# Learning Linear Algebra Using Models and Conceptual Activities

- 2 Citations
- 550 Downloads

## Abstract

In this chapter, an innovative approach, including challenging modeling situations and tasks sequences to introduce linear algebra concepts is presented. The teaching approach is based on Action, Process, Object, Schema (APOS) Theory. The experience includes the use of several modeling situations designed to introduce some of the main linear algebra concepts. Results obtained in several experiences involving different concepts are presented focusing on crucial moments where students develop new strategies, and on success in terms of student’s understanding of linear algebra concepts. Conclusions related to the success of the use of the approach in promoting student’s understanding are discussed.

## Keywords

APOS theory Schemas Systems of equations Linear independence Eigenvectors## Notes

### Acknowledgements

Work funded by Asociación Mexicana de Cultura A.C. and ITAM.

## References

- Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Roa Fuentes, S., Trigueros, M., & Weller, K. (2014).
*APOS Theory: A framework for research and curriculum development in*mathematics education. New York: Springer Verlag.Google Scholar - Bardini, C., & Stacey, K. (2006). Students’ conceptions of m and c: How to tune a linear function. In J. Novotna, H. Moraova, M. Kratka & N. Stehlikova (Eds.),
*Procedings of the 30th conference of the international group for the psychology of mathematics education*(Vol. 2, pp. 113–120). Prague, Czech Republic: Charles University.Google Scholar - Dogan, H. (2010). Linear Algebra Students’ Modes of Reasoning: Geometric Representations.
*Linear Algebra and Its Applications*, 432, 2141–2159.Google Scholar - Dogan-Dunlap, H. (2006). Lack of Set Theory-Relevant Prerequisite Knowledge.
*International Journal of Mathematical Education in Science and Technology*(IJMEST).*37*(4), 401–410.Google Scholar - Dorier, J. L., & Sierpinska, A. (2001).
*Research into the teaching and learning of linear algebra*. In D. Holton, M. Artigue, U. Krichgraber, J. Hillel, M. Niss, & A. Schoenfeld (Eds.), The Teaching and Learning of Mathematics at University Level: An ICMI Study (pp. 255–273). Dordrecht, Netherlands: Kluwer Academic Publishers.Google Scholar - Gol, S. (2012). Dynamic geometric representation of eigenvector. In S. Brown, S. Larsen, K. Marrongelle, & M. Oehrtman (Eds.),
*Proceedings of the 15th annual conference on research in undergraduate mathematics education*(pp. 53–58). Portland, Oregon.Google Scholar - Gueudet, G. (2004). Should we teach linear algebra through geometry?
*Linear Algebra and its Applications*, 379, 491–501.Google Scholar - Harel, G. (1999). Students’ understanding of proofs: A historical analysis and implications for the teaching of geometry and linear algebra.
*Linear Algebra and Its Applications*, 302–303, 601–613.Google Scholar - Larson, C., Rasmussen, C., Zandieh, M., Smith, M., & Nelipovich, J. (2007). Modeling perspectives in linear algebra: a look at eigen-thinking. 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. Paper presented at the 11 Research in Undergraduate Mathematics Education Conference, San Diego https://www.researchgate.net/profile/Chris_Rasmussen/publication/253936179.
- Malisani, E., & Spagnolo, F. (2009). From arithmetical thought to algebraic thought: The role of the “variable”.
*Educational Studies in Mathematics*, 71, 19–41.Google Scholar - Maracci, M. (2008). Combining different theoretical perspectives for analyzing students’ difficulties in vector spaces theory.
*ZDM*, 40, 265–276.Google Scholar - Oktaç, A., & Trigueros, M. (2010). ¿Cómo se aprenden los conceptos de álgebra lineal?
*Revista Latinoamericana de Investigación en Matemática Educativa*. 13, 373–385.Google Scholar - Possani, E., Trigueros, M., Preciado, G., & Lozano, M. D. (2010). Use of models in the Teaching of Linear Algebra.
*Linear Algebra and its Applications. 432*(8), 2125–2140.Google Scholar - Salgado, H., & Trigueros, M. (2014). Una experiencia de enseñanza de los valores, vectores y espacios propios basada en la Teoría APOE. Educación Matemática 26, 75–107.Google Scholar
- Sierpinska, A. (2000). On some aspects of students’ thinking in Linear Algebra. In J. Dorier (Ed.),
*On the Teaching of Linear Algebra*. 209–246.Google Scholar - Stewart, S., & Thomas, M. (2007). Eigenvalues and eigenvectors: formal, symbolic, and embodied thinking.
*The 10th CRUME (RUME),*275–296.Google Scholar - Thomas, M., & Stewart, S. (2011). Eigenvalues and eigenvectors: embodied, symbolic and formal thinking.
*Mathematics Education Research Group of Australasia.*23, 275–296.Google Scholar - Trigueros, M. (2014). Vínculo entre la modelación y el uso de representaciones en la comprensión de los conceptos de ecuación diferencial de primer orden y de solución
*. Educación Matemática*. 25 años (número especial) 207–226.Google Scholar - Trigueros, M. & Jacobs, S. (2008). On Developing a Rich Conception of Variable. In M. P. Carlson & C. Rasmussen (Eds.)
*Making the Connection: Research and Practice in Undergraduate Mathematics*. MAA Notes#73, Mathematical Association of America, pp. 3–14.Google Scholar - Trigueros, M., & Lozano, M. D. (2010). Learning linear independence through modelling. In M. F. Pinto & T. F. Kawasaki (Eds.),
*Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 4, pp. 233–240). Belo Horizonte, Brazil: PME.Google Scholar - Trigueros, M., & Possani, E. (2013). Using an economics model for teaching linear algebra.
*Linear Algebra and its Applications*. 438, pp. 1779–1792.Google Scholar - Trigueros, M., Oktaç, A., & Manzanero, L. (2007). Understanding of systems of equations in linear algebra. In
*Proceedings of the 5th CERME*, pp. 2359–2368.Google Scholar - Ursini, S., & Trigueros, M. (1997). Understanding of Different Uses of Variable: A Study with Starting College Students. Proceedings of the XXI PME International Conference, vol. 4, pp. 254–261.Google Scholar
- Wawro, M., Rasmussen, C., Zandieh, M., Larson, C., & Sweeney, G. (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.Google Scholar