Use of models in the teaching of linear algebra. (English) Zbl 1185.00018
Summary: We report results on an approach to teaching linear algebra using models. In particular we are interested in analyzing the use of two theories of mathematics education, namely, Models and Modeling and APOS in the design of a teaching sequence that starts with the proposal of a “real life” decision making problem to the students. We briefly illustrate the possibilities of this methodology through the analysis and description of our classroom experience on a problem related to traffic flow that elicits the use of a system of linear equations and different parameterizations of this system to answer questions on traffic control. We describe cycles of students’ work on the problem and discuss the advantages of this approach in terms of students’ learning and the possibilities of extending it to other problems and linear algebra concepts.
MSC:
| 00A35 | Methodology of mathematics |
| 97H60 | Linear algebra (educational aspects) |
| 97M10 | Modeling and interdisciplinarity (aspects of mathematics education) |
References:
| [1] | J.C. Barbosa, Mathematical modeling and parallel discussions, in: Proceedings of the Congress of the European Society for Research in Mathematics Education, 5. Larnaca: European Society for Research in Mathematics Education, 2007.; J.C. Barbosa, Mathematical modeling and parallel discussions, in: Proceedings of the Congress of the European Society for Research in Mathematics Education, 5. Larnaca: European Society for Research in Mathematics Education, 2007. |
| [2] | (Blum, W.; Galbraith, P. L.; Henn, H. W.; Niss, M., Modelling and Applications in Mathematics Education: The 14th ICMI study. Modelling and Applications in Mathematics Education: The 14th ICMI study, New ICMI Study Series, vol. 10 (2007), Springer: Springer New York) · Zbl 1105.97301 |
| [3] | English, L., Mathematical modeling with young learners, (Lamon, J.; Parker, W. A.; Houston, S. K., Mathematical Modelling: A Way of Life, ICTMA11 (2003), Horwood Publishing: Horwood Publishing Chichester), 3-17 |
| [4] | Kaiser, G.; Blomhoj, M.; Sriraman, B., Towards a didactical theory for mathematical modeling, ZDM, 38, 2, 82-85 (2006) |
| [5] | G. Stillman, P. Galbraith, J. Brown, I. Edwards, A framework for success in implementing mathematical modelling in the secondary classroom, Mathematics: essential research, essential practice, in: J. Watson, K. Beswick (Eds.), Proceedings of the 30th Annual Conference of the Mathematics Education Research Group of Australasia. vol. 2, 2007, pp. 688-690.; G. Stillman, P. Galbraith, J. Brown, I. Edwards, A framework for success in implementing mathematical modelling in the secondary classroom, Mathematics: essential research, essential practice, in: J. Watson, K. Beswick (Eds.), Proceedings of the 30th Annual Conference of the Mathematics Education Research Group of Australasia. vol. 2, 2007, pp. 688-690. |
| [6] | Zbiek, R. M.; Conner, A., Beyond motivation: exploring mathematical modeling as a context for deepening students’ understandings of curricular mathematics, Ed. Stud. Math., 63, 1, 89-112 (2006) |
| [7] | Lesh, R.; Doerr, H., (Hillsdale, B., Beyond Constructivist: A Models & Modelling Perspective on Mathematics Teaching, Learning, and Problems Solving (2003), Lawrence Erlbaum Associates: Lawrence Erlbaum Associates NH) · Zbl 1013.00006 |
| [8] | Lesh, R.; English, L., Trends in the evolution of models & modeling perspectives on mathematical learning and problem solving, ZDM Analyses, 37, 6, 487-489 (2005) |
| [9] | Sriraman, B.; Lesh, R., Beyond Traditional conceptions of modelling, Zentralblatt fuer Didaktik der Mathematik, 38, 3, 247-254 (2006) |
| [10] | Bosch, M.; García, F. J.; Gascón, J.; Ruiz Higueras, L., La modelización matemática y el problema de la articulación de la matemática escolar, Una propuesta desde la teoría antropológica de lo didáctico, Educación Matemática, 18, 2, 37-74 (2006), Editorial Santillana, México |
| [11] | M. Carlson, S. Larsen, R. Lesh, Integrating a Models and Modeling Perspective with Existing Research and Practice, 2005, pp. 465-478 (Chapter 25).; M. Carlson, S. Larsen, R. Lesh, Integrating a Models and Modeling Perspective with Existing Research and Practice, 2005, pp. 465-478 (Chapter 25). |
| [12] | C. Larson, J. Nelipovich, C. Rasmussen, M. Smith, M. Zandieh, Modeling perspectives in linear algebra, in: Electronic Proceedings for the Tenth Special Interest Group of the Mathematical Association of America on Research in Undergraduate Mathematics Education, 2007. <http://www.rume.org/crume2007/eproc.html>; C. Larson, J. Nelipovich, C. Rasmussen, M. Smith, M. Zandieh, Modeling perspectives in linear algebra, in: Electronic Proceedings for the Tenth Special Interest Group of the Mathematical Association of America on Research in Undergraduate Mathematics Education, 2007. <http://www.rume.org/crume2007/eproc.html> |
| [13] | Toumasis, C., A mathematical diet model, Teach. Math. Appl., 23, 4, 165-171 (2004) |
| [14] | M. Trigueros, Modeling in a dynamical system course, in: Electronic Proceedings for the Eleventh Special Interest Group of the Mathematical Association of America on Research in Undergraduate Mathematics Education, 2008. <http://www.rume.org/crume2008/eproc.html>; M. Trigueros, Modeling in a dynamical system course, in: Electronic Proceedings for the Eleventh Special Interest Group of the Mathematical Association of America on Research in Undergraduate Mathematics Education, 2008. <http://www.rume.org/crume2008/eproc.html> |
| [15] | D. Carlson, C.R. Johnson, D.C. Lay, A.D. Porter, A.E. Watkins, W. Watkins (Eds.), Resources for Teaching Linear Algebra, MAA Notes, vol. 42, 1997.; D. Carlson, C.R. Johnson, D.C. Lay, A.D. Porter, A.E. Watkins, W. Watkins (Eds.), Resources for Teaching Linear Algebra, MAA Notes, vol. 42, 1997. · Zbl 0868.15004 |
| [16] | Sierpinska, A., On some aspects of students thinking in linear algebra, (Dorier, J.-L., On The Teaching Linear Algebra (2000), Kluwer Academic Publishers), 209-246 |
| [17] | A. Sierpinska, A. Nnadozie, A. Oktac, A study of relationships between theoretical thinking and high achievement in Linear Algebra Concordia University: Montreal, 2002. <http://alcor.concordia.ca/ sierp/downloadpapers.html>; A. Sierpinska, A. Nnadozie, A. Oktac, A study of relationships between theoretical thinking and high achievement in Linear Algebra Concordia University: Montreal, 2002. <http://alcor.concordia.ca/ sierp/downloadpapers.html> |
| [18] | E. Dubinsky, Some thoughts on a first course in linear algebra at the college level in resources for teaching linear algebra, in: D. Carlson, C.R. Johnson, D.C. Lay, R.D. Porter, A. Watkins (Eds.), MAA Notes, vol. 42, 1997, pp. 85-105.; E. Dubinsky, Some thoughts on a first course in linear algebra at the college level in resources for teaching linear algebra, in: D. Carlson, C.R. Johnson, D.C. Lay, R.D. Porter, A. Watkins (Eds.), MAA Notes, vol. 42, 1997, pp. 85-105. |
| [19] | M. Trigueros, A. Oktac, L. Manzanero, Understanding of system of equations in algebra, in: Proceedings of the Congress of the European Society for Research in Mathematics Education, 5. Larnaca: European Society for Research in Mathematics Education, 2007.; M. Trigueros, A. Oktac, L. Manzanero, Understanding of system of equations in algebra, in: Proceedings of the Congress of the European Society for Research in Mathematics Education, 5. Larnaca: European Society for Research in Mathematics Education, 2007. |
| [20] | L. Manzanero, Sistemas de ecuaciones lineales: una perspectiva desde la teoría APOE, MSc. Thesis, Departamento de Matemática Educativa. CINVESTAV, Mexico, 2007.; L. Manzanero, Sistemas de ecuaciones lineales: una perspectiva desde la teoría APOE, MSc. Thesis, Departamento de Matemática Educativa. CINVESTAV, Mexico, 2007. |
| [21] | Dubinsky, E., The constructive aspects of reflective abstraction in advanced mathematics, (Steffe, L. P., Epistemological Foundations of Mathematical Experiences (1991), Springer-Verlag: Springer-Verlag New York) |
| [22] | E. Dubinsky, M. McDonald, A constructivist theory of learning in undergrad mathematics education research, in: D. Holton et al. (Eds.), The Teaching and Learning of Mathematics at University Level: An ICMI Study, Kluwer Academic Publishers, 2001, pp. 273-280.; E. Dubinsky, M. McDonald, A constructivist theory of learning in undergrad mathematics education research, in: D. Holton et al. (Eds.), The Teaching and Learning of Mathematics at University Level: An ICMI Study, Kluwer Academic Publishers, 2001, pp. 273-280. |
| [23] | Baker, B.; Cooley, B.; Trigueros, M., A calculus graphing schema, J. Res. Math. Ed., 31, 5, 557-578 (2000) |
| [24] | Cooley, L.; Trigueros, M.; Baker, B., Schema thematization: a framework and an example, J. Res. Math. Ed., 38, 4, 370-392 (2007) |
| [25] | K. Weller, A. Montgomery, J. Clark, J. Cottrill, M. Trigueros, I. Arnon, E. Dubinsky, Learning Linear Algebra with ISETL, 2002. <http://pc75666.math.cwu.edu/ montgomery/scholar/2002/0731-b-llawi.pdf><http://homepages.ohiodominican.edu/ cottrilj/datastore/linear-alg/LLAWI-P3.pdf>; K. Weller, A. Montgomery, J. Clark, J. Cottrill, M. Trigueros, I. Arnon, E. Dubinsky, Learning Linear Algebra with ISETL, 2002. <http://pc75666.math.cwu.edu/ montgomery/scholar/2002/0731-b-llawi.pdf><http://homepages.ohiodominican.edu/ cottrilj/datastore/linear-alg/LLAWI-P3.pdf> |
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