ZDM

, Volume 46, Issue 3, pp 389–406 | Cite as

Student reasoning about the invertible matrix theorem in linear algebra

Original Article

Abstract

I report on how a linear algebra classroom community reasoned about the invertible matrix theorem (IMT) over time. The IMT is a core theorem that connects many fundamental concepts through the notion of equivalency. As the semester progressed, the class developed the IMT in an emergent fashion. As such, the various equivalences took form and developed meaning as students came to reason about the ways in which key ideas involved were connected. Microgenetic and ontogenetic analyses (Saxe in J Learn Sci 11(2–3):275–300, 2002) framed the structure of the investigation. The results focus on shifts in the mathematical content of argumentation over time and the centrality of span and linear independence in classroom argumentation.

Keywords

Linear algebra Adjacency matrices Collective activity 

References

  1. Aberdein, A. (2009). Mathematics and argumentation. Foundations of Science, 14(1–2), 1–8.CrossRefGoogle Scholar
  2. Cobb, P. (1999). Individual and collective mathematical development: The case of statistical data analysis. Mathematical Thinking and Learning, 1(1), 5–43.CrossRefGoogle Scholar
  3. Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist, 31, 175–190.CrossRefGoogle Scholar
  4. Cole, R., Becker, N., Towns, M., Sweeney, G., Wawro, M., & Rasmussen, C. (2012). Adapting a methodology from mathematics education research to chemistry education research: Documenting collective activity. International Journal of Science and Mathematics Education, 10, 193–211.CrossRefGoogle Scholar
  5. Denzin, N. K. (1978). The research act: A theoretical introduction to research methods. New York: McGraw-Hill.Google Scholar
  6. Dreyfus, T., Hillel, J., & Sierpinska, A. (1998). Cabri-based linear algebra: Transformations. In I. Schwank (Ed.), Proceedings of the first conference of the European society for research in mathematics education (Vol. I + II, pp. 209–221). Osnabrueck: Forschungsinstitut fuer Mathematikdidaktik.Google Scholar
  7. Elbers, E. (2003). Classroom interaction as reflection: Learning and teaching mathematics in a community of inquiry. Educational Studies in Mathematics, 54(1), 77–99.CrossRefGoogle Scholar
  8. Ercikan, K., & Roth, W. M. (2006). What good is polarizing research into qualitative and quantitative? Educational Researcher, 35(5), 14–23.CrossRefGoogle Scholar
  9. Freudenthal, H. (1991). Revisiting mathematics education. Dordrecht: Kluwer.Google Scholar
  10. Harel, G. (1989). Learning and teaching linear algebra: Difficulties and an alternative approach to visualizing concepts and processes. Focus on Learning Problems in Mathematics, 11(1), 139–148.Google Scholar
  11. Harel, G., & Brown, S. (2008). Mathematical induction: Cognitive and instructional considerations. In M. Carlson & C. Rasmussen (Eds.), Making the connection: Research and practice in undergraduate mathematics (pp. 111–123). Washington: Mathematical Association of America.CrossRefGoogle Scholar
  12. Hillel, J. (2000). Modes of description and the problem of representation in linear algebra. In J.-L. Dorier (Ed.), On the teaching of linear algebra (pp. 191–207). Dordrecht: Kluwer.Google Scholar
  13. Hoyles, C. (1997). The curricular shaping of students’ approaches to proof. For the Learning of Mathematics, 17(1), 7–16.Google Scholar
  14. Larson, C., Zandieh, M., & Rasmussen, C. (2008). A trip through eigen-land: Where most roads lead to the direction associated with the largest eigenvalue. In Proceedings of the eleventh special interest group of the mathematical association of America on research in undergraduate mathematics education, San Diego, CA.Google Scholar
  15. Lay, D. C. (2003). Linear algebra and its applications (3rd ed.). Reading: Addison-Wesley.Google Scholar
  16. Portnoy, N., Grundmeier, T. A., & Graham, K. J. (2006). Students’ understanding of mathematical objects in the context of transformational geometry: Implications for constructing and understanding proofs. Journal of Mathematical Behavior, 25(3), 196–207.CrossRefGoogle Scholar
  17. Possani, E., Trigueros, M., Preciado, J. G., & Lozano, M. D. (2010). Use of models in the teaching of linear algebra. Linear Algebra and its Applications, 432(8), 2125–2140.CrossRefGoogle Scholar
  18. Rasmussen, C., & Stephan, M. (2008). A methodology for documenting collective activity. In A. E. Kelly, R. A. Lesh, & J. Y. Baek (Eds.), Handbook of innovative design research in science, technology, engineering, mathematics (STEM) education (pp. 195–215). New York: Taylor and Francis.Google Scholar
  19. Saxe, G. B. (2002). Children’s developing mathematics in collective practices: A framework for analysis. Journal of the Learning Sciences, 11(2–3), 275–300.CrossRefGoogle Scholar
  20. Saxe, G. B., & Esmonde, I. (2005). Studying cognition in flux: A historical treatment of fu in the shifting structure of Oksapmin mathematics. Mind, Culture, and Activity, 12(3–4), 171–225.Google Scholar
  21. Saxe, G., Gearhart, M., Shaughnessy, M., Earnest, D., Cremer, S., Sitabkhan, Y., et al. (2009). A methodological framework and empirical techniques for studying the travel of ideas in classroom communities. In B. Schwarz, T. Dreyfus, & R. Hershkowitz (Eds.), Transformation of knowledge through classroom interaction (pp. 203–222). New York: Routledge.Google Scholar
  22. Selden, A., & Selden, J. (1987). Errors and misconceptions in college level theorem proving. In J. D. Novake (Ed.), Proceedings of the second international seminar on misconceptions and educational strategies in science and mathematics (Vol. III, pp. 457–470). New York: Cornell University.Google Scholar
  23. Selinski, N., Rasmussen, C., Wawro, M., & Zandieh, M. (2013). A methodology for using adjacency matrices to analyze the connections students make between concepts in linear algebra. Manuscript submitted for publication.Google Scholar
  24. Sfard, A. (2007). When the rules of discourse change but nobody tells you: making sense of mathematics learning from a commognitive standpoint. The Journal of the Learning Sciences, 16(4), 565–613.CrossRefGoogle Scholar
  25. Sierpinska, A. (2000). On some aspects of students’ thinking in linear algebra. In J.-L. Dorier (Ed.), On the teaching of linear algebra (pp. 209–246). Dordrecht: Kluwer.Google Scholar
  26. Stephan, M., Cobb, C., & Gravemeijer, K. (2003). Coordinating social and individual analyses: Learning as participation in mathematical practices. Journal for Research in Mathematics Education. Monograph, 12, 67–102.Google Scholar
  27. Stephan, M., & Rasmussen, C. (2002). Classroom mathematical practices in differential equations. The Journal of Mathematical Behavior, 21(4), 459–490.CrossRefGoogle Scholar
  28. Strom, D., Kemeny, V., Lehrer, R., & Forman, E. (2001). Visualizing the emergent structure of children’s mathematical argument. Cognitive Science, 25(5), 733–773.CrossRefGoogle Scholar
  29. Tiberghien, A., & Malkoun, L. (2009). The construction of physics knowledge in the classroom from different perspectives: The classroom as a community and the students as individuals. In B. Schwarz, T. Dreyfus, & R. Hershkowitz (Eds.), Transformation of knowledge through classroom interaction (pp. 42–55). New York: Routledge.Google Scholar
  30. Toulmin, S. (1969). The uses of argument. Cambridge: Cambridge University Press.Google Scholar
  31. Wawro, M., Rasmussen, C., Zandieh, M., Sweeney, G., & 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.CrossRefGoogle Scholar
  32. Wawro, M., Sweeney, G., & Rabin, J. M. (2011). Subspace in linear algebra: Investigating students’ concept images and interactions with the formal definition. Educational Studies in Mathematics, 78(1), 1–19.CrossRefGoogle Scholar
  33. Weber, K. (2001). Student difficulty in constructing proofs: the need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101–119.CrossRefGoogle Scholar

Copyright information

© FIZ Karlsruhe 2014

Authors and Affiliations

  1. 1.Department of MathematicsVirginia TechBlacksburgUSA

Personalised recommendations