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Years 2 to 6 Students’ Ability to Generalise: Models, Representations and Theory for Teaching and Learning

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Book cover Early Algebraization

Part of the book series: Advances in Mathematics Education ((AME))

Abstract

Over the last three years, in our Early Algebra Thinking Project, we have been studying Years 3 to 5 students’ ability to generalise in a variety of situations, namely, compensation principles in computation, the balance principle in equivalence and equations, change and inverse change rules with function machines, and pattern rules with growing patterns. In these studies, we have attempted to involve a variety of models and representations and to build students’ abilities to switch between them (in line with the theories of Dreyfus 1991, and Duval 1999). The results have shown the negative effect of closure on generalisation in symbolic representations, the predominance of single variance generalisation over covariant generalisation in tabular Representations, and the reduced ability to readily identify commonalities and relationships in enactive and iconic representations. This chapter uses the results to explore the interrelation between generalisation, and verbal and visual comprehension of context. The studies evidence the importance of understanding and communicating aspects of representational forms which allowed commonalities to be seen across or between representations. Finally the chapter explores the implications of the results for a theory that describes a growth in integration of models and representations that leads to generalisation.

This chapter is a revised version of an article published in ZDM—International Reviews on Mathematical Education, 40(1), 23–37. DOI 10.1007/s11858-007-0066-8.

The research for this paper was funded by Australian Research Council Linkage grant: LP0348820.

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Author information

Authors and Affiliations

  1. Queensland University of Technology, Brisbane, Australia

    Tom J. Cooper

  2. Australian Catholic University, Brisbane, Australia

    Elizabeth Warren

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  1. Tom J. Cooper
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Correspondence to Tom J. Cooper .

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Editors and Affiliations

  1. Dept. Mathematical Sciences, University of Delaware, Ewing Hall 501, Newark, 19716, Delaware, USA

    Jinfa Cai

  2. University of Wisconsin-Madison, Teacher Education Building, 225 N. Mills Street, Madison, 53706, Wisconsin, USA

    Eric Knuth

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Cooper, T.J., Warren, E. (2011). Years 2 to 6 Students’ Ability to Generalise: Models, Representations and Theory for Teaching and Learning. In: Cai, J., Knuth, E. (eds) Early Algebraization. Advances in Mathematics Education. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17735-4_12

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