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

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## 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.

## Keywords

Learning linear algebra Linear transformations Dynamic geometry environment Semiotic mediation## Notes

### 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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