Skip to main content

Drawing space: mathematicians’ kinetic conceptions of eigenvectors

Educational Studies in Mathematics volume 74, pages 223–240 (2010)Cite this article

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.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price includes VAT (USA)
Tax calculation will be finalised during checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Notes

  1. The distinction between ideational and conceptual mathematics resembles Tall and Vinner’s (1981) distinction between concept image and concept definition, respectively.

  2. This observation, made by Mariolina Bartolini-Bussi (personal communication), can be seen in Archimedes’ and Cavalieri’s development of infinitesimals, as well as Desargues’s founding of projective geometry.

  3. Indeed, based on her interviews with over 70 mathematicians, Burton (2004) identifies three styles of thinking of mathematicians, including one that she calls visual, and to which she adds (parenthetically), sometimes dynamic. Her analyses focus entirely on verbal utterances made by mathematicians, and unfortunately, she provides no examples of what she counts as a visual, dynamic style.

  4. Furthermore, there are several examples of blind people doing mathematics who seem to draw on kinetic ways of thinking. For example, Helen Keller (1969), who was both deaf and blind, described straight lines as “I feel as if I were going forward in a straight line, bound to arrive somewhere, or go on forever without swerving to the right or to the left”. Also, Healy (2009) reports a blind student conceptualising a pyramid in terms of a gesture that involves wrapping his fingers around the base and tracing up the sides until his fingers meet at a point.

  5. For example, Courant and Robbins (1978) write “The hyperbola approaches more and more nearly the two straight lines qx ± py = 0 as we go to farther and farther from the origin, but it never actually reaches these lines. They are called the asymptotes of the parabola” (p. 76). In using the active verbs in the following phrases “approaching more and more” and “go out farther and farther” and “reaches”, the authors seem to be describing moving lines and changing distances.

  6. We included commutativity (along with area, distance and symmetry) so as to have examples of concepts that do not point back in time to a spatiotemporal conception—this based on Harris’s (2008) critique of Núñez’s work which used limit and continuity, both arguably already imbued with temporal conceptions,

  7. Ochs, Gonzales, and Jacobyet (1996)) note a similar phenomenon with physicists who, even in their spoken language, associated themselves with the physical phenomena under consideration, for example: “When I come down I’m in the domain state.”

  8. He also sees the metaphor as another mediating link between the body and mathematics.

  9. See also Sinclair (2010) for a discussion of the role of “covert” thinking in mathematics, which includes not only the surface intuitions mentioned by Châtelet, by also the more subconscious (irrational?) ways of knowing hinted at by Weil.

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].

  • 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.

    Article  Google 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.

    Book  Google 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.

    Chapter  Google 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.

    Chapter  Google Scholar 

  • Palmieri, P. (2009). Superposition: On Cavalieri’s practice of mathematics. Archive for History of Exact Sciences, 63, 471–495.

    Article  Google 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.

    Article  Google 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.

    Article  Google Scholar 

  • Seitz, J. A. (2000). The bodily basis of thought. New Ideas in Psychology, 18, 23–40.

    Article  Google Scholar 

  • Seitz, J. A. (2005). The neural, evolutionary, developmental, and bodily basis of metaphor. New Ideas in Psychology, 23, 74–95.

    Article  Google Scholar 

  • Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  • Sfard, A. (2009). What’s all the fuss about gestures? A commentary. Educational Studies in Mathematics, 70(2), 191–200.

    Article  Google 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.

    Article  Google Scholar 

  • Solomon, Y., & O’Neill, J. (1998). Mathematics and narrative. Language and Education, 12(3), 210–221.

    Article  Google 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.

    Article  Google 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.

    Article  Google 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.

    Article  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Simon Fraser University, Burnaby, BC, Canada

    Nathalie Sinclair & Shiva Gol Tabaghi

Authors
  1. Nathalie Sinclair
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shiva Gol Tabaghi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nathalie Sinclair.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Sinclair, N., Gol Tabaghi, S. Drawing space: mathematicians’ kinetic conceptions of eigenvectors. Educ Stud Math 74, 223–240 (2010). https://doi.org/10.1007/s10649-010-9235-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10649-010-9235-8

Keywords

  • Language
  • Metaphor
  • Gesture
  • Diagram
  • Motion
  • Time
  • Conceptual mathematics
  • Linear algebra