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Introduction to the papers of TWG14: University
mathematics education
Irene Biza, Viirman Olov, Ignasi Florensa, Ghislaine Gueudet, Mathilde Hitier, Athina Thoma
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Irene Biza, Viirman Olov, Ignasi Florensa, Ghislaine Gueudet, Mathilde Hitier, et al.. Introduction to the papers of TWG14: University mathematics education. Proceedings of the Fourteentlr Congress of the European Society for Researclr in Mathematics Education (CERME14), Free University of Bozen-Bolzano; ERME, Feb 2025, Bozen-Bolzano, Italy, hal-05334297
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Introduction to the papers of TWG14:
University mathematics education
Irene Biza1, Olov Viirman2, Ignasi Florensa3, Ghislaine Gueudet4, Mathilde Hitier5 and Athina
Thoma6
'School of Education and Lifelong learning, University of East Anglia, UK; i.biza@uea.ac.uk
2Department of Education, Uppsala University, Sweden
3Escola Universitària Salesiana de Sarria, Universitat Autònoma de Barcelona, Spain
4UR EST, Université Paris-Saclay, France
5 Département de Didactique, Faculté des Sciences de l’Éducation, Université de Montréal, Canada
6Southampton Education School, University of Southampton, UK
This paper summarises in five themes the 25 papers and 9 posters contributed to the TWG14: University Mathematics Education (UME) at CERME14, and presents an overview of the three thematic discussions conducted in the TWG, before concluding with reflections and suggestions for future research.
Keywords: University mathematics learning and teaching, resources, teaching interventions, mathematics for non-mathematicians, mathematicians and mathematical practices.
Introduction
We opened the activities of the TWG14 at CERME14 by honouring recently departed colleagues who contributed significantly to what we today call UME research: Guy Brousseau, Ed Dubinsky, Simon Goodchild, Anna Sierpihska and David Tall. We also revisited the activities of the group since its launch at CERME7 in 2011, concentrating on an overview of achievements between CERME13 (July 2023) and CERME14 (February 2025). This period saw five special issues related to UME - three in the International Journal of Research in Undergraduate Mathematics Education (Dreyfus et al., 2023; Ely & Jones, 2023; Geraniou et al., 2023), which celebrated a decade of contribution to UME research (Dreyfus et al. 2025), one in Educational Studies in Mathematics (Martino et al., 2023) and one in ZDM - Mathematics Education (Engelbrecht & Borba, 2024); specialised conferences - e.g., the fifth ERME topic conference of the International Network for Didactic Research in University Matheniatics (INDRUM2024); a book (Bridoux et al., 2023); and, a book chapter (Winslow et al. 2023). This year, TWG14 received 37 paper and 11 poster submissions. The review process resulted in 25 papers and 9 posters being presented at the conference, all of them published in CERME14 proceedings. This year, we noticed a decrease in the number of contributions compared to previous years, with 36 papers and 17 posters in the CERME13 proceedings. Such decrease was expected due to the emergence of the new TWG25 on “Calculus Teaching and Learning” as well as to the new policy on registration numbers. Contributions address at least one of five themes: teachers and teaching at school and university level; digital resources; mathematics and non-mathematics specialists; learning and teaching mathematical topics and practices; and, mathematicians and mathematical practices. Activities included a combination of parallel sessions for paper discussion and plenary sessions for poster teasers and thematic discussions. The three thematic sessions covered: digital resources; teachers and teaching at school and university; and research methodologies and methods. In what follows, we summarise contributions in each theme, and the thematic discussions, before concluding with reflections and potential ways forward.
Paper and poster contributions
Teachers and teaching at school and university level
Eleven papers and three posters were classified in this group that proved to be the most popular. Gueudet and Doukhan present a literature review about Didactic Conlraclfs)' in UM teaching, particularly teaching characteristics and their impact on students’ learning. The review identified three categories of studies that focus on changes of didactic contract at the secondary-tertiary transition; limitations in the students’ mathematical activity due to didactic contract rules; and, interventions proposing an ‘alternative contract’. Still with attention to teaching environment, and through a commognitive (Theory of Commognition, ToC) and Communities of Practice (CoP) perspective, Hooper et al. explore how an inter-community team of mathematicians and mathematics education researchers can redesign an introduction to university mathematics course to facilitate shifts in the lecturer’s pedagogical and students’ mathematical discourses. Witty et al. discuss the same course from the perspective of the lecturer aiming to redesign the course. Teaching actions towards students’ enculturation to university mathematics discourse is the focus of Karavi and Schuler-Meyer, who are interested in Opportunities to Learn offered in first-year teaching by four lecturers. The paper exemplifies such opportunities in a session about mathematical proof. Teaching practices towards first year students’ introduction to fundamental concepts of Analysis is also the topic of Petropoulou et al. The analysis draws on Radford’s dialectical nature of tensions to highlight how the lecturer’s background in both mathematics and mathematics education contributes to the development of an alternative teaching culture. Butler and Buchbinder use the perspective of teacher noticing to study the development of a learning assistant in Calculus over two years. They observe that his noticing shifts from describing particular aspects of student work to reflecting on mathematics generally. With attention to teaching preparation, Canogullari and Radmehr draw on the Anthropological Theory of the Didactic (ATD) to identify conditions and constraints experienced by lecturers when designing, adapting or selecting tasks. Thorup Eich-Hoy analyses mathematics lectures as a way to teach mathematics. Using an original approach based on ethnographic methods and psychoanalytic theory, the paper states that lectures can create situations of resonance, embodiment, and durational pleasure in contrast to the common critiques of lectures as outdated or passive.
This year we welcomed a body of contributions related to the role of mathematics courses (and mathematicians) in mathematics teacher education. Pinto investigates the practices of mathematicians and secondary mathematics teachers when comparing and discussing alternative proofs of the same
1 Due to space limitations, we do not provide explanation and references to theories, e.g. Didactic Contract, Theory of Commognition (ToC), Community of Practice (CoP), Anthropological Theory of the Didactic (ATD), Cultural Historical Activity Theory (CHAT), etc. We would invite readers to find more in the corresponding contributions in the proceedings. statement. The study highlights the importance of integrating both mathematical and pedagogical practices in secondary mathematics teaching and showcases how contrasting approaches offer further insights for teacher education. Weith et al., with attention to learning materials, examine the design of resources for pre-service teachers that incorporate theoretical considerations and aim to connect university and school mathematics. The paper presents the design process along with provisional findings on how pre-service teachers draw on their mathematical knowledge to inform their teaching decisions. Viirman et al. bring an international perspective on the role of mathematicians in secondary mathematics teacher education by reporting on the design of a large-scale survey on mathematicians’ views and practices towards strengthening mathematics education of prospective teachers. Marchionne et al. examine how prospective secondary teachers judge the value of instructional videos and STACK (System for Teaching and Assessment using Computer algebra Kernel) tasks in learning about secondary students’ misconceptions about derivatives. Still in a teacher education context, Hoffman observes how changing the response format of an interface task, from email answers to scriptwriting, led preservice secondary teachers to use overarching lines of argumentation that are more conducive to learning. Finally, with attention to learning materials, Morcos and Gonzalez- Martin identify inconsistencies in how permutations with indistinguishable objects are presented in secondary and tertiary mathematics textbooks.
Digital resources
Two papers and three posters were classified in this group. Within the emerging field of research regarding the use of genAI (Generative Artificial Intelligence) in UME, lannone et al. propose a potential research agenda. Using a Delphi method, they collected and classified the research questions considered as most urgent by academics. Miskei-Vàradi et al. examine the integration of ChatGPT into a second-year undergraduate numerical methods course and assessed students’ perceived usefulness of ChaGPT before and after the course. Findings indicate mixed perceptions on the utility of ChatGPT and its limitations in addressing misconceptions, including risks in accepting false proofs. Du et al. explore university students’ mathematical digital competencies in communication in computer-supported collaborative learning and propose a frame that addresses collaborative engagement. Regarding other digital resources, Thoma and lannone explore a lecturer’s perspective on using Lean, an Interactive Theorem Prover, and identify adaptations the lecturer made to his course, the constraints he faced, and the lessons learned in terms of teaching proof with Lean, but also teaching proof more generally. Finally, Markefka et al. describe the development and evaluation of STACK tasks for introductory courses in university mathematics in collaboration with lecturers.
Mathematics and non-mathematics specialists
Five papers and two posters were classified in this group. Dahl et al. explore the integration of virtual reality and problem-based learning to foster engineering students’ understanding of the vector cross product. Students reported positive learning experiences, although some of them identified usability issues or/and reported motion sickness. Norris and Noyes compare pre-university mathematics curricula followed by university students of geography, psychology and sociology in England in datasets from 2016-17 to 2019-20, concerning over 90,000 students. Analysis indicates significant differences in the opportunities to study mathematics both between and within the three disciplines, leading to a complex situation when designing mathematical courses for incoming students.
Several studies in this group take an ATD perspective. Sipus et al. analyse mathematical practices in advanced courses for electrical engineers through interviews with teachers and one master student who point out differences between mathematics and engineering practice. Hitier and Gonzalez-Martin investigate science students’ strategies when solving a non-familiar calculus task, with particular attention to praxeologies from mechanics. Students who employed mechanics’ or mixed praxeologies were generally more successful in solving the task. Florensa et al., also from the ATD perspective, characterise the views of mathematics and engineering teachers about mathematics in the first years of mathematical courses in engineering degrees. Preliminary findings indicate differences in the type of tasks mathematicians and engineers would prioritise (e.g., mathematics-oriented vs practice- oriented tasks). Fernandez-Ruano et al. discuss active learning teaching by drawing on Study and Research Paths (SRPs). The study compares the implementation of SRPs instigated by the same problem in three different university-level statistical courses and identifies differences in the ways students engage with statistical techniques. Finally, Kristinsdóttir and Gislason report on an Icelandic project on the secondary-tertiary transition, aimed at increasing the number of STEM graduates. They discuss similarities and differences in expectations between upper-secondary and university-level mathematics teachers regarding students’ mathematical preparation for STEM studies.
Learning and teaching of mathematical topics and practices
Four papers and one poster were classified in this group. Wawro et al. present the design, implementation and analysis of a task introducing the notion of ‘least squares’ in an introductory linear algebra course. Framed within the Realistic Mathematics Education framework the designed task contributes towards student development of graphical intuition with planes and vectors. De la Cruz Benito and Andrews-Larson investigate students’ proof methods in the context of linear independence after a course using Desmos, observing that a significant portion of students successfully construct deductive proofs, but others still use empirical proofs. Meanwhile, Wallach and Kontorovich propose a conceptualisation of problem solving based on the ToC, distinguishing between solution generation routines and solution conventionalisation routines. Rogovchenko and Rogovchenko discuss students’ difficulties with differential equations in relation to the general solution. They review influential classical textbooks and suggest an introduction to the notion of the general solution of differential equations as a family of functions. Finally, Veuillez-Mainard studies the teaching and learning of polynomials at the secondary-tertiary transition in France by focusing on students’ conceptions of polynomial functions and their graphs.
Mathematicians and mathematical practices
Three papers were classified in this group. Olsen et al. investigate the epistemic tensions associated with the role of programming in the research practices of mathematicians. Drawing on interview data from two mathematicians, one with a background in pure mathematics and the other in applied mathematics, the authors use Cultural Historical Activity Theory (CHAT) to examine the rules and motives shaping their research practices and the role programming plays within those practices. Goor and Pinto study positioning acts of mathematicians and secondary school teachers discussing the practices of a teacher presenting a proof in class, observing that the mathematicians can move from disciplinary authorities to fellow educators. Finally, L’ltalien-Bruneau et al. draw on the ToC to study the research discourse of doctoral students in mathematics. They draw on interviews with three such students to describe patterns characterising their specialised mathematics research discourse.
Thematic Discussions
Digital resources: Revisiting UM teaching, learning and assessment
In eight TWG14 contributions, digital resources were the focus or the context of the study. These resources included Virtual Reality, genAI, Programming, Interactive Theorem Provers, Videos, Desmos, Geogebra and STACK. Such attention to digital resources reflects broader trends within UME, as seen in IJRUME’s recent special issue on digital experiences (Geraniou et al., 2024), and more generally in mathematics education - especially since COVID-19 accelerated the use of digital resources (Engelbrecht & Borba, 2024). In the discussion, we focused on three digital environments: Interactive Theorem Provers, genAI and STACK. For each of these, we considered: (1) challenges and opportunities they present for learning, teaching and the development of students’ employability skills; (2) their influence on assessment, particularly in terms of assessment design; and (3) theoretical frameworks and methodologies that may offer deeper insights into the use of these digital tools.
Concerning the first point, we identified several opportunities and challenges from both student and teacher perspectives. For students, we discussed opportunities for practice, providing applications across different disciplines (e.g., computer science and mathematics), and the development of transferable skills linked to employability. However, concerns were also raised about students potentially becoming overly reliant on digital resources (e.g., genAI) and lacking the appropriate critical engagement allowing for an efficient and effective use of those resources. From the teachers’ perspective, the discussion was around the need for targeted professional development and the creation of appropriate teaching materials. It was also noted that support for the teaching implementation of specific digital resources is currently limited, and that some of those resources demand a steep learning curve (e.g., Interactive Theorem Provers). Insufficient availability of appropriate and adaptable teaching and learning materials raises concerns about the practical implementations of these tools in teaching, particularly in contexts where instructors’ and students’ needs may differ considerably. In terms of assessment practices, we considered tensions between checking correctness and redefining assessment criteria, as well as the contrast between evaluating the final product and placing greater emphasis on the learning process. We also highlighted the importance of rethinking assessment methods to offer a variety of methods but, also, to align better with employability skills. Furthermore, we reflected on potential theoretical frameworks to support research in this area, such as Instrumental Approach, CHAT and ToC. We also discussed how genAI could be implemented within our research process, e.g., as a tool, an agent or a collaborator? Finally, mixed-methods approaches were considered well suited to capture resource-use broadly, as they can provide valuable insights into usage patterns, accuracy, speed and feedback provision.
Teachers and teaching at school and university: A cross-community perspective
in recent CERMEs, TWG14 has seen an increase in the number of contributions related to teachers and teaching at school and university level, and it was a popular theme among contributions also at CERME14 (see earlier sections). This growing interest is also seen at the INDRUM2024 conference and in recent special issues (e.g., Buchbinder et al., 2023). Given the focus in several TWG14 contributions on cross-community collaboration, for instance between mathematicians and mathematics educators, for the second thematic discussion we chose to take a cross-community perspective on teaching at school and university level, including university-based preparation of school teachers and university teacher professional development. The discussion was structured in two parts: (1) reflecting on the affordances and challenges of working with multiple communities in teaching and teacher preparation; (2) considering how can we draw on these affordances and overcome the challenges. Among the studies presented in TWG14 this year, the two main affordances we identified were co-learning partnerships, where members from different communities gain from collaborating on planning and conducting teaching, and benefits for students from cross-community teaching collaboration. However, several challenges were also discussed. Institutional constraints, such as university organisation (e.g., mathematicians and mathematics educators working in different departments or even faculties) might, for instance, lead to a lack of awareness of communities’ willingness to collaborate. We also noted potential communication difficulties, stemming, for instance, from different perspectives on teaching and learning in different communities. Finally, the dependence of the ‘success stories’ on specific individuals involved was pointed out, raising the question of how to make initiatives sustainable. One proposed answer in the second part of our discussion was to conduct research at the ‘meta level’ on the conditions making cross-community teaching collaboration productive and sustainable. Other proposed strategies are making UME research publications accessible to mathematicians (e.g., through implications for practitioners), and collaborative research on teaching with mathematicians (or other specialists) as co-researchers.
Research methodologies and methods: Specificities, challenges and shared practices in UME
Also, we opted for a discussion on research methodologies and methods, which was motivated by the observation of an increased diversity of methods, including the introduction of methods uncommon in the group, such as a Delphi study and the use of AI tools for coding interview data (see the digital resources theme). The discussion was organised around different interconnected aspects related to methodologies and methods. Discussion revolved around the crucial role methods play, not only in the way research develops, but also in how results are shared and disseminated with close attention to the relation between theoretical frameworks, objects of study, and methods.
One of the issues that emerged from our discussion was the sharing of methodological tools (questionnaires, interview guidelines) within the UME research community, which can become a repository, but which is not sufficiently developed yet. Another topic that was discussed was the need for dissemination of task analysis instruments and approaches, and the possibility for such instruments to be shared with mathematicians - another opportunity for cross-community collaboration between mathematicians and mathematics educators. A further aspect that emerged during the discussion was the international movement towards open data, which may raise issues of anonymity and confidentiality in our field (typically, for classroom videos). The difficulty of sharing methods and instruments when different theories are employed was discussed as well, a concern that led us to express a need for theory networking towards shared methodologies in UME research. We also discussed the changes in methodological approaches in works presented at TWG14 across CERMEs, noticing a significant development of mixed methods research designs in recent years compared to a dominance of qualitative approaches in the past. Still, access to representative samples for quantitative studies remains an ongoing challenge in UME research.
Many of the issues above could apply to other CERME TWGs. However, specifically to TWG14, UME researchers might be UM teachers as well, which raises issues of balancing priorities and controlling potential biases. For example, a UME researcher who is also a UM teacher might need to reflect on or detach from the teacher’s position when doing research. Cross-community collaboration and international/comparative studies were seen as means that might be helpful in this regard. In the latter case, the design of methodological tools that can be used in different cultural and institutional contexts might be an additional challenge. Finally, replication studies were mentioned as an opportunity for transferability, testing and evaluation of methods in different contexts.
Reflections and ways forward
Summarising some of the tendencies we have seen at CERME14 TWG14 compared to previous CERMEs, there is, for instance, an increase in studies focusing on mathematicians’ involvement in schoolteacher preparation, and (unsurprisingly, given recent developments in technology and society as a whole) studies on genAI and integration of digital resources more widely. The secondary-tertiary transition is a steady area of interest, as is research on teachers and teaching practices, while this year saw a decrease in papers focusing on the teaching and learning of particular content, and on mathematics for non-mathematics specialist students. This can probably be at least partly explained by the already mentioned new TWG25 on Calculus, as well as by the further consolidation of the TWG26 on mathematics in STEM. This year we returned to contributions around mathematicians’ research and teaching practices as well as to their preparation (and pedagogical support) for teaching.
Concerning theoretical perspectives, ATD and ToC are still widely used in studies presented in the TWG14, while we saw a slight increase in ToC papers this year. Regarding methods, we observed the development of mixed methods, and of innovative methods such as the use of genAI for coding data. There is still a definite lack of contributions focusing on issues of equity and inclusion, despite this being a topic of crucial importance for UME. This lack has been discussed in the group over the years, and one tentative explanation is that researchers focusing on these types of issues, even if their research is conducted at the university level, see themselves more as parts of other research communities rather than of the UME community. This is purely speculative, however, and we would be keen on learning more about how we could attract more contributions from these areas.
In the final session, discussion time was devoted to looking ahead, and thoughts were raised that might serve as inspiration for future work. For instance, for quite some time now, a lot of attention has been paid to the utility aspects of higher mathematics, and its role in other disciplines. Perhaps it is time to also attend to the role of pleasure and enjoyment of pure mathematics? And in connection to this, perhaps it would be fruitful to pay more attention to students’ engagement in mathematical activities as enculturation into mathematical practices? We also discussed the dialectic relation between intuition and rigour. This can be understood both in relation to our common topic of study, namely university mathematics, but also in relation to our own practices as UME researchers. Generally, there seemed to be an interest in further work in the networking of theories and methodologies, in order to see further issues and gain deeper insights.
References
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Buchbinder, O., Wasserman, N. H., & Buchholtz, N. (Eds.). (2023). Exploring and strengthening university mathematics courses for secondary teacher preparation [Special issue]. ZDM - Mathematics Education, 55(4). https://link.springer.com/journal/11858/volumes-and-issues/55-4
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