A set of design principles for exercising mathematical communication competency when using a DGE

Authors

Cecilie Carlsen Bach

Abstract

In design research, design principles involve the development of theory and practice. This paper refines a set of humble design heuristics into a set of design principles in the third iteration of a design research project. The set of design principles aims to exercise (meaning ”to put into practice”) students’ mathematical communication competency when using a dynamic geometry environment (DGE). Based on an analysis, which includes perspectives on the instrumental approach, semiotic registers and mathematical language, the set of design principles is refined by transforming an analysis of two 9th grade (15–16 years old) students’ interactions with the task design into prescriptive principles. The overall principle of separate – join – new separate indicates that it is crucial to relate mathematical representations across registers in the different steps, individually and in collaboration.

References

Akker, J. van den (1999). Principles and methods of development research. In J. v. d. Akker, R. M. Branch, K. Gustafson, N. Nieveen & T. Plomp (Eds.), Design approaches and tools in education and training (pp. 1-14). Kluwer. https://doi.org/10.1007/978-94-011-4255-7_1

Artigue, M. (2005). The integration of symbolic calculators into secondary education: some lessons from didactical engineering. In D. Guin, K. Ruthven & L. Trouche (Eds.), The didactical challenge of symbolic calculators: turning a computational device into a mathematical instrument (pp. 231-294). Springer. https://doi.org/10.1007/0-387-23435-7_10

Artigue, M. & Trouche, L. (2021). Revisiting the French didactic tradition through technological lenses. Mathematics MDPI, 9 (6), 1-19. https://doi.org/10.3390/math9060629

Arzarello, F., Olivero, F., Paola, D. & Robutti, O. (2002). A cognitive analysis of dragging practices in Cabri environments. ZDM, 34 (3), 66-72. https://doi.org/10.1007/BF02655708

Bach, C. C. (submitted). Adapting profiles for CASs to students' use of DGEs: through a transition perspective.

Bach, C. C. & Bergqvist, E. (submitted). Students' mathematical communication when using digital tools.

Bach, C. C., Bergqvist, E. & Jankvist, U. T. (2022a). Mathematical communication when using DGE: balancing between object and representations. In H.-G. Weigand, A. Donevska-Todorova, E. Faggiano, P. Iannone; J. Medová et al. (Eds.), Proceedings of ETC13 (pp. 88-95). Constantine the Philosopher University in Nitra.

Bach, C. C. & Bikner-Ahsbahs, A. (in press). Activating mathematical communication competency when using DGE - Is it possible? In U. T. Jankvist & E. Geraniou (Eds.), Mathematical competencies in the digital era - connecting theoretical perspectives. Springer.

Bach, C. C. & Bikner-Ahsbahs, A. (2020). Students' experiences with dynamic geometry software and its mediation on mathematical communication competency. In A. Donevska-Todorova, E. Faggiano, J. Trgalova, Z. Lavicza, R. Weinhandl et al. (Eds.), Proceedings of ECT 10 on Mathematics Education in the Digital Age (pp. 427-434). Johannes Kepler University.

Bach, C. C. & Bikner-Ahsbahs, A. (2022). When a digital tool guides mathematical communication. In U. T. Jankvist, R. Elicer, A. Clark-Wilson, H.-G. Weigand & M. Thomsen (Eds.), Proceedings of ICTMT 15 (pp. 224- 231). Aarhus University.

Bach, C. C., Gregersen, R. M., Pedersen, M. K. & Jankvist, U. T. (2022b). Networking practices in design research: refining design principles. In J. Hodgen, E. Geraniou, G. Bolondi & F. Ferretti (Eds.), Proceedings of CERME12 (pp. 2922-2929). Free University of Bozen-Bolzano and ERME.

Cobb, P., Confrey, J., DiSessa, A., Lehrer, R. & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32 (1), 9-13. https://doi.org/10.3102/0013189X032001009

Davydov, V. V. (1990). Types of generalisation in instruction: logical and psychological problems in the structuring of school curricula. NCTM.

Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61 (1-2), 103-131. https://doi.org/10.1007/s10649-006-0400-z

Duval, R. (2017). Understanding the mathematical way of thinking - the registers of semiotic representation. Springer. https://doi.org/10.1007/978-3-319-56910-9

Drijvers, P., Ball, L., Barzel, B., Heid, M. K., Cao, Y. & Maschietto, M. (2016). Uses of technology in lower secondary mathematics education - a concise topical survey. Springer. https://doi.org/10.1007/978-3-319-33666-4

Drijvers, P., Kieran, C., Mariotti, M. A., Ainley, J., Andresen, M. et al. (2009). Integrating technology into mathematics education: theoretical perspectives. In C. Hoyles & J. B. Lagrange (Eds.), Mathematics education and technology - rethinking the terrain (pp. 89-132). Springer. https://doi.org/10.1007/978-1-4419-0146-0_7

Flyvbjerg, B. (2006). Five misunderstandings about case-study research. Qualitative Inquiry, 12 (2), 219-245. https://doi.org/10.1177/1077800405284363

Freiman, V. (2020). Types of technology in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 869-879). Springer. https://doi.org/10.1007/978-3-030-15789-0_158

Gibbons, P. (2015). Scaffolding language, scaffolding learning, second edition - teaching english language learners in the mainstream classroom. Heinemann.

Guin, D. & Trouche, L. (1998). The complex process of converting tools into mathematical instruments: the case of calculators. International Journal of Computers for Mathematical Learning, 3 (3), 195-227. https://doi.org/10.1023/A:1009892720043

Højsted, I. H. (2020). Guidelines for utilising affordances of dynamic geometry environments to support development of reasoning competency. Nordic Studies in Mathematics Education, 25 (2), 71-98.

Johnson, H. L. & McClintock, E. (2018). A link between students' discernment of variation in unidirectional change and their use of quantitative variational reasoning. Educational Studies in Mathematics, 97, 299-316. https://doi.org/10.1007/s10649-017-9799-7

Kaur, H. (2015). Two aspects of young children's thinking about different types of dynamic triangles: prototypicality and inclusion. ZDM, 47 (3), 407-420. https://doi.org/10.1007/s11858-014-0658-z

Lagrange, J.B. (1999). Complex calculators in the classroom: theoretical and practical reflections on teaching pre-calculus. International Journal of Computers for Mathematics Learning, 4 (1), 51-81. https://doi.org/10.1023/A:1009858714113

Leung, A. (2011). An epistemic model of task design in dynamic geometry environment. ZDM, 43 (3), 325-336. https://doi.org/10.1007/s11858-011-0329-2

Leung, A. (2017). Exploring techno-pedagogic task design in the mathematics classroom. In A. Leung & A. Baccaglini-Frank (Eds.), Digital technologies in designing mathematics education tasks (pp. 3-16). Springer. https://doi.org/10.1007/978-3-319-43423-0_1

Ministry of Education (2019). Matematik. Fælles mål [Mathematics. Common goals]. Ministry of Education. https://emu.dk/sites/default/files/2020-09/GSK_F%C3%A6llesM%C3%A5l_Matematik.pdf

Ng, O. L. (2016) Comparing calculus communication across static and dynamic environments using a multimodal approach. Digital Experiences in Mathematics Education, 2 (2), 115-141. https://doi.org/10.1007/s40751-016-0014-8

Ng, O. L. (2019). Examining technology-mediated communication using a commognitive lens: the case of touchscreen-dragging in dynamic geometry environments. International Journal of Science and Mathematics Education, 17 (6), 1173-1193. https://doi.org/10.1007/s10763-018-9910-2

Niss, M. & Højgaard, T. (2011). Competencies and mathematical learning ideas and inspiration for the development of mathematics teaching and learning in Denmark. IMFUFA, Roskilde University.

Niss, M. & Højgaard, T. (2019). Mathematical competencies revisited. Educational Studies in Mathematics, 102 (1), 9-28. https://doi.org/10.1007/s10649-019-09903-9

Oner, D. (2016). Tracing the change in discourse in a collaborative dynamic geometry environment: from visual to more mathematical. International journal of computer supported collaborative learning, 11 (1), 59-88. https://doi.org/10.1007/s11412-016-9227-5

Prediger, S. (2019). Theorizing in design research. AIEM - Avances de Investigación en Educación Matemática, 15, 5-27. https://doi.org/10.35763/aiem.v0i15.265

Prediger, S., Clarkson, P. & Bose, A. (2016). Purposefully relating multilingual registers: building theory and teaching strategies for bilingual learners based on an integration of three traditions. In R. Barwell, P. Clarkson, A. Halai, M. Kazima, J. Moschkovich et al. (Eds.), Mathematics education and language diversity (pp. 193-215). Springer. https://doi.org/10.1007/978-3-319-14511-2_11

Prediger, S. & Neugebauer, P. (2021). Can students with different language backgrounds profit equally from a language-responsive instructional approach for percentages? Differential effectiveness in a field trial. Mathematical Thinking and Learning, 25 (1), 1-21. https://doi.org/10.1080/10986065.2021.1919817

Prediger, S. & Wessel, L. (2011). Relating registers for fractions - multilingual learners on their way to conceptual understanding. In M. Setati, T. Nkambule & L. Goosen (Eds.), Proceedings of the ICMI Study 21 - mathematics and language diversity (pp. 324-333). ICME.

Prediger, S. & Wessel, L. (2013). Fostering German language learners' constructions of meanings for fractions - design and effects of a language- and mathematics-integrated intervention. Mathematics Education Research Journal, 25 (3), 435-456. https://doi.org/10.1007/s13394-013-0079-2

Rojano, T. & Sutherland, R. (2020). Technology and curricula in mathematics education. In S. Lerman (Ed.), Encyclopaedia of mathematics education (pp. 849-853). Springer. https://doi.org/10.1007/978-3-030-15789-0_154

Schacht, F. (2015). Student documentations in mathematics classrooms using digital tools: theoretical considerations and empirical findings. The Electronic Journal of Mathematics and Technology, 9 (5), 320-339.

Schacht, F. (2018) Between the conceptual and the signified: how language changes when using dynamic geometry software for construction tasks. Digital Experiences in Mathematics Education, 4, 20-47. https://doi.org/10.1007/s40751-017-0037-9

Schleppegrell, M. J. (2004). The language of schooling: a functional linguistics perspective. Lawrence Erlbaum. https://doi.org/10.4324/9781410610317

Shvarts, A., Doorman, M. & Alberto, R. (2022). Concrete-abstract-new-concrete: Freudenthal and Davydov in advancing embodied design framework. Paper presented at CERME12, Bozen-Bolzano.

Sutherland, R. & Rojano, T. (2014) Technology and Curricula in Mathematics Education. In S. Lerman (Ed.), Encyclopaedia of mathematics education (p. 601). Springer. https://doi.org/10.1007/978-94-007-4978-8_154

Toulmin, S. E. (1969). The uses of argument. Cambridge University Press.

Trouche, L., Drijvers, P., Gueudet, G. & Sacristán, A. I. (2013). Technology driven developments and policy implications for mathematics education. In M. A. Clements, A. J. Bishop, C. Keitel, J. Kilpatrick & F. K. S. Leung (Eds.), Third international handbook of mathematics education (pp. 753-789). Springer. https://doi.org/10.1007/978-1-4614-4684-2_24

Trouche, L. (2005a). An instrumental approach to mathematics learning in symbolic calculators environments. In D. Guin, K. Ruthven & L. Trouche (Eds.), The didactical challenge of symbolic calculators: turning a computational device into a mathematical instrument (pp. 137-162). Springer. https://doi.org/10.1007/0-387-23435-7_7

Trouche, L. (2005b). Instrumental genesis, individual and social aspects. In D. Guin, K. Ruthven & L. Trouche (Eds.), The didactical challenge of symbolic calculators: turning a computational device into a mathematical instrument (pp. 197-230). Springer. https://doi.org/10.1007/0-387-23435-7_9

Trouche, L. (2020a). Instrumentalisation in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 392-403). Springer. https://doi.org/10.1007/978-3-030-15789-0_100013

Trouche L. (2020b). Instrumentation in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 404-412). Springer. https://doi.org/10.1007/978-3-030-15789-0_80

Trouche, L. & Drijvers. P. (2010). Handheld technology for mathematics education: flashback into the future. ZDM, 42, 667-681. https://doi.org/10.1007/s11858-010-0269-2

Vérillon, P. & Rabardel, P. (1995). Cognition, and artifact: a contribution to the study of thought in relation to instrumented activity. European Journal of Psychology in Education, 10 (1), 77-101. https://doi.org/10.1007/BF03172796

Downloads

Published

2023-12-01

How to Cite

Bach, C. C. (2023). A set of design principles for exercising mathematical communication competency when using a DGE. NOMAD Nordic Studies in Mathematics Education, 28(3-4), 125–155. Retrieved from https://tidsskrift.dk/NOMAD/article/view/149265