Teaching for Abstraction: A Model

This article outlines a theoretical model for teaching elementary mathematical concepts that we have developed over the past 10 years. We begin with general ideas about the abstraction process and differentiate between abstract-general and abstract-apart concepts. A 4-phase model of teaching, called Teaching for Abstraction, is then proposed that is explicitly designed to promote abstract-general learning. Studies investigating the model with 4 different topics (angles, decimals, percentages, and ratios) are reported, and the 4 phases are further elucidated. The article concludes with a discussion of the effectiveness of the model and its applicability to other mathematical concepts.

[1]  Alan Bell.  Some experiments in diagnostic teaching , 1993 .

[2]  Michael Mitchelmore,et al.  Teaching Angles by Abstraction from Physical Activities with Concrete Materials. , 2003 .

[3]  K.P.E. Gravemeijer,et al.  Developing realistic mathematics education , 1994 .

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

[5]  A. Bell,et al.  Principles for the design of teaching , 1993 .

[6]  H.J.M. van Oers,et al.  Contextualisation for abstraction , 2001 .

[7]  Paul White,et al.  ABSTRACTION IN MATHEMATICS AND MATHEMATICS LEARNING , 2004 .

[8]  Kaye Stacey,et al.  The effect of epistemic fidelity and accessibility on teaching with physical materials: A comparison of two models for teaching decimal numeration , 2001 .

[9]  Paul White,et al.  Conceptual Knowledge in Introductory Calculus , 2007 .

[10]  Paul White,et al.  Teaching and Learning in Mathematics by Abstraction , 2002 .

[11]  Hans Freudenthal,et al.  Revisiting mathematics education : China lectures , 1991 .

[12]  Paul White,et al.  Critical numeracy and abstraction: percentages , 2009 .

[13]  Paul White,et al.  Abstraction in mathematics: Conflict, resolution and application , 1995 .

[14]  Graeme S. Halford,et al.  Mathematics Education: Models and Processes , 1995 .

[15]  J. Lave.  Cognition in Practice: Outdoors: a social anthropology of cognition in practice , 1988 .

[16]  Tommy Dreyfus,et al.  Advanced Mathematical Thinking Processes , 2002 .

[17]  David Tall,et al.  Ambiguity and flexibility: A proceptual view of simple arithmetic , 1983 .

[18]  Paul White,et al.  Teaching ratio and rates for abstraction , 2007 .

[19]  D. D. Foxman,et al.  Mathematical development : secondary survey report , 1980 .

[20]  H. Freudenthal.  Why to teach mathematics so as to be useful , 1968 .

[21]  L. Vygotsky.  Thinking and Speech , 1987 .

[22]  A. Sfard.  On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin , 1991 .

[23]  Paul White,et al.  Development of angle concepts by progressive abstraction and generalisation , 2000 .

[24]  J. Hiebert,et al.  Conceptual and Procedural Knowledge in Mathematics: An Introductory Analysis , 1986 .

[25]  Baruch B. Schwarz,et al.  Abstracting processes, from individuals’ constructing of knowledge to a group’s “shared knowledge” , 2007 .

[26]  B. Schwarz,et al.  Abstraction in Context: Epistemic Actions , 2001 .

[27]  R. Skemp.  The psychology of learning mathematics , 1979 .

[28]  Michael Mitchelmore.  The role of abstraction and generalisation in the development of mathematical knowledge , 2002 .

[29]  Michael Mitchelmore,et al.  The Role of abstraction in learning about rates of change , 2006 .

[30]  Paul White,et al.  Teaching for abstraction: reconstructing constructivism , 2000 .