## 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.

This is a preview of subscription content, log in to check access.

## Notes

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

- 4.
The teacher never mentions a matrix; the students relate the situation to functions and matrices under the teacher’s orientation.

- 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.

## References

Andrews-Larson, C., Wawro, M., & Zandieh, M. (2017). A hypothetical learning trajectory for conceptualizing matrices as linear transformations.

*International Journal of Mathematical Education in Science and Technology, 48*(6), 809–929.Anton, H. (1981).

*Elementary linear algebra*(3rd ed.). New York: Wiley.Anton, H., & Rorres, C. (2014).

*Elementary linear algebra*(11th ed.). New Jersey: Wiley.Bagley, S., Rasmussen, C., & Zandieh, M. (2015). Inverse, composition, and identity: The case of function and linear transformation.

*The Journal of Mathematical Behavior, 37*, 36–47.Bakker, A., & van Eerde, D. (2015). An introduction to design–based research with an example from statistics education. In A. Bikner-Ahsbahs, C. Knipping & N. Presmeg (Eds.),

*Approaches to qualitative research in mathematics education: Examples of methodology and methods*(pp. 429–466). Dordrecht: Springer.Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: Artifacts and signs after a Vygotskian perspective. In L. English, M. Bartolini Bussi, G. Jones, R. Lesh & D. Tirosh (Eds.),

*Handbook of international research in mathematics education*(Vol. 2, pp. 746–783). Mahwah: Erlbaum.Carlson, D. (1997). Teaching linear algebra: Must the fog always roll in? In D. Carlson, C. R. Johnson, D. C. Lay, A. D. Porter, A. Watkins & W. Watkins (Eds.),

*Resources for teaching linear algebra. MAA notes*(Vol. 42, pp. 39–51). Washington: Mathematical Association of America.Confrey, J., & Smith, E. (1991). A framework for functions: Prototypes, multiple representations, and transformations. In R. G. Underhill (Ed.),

*Proceedings of the 13th annual meeting North American chapter of the international group for the psychology of mathematics education*(Vol. 1, pp. 57–63). Virginia: Christiansburg Printing Company Inc.Dorier, J.-L., Robert, A., Robinet, J., & Rogalski, M. (2000). The obstacle of formalism in linear algebra. In J.-L. Dorier (Ed.),

*On the teaching of linear algebra*(pp. 85–124). Dordrecht: Kluwer Academic Publishers.Falcade, R., Laborde, C., & Mariotti, M. A. (2007). Approaching functions: Cabri tools as instruments of semiotic mediation.

*Educational Studies in Mathematics, 66*(3), 317–333.Figueroa, A. P., Possani, E., & Trigueros, M. (2018). Matrix multiplication and transformations: An APOS approach.

*The Journal of Mathematical Behavior, 52*, 77–91.Gueudet-Chartier, G. (2004). Should we teach linear algebra through geometry?

*Linear Algebra and Its Applications, 379*, 491–501.Harel, G. (2000). Three principles of learning and teaching mathematics. In J.-L. Dorier (Ed.),

*On the teaching of linear algebra*(pp. 177–189). Dordrecht: Kluwer Academic Publishers.Harel, G. (2017). The learning and teaching of linear algebra: Observations and generalizations.

*The Journal of Mathematical Behavior, 46*, 69–95.Hazzan, O., & Goldenberg, E. P. (1997). Students’ understanding of the notion of function in dynamic geometry environments.

*International Journal of Computers for Mathematical Learning, 1*(3), 263–291.Hollebrands, K. F. (2003). High school students’ understandings of geometric transformations in the context of a technological environment.

*The Journal of Mathematical Behavior, 22*(1), 55–72.Kolman, B., & Hill, D. R. (2007).

*Elementary linear algebra with applications*(9th ed.). New York: Pearson Education Inc.Larson, C., & Zandieh, M. (2013). Three interpretations of the matrix equation Ax = b.

*For the Learning of Mathematics, 33*(2), 11–17.Lay, D. C. (2006).

*Linear algebra and its applications*(3rd ed.). Boston: Pearson Addison-Wesley.Mariotti, M. A. (2009). Artifacts and signs after a Vygotskian perspective: The role of the teacher.

*ZDM - The International Journal on Mathematics Education, 41*(4), 427–440.Mariotti, M. A. (2014). Transforming images in a DGS: The semiotic potential of the dragging tool for introducing the notion of conditional statement. In S. Rezat, M. Hattermann & A. Peter-Koop (Eds.),

*Transformation—a fundamental idea of mathematics education*(pp. 155–172). New York: Springer.MEB. (2013). Ortaöğretim Matematik Dersi (9. (10., 11 ve 12. sınıflar) Öğretim Programı [Mathematics Curricula for 9., 10., 11). and 12. Grades]. Ankara: Talim Terbiye Kurulu Başkanlığı.

Monaghan, J., & Trouche, L. (2016). Introduction to the book. In

*Tools and mathematics*(pp. 3–12). Cham: Springer International Publishing.Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity.

*Educational Studies in Mathematics, 12*(2), 151–169.Thompson, P. W. (1994). Students, functions, and the undergraduate curriculum. In E. Dubinsky, A. H. Schoenfeld & J. J. Kaput (Eds.),

*Research in collegiate mathematics education*(Vol. 4, pp. 21–44). Providence: American Mathematical Society.Turgut, M. (2017). Students’ reasoning on linear transformations in a DGS: A semiotic perspective. In T. Dooley & G. Gueudet (Eds.),

*Proceedings of the 10th congress of European society for research in mathematics education*(pp. 2652–2659). Dublin: Institute of Education, Dublin City University, Ireland, and ERME.Turgut, M. (2018). How does a dynamic geometry system mediate students’ reasoning on 3D linear transformations? In S. Stewart, C. Andrews-Larson, A. Berman & M. Zandieh (Eds.),

*Challenges and strategies in teaching linear algebra*(pp. 241–259). Cham: Springer International Publishing.YÖK. (2007). Eğitim Fakültesi Öğretmen Yetiştirme Lisans Programları [Teacher Bachelor Curriculas of Educational Faculties of Turkey]. Ankara: YÖK. http://goo.gl/1GiUuQ. Accessed 5 May 2018.

Zandieh, M., Ellis, J., & Rasmussen, C. (2012). Student concept images of function and linear transformation. In S. Brown, S. Larsen, K. Marrongelle & M. Oehrtman (Eds.),

*Proceedings of the 15th annual conference on research in undergraduate mathematics education*(pp. 320–328). Portland: SIGMAA.Zandieh, M., Ellis, J., & Rasmussen, C. (2017). A characterization of a unified notion of mathematical function: The case of high school function and linear transformation.

*Educational Studies in Mathematics, 95*(1), 21–38.

## 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).

## Author information

### Affiliations

### Corresponding author

## Rights and permissions

## About this article

### Cite this article

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

Accepted:

Published:

Issue Date:

### Keywords

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