IMPACT OF EXPLICIT PRESENTATION OF SLOPES IN THREE DIMENSIONS ON STUDENTS’ UNDERSTANDING OF DERIVATIVES IN MULTIVARIABLE CALCULUS
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In two dimensions (2D), representations associated with slopes are seen in numerous forms before representations associated with derivatives are presented. These include the slope between two points and the constant slope of a linear function of a single variable. In almost all multivariable calculus textbooks, however, the first discussion of slopes in three dimensions (3D) is seen with the introduction of partial derivatives. The nature of the discussions indicates that authors seem to assume that students are able to naturally extend the concept of a 2D slope to 3D and correspondingly it is not necessary to explicitly present slopes in 3D. This article presents results comparing students that do not explicitly discuss slopes in 3D with students that explicitly discuss slopes in 3D as a precursor to discussing derivatives in 3D. The results indicate that students may, in fact, have significant difficulty extending the concept of a 2D slope to a 3D slope. And that the explicit presentation of slopes in 3D as a precursor to the presentation of derivatives in 3D may significantly improve student comprehension of topics of differentiation in multivariable calculus.
Key wordsdifferentiation multivariable calculus representations semiotic registers slopes
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- Arnon, I., Cotrill, J., Dubinsky, E., Octaç, A., Roa Fuentes, S., Trigueros, M. & Weller, K. (2013). APOS theory: A framework for research and curriculum development in mathematics education. New York: Springer.Google Scholar
- Asiala, M., Brown, A., DeVries, D., Dubinsky, E., Mathews, D. & Thomas, K. (1996). A framework for research and curriculum development in undergraduate mathematics education. Research in Collegiate Mathematics Education, 2(3), 1–32.Google Scholar
- Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 95–126). Dordrecht: Kluwer.Google Scholar
- Duval, R. (1999). Representation, vision and visualization: Cognitive functions in mathematical thinking. Basic issues for learning. In F. Hitt & M. Santos (Eds.), Proceedings of the Twenty-first Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME), (Vol. 1, pp. 3-26), Mexico: PME.Google Scholar
- Edwards, H., and Penney, D. (2008). Calculus: Early Transcendentals 6th Edition, Prentice Hall.Google Scholar
- Erikson Institute, The Early Mathematics Collaborative (2013), Big Ideas of Early Mathematics: What Teachers of Young Children Need to Know, Pearson Education US, ISBN: 9780132946971.Google Scholar
- Glaser, B. G. (1998). Doing grounded theory: Issues and discussions. Mill Valley: Sociology.Google Scholar
- Huang, C. H. (2011). Engineering students’ conceptual understanding of the derivative in calculus. World Transactions on Engineering and Technology Education, 9, 209–214.Google Scholar
- Kant, I. (1929). Critique of Pure Reason. (N. K. Smith, Trans.). London: Macmillan. (Original work published 1781).Google Scholar
- Nagle, C. & Moore-Russo, D. (2014). The concept of slope: Comparing teachers’ concept images and instructional content. Investigations in Mathematics Learning, 6(2), 1–18.Google Scholar
- Piaget, J. (1971). Biology and knowledge. Chicago: University of Chicago Press. Original work published 1967; (B. Walsh, Trans.).Google Scholar
- Piaget, J. (1977). Understanding causality. New York: Norton (D. Miles and M. Miles, Trans.).Google Scholar
- Rodriguez, P. (2008). Calculus for the biological sciences. New York: Wiley.Google Scholar
- Roorda, G., Vos, P. & Goedhart, M. (2009). Derivatives and applications; development of one student’s understanding. Proceedings of CERME 6, January 28th-February 1st 2009, Lyon France. Working group 12. Available at http://www.inrp.fr/editions/cerme6. Accessed 1 May 2014.
- Stewart, J. (2006). Calculus: early transcendentals (6th ed.). Boston: Thomson-Brooks/Cole.Google Scholar
- Strauss, M., Bradley, G. & Smith, K. (2002). Calculus (3rd ed.). Upper Saddle River: Prentice Hall.Google Scholar
- Swokowski, E., Olinick, M. & Pence, D. (1992). Calculus (6th ed.). Boston: PWS.Google Scholar
- Tall, D. (2010). A sensible approach to the calculus. Plenary Session at The National and International Meeting on the Teaching of Calculus, 23–25th September 2010, Puebla, Mexico. Available at http://homepages.warwick.ac.uk/staff/David.Tall/downloads.html. Accessed 1 May 2014.
- Waner, S. & Costenoble, S. (2007). Finite mathematics and applied calculus (4th ed.). Boston: Thomson-Brooks/Cole.Google Scholar
- Zandieh MJ (2000). A theoretical framework for analyzing student understanding of the concept of derivative. CBMS Issues in Mathematics Education, 8, 103–122.Google Scholar