# Using Dynamic Geometry Software to Explore Eigenvectors: The Emergence of Dynamic-Synthetic-Geometric Thinking

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## Abstract

This article analyses students’ thinking as they interacted with a dynamic geometric sketch designed to explore eigenvectors and eigenvalues. We draw on the theory of instrumental genesis and, in particular, attend to the different dragging modalities used by the students throughout their explorations. Given the kinaesthetic and dynamic features of the environment, we also draw on theories of embodied cognition to analyse students’ emergent visual and kinaesthetic understandings. Our analysis suggests that, in contrast with the predominantly analytic-arithmetic mode of thinking (and the consequent procedural knowledge) reported in the literature, our students developed a synthetic-geometric mode of thinking, which researchers have pointed to as being essential to the understanding of linear algebra, but absent for most students. We also found that their synthetic-geometry mode of thinking strongly featured *motion*-*based* conceptions of eigenvectors and eigenvalues, thus leading us to characterise their thinking as *dynamic*-*synthetic*-*geometric*.

## Keywords

Linear algebra Dynamic geometric environments Instrumental genesis Dynamic and kinaesthetic imagery Embodiment Gestures## References

- Alves Dias, M., & Artigue, M. (1995). Articulation problems between different systems of symbolic representations in linear algebra.
*Proceedings of the 19th annual meeting of the international group for the psychology of mathematics education*(Vol. 3, no. 2, pp. 34–41). Brazil: Universidade Federal de Pernambuco.Google Scholar - Arzarello, F., Ferrara, F., Paola, D., Robutti, O., & Sabena, C. (2005). Shaping a multi-dimensional analysis of signs. Research Forum in H. L. Chick & J. L. Vincent (Eds.),
*Proceedings of the 29th conference of the international group for the psychology of mathematics education*(Vol. 1, pp. 126–130).Google Scholar - Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging practises in Cabri environment.
*ZDM,**34*(3), 66–72.CrossRefGoogle Scholar - Châtelet, J. (2000).
*Figuring space: Philosophy, mathematics and physics*. Translated by Robert Shore and Muriel Zagha. Dordrecht, Boston: Kluwer.Google Scholar - Edwards, L., Radford, L., & Arzarello, F. (2009). Gestures and multimodality in the construction of mathematical meaning [Special Issue].
*Educational Studies in Mathematics,**70*(2), 175–189.CrossRefGoogle Scholar - Hillel, J., & Sierpinska, A. (1994). On one persistent mistake in linear algebra.
*Proceedings of the 18th conference on psychology of mathematics education*(pp. 65–72). Portugal: University of Lisbon.Google Scholar - Jackiw, N. (1989).
*The geometer’s sketchpad*. Berkeley, CA: Key Curriculum Press.Google Scholar - Lay, D. C. (2006).
*Linear Algebra and Its Applications*(3rd ed.). USA: Addison-Wesley publisher.Google Scholar - Núñez, R. (2006). Do real numbers really move? Language, thought, and gesture: The embodied cognitive foundations of mathematics. Reprinted in R. Hersh (Ed.),
*18 Unconventional Essays on the Nature of Mathematics*(pp. 160–181). New York: Springer.Google Scholar - Presmeg, N. C. (1986). Visualization in high school mathematics.
*For the learning of Mathematics,**6*(3), 42–46.Google Scholar - Radford, L. (2009). Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings.
*Educational Studies in Mathematics,**70*(2), 111–126.CrossRefGoogle Scholar - Seitz, J. A. (2000). The bodily basis of thought.
*New Ideas in Psychology: An International Journal of Innovative Theory in Psychology,**18*(1), 23–40.CrossRefGoogle Scholar - Sheets-Johnston, M. (2009). Animation: the fundamental, essential, and properly descriptive concept.
*Continental Philosophy Review,**42*, 375–400.CrossRefGoogle Scholar - Sierpinska, A. (2000). On some aspects of students’ thinking in linear algebra. In J. Dorier (Ed.),
*On the teaching of linear algebra*(pp. 209–246). Netherlands: Kluwer Academic publishers.Google Scholar - Sierpinska, A., Deyfus, T., & Hillel, J. (1999). Evaluation of a teaching design in linear algebra: the case of linear transformations.
*Recherches en Didactique des Mathematique,**19*(1), 7–41.Google Scholar - Sinclair, N., & Gol Tabaghi, S. (2010). Drawing space: mathematicians’ kinetic conceptions of eigenvectors.
*Educational Studies in Mathematics,**74*(3), 223–240.CrossRefGoogle Scholar - Stewart, S. (2008). Understanding linear algebra concepts through the embodied symbolic and formal worlds of mathematical thinking.
*Doctoral thesis*: Auckland University. Available from http://hdl.handle.net/2292/2912. - Thurston, W. P. (1994). On proof and progress in mathematics.
*The American Mathematical Society,**30*(2), 161–177.Google Scholar - Trouche, L. (2005). An instrumental approach to mathematics learning in symbolic calculators environments. In D. Guin, K. Ruthven, & L. Trouche (Eds.),
*The didactical challenge of symbolic calculators*(pp. 137–163). New York: Springer.CrossRefGoogle Scholar - Verillon, 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,**9*(3), 77–101.CrossRefGoogle Scholar