Advertisement

Stretch Directions and Stretch Factors: A Sequence Intended to Support Guided Reinvention of Eigenvector and Eigenvalue

  • David PlaxcoEmail author
  • Michelle Zandieh
  • Megan Wawro
Chapter
Part of the ICME-13 Monographs book series (ICME13Mo)

Abstract

In this chapter, we document the reasoning students exhibited when engaged in an instructional sequence designed to support student development of notions of eigenvectors, eigenvalues, and the characteristic polynomial. Rooted in the curriculum design theory of Realistic Mathematics Education (RME; Gravemeijer, 1999), the sequence builds on student solution strategies from each problem to the next. Students’ used their knowledge of how matrix multiplication transforms space to engage in problems involving stretch factors and stretch directions. In working through these problems students reinvented general strategies for determining eigenvectors, eigenvalues, and the characteristic polynomial.

Keywords

Linear algebra Eigenvector Eigenvalue Realistic mathematics education Inquiry oriented curriculum 

References

  1. 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, 1–21. https://doi.org//10.1080/0020739X.2016.1276225.
  2. Freudenthal, H. (1991). Revisiting mathematics education. Dordrecht: Kluwer Academic Publishers.Google Scholar
  3. Gol Tabaghi, S., & Sinclair, N. (2013). Using dynamic geometry software to explore eigenvectors: The emergence of dynamic-synthetic-geometric thinking. Technology, Knowledge and Learning, 18(3), 149–164.Google Scholar
  4. Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1, 155–177.Google Scholar
  5. Henderson, F., Rasmussen, C., Zandieh, M., Wawro, M., & Sweeney, G. (2010). Symbol sense in linear algebra: A start toward eigen theory. Proceedings of the 13th Annual Conference on Research in Undergraduate Mathematics Education. Raleigh, N.C. Retrieved from: http://sigmaa.maa.org/rume/crume2010.
  6. Larson, C., Zandieh, M, Rasmussen, C., & Henderson, F. (2009). Student Interpretations of the Equal Sign in Matrix Equations: The Case of Ax = 2x. Proceedings for the Twelfth Conference On Research In Undergraduate Mathematics Education. Google Scholar
  7. Larson, C., & Zandieh, M. (2013). Three interpretations of the matrix equation Ax = b. For the Learning of Mathematics, 33(2), 11–17.Google Scholar
  8. Rasmussen, C., Zandieh, M., & Wawro, M. (2009). How do you know which way the arrows go? The emergence and brokering of a classroom mathematics practice. In W.M. Roth (Ed.), Mathematical representation at the interface of body and culture (pp. 171–218). Charlotte, NC: Information Age Publishing.Google Scholar
  9. Salgado, H., & Trigueros, M. (2015). Teaching eigenvalues and eigenvectors using models and APOS Theory. The Journal of Mathematical Behavior, 39, 100–120.Google Scholar
  10. Sinclair, N., & Gol Tabaghi, S. (2010). Drawing space: Mathematicians’ kinetic conceptions of eigenvectors. Educational Studies in Mathematics, 74, 223–240.Google Scholar
  11. Stewart, S. & Thomas, M. O. J. (2006). Process-object difficulties in linear algebra: Eigenvalues and eigenvectors. In Novotná, J., Moraová, H., Krátká, M. & Stehlíková, N. (Eds.), Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 5, pp. 185–192). Prague: PME.Google Scholar
  12. Thomas, M. O. J., & Stewart, S. (2011). Eigenvalues and eigenvectors: Embodied, symbolic and formal thinking. Mathematics Education Research Journal, 23(3), 275–296.Google Scholar
  13. Wawro, M., Larson, C., Zandieh, M., & Rasmussen, C. (2012). A hypothetical collective progression for conceptualizing matrices as linear transformations. In S. Brown, S. Larsen, K. Marrongelle, and M. Oehrtman (Eds.), Proceedings of the 15th Annual Conference on Research in Undergraduate Mathematics Education (pp. 1-465–1-479), Portland, OR.Google Scholar
  14. Wawro, M., Rasmussen, C., Zandieh, M., & Larson, C. (2013). Design research within undergraduate mathematics education: An example from introductory linear algebra. In T. Plomp & N. Nieveen (Eds.), Educational design research—Part B: Illustrative cases (pp. 905–925). Enschede: SLO.Google Scholar
  15. Wawro, M., Rasmussen, C., Zandieh, M., Sweeney, G. F., & Larson, C. (2012). An inquiry-oriented approach to span and linear independence: The case of the magic carpet ride sequence. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 22(8), 577–599. https://doi.org//10.1080/10511970.2012.667516.
  16. Zandieh, M., Wawro, M., & Rasmussen, C. (2017). An Example of Inquiry in Linear Algebra: The Roles of Symbolizing and Brokering, PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 27(1), 96–124. https://doi.org//10.1080/10511970.2016.1199618.

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.Clayton State UniversityMorrowUSA
  2. 2.Arizona State UniversityTempeUSA
  3. 3.Virginia TechBlacksburgUSA

Personalised recommendations