Login | Register

The Bare Necessities for Doing Undergraduate Multivariable Calculus

Title:

The Bare Necessities for Doing Undergraduate Multivariable Calculus

Brandes, Hadas (2017) The Bare Necessities for Doing Undergraduate Multivariable Calculus. Masters thesis, Concordia University.

[img]
Preview
Text (application/pdf)
Brandes_MSc_S2018.pdf - Accepted Version
Available under License Spectrum Terms of Access.
17MB

Abstract

Students in two mathematics streams at Concordia University start their programs on similar footing in terms of pre-requisite courses; their paths soon split in the two directions set by the Pure and Applied Mathematics (MATH) courses and the Major in Mathematics and Statistics (MAST) courses. In particular, likely during their first year of studies, the students set out to take a two-term arrangement of Multivariable Calculus in the form of MAST 218 – 219 and MATH 264 – 265, respectively. There is an ongoing discussion about the distinction between the MAST and MATH courses, and how it is justified. This thesis seeks to address the matter by identifying the mathematics that is essential for students to learn in order to succeed in each of these courses. We apply the Anthropological Theory of the Didactic (ATD) in order to model the knowledge to be taught and to be learned in MAST 218 and MATH 264, as decreed by the curricular documents and course assessments. The ATD describes units of mathematical knowledge in terms of a practical block (tasks to be done and techniques to accomplish them) and a theoretical block that frames and justifies the practical block. We use these notions to model the knowledge to be taught and learned in each course and reflect on the implications of the inclusion and exclusion of certain units of knowledge in the minimal core of what students need to learn. Based on these models, we infer that the learning of Multivariable Calculus in both courses follows in a tradition observed in single-variable calculus courses, whereby students develop compartmentalized units of knowledge. That is, we find that it is necessary for students in MAST 218 and MATH 264 to specialize in techniques that apply to certain routine tasks, and to this end, it suffices to learn bits and pieces of theoretical knowledge that are not unified in a mathematically-informed way. We briefly consider potential implications of such learning in the wider context of the MATH and MAST programs.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Thesis (Masters)
Authors:Brandes, Hadas
Institution:Concordia University
Degree Name:M. Sc.
Program:Mathematics
Date:1 September 2017
Thesis Supervisor(s):Hardy, Nadia
Keywords:Anthropological Theory of the Didactic Multivariable Calculus
ID Code:983041
Deposited By: HADAS BRANDES
Deposited On:16 Nov 2017 17:33
Last Modified:18 Jan 2018 17:56

References:

Barbé, J., Bosch, M., Espinoza, L., & Gascón, J. (2005). Didactic Restrictions on the Teacher’s Practice: The Case of Limits of Functions in Spanish High Schools. In Beyond the Apparent Banality of the Mathematics Classroom (pp. 235-268). Boston, MA: Springer.
Barquero, B., Bosch, M., & Gascón, J. (2008). Using research and study courses for teaching mathematical modelling at university level. In D. Pitta-Pantazi & G. Pilippou (Eds.), Proceedings of the fifth congress of the European society for research in mathematics education (pp. 2050– 2059). Larnaca: University of Cyprus.
Bergé, A. (2008). The completeness property of the set of real numbers in the transition from calculus to analysis. Educational Studies in Mathematics, 67(3), 217-235. doi:10.1007/s10649-007-9101-5
Bosch, M., Chevallard, Y., & Gascón, J. (2005). Science or magic? The use of models and theories in didactics of mathematics. Proceedings of CERME4.
Bosch, M., & Gascón, J. (2006). Twenty-five years of the didactic transposition. ICMI Bulletin, 58, 51–63.
Chevallard, Y. (1985). La transposition didactique, du savoir savant au savoir enseigné [The didactic transposition, from the learned knowledge to the taught knowledge]. Grenoble: La Pensée Sauvage.
Chevallard, Y. (1999). L’analyse des pratiques enseignantes en théorie anthropologique du didactique [The analysis of teaching practice in the anthropological theory of the didactic]. Recherches en Didactique des Mathématiques, 19, 221–266.
Chevallard, Y. (2002a). Organiser l’étude 1. Structures & function [To organise the study, 1]. In J.-L. Dorier, M. Artaud, M. Artigue, R. Berthelot, & R. Floris (Eds.), Actes de la XIe école d’été de didactique des mathématiques [Proceedings of the 11th summer school on didactics of mathematics] (pp. 3–32). Grenoble: La Pensée Sauvage.
Chevallard, Y. (2002b). Organiser l’étude 3 [To organise the study, 3]. Ecologie & régulation. In J.-L. Dorier, M. Artaud, M. Artigue, R. Berthelot, & R. Floris (Eds.), Actes de la XIe école d’été de didactique des mathématiques [Proceedings of the 11th summer school on didactics of mathematics] (pp. 41–56). Grenoble: La Pensée Sauvage.
Cornu, B. (2002). Limits. In D. Tall (Ed.), Mathematics Education Library: Vol. 11. Advanced Mathematical Thinking. Dordrecht: Springer.
Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K., & Vidakovic, D. (1996). Understanding the limit concept: Beginning with a coordinated process scheme. The Journal of Mathematical Behavior, 15(2), 167-192. doi:10.1016/s0732-3123(96)90015-2
De Vleeschouwer, M. (2010). An institutional point of view of the secondary–university transition: The case of duality. International Journal of Mathematical Education in Science and Technology, 41, 155–171. doi:10.1080/00207390903372445
Eisenhart, L. P. (1909). A treatise on the differential geometry of curves and surfaces. Boston, New York: Ginn and Company.
Giovanniello, S. (2017). What Algebra Do Calculus Students Need to Know? (Unpublished master's thesis). Concordia University.
Gray, S., Loud, B., & Sokolowski, C. (2009). Calculus Students Use and Interpretation of Variables: Algebraic vs. Arithmetic Thinking. Canadian Journal of Science, Mathematics and Technology Education, 9(2), 59-72. doi:10.1080/14926150902873434
Hardy, N. (2009a). Students' models of the knowledge to be learned about limits in college level Calculus courses. The influence of routine tasks and the role played by institutional norms (Doctoral dissertation, Concordia University, Montreal, Canada). Retrieved from http://spectrum.library.concordia.ca/976385/
Hardy, N. (2009b). Students’ perceptions of institutional practices: The case of limits of functions in college level Calculus courses. Educational Studies in Mathematics, 72, 341–358. doi:10.1007/s10649-009-9199-8
Lithner, J. (2000). Mathematical reasoning in task solving. Educational studies in mathematics, 41, 165–190.
Lithner, J. (2004). Mathematical reasoning in calculus textbook exercises. The Journal of Mathematical Behavior, 23(4), 405-427. doi:10.1016/j.jmathb.2004.09.003
Marsden, J. E., Tromba, A. J., & Weinstein, A. (1993). Basic multivariable calculus. New York: W.H. Freeman / Springer-Verlag.
Pelczer, I. (2017). A Praxeological Model of Future Elementary Teachers’ Envisioned Practice of Teaching Geometric Transformations (Unpublished doctoral dissertation). Concordia University.
Schoenfeld, Alan H., Mathematical problem solving, Academic Press, Orlando, FL, 1985.
Selden, J., Mason, A., & Selden, A. (1989). Can average calculus students solve nonroutine problems? Journal of Mathematical Behavior, 8, 45-50.
Selden, A., Selden, J., Hauk, S., & Mason, A. (1999). Do calculus students eventually learn to solve non-routine problems? (Rep. No. 1999-5). Cookeville, TN: Tennessee Technological University.
Sierpinska, A. (1990). Some remarks on understanding in mathematics. For the Learning of Mathematics, 10(3), 24-36.
Sierpinska, A. (1994). Understanding in Mathematics. Bristol, PA: The Falmer Press, Taylor & Francis Inc.
Sierpinska, A., Bobos, G., & Knipping, C. (2008). Sources of students’ frustration in pre-university level, prerequisite mathematics courses. Instructional Science, 36, 289–320. doi:10.1007/s11251- 007-9033-6
Stewart, J. (2015). Multivariable calculus (8th ed.). Boston, MA: Cengage Learning.
Swinyard, C., & Larsen, S. (2012). Coming to Understand the Formal Definition of Limit: Insights Gained From Engaging Students in Reinvention. Journal for Research in Mathematics Education, 43(4), 465-493. doi:10.5951/jresematheduc.43.4.0465
Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151-169. doi:10.1007/bf00305619
Thompson, P. W. (1994). Images of Rate and Operational Understanding of the Fundamental Theorem of Calculus. Educational Studies in Mathematics, 26, 229-274.
Vinner, S. (1987). Continuous functions: Images and reasoning in college students. Proceedings of the Annual Meeting of the International Group for the Psychology of Mathematics Education.
White, P., & Mitchelmore, M. (1996). Conceptual Knowledge in Introductory Calculus. Journal for Research in Mathematics Education, 27(1), 79-95. doi:10.2307/749199
Winsløw, C., Barquero, B., Vleeschouwer, M. D., & Hardy, N. (2014). An institutional approach to university mathematics education: from dual vector spaces to questioning the world. Research in Mathematics Education, 16(2), 95-111. doi:10.1080/14794802.2014.918345
All items in Spectrum are protected by copyright, with all rights reserved. The use of items is governed by Spectrum's terms of access.

Repository Staff Only: item control page

Downloads per month over past year

Back to top Back to top