Agents of Change in Promoting Reflective Abstraction: A Quasi-Experimental, Study on Limits in College Calculus

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.

[1]  Ivy Kidron,et al.  Abstraction and consolidation of the limit procept by means of instrumented schemes: the complementary role of three different frameworks , 2008 .

[2]  Phillip Kent,et al.  Mathematics thinking and learning at post-secondary level , 2007 .

[3]  Nadia Hardy,et al.  Students’ perceptions of institutional practices: the case of limits of functions in college level Calculus courses , 2009 .

[4]  Jacob Cohen.  Statistical Power Analysis for the Behavioral Sciences , 1969, The SAGE Encyclopedia of Research Design.

[5]  P. Lachenbruch.  Statistical Power Analysis for the Behavioral Sciences (2nd ed.) , 1989 .

[6]  William G. McCallum,et al.  Calculus: Single Variable , 1994 .

[7]  P. Cobb,et al.  Reflective Discourse and Collective Reflection , 1997 .

[8]  R. B. Davis,et al.  The Notion of Limit: Some Seemingly Unavoidable Misconception Stages. , 1986 .

[9]  Shlomo Vinner,et al.  The Role of Definitions in the Teaching and Learning of Mathematics , 2002 .

[10]  David Tall,et al.  Constructing Different Concept Images of Sequences & Limits by Programming , 1999 .

[11]  A. Sierpińska.  Humanities students and epistemological obstacles related to limits , 1987 .

[12]  Evert W. Beth,et al.  Mathematical Epistemology And Psychology , 1966 .

[13]  Michael Oehrtman,et al.  Collapsing dimensions, physical limitation, and other student metaphors for limit concepts : an instrumentalist investigation into calculus students' spontaneous reasoning , 2009 .

[14]  Ed Dubinsky,et al.  Reflective Abstraction in Advanced Mathematical Thinking , 2002 .

[15]  Robert W. Cappetta.  Reflective abstraction and the concept of limit : a quasi-experimental study to improve student performance in college calculus by promoting reflective abstraction through individual, peer, instructor and curriculum initiates , 2007 .

[16]  David Tall,et al.  Concept image and concept definition in mathematics with particular reference to limits and continuity , 1981 .

[17]  Steven R. Williams.  Models of Limit Held by College Calculus Students. , 1991 .

[18]  David Tall.  Concept Image and Concept Definition , 1988 .

[19]  Aline Robert,et al.  L'acquisition de la notion de convergence des suites numériques dans l'enseignement supérieur. , 1982 .