# An epistemic model of task design in dynamic geometry environment

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## Abstract

Dynamic geometry environment (DGE) has been a catalytic agent driving a paradigm shift in the teaching and learning of school geometry in the past two decades. It opens up a pedagogical space for teachers and students to engage in mathematical explorations that niche across the experimental and the theoretical. In particular, the drag-mode in DGE has been a unique pedagogical tool that can facilitate and empower students to experiment with dynamic geometrical objects which can lead to generation of mathematical conjectures. Furthermore, the drag-mode seems to open up a new methodology and even a new discourse to acquire geometrical knowledge alternative to the traditional Euclidean deductive reasoning paradigm. This discussion paper proposes an epistemic model of *techno-pedagogic mathematic task design* which serves as a theoretical combined-lens to view mathematics knowledge acquisition. Three epistemic modes for techno-pedagogic mathematical task design are proposed. They are used to conceptualize design of dynamic geometry tasks capitalizing the unique drag-mode nature in DGE that opens up an explorative space for learners to acquire mathematical knowledge.

## Keywords

Task design Dynamic geometry environment Drag-mode## References

- Ainley, J., Pratt, D., & Hansen, A. (2006). Connecting engagement and focus in pedagogic task design.
*British Educational Research Journal,**32*(1), 21–36.CrossRefGoogle Scholar - Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practices in Cabri environments.
*Zentralblatt für Didaktik der Mathematik,**34*(3), 66–72.CrossRefGoogle Scholar - Baccaglini-Frank, A., & Mariotti, M. A. (2010). Generating conjectures in dynamic geometry: The maintaining dragging model.
*International Journal of Computers for Mathematical Learning,**15*, 225–253.CrossRefGoogle Scholar - Baccaglini-Frank, A., Mariotti, M. A., & Antinini, S. (2009). Different perceptions of invariants and generality of proof in dynamic geometry. In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.),
*Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 89–96). Thessaloniki, Greece: PME.Google Scholar - Brown, J. (2005). Affordances of a technology-rich teaching and learning environment. In P. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce, M. Horne, & A. Roche (Eds.),
*Building connections: Theory, research, and practice*,*Proceedings of the 28th annual conference of the Mathematics Education Research Group of Australasia*,*Melbourne*(Vol. 1, pp. 177–184). Sydney: MERGA.Google Scholar - Duval, R. (1995). Geometrical pictures: Kinds of representation and specific processing. In R. Sutherland & J. Mason (Eds.),
*Exploiting mental imagery with computers in mathematics education*(pp. 142–157). New York: Springer.Google Scholar - Freudenthal, H. (1968). Why to teach mathematics so as to be useful.
*Educational Studies in Mathematics,**1*, 3–8.CrossRefGoogle Scholar - Freudenthal, H. (1991).
*Revisiting mathematics education*. Dordrecht: Kluwer Academic Publishers.Google Scholar - Gibson, J. J. (1979).
*The ecological approach to visual perception*. Boston: Houghton Mifflin.Google Scholar - Laborde, C. (2005). Robust and soft constructions: Two sides of the use of dynamics geometry environments. In
*Proceedings of the Tenth Asian Technology Conference in Mathematics*(pp. 22–35). Korea National University of Education, Cheong-Ju, South Korea.Google Scholar - Leung, A. (2003) Dynamic geometry and the theory of variation. In N. A. Pateman, B. J. Dougherty, & J. Zilliox (Eds.),
*Proceedings of PME 27: Psychology of Mathematics Education 27th International Conference*(Vol. 3, pp. 197–204). Honolulu, USA.Google Scholar - Leung, A. (2008). Dragging in a dynamic geometry environment through the lens of variation.
*International Journal of Computers for Mathematical Learning,**13*, 135–157.CrossRefGoogle Scholar - Leung, A. (2009). Written proof in dynamic geometry environment: Inspiration from a student’s work. In
*Proceedings of the ICMI 19 Study Conference: Proof and Proving in Mathematics Education*(Vol. 2, pp. 15–20). Taipei, Taiwan.Google Scholar - Leung, A., Chan, Y. C., & Lopez-Real, F. (2006). Instrumental genesis in dynamic geometry environments. In
*Proceedings of the ICMI 17 Study Conference: Technology Revisited, Part 2*(pp. 346–353). Hanoi, Vietnam.Google Scholar - Leung, A., & Lopez-Real, F. (2002). Theorem justification and acquisition in dynamic geometry: A case of proof by contradiction.
*International Journal of Computers for Mathematical Learning,**7*, 145–165.CrossRefGoogle Scholar - Leung, A., & Or, A. (2009). Cognitive apprehension in Cabri 3D environment. In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.),
*Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education*(Vol. 1, p. 417). Thessaloniki, Greece: PME.Google Scholar - Lopez-Real, F., & Leung, A. (2006). Dragging as a conceptual tool in dynamic geometry environments.
*International Journal of Mathematical Education in Science and Technology,**37*(6), 665–679.CrossRefGoogle Scholar - Mariotti, M. A. (2002). Influence of technologies advances on students’ maths learning. In L. English, M. Bartolini Bussi, G. Jones, R. Lesh, & D. Tirosh (Eds.),
*Handbook of international research in mathematics education*(pp. 695–721). New Jersey: Lawrence Erlbaum Associates.Google Scholar - Mason, J., & Johnston-Wilder, S. (2006).
*Designing and using mathematical tasks*. St. Albans: Tarquin Publications.Google Scholar - Noss, R., & Hoyles, C. (1996).
*Windows on mathematical meanings*. Dordrecht: Kluwer Academic Publishers.Google Scholar - Pratt, D., & Noss, R. (2010). Designing for mathematical abstraction.
*International Journal of Computers for Mathematical Learning,**15*, 81–97.CrossRefGoogle Scholar - Restrepo, A. M. (2008).
*Genese instrumentale du deplacement en geometrie dynamique chez des eleves de 6eme*. Unpublished Ph.D. Thesis. Universite Joseph Fourier.Google Scholar - Talmon, V., & Yerushalmy, M. (2004). Understanding dynamic behaviour: Parent–child relations in dynamic geometry environments.
*Educational Studies in Mathematics,**57*, 91–119.CrossRefGoogle Scholar - Vérillon, P., & Rabardel, P. (1995). Cognition and artifacts: A contribution to the study of thought in relation to instrumented activity.
*European Journal of Psychology of Education,**10*(1), 77–101.CrossRefGoogle Scholar