Philosophy of mathematical practice: a primer for mathematics educators
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ORIGINAL ARTICLE
Philosophy of mathematical practice: a primer for mathematics educators Yacin Hamami1 · Rebecca Lea Morris2 Accepted: 10 April 2020 © FIZ Karlsruhe 2020
Abstract In recent years, philosophical work directly concerned with the practice of mathematics has intensified, giving rise to a movement known as the philosophy of mathematical practice. In this paper we offer a survey of this movement aimed at mathematics educators. We first describe the core questions philosophers of mathematical practice investigate as well as the philosophical methods they use to tackle them. We then provide a selective overview of work in the philosophy of mathematical practice covering topics including the distinction between formal and informal proofs, visualization and artefacts, mathematical explanation and understanding, value judgments, and mathematical design. We conclude with some remarks on the potential connections between the philosophy of mathematical practice and mathematics education.
1 Introduction A wide variety of fields are concerned with the study of mathematics as a human practice, including anthropology, history, pedagogy, psychology, and sociology. Many philosophers have also engaged deeply with the mathematical practices of their time, from Plato, Descartes, Leibniz, Berkeley, and Kant to Frege, Russell, Hilbert, and Wittgenstein. In recent years, philosophical work directly concerned with the practice of mathematics has intensified, giving rise to a movement known as the philosophy of mathematical practice. If one accepts the premise of this special issue— that mathematicians’ practice matters to mathematical instruction—then work in the philosophy of mathematical practice is potentially relevant to mathematics education research. To foster interactions between philosophical and pedagogical approaches to the study of mathematical practice, we provide in this paper an overview of the philosophy of mathematical practice by (1) highlighting its driving questions and methods and (2) surveying some of its recent developments.
* Rebecca Lea Morris [email protected] 1
Centre for Logic and Philosophy of Science, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium
Minneapolis, MN, USA
2
Perhaps the first steps towards a philosophy of mathematical practice were taken by Wilder (1950, 1981), who argued that mathematics is a cultural system, and Pólya (1945, 1954, 1962) who investigated mathematical problem solving, heuristics, and discovery. Lakatos (1976) dedicated his famous dialogue Proofs and Refutations to Pólya, as well as to Popper, the philosopher of science. In this work, he argued that mathematical knowledge grows not via the continuous production of formal derivations but by a dynamic process which involves proposing “proofs” which are then refuted and subsequently refined. Kitcher (1984) was also concerned with the growth of mathematical knowledge, arguing that modern mathematics evolved via a series of rational transitions from earlier practices. In the past 20
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