Skip to main content
SpringerLink
Log in

Eigenvalues and eigenvectors: embodied, symbolic and formal thinking

  • Original Article
  • Published:
Mathematics Education Research Journal Aims and scope Submit manuscript

Abstract

Many beginning university students struggle with the new approaches to mathematics that they find in their courses due to a shift in presentation of mathematical ideas, from a procedural approach to concept definitions and deductive derivations, and ideas building upon each other in quick succession. This paper highlights this struggle by considering some conceptual processes and difficulties students find in learning about eigenvalues and eigenvectors. We use the theoretical framework of Tall’s three worlds of mathematical thinking, along with perspectives from Dubinsky’s APOS (action, process, object, schema) theory and Thomas’s representational versatility. The results of the study describe thinking about these concepts by several groups of first- and second-year university students. In particular the obstacles they faced, and the emerging links some were constructing between parts of their concept images formed from the embodied, symbolic, and formal worlds are presented. We also identify some fundamental problems with student understanding of the definition of eigenvectors that lead to implementation problems, and some of the concepts underlying such difficulties.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  • Anton, H., & Busby, R. C. (2003). Contemporary linear algebra. New York: Wiley.

    Google Scholar 

  • Asiala, M., Dubinsky, E., Mathews, D., Morics, S., & Oktac, A. (1997). Student understanding of cosets, normality, and quotient groups. The Journal of Mathematical Behavior, 16(3), 241–309.

    Article  Google Scholar 

  • Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K., & Vidakovic, D. (1996). Understanding the limit concept: beginning with a coordinated process scheme. The Journal of Mathematical Behavior, 15, 167–192.

    Article  Google Scholar 

  • Dorier, J.-L., & Sierpinska, A. (2001). Research into the teaching and learning of linear algebra. In D. Holton (Ed.), The teaching and learning of mathematics at university level: An ICMI study (pp. 255–273). Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Dorier, J.-L., Robert, A., Robinet, J., & Rogalski, M. (2000). On a research program concerning the teaching and learning of linear algebra in the first-year of a French science university. International Journal of Mathematics Education, Science and Technology, 31(1), 27–35.

    Article  Google Scholar 

  • Dorier, J.-L., Robert, A., & Rogalski, M. (2003). Some comments on “The role of proof in comprehending and teaching linear algebra” by F. Uhlig. Educational Studies in Mathematics, 51, 185–191.

    Article  Google Scholar 

  • Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. O. Tall (Ed.), Advanced mathematical thinking (pp. 95–123). Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Dubinsky, E. (1997). Some thoughts on a first course in linear algebra at the college level. In D. Carlson, C. Johnson, D. Lay, A. D. Porter, A. Watkins, & W. Watkins (Eds.), Resources for teaching linear algebra (pp. 85–106). MAA notes volume 42, Washington: The Mathematical Association of America.

    Google Scholar 

  • Dubinsky, E., & McDonald, M. (2001). APOS: a constructivist theory of learning. In D. Holton (Ed.), The teaching and learning of mathematics at university level: An ICMI study (pp. 275–282). Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61, 103–131.

    Article  Google Scholar 

  • Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity and flexibility: a proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 26(2), 115–141.

    Google Scholar 

  • Greer, B., & Harel, G. (1998). The role of isomorphisms in mathematical cognition. The Journal of Mathematical Behavior, 17(1), 5–24.

    Article  Google Scholar 

  • Harel, G. (2000). Three principles of learning and teaching mathematics. In J.-L. Dorier (Ed.), The teaching of linear algebra in question (pp. 177–189). Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Hillel, J. (2000). Modes of description and the problem of representation in linear algebra. In J.-L. Dorier (Ed.), The teaching of linear algebra in question (pp. 191–207). Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Lesh, R. (1999). The development of representational abilities in middle school mathematics. In I. E. Sigel (Ed.), Development of mental representation: Theories and application (pp. 323–350). Hillsdale: Lawrence Erlbaum Associates.

    Google Scholar 

  • Lesh, R. (2000). What mathematical abilities are most needed for success beyond school in a technology based age of information? In M. O. J. Thomas (Ed.), Proceedings of TIME 2000 an International Conference on Technology in Mathematics Education (pp. 72–82). Auckland: Auckland University.

    Google Scholar 

  • Mason, J. (1992). Doing and construing mathematics in screenspace. In B. Southwell, B. Perry, & K. Owens (Eds.), Space the first and final frontier (Proceedings of the Fifteenth Annual Conference of the Mathematics Education Research Group of Australasia pp. 1–17). Sydney: University of Western Sydney.

  • Mason, J. (1995). Less may be more on a screen. In L. Burton & B. Jaworski (Eds.), Technology in mathematics teaching: A bridge between teaching and learning (pp. 119–134). London: Chartwell-Bratt.

    Google Scholar 

  • Schonefeld, S. (1995). Eigenpictures: picturing the eigenvector problem. The College Mathematics Journal, 26(4), 316–319.

    Article  Google Scholar 

  • Sierpinska, A. (2000). On some aspects of students’ thinking in linear algebra. In J.-L. Dorier (Ed.), The teaching of linear algebra in question (pp. 209–246). Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Skemp, R. R. (1971). The psychology of learning mathematics. Harmondsworth: Penguin.

    Google Scholar 

  • Stewart, S. (2008). Understanding linear algebra concepts through the embodied, symbolic and formal worlds of mathematical thinking. Unpublished PhD thesis. The University of Auckland.

  • Stewart, S., & Thomas, M. O. J. (2008). Linear algebra thinking: Embodied, symbolic and formal aspects of linear independence. Proceedings of the 11th Annual Conference of the Special Interest Group of the Mathematical Association of America (SIGMAA) on Research in Undergraduate Mathematics Education, San Diego, USA. Available from electronic proceedings http://cresmet.asu.edu/crume2008/ Proceedings/RUME08_Stewart_Thomas.doc.

  • Tall, D. O. (1998). Information technology and mathematics education: enthusiasms, possibilities & realities. In C. Alsina, J. M. Alvarez, M. Niss, A. Perez, L. Rico, & A. Sfard (Eds.), Proceedings of the 8th International Congress on Mathematical Education (pp. 65–82). Seville: SAEM Thales.

    Google Scholar 

  • Tall, D. O. (2004a). Building theories: the three worlds of mathematics. For the Learning of Mathematics, 24(1), 29–32.

    Google Scholar 

  • Tall, D. O. (2004b). Thinking through three worlds of mathematics. In M. J. Hoines & A. B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 281–288). Bergen: Bergen University College.

    Google Scholar 

  • Tall, D. O. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal, 20(2), 5–24.

    Article  Google Scholar 

  • Tall, D. O., & Thomas, M. O. J. (1991). Encouraging versatile thinking in algebra using the computer. Educational Studies in Mathematics, 22, 125–147.

    Article  Google Scholar 

  • Thomas, M. O. J. (2008). Developing versatility in mathematical thinking. Mediterranean Journal for Research in Mathematics Education, 7(2), 67–87.

    Google Scholar 

  • Thomas, M. O. J., & Hong, Y. Y. (2001). Representations as conceptual tools: process and structural perspectives. In M. van den Heuvel-Panhuizen (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 257–264). Utrecht: Utrecht University.

    Google Scholar 

  • Uhlig, F. (2003a). The role of proof in comprehending and teaching elementary linear algebra. Educational Studies in Mathematics, 50(3), 335–346.

    Article  Google Scholar 

  • Uhlig, F. (2003b). Author’s response to the comments on “The role of proof in teaching elementary linear algebra”, by Frank Uhlig, this journal, Vol. 50.3 (2002), pp. 335–346. Educational Studies in Mathematics, 53, 271–274.

  • Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. O. Tall (Ed.), Advanced mathematical thinking (pp. 65–81). Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Zaslavsky, O., & Shir, K. (2005). Students’ conceptions of a mathematical definition. Journal for Research in Mathematics Education, 36(4), 317–346.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The University of Auckland, Auckland, New Zealand

    Michael O. J. Thomas & Sepideh Stewart

Authors
  1. Michael O. J. Thomas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sepideh Stewart
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael O. J. Thomas.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Thomas, M.O.J., Stewart, S. Eigenvalues and eigenvectors: embodied, symbolic and formal thinking. Math Ed Res J 23, 275–296 (2011). https://doi.org/10.1007/s13394-011-0016-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13394-011-0016-1

Keywords

Search

Navigation