Sense-making regarding matrix representation of geometric transformations in \({{\mathbb{R}}^2}\): a semiotic mediation perspective in a dynamic geometry environment

Abstract

The aim of this research is to analyse students’ sense-making regarding matrix representation of geometric transformations in a dynamic geometry environment (DGE) within the perspective of semiotic mediation. In particular, the focus is on students’ reasoning on the transition from the notion of function to transformation and to matrix representation of geometric transformations in \({{\mathbb{R}}^2}\). Along these lines, the theory of semiotic mediation is referred to as a theoretical framework in both the design of a teaching and learning environment and the emergence of mathematical thinking. Epistemological analysis was employed to elaborate the semiotic potential of the DGE and, thereafter, two specific tasks were considered. Task-based interviews were conducted with a pair of undergraduate linear algebra students, with the students working in front of a computer installed with a specific DGE: GeoGebra. The data sources are video-recorded interviews, screen recorder software, field notes and the students’ production analysed through a semiotic lens. According to the results, the dragging tool evokes a sense of understanding of covariation and independent/dependent variables. In addition, the simultaneous use of the dragging tool and grid function evokes a sense of the geometric transformation and the notion of matrix representation of geometric transformations, while the ApplyMatrix construction command plays a key role in linking the notions of function, transformation and matrix transformation.

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Notes

  1. 1.

    In the rest of the paper, we consider an initial (independent) figure as drawing, and new figures obtained through DGE’s tools and commands applied to drawing, which cannot be moved freely, as construction.

  2. 2.

    Here, an artefact has a general meaning. An artefact can be considered as a material (not necessarily physical, it could be a digital entity) that is produced by humans to do something (Mariotti 2009), such as a compass. Therefore, specific functions and tools of DGE or DGE in itself are considered as artefacts in this study. Regarding the distinction between an artefact and a tool, for instance, a compass as an artefact “becomes a tool when it is used to draw a circle (which is its intended purpose); the same artefact becomes a different tool when it is used to stab someone” (Monaghan and Trouche 2016, p. 6).

  3. 3.

    It should be noted that it is possible to change the matrix under consideration by over-typing it on the Algebra side view.

  4. 4.

    The teacher never mentions a matrix; the students relate the situation to functions and matrices under the teacher’s orientation.

  5. 5.

    The elementary mathematics teacher education program is designed by the Higher Education Council of Turkey for training future lower secondary (grades 5–8) mathematics teachers. In this four-year program, pre-service teachers receive fundamental mathematical courses, such as calculus and linear algebra, in addition to pedagogical courses, such as development and learning, and methods of teaching mathematics and related courses.

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Acknowledgements

I would particularly like to thank Maria Alessandra Mariotti (University of Siena) and my supervisor Paul Drijvers (Freudenthal Institute, Utrecht University) for their numerous readings of this paper and for making constructive improvements to the text. Special thanks must go to the Reviewers, Norma Presmeg and the Editor for their careful readings of the paper and for making constructive suggestions. Funding was provided by Scientific and Technological Research Council of Turkey (TUBITAK) (Grant no. 1059B191401098).

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Turgut, M. Sense-making regarding matrix representation of geometric transformations in \({{\mathbb{R}}^2}\): a semiotic mediation perspective in a dynamic geometry environment. ZDM Mathematics Education 51, 1199–1214 (2019). https://doi.org/10.1007/s11858-019-01032-0

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Keywords

  • Learning linear algebra
  • Linear transformations
  • Dynamic geometry environment
  • Semiotic mediation