On Eigenvalue And Eigenvector Perceptions of Undergraduate Pre-service Mathematics Teachers

Year 2022, Volume: 16 Issue: 1, 172 - 188, 30.06.2022

Şerife Yılmaz

https://doi.org/10.17522/balikesirnef.1076124

Cited By: 1

Abstract

Analysis of systems that can be expressed in matrices is very important in the field of application. Whether such a system works properly is determined by eigenvalues of the matrix representing the system. Eigenvalue and eigenvector concepts are taught within the scope of linear algebra course at undergraduate level. In this study, perceptions of undergraduate students who took the linear algebra course about the eigenvalue-eigenvector concepts are investigated. The research is conducted with the participation of 95 students from the Faculty of Education, Department of Mathematics Education. A scale measuring the students’ approached about eigen theory was developed. For the reliability of the scale, Kuder-Richardson 20 (KR-20) reliability analysis was done and 0,72 was obtained. To see the relationship between learning outcomes and academic achievement is used the chi-square test and descriptive analysis are made in the study. Problems arising in the perception of eigenvalue and solutions are presented.

Keywords

linear algebra, matrices, eigenvalue, eigenvector

References

• Bretscher, O. (2013). Linear algebra with applications (5th ed.), Pearson.
• Brousseau, G. (1998). La théorie des situations didactiques. La Pensée sauvage éditeur, Grenoble, France.
• Burden, Richard L., & Douglas Faires, J. (1993). Numerical analysis (8th ed.), Boston: Prindle, Weber and Schmidt.
• Carlson, D. (1993). Teaching linear algebra: must the fog always roll in? College Mathematics Journal, 24(1), 29-40.
• Dorier, J. L. (1998). The role of formalism in the teaching of the theory of vector spaces. Linear Algebra and its Applications 275-276, 141-160.
• Dorier, J. L., Robert, A., Robinet, J., & Rogalski, M. (2000). The obstacle of formalism in linear algebra. In Dorier J.L. (Ed.), On the teaching of linear algebra (pp. 85-94). The Netherlands: Kluwer Academic Publishers.
• 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 Dordrecht: Kluwer Academic Publishers.
• Dorier, J. L. (2002). Teaching linear algebra at university. In the Proceedings of the International Congress of Mathematicians, 3, 875-884, China.
• Fraenkel, J.R. & Wallen, N.E. (2008). How to design and evaluate research in education (7th ed.), New York: McGraw-Hill.
• Gol, S. (2012). Dynamic geometric representation of eigenvector. In S. Brown, S. Larsen, K. Marrongelle, & M. Oehrtman (Eds.), Proceedings of 15th annual conference on research in undergraduate Mathematics Education, 53-58, Portland, OR.
• 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, 149-164. https://doi.org/10.1007/s10758-013-9206-0
• Gueudet-Chartier, G. (2004). Should we teach linear algebra through geometry. Linear Algebra and its Applications, 379, 491-501. https://doi.org/10.1016/S0024-3795(03)00481-6
• Harel, G. (1989). Learning and teaching linear algebra: diffuculties and an alternative approach to visualizing concepts and processes. Focus on Learning Problems in Mathematics, 11(2), 139-148.
• Hillel, J., & Sierpinska, A. (1994). On one persistent mistake in linear algebra. Proceedings of the 18th conference on psychology of mathematics education, 65-72, University of Lisbon , Portugal.
• Horn, R. A., & Johnson C. R. (2013). Matrix analysis. Cambridge: Cambridge University Press.
• Kreyszig, E. (2006). Advanced engineering mathematics (9th ed.), Wiley.
• Larson, C., Rasmussen, C., Zandieh, M., Smith, M., & Nelipovich, J. (2007). Modelling perspectives in linear algebra: A look at eigen-thinking. Last accessed on August 3, 2014 from (http://www.rume.org/crume2007/papers/larson-rasmussen-zandieh-smith-nelipovich.pdf).
• 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 11th annual conference on research in undergraduate mathematics education, SIGMAA on RUME. Last accessed on August 3, 2014 from (http://sigmaa.maa.org/rume2008/Proceedings/Larson%20SHORT.pdf).
• Meyer, C. D. (2000). Matrix analysis and applied linear algebra. SIAM, Philadelphia.
• Salgado, H., & Trigueros, M. (2015). Teaching eigenvalues and eigenvectors using models and APOS theory. The Journal of Mathematical Behavior, 39, 100-120.
• Sinclair, N., & Gol Tabaghi, S. (2010). Drawing space: Mathematicians’ kinetic conceptions of eigenvectors. Educational Studies in Mathematics, 74(3), 223-240. https://doi.org/10.1007/s10649-010-9235-8.
• Stewart, S., & Thomas, M.O.J., (2011). Embodied, symbolic and formal thinking in linear algebra. International Journal of Mathematical Education in Science and Technology, 38(7), 927-937. https://doi.org/10.1080/00207390701573335.
• Szabo, F. (2000). Linear algebra: An introduction using mathematica. London: Elsevier.
• Thomas, M. & Stewart, S. (2011), Eigenvalues and eigenvectors: Embodied, symbolic and formal thinking. Mathematics Education Research Group of Australasia, 23, 275-296.