# Collaborative reflection on embodied abstraction for geometry: Questioning the research-teaching interaction

Benjamin Nicolas-Noir, Karel Boonstra

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Benjamin Nicolas-Noir, Karel Boonstra. Collaborative reflection on embodied abstraction for geometry: Questioning the research-teaching interaction. 2024. hal-04901342

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# Collaborative reflection on embodied abstraction for geometry: Questioning the research-teaching interaction

Benjamin Nicolas-Noir¹ and Karel Boonstra²

¹HEP Vaud, Switzerland, and Laval University, Canada; beninl@ulaval.ca

²University of applied sciences Utrecht, Utrecht, The Netherlands

Keywords: Embodied abstraction, teacher-researcher collaboration, geometry, primary school.

## Introduction

A growing body of research supports the recognition of embodied processes in the learning and teaching of mathematics (Abrahamson et al., 2020). However, as stated in the call for contributions, sharing these insights with a broader community remains challenging. We aim here to investigate this methodological issue by problematising the interaction between research on embodied processes and teaching practices. Considering the development of embodied designs, Abrahamson et al. (2020) frame the relationship between research and teaching as a formative one: “the time is ripe to educate prospective teachers, in-service teachers, and collegiate instructors regarding embodiment” (p. 17). While we recognise the importance of understanding how research can *inform* teaching, we aim to investigate how teaching practices can also *inform* research. Carrying the call to “cross traditional boundaries” (ibid., p. 20), we therefore consider in-service teachers’ perspective on embodiment in teaching and learning mathematics and focus on how researchers’ and teachers’ perspectives *interact*. Thus, our research question reads: what knowledge arises when researchers and teachers commonly investigate the role of embodiment in teaching and learning mathematics?

## Context and methods

We investigate the problem of research-teaching interaction by focusing on a specific mathematical content and a localised teaching context. To do so, we build on previous research conducted by Boonstra et al. (2023) on the fostering of *embodied abstraction* in primary school geometry. This work recognized a tension between the embodied abstraction approach on geometry and the teaching of geometry at the primary level in the Netherlands. Thus, a series of tasks was developed for teachers to experience and reflect on emerging embodied processes, focusing on reflectional symmetry. In the Vaud canton of Switzerland, recommendations for primary teachers also contrast with the approach of embodied abstraction. They focus on the use of manipulatives for building “mental images”, on sensitive experiences of physical quantities (weight, length, area...) and on changes of spatial scales (the sheet of paper, the classroom or the building). Thus, the tasks developed by Boonstra et al. (2023) are likely to come into *tension* with the practices of local primary teachers, bringing the perspectives of research and teaching practices to interact. Using tasks on equidistance between reflective symmetric points, we investigate with Swiss primary teachers how the action-based design and its underlying principles (see Abrahamson, 2014) could relate to their teaching practices.

The methods (and, admittedly, the reasons) for conducting this investigation ground in the principles of *collaborative research* (Savoie-Zajc & Descamps-Bednarz, 2007). This approach frames the teacher-researcher relationship as *co-constructive* for knowledge about teaching practices. To co-
construct such knowledge, the setting of an activity that *reflects* teachers’ practices and allows to reflect on them is required. Thus, we investigated the selected tasks in relation to the rationale of action-based design for mathematics (Abrahamson, 2014); then, we experienced the tasks, discussing their use in their respective classes; finally, teachers could try the tasks in class and reflect on the experience with the first author. Results from the first two phases are presented, providing methodological insights on possible interactions of teachers’ and researchers’ perspectives.

## Insights and opening question

Discussions display a research-teaching tension (resonances and divergences) concerning the epistemological value of bodily experience in mathematics. In resonance with the tasks’ rationale, teachers differentiated types of tasks involving the body (Boonstra et al., 2023, p. 7): including the body as a resource for solving a mathematical task was valued, rather than a medium for expressing the solution of a mathematical task. However, diverging with the tasks’ rationale, teachers differentiated tasks depending on the scale of embodiment: teachers deemed important to involve the body on a larger scale, asking pupils to reflect the displacements of a fellow learner. This divergence can be related to their institutional context of practice, emphasising various spatial scales in mathematics rather than sensorimotor challenges. However, spatial scales in geometry are also addressed in research on embodiment (Marchand et al., 2024). Thus, how to build on the diversity of “embodied research strands” (Gerofsky, 2014) to interact with teachers’ perspectives?

## References

* Abrahamson, D. (2014). Building educational activities for understanding: An elaboration on the embodied-design framework and its epistemic grounds. *International Journal of Child-Computer Interaction*, 2(1), 1–16. https://doi.org/10.1016/j.ijcci.2014.07.002

* Abrahamson, D., Nathan, M. J., Williams-Pierce, C., Walkington, C., Ottmar, E. R., Soto, H., & Alibali, M. W. (2020). The future of embodied design for mathematics teaching and learning. *Frontiers in Education*, 5. https://doi.org/10.3389/feduc.2020.00147

* Boonstra, K., Kool, M., Shvarts, A., & Drijvers, P. (2023). Theories and practical perspectives on fostering embodied abstraction in primary school geometry education. *Frontiers in Education*, 8. https://doi.org/10.3389/feduc.2023.1162681

* Gerofsky, S. (2014). Making sense of the multiple meanings of “embodied mathematics learning”. In P. Liljedahl, C. Nicol, S. Oesterle, & D. Allan (Eds.), *Proceedings of the Joint Meeting of PME 38 and PME-NA 36* (Vol. 1, pp. 145–152). Vancouver, Canada: PME.

* Marchand, P., Sinclair, N., Julien, N., & Forest, M. (2024). A dynamic and spatial approach to enrich the teaching and learning of geometry in primary school. In T. Lowrie, A. Gutiérrez, & F. Emprin (Eds.), *Proceedings of the 26th ICMI Study Conference (Advances in Geometry Education) (pp. 85–92)*. ICMI.

* Savoie-Zajc, L., & Descamps-Bednarz, N. (2007). Action research and collaborative research: Their specific contributions to professional development. *Educational Action Research*, 15(4), 577–596. https://doi.org/10.1080/09650790701664013