Local Instruction Theories at the University Level: An Example in a Linear Algebra Course

Andrea Cárcamo; Claudio Fuentealba; Diego Garzón

doi:10.29333/ejmste/108648

Full Text (PDF)

Andrea Cárcamo ¹ ^* , Claudio Fuentealba ¹, Diego Garzón ²

More Detail

¹ Facultad de Ciencias de la Ingeniería, Universidad Austral de Chile, CHILE
² Instituto de Educación y Pedagogía, Universidad del Valle, COLOMBIA
^* Corresponding Author

Abstract

In order to promote the design of innovative instructional activities at the Linear Algebra, we perform a design-based research project to explore how to teach Linear Algebra at the university level. In this article, we present the results of three cycles of a teaching experiment that we carried out to design, try out, and improve a local instruction theory (LIT) on the teaching of the concepts of spanning set and span in Linear Algebra with first-year engineering students. In a retrospective analysis, we looked for patterns in the data set of all the experiments, and we identified key learning moment of the students. Based on these patterns, we formulated a LIT to support the construction of the concepts of spanning set and span.

Keywords

emergent models
instructional design
linear algebra
local instruction theory

License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

https://doi.org/10.29333/ejmste/108648

EURASIA J Math Sci Tech Ed, 2019 - Volume 15 Issue 12, Article No: em1781

Publication date: 05 Jul 2019

Article Views: 1672

Article Downloads: 902

Open Access References How to cite this article

References

Bakker, A., & Van Eerde, H. A. A. (2015). An introduction to design based research with an example from statistics education. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Doing qualitative research: Methodology and methods in mathematics education (pp. 429–466). New York, USA: Springer. https://doi.org/10.1007/978-94-017-9181-6_16.
Ball, G., Stephenson, B., Smith, G., Wood, L., Coupland, M., & Crawford, K. (1998). Creating a diversity of mathematical experiences for tertiary students. International Journal of Mathematical Education in Science and Technology, 29(6), 827-841. https://doi.org/10.1080/0020739980290605.
Blum W., & Leiss D. (2007). How do students and teachers deal with modelling problems? In C. Haines, P. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modelling (ICTMA12): Education, Engineering and Economics (pp. 222-231). Chichester, UK: Horwood Publishing. https://doi.org/10.1533/9780857099419.5.221.
Cárcamo, A., Fortuny J., & Gómez, J. (2017). Mathematical modelling and the learning trajectory: tools to support the teaching of linear algebra. International Journal of Mathematical Education in Science and Technology, 48(3),338-352. https://doi.org/10.1080/0020739X.2016.1241436.
Cárcamo, A., Fortuny, J., & Fuentealba, C. (2018). The emergent models in linear algebra: an example with spanning set and span. Teaching Mathematics and its Applications: An International Journal of the IMA, 37(4), 202-217. https://doi.org/10.1093/teamat/hrx015.
Cárcamo, A., Gómez, J., & Fortuny, J. (2016). Mathematical Modelling in Engineering: A Proposal to Introduce Linear Algebra Concepts. Journal of Technology and Science Education, 6(1), 62-70. https://doi.org/10.3926/jotse.177.
Carlson, D. (1997). Teaching linear algebra: Must the fog always roll in? In D. Carlson, C. R. Johnson, D. C. Lay, A. D. Porter, A. Watkins & W. Watkins (Eds.), Resources for Teaching Linear Algebra, MAA Notes (Vol. 42, pp. 39-51). Washington, USA: Mathematical Association of America. https://doi.org/10.1080/03081089708818529.
Carlson, D. (2004). The Teaching and Learning of Tertiary Algebra. In K. Stacey, H. Chick, M. Kendal, B. Barton, & J. C. e Silva (Eds.), The Future of the Teaching and Learning of Algebra, the 12th ICMI Study (Vol. 8, pp. 293–309). Dordrecht, Netherlands: Kluwer Academic Publishers.
Cobb, P., & Gravemeijer, K. (2008). Experimenting to support and understand learning processes. In A.E Kelly, R.A. Lesh, & J.Y. Baek (Eds.). Handbook of design research methods in education: Innovations in science, technology, engineering, and mathematics learning and teaching (pp. 68-95). Mahwah, USA: Lawrence Erlbaum Associates.
de Beer, H., Gravemeijer, K., & van Eijck, M. (2017). A proposed local instruction theory for teaching instantaneous speed in grade five. The Mathematics Enthusiast, 14(1), 435-468.
Dierdorp, A., Bakker, A., Eijkelhof, H., & van Maanen, J. (2011). Authentic practices as contexts for learning to draw inferences beyond correlated data. Mathematical Thinking and Learning, 13(1-2), 132-151. https://doi.org/10.1080/10986065.2011.538294.
Doorman, M., & Gravemeijer, K. (2009). Emergent modeling: discrete graphs to support the understanding of change and velocity. ZDM Mathematics Education, 41(1-2), 199-211. https://doi.org/10.1007/s11858-008-0130-z.
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. 255–274). Dordrecht, Netherlands: Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47231-7_24.
Drijvers, P. (2003). Learning algebra in a computer algebra environment: Design research on the understanding of the concept of parameter (Doctoral dissertation). Utrecht University, Netherlands.
Galbraith, P. (2007). Authenticity and goals – Overview. In W. Blum, P. L. Gal¬braith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education: The 14th ICMI study (pp. 181–184). New York, USA: Springer. https://doi.org/10.1007/978-0-387-29822-1_17.
Gravemeijer, K. (1994). Developing Realistic Mathematics Education: Ontwikkelen Van Realistisch Reken/wiskundeonderwijs. CD-[beta] Press.
Gravemeijer, K. (1998). Developmental research as a research method. In J. Kilpatrick &A. Sierpinska (Eds.), Mathematics education as a research domain: A search for identity: An ICMI study (Vol. 2, pp. 277–295). Dordrecht, Netherlands: Kluwer Academic. https://doi.org/10.1007/978-94-011-5196-2_3.
Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1(2), 155–177. https://doi.org/10.1207/s15327833mtl0102_4.
Gravemeijer, K. (2002, july). Emergent modeling as the basis for an instructional sequence on data analysis. Paper presented at the 6th International Conference on Teaching Statistics (ICOTS-6), Cape Town, South Africa.
Gravemeijer, K. (2004a). Local instruction theories as means of support for teachers in reform mathematics education. Mathematical thinking and learning, 6(2), 105-128. https://doi.org/10.1207/s15327833mtl0602_3.
Gravemeijer, K. (2004a, July). Creating opportunities for students to reinvent mathematics. Paper presented at the 10th International Congress on Mathematical Education (ICME), Copenhagen, Denmark.
Gravemeijer, K. (2007, December). Emergent modeling and iterative processes of design and improvement in mathematics education. Paper presented at Tsukuba International Conference III (APEC), Tokyo Kanazawa & Kyoto, Japan.
Gravemeijer, K., & Cobb, P. (2013). Design research from the learning design perspective. In Plomp T., & N. Nieveen (Eds.), Educational Design research. Part A: An introduction (pp. 72-113). Enschede: Netherlands: SLO.
Gravemeijer, K., & Stephan, M. (2011). Emergent models as an instructional design heuristic. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolizing modeling and tool use in mathematics education (pp. 145–169). Dordrecht: Kluwer Academic Publishers. https://doi.org/10.1007/978-94-017-3194-2_10.
Gravemeijer, K., & van Eerde, D. (2009). Design research as a means for building a knowledge base for teachers and teaching in mathematics education. The Elementary School Journal, 109(5), 510-524. https://doi.org/10.1086/596999.
Gravemeijer, K., Cobb, P., Bowers, J., & Whitenack, J. (2000). Symbolizing, modeling and instructional design. In P. Cobb, E. Yackel, & K. McLain (Eds.), Symbolizing and communicating in mathematics classrooms (pp. 225–273). Mahwah, USA: Lawrence Erlbaum Associates.
Grossman, S. (1996). Álgebra Lineal [Linear Algebra]. México: Mc Graw Hill (5ª Edición).
Julie, C., & Mudaly, V. (2007). Mathematical modelling of social issues in school mathematics in South Africa. In W. Blum, P. L. Galbraith, H.-W. Henn, & M. Niss (Eds.), Modelling and applications in mathematics education: the 14th ICMI study (pp. 503–510). New York, USA: Springer. https://doi.org/10.1007/978-0-387-29822-1_58.
Kaiser, G., & Schwarz, B. (2010). Authentic modelling problems in mathematics education—examples and experiences. Journal für Mathematik-Didaktik, 31(1), 51-76. https://doi.org/10.1007/s13138-010-0001-3.
Kú (2012). Análisis sobre la comprensión de los conceptos Conjunto Generador y Espacio Generado desde la mirada de la teoría APOE [Analysis on the understanding of the concepts spanning set and span from the perspective of the APOS theory]. (Doctoral dissertation). Centro de Investigación y Estudios Avanzados del Instituto Politécnico Nacional, Mexico.
Kú, D., Trigueros, M., & Oktaç, A. (2008). Comprensión del concepto de base de un espacio vectorial desde el punto de vista de la teoría APOE [Understanding the basis concept of a vector space from the point of view of the APOS theory]. Revista Educación Matemática, 20(2), 65-89.
Larsen, S. P. (2013). A local instructional theory for the guided reinvention of the group and isomorphism concepts. The Journal of Mathematical Behavior, 32(4), 712-725. https://doi.org/10.1016/j.jmathb.2013.04.006.
Lay, D. (2007). Álgebra lineal y sus aplicaciones [Linear algebra and its applications]. México: Pearson (3ª edición).
Leikin, R., & Dinur, S. (2003, February). Patterns of flexibility: Teachers’ behavior in mathematical discussion. Paper presented at Third Conference of the European Society for Research in Mathematics Education, Bellaria, Italy.
Lesh, R., & Caylor, B. (2007). Introduction to the special issue: Modeling as application versus modeling as a way to create mathematics. International Journal of computers for mathematical Learning, 12(3), 173-194. https://doi.org/10.1007/s10758-007-9121-3.
Lesh, R., & English, L. D. (2005). Trends in the evolution of models y modeling perspectives on mathematical learning and problem solving. ZDM Mathematics Education, 37(6), 487-489. https://doi.org/10.1007/BF02655857.
Lewis, J. M., & Blunk, M. L. (2012). Reading between the lines: Teaching linear algebra. Journal of Curriculum Studies, 44(4), 515-536. https://doi.org/10.1080/00220272.2012.716975.
Lipschutz, S. (1992). Álgebra Lineal [Linear Algebra]. Spain: McGraw Hill (2nd Ed.).
Molina, M. (2006). Desarrollo de pensamiento relacional y comprensión del signo igual por alumnos de tercero de educación primaria [Development of relational thinking and understanding of the equal sign by third-year primary school students] (Doctoral dissertation). Universidad de Granada, Spain.
Nardi, E. (1997). El encuentro del matemático principiante con la abstracción matemática: Una imagen conceptual de los conjuntos generadores en el análisis vectorial [The encounter of the beginner mathematician with the mathematical abstraction: A conceptual image of the spanning sets in the vector analysis]. Educación Matemática, 9(1), 47-60.
Nickerson, S. D., & Whitacre, I. (2010). A local instruction theory for the development of number sense. Mathematical Thinking and Learning, 12(3), 227–252. https://doi.org/10.1080/10986061003689618.
Niss, M. (2012). Models and modelling in mathematics education. Newsletter of the European Mathematical Society, 86, 49-52.
Plomp, T. (2013). Educational design research: An introduction. In T. Plomp, & N. Nieveen (Eds.), Educational design research – Part A: An introduction (pp. 10-51). Enschede, Netherlands: SLO.
Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26(2), 114-145. https://doi.org/10.2307/749205.
Stewart, S., & Thomas, M. O. (2010). Student learning of basis, span and linear independence in linear algebra. International Journal of Mathematical Educa¬tion in Science and Technology, 41(2), 173-188. https://doi.org/10.1080/00207390903399620.
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, 22(8), 577-599. https://doi.org/10.1080/10511970.2012.667516.

How to cite this article

APA

Cárcamo, A., Fuentealba, C., & Garzón, D. (2019). Local Instruction Theories at the University Level: An Example in a Linear Algebra Course. Eurasia Journal of Mathematics, Science and Technology Education, 15(12), em1781. https://doi.org/10.29333/ejmste/108648

Vancouver

Cárcamo A, Fuentealba C, Garzón D. Local Instruction Theories at the University Level: An Example in a Linear Algebra Course. EURASIA J Math Sci Tech Ed. 2019;15(12):em1781. https://doi.org/10.29333/ejmste/108648

AMA

Cárcamo A, Fuentealba C, Garzón D. Local Instruction Theories at the University Level: An Example in a Linear Algebra Course. EURASIA J Math Sci Tech Ed. 2019;15(12), em1781. https://doi.org/10.29333/ejmste/108648

Chicago

Cárcamo, Andrea, Claudio Fuentealba, and Diego Garzón. "Local Instruction Theories at the University Level: An Example in a Linear Algebra Course". Eurasia Journal of Mathematics, Science and Technology Education 2019 15 no. 12 (2019): em1781. https://doi.org/10.29333/ejmste/108648

Harvard

Cárcamo, A., Fuentealba, C., and Garzón, D. (2019). Local Instruction Theories at the University Level: An Example in a Linear Algebra Course. Eurasia Journal of Mathematics, Science and Technology Education, 15(12), em1781. https://doi.org/10.29333/ejmste/108648

MLA

Cárcamo, Andrea et al. "Local Instruction Theories at the University Level: An Example in a Linear Algebra Course". Eurasia Journal of Mathematics, Science and Technology Education, vol. 15, no. 12, 2019, em1781. https://doi.org/10.29333/ejmste/108648