Agents of Change in Promoting Reflective Abstraction: A Quasi-Experimental, Study on Limits in College Calculus
Downloads
Abstract
We measured student performance on the concept of limit by promoting reflection through four agents of change: instructor, peer, curriculum and individual. It is based on Piaget’s four constructs of reflective abstraction: interiorization, coordination, encapsulation, and generalization, and includes the notion of reversal, as refined into a construct by Dubinsky. Our quasi-experimental study examined the performance of two sections of first-semester calculus students at a midwestern community college. Scores by students in the experimental section were significantly higher than scores by students in the control traditional section on a posttest of limits. A deeper examination of a three-tiered subgroup showed the reflective abstraction section had moderate effect size on mathematical knowledge and strategic knowledge and a large effect size on explanation.
Downloads
References
Artigue, M. (1992). Functions from an algebraic and graphic point of view: Cognitive difficulties and teaching practices. In The concept of function: Aspects of epistemology and pedagogy. (pp. 109-132). MAA Notes No. 28.
Google Scholar CrossrefArtigue, M., Batanero, C., & Kent, P. (2007). Mathematics teaching and learning at post-secondary level. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 1011-1050). Reston, VA: NCTM.
Google Scholar CrossrefAuthor. (2007). Article name. Journal name.
Google Scholar CrossrefAuthors. (2009). Article name. Journal name.
Google Scholar CrossrefBeth, E. and Piaget, J. (1966). Mathematical epistemology and psychology. D. Reidel, Dordrecht, The Netherlands: D. Reidel.
Google Scholar CrossrefCobb, P., Boufi, A., McClain, K., & Whitenack, J. (1997). Reflective discourse and collective reflection. Journal for Research in Mathematics Education, 28(3), 258-277.
Google Scholar CrossrefCobb, P., Jaworski, B., & Presmeg, N. (1996). Emergent and sociocultural views of mathematical activity. In L. Steffe, P. Nesher, P. Cobb, G. Goldin, & B. Greer Theories of mathematical learning (pp. 3-19). Mahwah, N.J.: Lawrence Erlbaum.
Google Scholar CrossrefCohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Earlbaum Associates.
Google Scholar CrossrefCornu B. (1981) Learning the concept of limit: Models and spontaneous modes. In Proceedings of the fifth conference for the Psychology of Mathematics Education (pp. 322-326). Grenoble, France: Psychology of Mathematics Education.
Google Scholar CrossrefCornu, B. (1991). Limits. In D. Tall (Ed.), Advanced mathematical thinking. (pp. 153-166). Boston: Kluwer.
Google Scholar CrossrefDavis, B., Porta, H., & Uhl, J. (1994). Calculus and Mathematica. Addison-Wesley: Reading, MA.
Google Scholar CrossrefDavis, R. & Vinner, S. (1986). The notion of limit: Some seemingly unavoidable misconception stages. Journal of Mathematical Behavior, 5, 281-303.
Google Scholar CrossrefDubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking. (pp. 95-126). Boston: Kluwer.
Google Scholar CrossrefHardy, N. (2009). Students' perceptions of instructional practices: The case of limits and functions in college level calculus courses. Educational Studies in Mathematics, 72(3), 341-358.
Google Scholar CrossrefHughes-Hallett, D. (1997). Calculus single variable. New York: John Wiley and Sons.
Google Scholar CrossrefIllinois State Board of Education (2005a). Extended response rubric. Retrieved May 22, 2007 from http://www.isbe.net/assessment/docs/ERMathRubric.rtf
Google Scholar CrossrefIllinois State Board of Education (2005b). Short response rubric. Retrieved May 22, 2007 from http://www.isbe.net/assessment/docs/SRMathRubric.rtf
Google Scholar CrossrefJuter, K. (2005). Limits of functions: Traces of students’ concept images. Nordic Studies in Mathematics Education, 10(3-4), 65-82.
Google Scholar CrossrefJuter, K. (2007). Students’ conceptions of limits: high achievers versus low achievers. The Montana Mathematics Enthusiast, 4(1), 53-65.
Google Scholar CrossrefKidron, I. (2008). Abstraction and consolidation of the limit procept by means of instructional schemes: The complementary role of three different frameworks. Educational Studies in Mathematics, 69(3), 197–216.
Google Scholar CrossrefLarson, R., Hostetler, R. & Edwards, B. (2006). Calculus. Boston: Houghton Mifflin.
Google Scholar CrossrefLi, L., & Tall, D. (1993). Constructing different concept images of sequences and limits by programming. In Proceedings of the seventeenth conference for the Psychology of Mathematics Education (pp. 41-48). Tsukuba, Japan: Psychology of Mathematics Education.
Google Scholar CrossrefMonaghan, J., Sun, S. & Tall, D. (1994). Construction of the limit concept with a computer algebra system, In Proceedings of the eighteenth conference for the Psychology of Mathematics Education (pp. 279-286). Lisbon, Spain: Psychology of Mathematics Education.
Google Scholar CrossrefOehrtman, M. (2009). Collapsing dimensions, physical limitation, and other student metaphors
Google Scholar Crossreffor limit concepts. Journal for Research in Mathematics Education, 40 (4), 396-426.
Google Scholar CrossrefPiaget, J. (1967). Genetic epistemology, a series of lectures delivered by Piaget at Columbia University. New York: Columbia University Press.
Google Scholar CrossrefRobert, A. (1982). L’Acquisition de la notion de convergence des suites numériques dans l’Enseignement Supérieur, Reserches en didactique des mathématiques, 3 (3), 307-341.
Google Scholar CrossrefSierpinska, A. (1987). Humanities students and epistemological obstacles related to limits. Educational Studies in Mathematics, 18, 371-397.
Google Scholar CrossrefSmith, D. & Moore, L. (1991). Project CALC: An integrated laboratory course. In L. C. Leinbach (Ed.), The laboratory approach to teaching calculus (pp. 81-92). Washington, DC: The Mathematical Association of America.
Google Scholar CrossrefTall, D. (1992). The transition to advanced mathematical thinking: Functions, limits, infinity and proof. In D. Grouws (Ed.) Handbook of research on mathematics teaching and learning (pp. 495-511). New York: Macmillan.
Google Scholar CrossrefTall, D. & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151-169.
Google Scholar CrossrefVinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.), Advanced mathematical thinking (pp. 65-81). Dordrecht: Kluwer Academic Publishers.
Google Scholar CrossrefVernez, G. & Zimmer, R. (2007). Interpreting the effects of Title I supplemental educational services. Retrieved March 29, 2011 from http://www2.ed.gov/rschstat/eval/choice/implementation/achievementanalysis-sizes.pdf
Google Scholar CrossrefWeller, K., Clark, J., Dubinsky, E., Loch, S., McDonald, M., & Merkovsky, R. (2003). Student performance and attitudes in courses based on APOS Theory and the ACE Teaching Cycle. In A. Selden, E. Dubinsky, G. Harel, & F. Hitt (Eds.), Research in collegiate mathematics education V (pp. 97-131). Providence, RI: American Mathematical Society.
Google Scholar CrossrefWilliams, S. (1991). Models of limit held by college calculus students. Journal for Research in Mathematics Education, 22(3), 219-236.
Google Scholar CrossrefDownloads
Published
Almetric
Dimensions
Issue
Section
License
Authors retain copyright and grant the journal the right of first publication but allow anyone to share: (unload, , reprint, distribute and/or copy) and adapt (remix, transform reuse, modify,) for any proposition, even commercial, always quoting the original source.
