Understanding and describing mathematical knowledg

Authors: Andreas J Stylianides Deborah L Ball

Publish Date: 2008/03/12

Volume: 11, Issue: 4, Pages: 307-332

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Abstract

This article is situated in the research domain that investigates what mathematical knowledge is useful for and usable in mathematics teaching Specifically the article contributes to the issue of understanding and describing what knowledge about proof is likely to be important for teachers to have as they engage students in the activity of proving We explain that existing research informs the knowledge about the logicolinguistic aspects of proof that teachers might need and we argue that this knowledge should be complemented by what we call knowledge of situations for proving This form of knowledge is essential as teachers mobilize proving opportunities for their students in mathematics classrooms We identify two subcomponents of the knowledge of situations for proving knowledge of different kinds of proving tasks and knowledge of the relationship between proving tasks and proving activity In order to promote understanding of the former type of knowledge we develop and illustrate a classification of proving tasks based on two mathematical criteria 1 the number of cases involved in a task a single case multiple but finitely many cases or infinitely many cases and 2 the purpose of the task to verify or to refute statements In order to promote understanding of the latter type of knowledge we develop a framework for the relationship between different proving tasks and anticipated proving activity when these tasks are implemented in classrooms and we exemplify the components of the framework using data from third grade We also discuss possible directions for future research into teachers’ knowledge about proofWe wish to thank Terry Wood Steve Galovich † Gabriel Stylianides anonymous reviewers and the participants of the fourth Nuffield Seminar on Mathematical Knowledge in Teaching Cambridge UK January 2008 for useful comments on earlier versions of the article

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