Reflections on Justification and Proof
In this chapter, we explore how investigations into mathematicians’ practice can inform instruction on justification and proof. Each co-author of this practice presents an investigation of how mathematicians use justification and proof in their profession
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Reflections on Justification and Proof Justification and Proof in Mathematics and Mathematics Education Keith Weber Abstract In this chapter, we explore how investigations into mathematicians’ practice can inform instruction on justification and proof. Each co-author of this practice presents an investigation of how mathematicians use justification and proof in their professional practice and suggests pedagogical implications based upon insights from their investigations. Keywords Justification · Mathematicians’ practice · Proof
Introduction In 2000, Eric Knuth noted that there was a rebirth of proof in mathematics classrooms. In the 1990s, many researchers noted that the role of proof in mathematics was limited (e.g., Schoenfeld 1994; Wu 1996). But in the last decade, mathematics educators have advocated that justification and proof was expected to play an important role in all aspects of students’ mathematical learning (e.g., Knuth 2000). If we accept the premise that proof should be taught to all students of mathematics, this begs the questions: what role should proof play in the mathematics classroom and how should it be taught? Often proof is taught independently from other mathematical content, leading students to view proof as a pedantic ritual that they are required to engage in rather than as a tool for facilitating communication and advancing mathematical knowledge (e.g., Harel 1998; Schoenfeld 1994)—consequently students often see little value in the proof they observe or produce (e.g., Harel 1998; Healy and Hoyles 2000; Schoenfeld 1989).
With contributions by Gila Hanna, Ontario Institute for Studies in Education, University of Toronto, Toronto, Canada Guershon Harel, University of California-San Diego, La Jolla, CA, USA Ivy Kidron, Jerusalem College of Technology, Jerusalem, Israel Annie Selden and John Selden, New Mexico State University, Las Cruces, NM, USA K. Weber (B) Graduate School of Education, Rutgers University, 10 Seminary Place, New Brunswick, NJ 08901, USA e-mail: [email protected] M.N. Fried, T. Dreyfus (eds.), Mathematics & Mathematics Education: Searching for Common Ground, Advances in Mathematics Education, DOI 10.1007/978-94-007-7473-5_14, © Springer Science+Business Media Dordrecht 2014
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Addressing these shortcomings involves an important opportunity for mathematicians and mathematics educators to collaborate. An important source of insight into what justification and proof should mean and how these constructs should be used for students is mathematicians’ practice with justification and proof. Toward this end, mathematics educators have explored how justification and proof have been used in mathematical practice. This is based on the assumption that what motivates mathematicians to engage in justification and proof might similarly motivate students to do the same and that students may gain similar benefits from the kind of proving mathematicians do. Narrowing the gap between mathematical and classroom practice with respect to justification and proof h
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