# Drawing space: mathematicians’ kinetic conceptions of eigenvectors

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## Abstract

This paper explores how mathematicians build meaning through communicative activity involving talk, gesture and diagram. In the course of describing mathematical concepts, mathematicians use these semiotic resources in ways that blur the distinction between the mathematical and physical world. We shall argue that mathematical meaning of eigenvectors depends strongly on both time and motion—hence, on physical interpretations of mathematical abstractions—which are dimensions of thinking that are typically deliberately absent from formal, written definition of the concept. We shall also show how gesture and talk contribute differently and uniquely to mathematical conceptualisation and further elaborate the claim that diagrams provide an essential mediating role between the two.

## Keywords

Language Metaphor Gesture Diagram Motion Time Conceptual mathematics Linear algebra## Notes

### Acknowledgements

This research has been supported by the Social Sciences and Humanities Research Council of Canada. We would like to thank the mathematicians who participated in this study. We would also like to thank David Pimm for his feedback on previous versions. We extend our acknowledgements as well to three anonymous reviewers and to Norma Presmeg for her guidance in our revisions.

## References

- Balacheff, N. (1988). Aspects of proof in pupils’ practice of school mathematics (translated by Pimm D.). In D. Pimm (Ed.),
*Mathematics, teachers and children*(pp. 216–235). London: Hodder and Stoughton.Google Scholar - Burton, L. (2004).
*Mathematicians as enquirers: Learning about learning mathematics*. Dordrecht: Kluwer.Google Scholar - Châtelet, G. (1993).
*Les enjeux du mobile*. Paris: Seuil [English translation by R. Shore & M. Zagha:*Figuring space: Philosophy, mathematics and physics*, Dordrecht: Kluwer, 2000].Google Scholar - Courant, R., & Robbins, H. (1978).
*What is mathematics?*New York: Oxford University Press.Google Scholar - Goldin-Meadow, S. (2003).
*Gesture: How our hands help us think*. Cambridge: Harvard University Press.Google Scholar - Harris, M. (2008). Why mathematics? You might ask. In T. Gowers, J. Barrow-Green, & I. Leader (Eds.),
*The Princeton companion to mathematics*. Princeton: Princeton University Press.Google Scholar - Hawkins, T. (1975). Cauchy and the spectral theory of matrices.
*Historia Mathematica, 2*, 1–29.CrossRefGoogle Scholar - Healy, L. (2009). Relationships between sensory activity, cultural artefacts and mathematical cognition. In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.),
*Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education*. Thessaloniki: PME.Google Scholar - Keller, H. (1969).
*The story of my life*. New York: Collier-Macmillan.Google Scholar - Lakoff, G., & Núñez, R. (2000).
*Where mathematics come from: How the embodied mind brings mathematics into being*. New York: Basic Books.Google Scholar - Lay, D. C. (2006).
*Linear algebra and its application*. USA: Pearson Addison Wesley.Google Scholar - Mancosu, P. (1996).
*Philosophy of mathematics and mathematical practice in the seventeenth century*. NewYork: Oxford University Press.Google Scholar - Nemirovsky, R., & Borba, M. (2003). Perceptuo-motor activity and imagination in mathematics learning. In N. Pateman, B. Dougherty, & J. Zilliox (Eds.),
*Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education, 1*(pp. 103–135). Manoa: University of Hawaii.Google Scholar - Netz, R. (2009).
*Ludic mathematics: Greek mathematics and the Alexandrian aesthetic*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Núñez, R. (2006). Do real numbers really move? Language, thought, and gesture: The embodied cognitive foundations of mathematics. In R. Hersh (Ed.),
*18 Unconventional essays on the nature of mathematics*(pp. 160–181). New York: Springer.CrossRefGoogle Scholar - Ochs, E., Gonzales, P., & Jacobyet, S. (1996). “When I come down I’m in the domain state”: Grammar and graphic representation in the interpretive activity of physicists. In E. Ochs, E. A. Schegloff, & S. A. Thompson (Eds.),
*Interaction and grammar*(pp. 328–369). Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Palmieri, P. (2009). Superposition: On Cavalieri’s practice of mathematics.
*Archive for History of Exact Sciences, 63*, 471–495.CrossRefGoogle Scholar - Papert, S. (1980).
*Mindstorms: Children, computers and powerful ideas*. New York: Basic Books.Google Scholar - Pimm, D. (2006). Drawing on the image in mathematics and art. In N. Sinclair, D. Pimm, & W. Higginson (Eds.),
*Mathematics and the aesthetic: New approaches to an ancient affinity*(pp. 160–189). 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). No! He starts walking backward!: Interpreting motion graphs and the question of space, place and distance.
*ZDM the International Journal on Mathematics Education, 41*, 467–480.CrossRefGoogle Scholar - Robutti, O. (2006). Motion, technology, gesture in interpreting graphs.
*The International Journal for Technology in Mathematics Education, 13*(30), 117–126.Google Scholar - Rotman, B. (2008).
*Becoming beside ourselves: The alphabet, ghosts, and distributed human beings*. Durham: Duke University Press.Google Scholar - Russell, B. (1903).
*The principles of mathematics*. Cambridge: Cambridge University Press.Google Scholar - Schiralli, M., & Sinclair, N. (2003). A constructive response to where mathematics comes from.
*Educational Studies in Mathematics, 52*(1), 79–91.CrossRefGoogle Scholar - Seitz, J. A. (2000). The bodily basis of thought.
*New Ideas in Psychology, 18*, 23–40.CrossRefGoogle Scholar - Seitz, J. A. (2005). The neural, evolutionary, developmental, and bodily basis of metaphor.
*New Ideas in Psychology, 23*, 74–95.CrossRefGoogle Scholar - Sfard, A. (2008).
*Thinking as communicating: Human development, the growth of discourses, and mathematizing*. Cambridge: Cambridge University Press.CrossRefGoogle Scholar - Sfard, A. (2009). What’s all the fuss about gestures? A commentary.
*Educational Studies in Mathematics, 70*(2), 191–200.CrossRefGoogle Scholar - Sinclair, N. (2010). Knowing more than we can tell. In B. Sriraman & L. English (Eds.),
*Theories of mathematics education: Seeking new frontiers, advances in mathematics education*(pp. 591–608). New York: Springer.Google Scholar - Sinclair, N., Healy, L., & Sales, C. O. R. (2009). Time for telling stories: Narrative thinking with dynamic geometry.
*ZDM, 41*, 441–452.CrossRefGoogle Scholar - Solomon, Y., & O’Neill, J. (1998). Mathematics and narrative.
*Language and Education, 12*(3), 210–221.CrossRefGoogle Scholar - Tall, D. O., & Vinner, S. (1981). Concept image and concept definition in mathematics, with special reference to limits and continuity.
*Educational Studies in Mathematics, 12*, 151–169.CrossRefGoogle Scholar - Talmy, L. (1996). Fictive motion in language and “caption”. In P. Bloom, M. Peterson, L. Nadel, & M. Garrett (Eds.),
*Language and space*(pp. 212–273). Cambridge: MIT Press.Google Scholar - Thurston, W. P. (1994). On proof and progress in mathematics.
*The American Mathematical Society, 30*(2), 161–177.CrossRefGoogle Scholar - Wright, T. (2001). Karen in motion: The role of physical enactment in developing an understanding of distance, time, and speed.
*Journal of Mathematical Behavior, 20*, 145–162.CrossRefGoogle Scholar