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Calculus courses have been attracting the attention of mathematics education researchers over the last decades. Recent publications and special issues dedicated to calculus highlight the diverse roles of calculus courses in the pathways for STEM studies within and across educational levels, as well as issues related to the academic preparation of future STEM professionals, including mathematicians, non-mathematics professionals (NMPs) and teachers. In this paper, we review recent literature, mostly since 2015, to identify and characterize explicit references to such diverse roles of calculus courses. The review highlights an intrinsic tension between two institutional roles of calculus: scaffolding and filtering. We find nuanced yet substantial variation in how the dual role of calculus is experienced and managed across different groups of students and stakeholders. With regard studies on calculus for mathematics students, recent studies attend to changes in the rules underpinning calculus notions and processes, and highlight the scaffolding potential of calculus courses towards abstraction and reasoning. In contrast, studies on calculus for NMPs report that such courses often promote filtering rather than scaffolding and question transferability and relevance of those courses for NMPs. Finally, studies challenge the perception that traditional tertiary calculus courses scaffold school teaching and underline the need for empirical research that would explain the nature of this scaffolding and how it can be realized in practice. We conclude that further empirical research, in particular theory development, is needed to address the diverse roles of calculus at the intersection of institutions, disciplines and communities.

The secondary-tertiary transition (STT) in mathematics education is a longstanding concern. This study explores university mathematics lecturers’ perspectives on the challenges underlying STT and on the effectiveness of university-level coping measures currently employed. The analysis of 311 responses to an international survey suggests that there is considerable variability regarding the prevalent perspectives on STT among university lecturers. While most respondents recognized school-related factors, the coping measures they recommended were mainly university-related. The findings stress the need to improve communication, both between university mathematics lecturers and the school mathematics education community, and across universities, for promoting comprehensive initiatives to address STT.

This theoretical article explores an issue of developing education research competencies in mathematics teachers through their involvement in mathematics education research. We first argue that the development of education research competencies is beneficial for the teachers’ professional growth. We then identify opportunities for mathematics teachers to develop education research competencies through different modes of research-practice partnerships. In the main part of the paper, we present a particular theoretical-organizational framework for large-scope teacher-researcher collaborations in educational research. The framework is called Teacher-Researcher Alliance for Investigating Learning (TRAIL), and consists of a set of theoretically laden premises, design heuristics, and provisional partnerships.

Undergraduate mathematics students are expected to spend a substantial amount of time outside class studying proofs independently. Yet, little is known about how such expectations shape instruction in proof-oriented courses. This paper investigates this issue in a case study of an experimental teaching strategy, designed by a mathematics professor, for scaffolding student independent proof reading. On the basis of interviews and classroom observations we scrutinize the instructor’s course image – his envisioning of the teaching and learning activities, inside and outside class, throughout the course. We examine a representative case of proof presentation through the lens of the instructor’s course image. Our analysis reveals complex relationships between various teaching and learning activities in the course, and demonstrates that unpacking these relationships can provide valuable insights into specific teaching practices. Findings suggest that the concept of course image can be useful in enhancing our understanding of mechanisms underlying collegiate mathematics instruction.

In this paper we investigate task design in an unusual professional development course for elementary school teachers, conceived and taught by research mathematicians. Prior analysis singled out relevance for teaching as a critical design issue for engaging teachers in effective learning. The aim of the current research is to uncover how relevance for teaching was achieved without compromising mathematical rigor and depth. Findings are based on an analysis of three representative cases of task designing in which the authors where involved – one as instructor and task designer, the other as participant observer. Our analysis reveals a designing model that first addresses purely mathematical concerns and then refines tasks, taking into consideration a series of constraints imposed by the requirement of relevance for teaching. Using Schoenfeld’s Resources-Orientations-Goals framework for decision-making, we show how the mathematicians drew on their special knowledge of mathematical content to achieve such relevance in ingenious ways. We find that tasks were best aligned with Knowles’s principles of Adult Learning in cases where the designers appropriated the teachers’ point of view, no longer seeing the need for relevance as a constraining imposition, but rather as an opportunity to combine and merge knowledge specific for teaching and purely mathematical knowledge.

This paper investigates the variability in the content of lectures by different university mathematics instructors teaching concurrently in the same course. The instructors enacted a shared lesson-plan and coordinated their teaching to ensure students in different sections of the course will receive equivalent academic preparation. However, classroom observations and interviews with the instructors indicate that while explicitly discussing the same definitions and theorems and working out similar examples, the instructors were implicitly trying to convey different informal content. An analysis of the instructors’ mathematical discourse during the lectures and the interviews reveals that the instructors coordinated with each other content corresponding with object-level learning, while implicitly orienting their lectures towards fostering meta-level learning. This found gap highlights miscommunication between the instructors and points to the need of a shared and explicit discourse on meta-level learning at the collegiate level.

International audience

International audience; Research shows that lecturer feedback on students’ proofs is crucial for developing proof-comprehension in advanced mathematics courses, yet students often fail to comprehend lecturer feedback, and only rarely receive further feedback on their revisions. In this study we investigate the affordance of a novel formative-assessment scheme, designed and enacted by a mathematics professor, which employed multiple rounds of lecturer-feedback / student-revision. We analyze one such round, focusing on various facets of proof comprehension that underlie the lecturer’s feedback and the student’s subsequent revisions. On the basis of this analysis we discuss various ways in which lecturers can leverage feedback/revision cycles, not only for gaining insight into students’ comprehension, but also for fostering meta-level ideas, and affording opportunities for students to develop agency and holistic proof comprehension.