus had opportunities to refigure their identity as capable learners and doers of mathematics yet still viewed the subject as ancillary to their disciplinary interests. Students in physics calculus sustained their mathematical identity through strong peer collaborations. Finally, students in standard calculus shifted their identity as survivors of mathematics, conveying a sense of received knowing from their instructor. The ability of any particular course variation enactment to provide opportunities for refiguring mathematical identities resides within the goals, structures, students, and instructors that help to form these figured worlds. The results presented here provide insights into course variation instructional design and instructional practices that can positively affect students' mathematical identities, and in turn, their persistence in mathematics and science.

This paper reports a study of university students' experiences of learning mathematics in an introductory differential equations course that takes an inquiry approach to learning and teaching. The students are on a trajectory leading toward a bachelor's degree in mathematics to be followed by a year-long teacher education credential program. The analysis of students' end of course presentations and written portfolios is framed with community of practice theory, complemented with analytic categories from community of inquiry, including notions of inquiry stance and critical alignment. Our social theorization of university mathematics education adds a complementary perspective to the more conventional cognitive and discursive analyses and is necessitated by the very nature of inquiry approaches to education. The analysis of students' presentations and writing leads to exploring their development of the notion of critical stance, which the students are seen to take toward the mathematics, their learning, and approaches to teaching mathematics. We find that the notion of critical stance can be distilled into three components: awareness, self-evaluation, and agency.

This article relates a case study on how a conversation with materials and diagrams -- the actual use of materials and diagrams to think, imagine, explain, collaborate, design and build -- featured a certain kind of interplay between material and digital components. The physical components present in this setting included a water wheel, which is a wheel driven by flow of water whose rotational motion is a classic example of chaotic dynamics regulated by Lorenz equations. Digital components allowed for real-time graphical displays corresponding to the turning of the water wheel. We selected for this article a sequence of episodes from an interview with Jake, an undergraduate student majoring in engineering. Through a micro-ethnographic analysis, we reflect on how Jake combined the responsiveness of the digital displays with the tangibility of the water wheel to gain insight into some of the intricacies of oscillatory motion.

In this paper, we investigate leadership related to the instruction of lower division undergraduate courses at five university mathematics departments with strong calculus programs. We use social network analysis to identify patterns of influence on instruction, using the relations: advice seeking, instructional material sharing, discussion of instructional matters, and explicit influence on teaching approach. Data were gathered through social network surveys, which received over 60% response rate at each site. Considering both formal and informal phenomena, we identify variation in vertical and distributed instructional leadership structures within these departments. In these five communities, not all those with hierarchical authority have real influence over instructional practice, but those with the most influence over instruction do hold formally recognized positions. In light of these findings, we discuss implications for future research and practice in higher education.

While the number of studies of flipped classrooms has increased, they have primarily addressed the efficacy of using such an approach on student outcomes, often failing to account for the classroom activities and learning theories used to design the curriculum. This study begins to fill this gap in the literature by uniting the at-home video and in-class curricular components of the flipped classroom via design heuristics that empower students to critically think about mathematical problems individually before engaging with the task in a collective environment. To that end, we illustrate how elements of the instructional design theory of Realistic Mathematics Education and Culturally Responsive Pedagogy influenced the written and hidden curriculum and how those considerations were then experienced by the students as part of the enacted components of the curriculum. The context of the study is a 2-week classroom teaching experiment covering topics in trigonometry and vectors for 27 calculus students at a Norwegian university.

This research commentary argues for more research that attends to the processes of organizational change in mathematics departments. It outlines both the ways that research on organizational change can benefit scholarship in mathematics departments, and how mathematics education researchers are needed to develop theories of change that are contextualized to the teaching and learning of undergraduate mathematics. This commentary closes with a research agenda for moving this emergent field of study forward. This agenda involves applying change theories to historical, ongoing, and new change projects in mathematics departments, while simultaneously attending to issues of equity and social justice.

The networking of theories is an increasingly common and powerful approach to analyzing complex phenomena such as learning processes in classrooms. In this paper, we aim to advance the theoretical coordination of two approaches that we have previously combined to analyze individual, small group, and whole class mathematical progress. The theoretical advances we make are twofold. First, we identify and illuminate environmental and internal-theoretical commonalities across the two approaches, commonalities that contribute to the productivity of networking. Second, we propose an argumentative grammar for the networking, thus elevating the methodological logic and rationale of networking in this case and potentially in general.

Precalculus and single-variable calculus courses are a critical early juncture for the success of science, technology, engineering, and mathematics (STEM) students in the United States (US). The default course structure is often a required three-term sequence of precalculus, differential calculus, and integral calculus. In this paper we analyse variations to this structure, the nature of such variations, their frequency across US universities, and how instructional approach and performance rates in those variations compare to those in the standard course offerings. We found that course variations are relatively common, and are primarily targeting students with more math-preparation, less math preparation, or those majoring in a specific discipline. We found that for less prepared students, who are typically at the highest risk of failing, course variations resulted in similar passing rates, essentially levelling the playing field for these students. Although we had conjectured that disruptions to the standard offering would allow for greater use of active learning strategies, no such difference was observed. Ways in which these course variations can redress systemic educational inequalities, opportunities for design improvement, and future research are discussed. [Cowritten with the Progress through Calculus Team.]

This study furthers the theory of conceptual blending as a useful tool for revealing the structure and process of student reasoning in relation to the Sierpinski triangle (ST). We use conceptual blending to investigate students' reasoning, revealing how students engage with the ST and coordinate their understandings of its area and perimeter. Our analysis of ten individual interviews with mathematics education masters' student documents diverse ways in which students reason about this situation through the constituent processes of blending: composition, completion, and elaboration. This reveals that students who share basic understandings of the area and perimeter of the ST recruit idiosyncratic ideas to engage with and resolve the paradox of a figure with infinite perimeter and zero area.

In the United States (US) and elsewhere across the world, undergraduate mathematics instructors are increasingly aware of the value of inquiry-based instruction. In this research commentary, we describe the intellectual origins and development of two major strands of inquiry in US higher education, offer an explanation for apparent differences in these strands, and argue that they be united under a common vision of Inquiry-Based Mathematics Education (IBME). Central to this common vision are four pillars of IBME: student engagement in meaningful mathematics, student collaboration for sensemaking, instructor inquiry into student thinking, and equitable instructional practice to include all in rigorous mathematical learning and mathematical identity-building. We conclude this commentary with a call for a four-pronged agenda for research and practice focused on learning trajectories, transferable skills, equity, and a systems approach.

The authors present findings from a recently completed census survey of all mathematics departments in the United States that offer a graduate degree in mathematics. The authors report the viewpoints of departments about characteristics shown to support students' success as well as the extent to which these characteristics are being implemented in programs across the country. [The Progress through Calculus Team wrote this article. In addition to the authors listed, other personnel include Naneh Apkarian, Linda Braddy, Jessica Gehrtz, Dana Kirin, Matthew Voigt, and Kristen Vroom.]

This article provides an overview of a modeling sequence that culminates in student reinvention of a bifurcation diagram. The sequence is the result of years of classroom-based research and curriculum development grounded in the instructional design theory of Realistic Mathematics Education. The sequence of modeling tasks and examples of student work tendered offers a fresh perspective on modeling and how an inquiry-oriented sequence of tasks can lead to the reinvention of significant mathematics.