From ”chalk and talk” to active learning:
inter-community partnerships and shifts in pedagogical
and mathematical discourse
Owen Hooper, Elena Nardi, Irene Biza, George Witty
To cite this version:
Owen Hooper, Elena Nardi, Irene Biza, George Witty. From ”chalk and talk” to active learning: intercommunity partnerships and shifts in pedagogical and mathematical discourse. Proceedings of the Fourteenth Congress of the European Society for Research in Mathematics Education (CERME14), Free University of Bozen-Bolzano; ERME, Feb 2025, Bozen-Bolzano, Italy. hal-05237879
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Submitted on 3 Sep 2025
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From “chalk and talk” to active learning: inter-community
partnerships and shifts in pedagogical and mathematical discourse
Owen Hooper, Elena Nardi, Irene Biza and George Witty
University of East Anglia (UEA), Norwich, United Kingdom; O.Hooper@uea.ac.uk
Emergency remote teaching (ERT) during the Covid-19 pandemic exposed lecturers to new pedagogical challenges in which alternative active learning approaches were implemented to manage the absence of “chalk and talk”. We explore how an inter-community (Mathematics, Mathematics Education) team can redesign a university mathematics course titled Mathematical Skills (MS) to facilitate shifts in the lecturer’s pedagogical discourse and in students’ mathematical discourse. Through a commognitive/communities of practice analysis of research design team interviews, we present preliminary findings that evidence shifts: from object-level to meta-level learning features in tasks posed by the lecturer; in the explicitness of lecturer goal setting routines; and, towards tasks that encourage active student participation. We conclude with noting the potential benefits to the students’ learning experience emerging from the inter-community partnership.
Keywords: Active learning, Commognition, Communities of Practice, inter-community partnerships.
Introduction
In recent years, research related to Active learning (AL) pedagogies and practice orientated interventions in university mathematics has grown (Biehler et al., 2022). However, the adoption of AL approaches in tertiary mathematics has been gradual, with chalk and talk (C&T) remaining the pedagogical norm amongst mathematics lecturers (Cevikbas & Kaiser, 2023). The Covid-19 lockdowns forced universities to shift teaching online, labelled by Engelbrecht et al., (2023) as emergency remote teaching (ERT). In this new environment, lecturers needed to adapt their teaching approach to keep students engaged (Hooper & Nardi, 2021), as C&T pedagogies were incompatible with ERT. A shift to AL environments with flipped / blended pedagogy elements – including tasks such as problem posing, group work, presentations and time for class discussions – seemed to address some of the challenges of ERT (Hooper & Nardi, 2021), but a reported limitation was the lack of time and resources for effective planning and execution, with much left to trial and error (Engelbrecht et al., 2023). However, conditions are now more conducive to a shift towards AL pedagogies (Engelbrecht et al., 2023), with learned societies such as the Institute for Mathematics and its Applications in the UK funding professional development initiatives for mathematicians on said approaches (Thorogood, 2024). In this paper, we explore how establishing a research design team (RDT) can facilitate a collaborative inter-community partnership (ICP) between mathematicians and mathematics education researchers. The formation of an ICP serves two purposes– the first is to generate innovative pedagogical practice by redesigning a first year (Mathematical Skills, Sets, Numbers and Proof, thereafter MS) course that is infused with AL tasks. The second is to implement this redesign to facilitate shifts in the lecturers’ pedagogical discourse, and the students’ mathematical discourse. We present preliminary analysis of RDT data, reporting on the shifts in the lecturer’s pedagogical discourse and the potential impact these may have on the students’ mathematical discourse. We first outline key elements from the AL and RDT literature before introducing the theoretical framing and methodology of our study.
Literature Review
The relationship between mathematics education researchers and academic mathematicians has been described as fragile (Nardi, 2016). However, the structure and dynamics of this relationship are shifting, as evidenced by the growth in ICPs (Biehler et al., 2022; Nardi, 2016): see, for example, the implementation of a research design cycle towards upscaling an intervention in an abstract algebra course in Biehler et al. (2022). Our study aims to build effective ICPs between mathematicians and mathematics education researchers that are “non-deficit, non-prescriptive, context specific and example centred” (Nardi, 2016, p. 365) and have the potential to spark communications that are “jargon-free, even though their construction is fundamentally driven by mathematics education research findings” (Nardi, 2016, p. 372). By establishing this ICP, we aim to find common ground between the two communities and facilitate a redesign of aforementioned MS course.
Active learning is defined by (Freeman et al., 2014) as a pedagogy that “engages students in the process of learning through activities and/or discussion in class, as opposed to passively listening to an expert” (p. 8413). The broadness of this term is beneficial for adopting a flexible, non-prescriptive approach that may blend various tasks and environments. For example, the diversity of AL approaches is evident in research on flipped/blended classrooms (Cevikbas & Kaiser, 2023). Additionally, research has focused on specific types of AL tasks, such as example generation (Kinnear et al., 2023) or solution sharing (Biehler et al., 2022). The recent growth in AL approaches in university courses may potentially be attributed to exposure to alternative pedagogies during ERT, which highlighted the potential effectiveness of AL to improve student engagement in university mathematics (Engelbrecht & Borba, 2024). Research on AL environments has highlighted benefits regarding student engagement, mathematical understanding and class discussion. However, such environments are time consuming to plan and rely on student buy-in to be successful (Cevikbas & Kaiser, 2023). A critique from Deitrich & Evans (2022) is that the reported benefits have not been consistently demonstrated in practice yet. Additionally, there is a growing body of research exploring specific AL tasks. For example, Kinnear et al. (2023) explored how example generation tasks support students’ understanding of challenging concepts such as injective, bijective and surjective functions. Space constraints do not allow us to explore each type of task used in the project; yet, we note that a variety of AL strategies have been included in the redesign of the course to foster student understanding and engagement. Whilst not an exhaustive list, there are four types of tasks regularly incorporated into our course intervention: quizzes, example generation, group work and solution sharing. The ICP that our project is building is underpinned theoretically by networking two theories (Commognition, ToC; Communities of Practice, CoP) which we briefly outline in what follows.
Theoretical Framework
ToC, introduced by Sfard (2008), is a discursive approach to learning, with emphasis placed on the various acts of communication: with oneself, with peers, verbal and non-verbal (e.g. writing on the board). These communicative acts form part of a community’s discourse, with mathematics discourse defined by four characteristics: word use, visual mediators, endorsed narratives and routines. Sfard (2008) notes that mathematical learning can be at object-level (learning about the objects that make up the mathematical discourse, for example, learning the Fundamental Theorem of Arithmetic, what symbols such as ∃ and ∀ mean, and so on) and at metal-level (learning about how discursants are expected to behave within the discourse community in, for example, formulating and substantiating object-level narratives). As students transition to university mathematics courses, they shift from being governed by the rules of school mathematics to new, university mathematics rules. One such rule change students experience is the different expectations for what consistutes an acceptable mathematical argument (or proof). For students, just writing a solution with the correct final answer is insufficient at university level; the lecturer expects evidence of mathematical literacy, developed explanations and logical structure. At university, the rules of the mathematical game change, and engaging with university mathematics involves not merely learning about new mathematical objects (object-level learning) but, crucially, about these new rules (meta-level learning). To facilitate both types of learning, our MS course intervention incorporates variety of AL tasks which are designed to give students the opportunity to explore and engage with these new rules. The ToC was selected to capture shifts in student’s mathematical discourse (Sfard, 2008) and the lecturer’s mathematical pedagogical discourse (Hooper & Nardi, 2021). Using the ToC as a foundation to inspire our AL tasks, we can see from the examples below how well-designed tasks can facilitate shifts in the student discourse and create opportunities for our lecturer to undergo shifts in their mathematical pedagogical discourse. In this project, we utilise the ToC as a tool to analyse the data and inform the design of our intervention. This will be expanded upon in subsequent sections.
According to Wenger (1998), CoPs are established overtime where a group is defined by the ways members have a shared relationship, beliefs, values, and tasks. In the MS course, we define the community by the beliefs (in this case their view of mathematics, and what it means to them to be a mathematician), the pursuit of shared goals (e.g., gaining skills to be a successful mathematician at university) and the tasks that make up the course (e.g., using proof by induction, deciding if a function is injective or not). Within the CoP, we have two distinct players – the old-timers who are experienced, fully active members of the community (e.g., the lecturer) and the newcomers, those that are yet to engage with the community, its goals and tasks (e.g., the first-year students), but by participating in its practices they aim to become old-timers. In this CoP, Witty is the lecturer of the MS course with the CoP shaped by his previous teaching experiences, his beliefs, the goals he wants students to achieve and tasks. As a researcher in UME, Hooper’s role in shaping the CoP is to highlight relevant research literature (e.g., as mentioned in the literature review above) as well as experiences as an old-timer on a similar course. To facilitate the students’ transition the new rules of university mathematics, we implement what Wenger (1998) calls an inbound learning trajectory: this involves designing a sequence of tasks and goals for newcomers to engage with during the semester. The concept of a learning trajectory provides a framework for the MS intervention to scaffold tasks that support students from the periphery to the centre of the CoP.
As Biza et al. (2014) note, in CoP there is often a lack of detail regarding the fine-grained elements of the learning trajectory. Networking CoP and ToC provides the nuance needed to trace the aforementioned learning trajectory (CoP) in the terms of what ToC labels as discursive shifts. Defining the MS course community provides us with a boundary to operate within. We root our definition of the MS community in the meta- and object-level rules we essentially want the students to engage with throughout the semester. So, with the boundary now set, we help the newcomers navigate through it. The construction of a learning trajectory gives a structure for students to engage, peripherally at first, with this new community and its associated meta-rules. The learning trajectory consists of a sequence of tasks, with an underlying goal of shifting students’ mathematical discourse. We, therefore, design a learning trajectory which attends to the goals of the community whilst also facilitating students’ object-level and meta-level learning. We also use the four characteristics of mathematical discourse both to inform the tasks and to analyse the students shifts along the learning trajectory. By networking these two theories we achieve both a macro and micro level perspective to analyse and inform the redesign of the MS course.
Context, methods, and participants
The ICP is redesigning and implementing an intervention in a 12-week face to face, year 1 MS course with 100% summative coursework. The course offers students an introduction to concepts in Set theory, Number theory, Proofs and mathematical skills. The ICP has two key brokers: Hooper (doctoral researcher supervised by Nardi and Biza, representing the community of mathematics education researchers) and Witty (representing the community of mathematicians). The team held regular bi-weekly meetings lasting 60-90 minutes which were open ended, with discussion informed by goals set and tasks completed by each broker between meetings. Hooper gave a brief overview of ToC and CoP (to ensure transparency and openness on how the analysis of Witty’s teaching will take place). These meetings are the focus of the data analysis in this paper. The aim of these meetings is to provide opportunities for discussion, debate and decision making regarding the learning trajectory, AL tasks and the selection of resources. However, the “ultimate substantiator” (Sfard, 2008, p.243), in this case Witty, effectively made the final decision on the content of the learning trajectory. During the design phase, there were two central data collection points: the five RDT meetings (recorded and transcribed by Hooper) and resources uploaded on the shared RDT OneDrive. Hooper conducted a partial analysis of the transcripts using the ToC to identify shifts in the lecturer’s pedagogical discourse. Resources shared on the drive included draft session plans which Hooper also analysed. We present three examples in this paper that have emerged from our preliminary data analysis. Example one concerns the lecturer shifting the focus of tasks from object-level learning goals towards meta-level learning goals, with reference to an excerpt on quantifiers. Example two explores a shift in Witty’s goal setting routine, where session goals hitherto tacit to students are now made explicit. For example, Witty’s session plans began to include a written explanation of what students should achieve from engaging with the various tasks. Example three focuses on a pedagogical shift as Witty seeks to implement tasks to facilitate legitimate student participation in the lectures. In tandem, we try, where possible, to identify potential shifts these changes may have on students’ mathematical discourse.
Results
Example One: Shifting task focus from object-level to (also) meta-level learning
The partial analysis of Witty’s previous course materials revealed an emphasis on object-level learning, specifically on introducing new definitions (e.g., of a set), theorems (e.g., the Euclidean algorithm) and notation (e.g., ∃ and ∀). These are important mathematical objects for student participation in the community of university mathematics. However, such participation also requires attending to a new set of meta-level rules which Witty is responsible for introducing. Disproportionate attention to object-level learning may stifle student engagement with these new meta-level rules, potentially hinder students’ discursive shifts and limit opportunities for legitimate participation in the MS course. The example Hooper and Witty discussed concerned challenges with making sense of, and using, the notation ∀ (for all) and ∃ (there exists).
Witty: ...it’s weird the notation for all and there exists then just becomes sort of like I dunno a barrier, it’s something they have just seen and are like “I need to comprehend that and I’m trying to do some maths as well”.
This highlights challenges students have in engaging with new symbols, deciphering their meaning and use whilst the lecturer has already introduced more mathematical content. This density of content may be overwhelming for new community members. The new rules of the game demand students understand such notation. However, without the opportunity to explore and endorse these new narratives, students are prevented them shifting further than the periphery of the CoP (Wenger, 1998) in the MS course. Therefore – whilst they are engaging with new mathematical content (object-level, quantifiers), their participation in the mathematical discourse of university mathematics is hindered by not being able to utilise notation to solve mathematical problems. Student challenges with quantifiers is highlighted in research, for example, students attempting to solve problems that involve quantified statements without acknowledgement of the quantifiers (Sellers et al., 2021). We discussed this challenge in our RDT meetings as another lecturer (occasionally present at these meetings) stated:
…my biggest thing when I was an undergraduate especially in analysis was the fact, I literally learnt for all epsilon greater than 0… and I didn’t, I wasn’t registering the for all epsilon I was just registering the sound for all epsilon greater and not really understanding what it meant.
Discussing AL tasks that may address problems such as these, the lecturer drew on their experience and suggested utilising tasks such as “fill in the gaps” or “mix and match” style exercises. The aim of these tasks is for students to engage with the new symbols to explore their differences and the implications their use has for the mathematical meaning of a statement – as well as realise how communicating in this new way makes them more central participants in the practices of university mathematics. This example highlights the potential of the intercommunity partnership to identify challenges students may have with these new rules of the game and utilise research and literature to incorporate AL tasks to facilitate meta-level learning where previously the focus had been on objectlevel learning. By utilising Witty’s previous course materials, we leveraged a context driven, jargon free discussion that facilitated reflections, discussions, and modifications which informed the design of the learning trajectory.
Example Two: Shifting from tacit to explicit lecturer’s goal setting routines
Hooper: This [course] is titled maths skills but there is no…this is the point of this module…, this is why we are doing maths skills, these are the things you’ll be expected to engage with.
Despite being named Mathematical Skills, it was not explicit to students what these skills were, how they were expected to engage with them, or why they were important for newcomers to the community of university mathematics. If we want students to engage beyond the periphery of the community, it is important they have a clear and explicit trajectory to do so. The comment by Hooper was aimed to spark a discussion on the aims and goals of the course and how this could inform our learning trajectory and build our sequence of tasks. As a result of these discussions, a shift occurred in the lecturer’s pedagogical discourse: Witty’s goal setting routines shifted from tacit to explicit student goals within the MS course. In practice, this sparked an overhaul in the design of the opening lecture sequence. The above comment likely prompted a re-evaluation of the lecturer’s pedagogical choices (e.g., order of the sequence of tasks) and the importance to the students of clarity in terms of what is expected of them. As a result, Witty in Lecture 1 aimed to discuss and explore students’ meaning of mathematical skills and their importance.
Witty: for example, this exercise here leads into them submitting something as well, like having a goal ［…］ by the end of this you’re gonna like have solved these problems and written them up.
Additionally, the explicit goal setting extended to Witty’s own pedagogical routines as for each AL task included it had an associated goal (or purpose). As Witty notes,
there are so many things we could do…it’s trying to not like, I don’t know, just do them for the sake of it. But, also like make sure there is a purpose.
By our fourth meeting, these utterances had become routine and Witty’s desire for purpose in each task became regular goal setting. In the above extract, we see the lecturer develop a routine of selfquestioning and justification, related to their selection of tasks. As the lecturer engages in this intercommunity partnership, they have started adjusting their pedagogical routines. The discussion, sparked by Hooper’s CoP informed observations regarding the lack of explicit goals on the MS course, has helped facilitate a shift in the lecturer’s goal-setting routines. This shift is crucial for the intervention design, as these goals form the foundation of our learning trajectory that will guide students’ engagement throughout the semester. The engagement with this inter-community partnership has the potential to not just produce an intervention to facilitate students’ shifts in mathematical discourse but also present opportunities for lecturers to shift their pedagogical discourse as they are exposed to propositions emerging from mathematics education research.
Example Three: Pedagogical tasks to encourage legitimate student participation
As mentioned earlier, one motivation for redesigning the MS course was to help student engagement by shifting from a largely C&T approach to one infused with AL opportunities. Where deemed appropriate, Hooper proposed transformation of previously used teaching materials that were familiar to Witty (e.g., replacing giving examples to students with an example generation task). In response, on one occasion Witty commented that
Witty: with the quiz they have to think about it, but they are not writing it down and you know maybe that is quite a powerful thing that they need to be doing.
This is a key moment where Witty endorses a change in students’ engagement routines to encourage legitimate student participation in lectures. This concept of legitimate participation became a regular theme across our meetings with Witty reinforcing narratives on student participation in the lectures.
Witty: I guess for a quiz that they might interact with notation, but sometimes it could be we want you to write that down because that’s what you're going to be doing in a bit, writing it down. It's good to recognize it, but also like being able to, using it straight away allows you to go “I’m using this correctly.”
This provides insight into Witty’s aspirations to shift the students’ engagement routines from passive observers to legitimate participants in the MS course. The aim is that both object-level and meta-level learning emerges as students are allowed time for exploration and practice when introducing content.
Concluding Remarks
The examples above have highlighted how inter-community partnerships (between mathematics education researchers and mathematicians) can facilitate the design of an AL infused MS course through engagement with research findings, prior experiences and practice. Networking the ToC and CoP provide the theoretical foundations for the RDT, for designing the intervention and for data analysis. The ICP facilitates a shift in the lecturer’s pedagogical discourse from C&T to AL pedagogies, exemplified by the three examples above. Example One illustrates a shifting in the focus of tasks set by the lecturer from object-level to (also) meta-level learning. These discussions (in a jargon-free way) provided opportunities for the lecturer to re-evaluate the tasks (and their goals) that have shaped the course previously. Example Two illustrates shifts in the lecturer’s goal setting routines, ensuring the aims of the course were now explicit to the students. Example Three illustrates a shift towards designing tasks that encourage legitimate student participation. So far, we have seen how the ICP can facilitate a redesign of a mathematics course and the shifts this presents in the lecturer’s pedagogical discourse. Whether and how these changes result in shifts in students’ mathematical discourse is the next and imminent phase of our ongoing study.
Acknowledgment
This paper draws on Hooper’s UEA funded, ongoing PhD study. We extend our gratitude to the participants involved in this ongoing project.
References
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