Advances in Mathematics Education Research on Proof and Proving

This is one of several ICME-13 monographs published after the 13th quadrennial International Congress on Mathematical Education held in Hamburg, Germany in July 2016. The contents cover new trends and developments in mathematics education on proof and proving. Contributions come from researchers from 12 different countries including Canada, Chile, France, Germany, Hong Kong, Japan, Norway, Peru, the UK, and the USA. The volume is divided into four sections organized around the following themes: epistemological issues, classroom issues, cognitive and curricular issues, and the use of examples in proof and proving. Each section has four chapters and a concluding commentary written by an invited contributor. The book’s main chapters are revised and extended versions of papers presented at ICME-13 in the study group on Reasoning and Proof in Mathematics Education.

Chapter 1 was written by Gila Hanna of the Ontario Institute for Studies in Education at the University of Toronto, who is well known for making the distinction between “proofs that only prove” versus “proofs that explain”.  She discusses two distinct ways of considering explanatory proofs. The first from a philosophy of mathematics considers a proof explanatory if it helps account for a mathematical result by clarifying how it follows from others—it is not concerned with pedagogical factors. The second considers a proof explanatory if it helps convey mathematical insights to individuals in a pedagogically appropriate manner. She claims there are proofs which are explanatory in both senses. Examples include a proof of Pick’s Theorem, which gives a formula for calculating the area of a polygon in terms of the number of lattice points in its interior and the number of lattice points on the boundary, that was developed by middle- and high-school students at St. Mark’s Institute of Mathematics [MAA Focus, 30 (1), 2010].

Chapter 8 by Shiv Smith Karunakaran reports part of a larger study of the proving processes of five novice provers (undergraduates who had completed at least one proof-based course) versus five expert provers (doctoral students who had successfully passed their department’s doctoral qualifying examination). The participants were each given five real analysis tasks of the form, “Validate or refute the following statement.” One of the findings involves how the expert and novice provers differ in their organization and use of deductive logic. In contrast to the novice provers, the expert provers did not need to validate each step in order to move forward in their deductive reasoning—they set aside any questions temporarily to be resolved later. This finding reminds me of a result by Nachleili and Herbst [JRME, 40(4), 2009], who found that high school geometry teachers enact a norm that requires their students to give a reason for each step before continuing with a proof, even though they know that proofs are not usually constructed that way. This leads me to conjecture that the undergraduates might have been enacting a norm remembered from their high school geometry courses and not yet counteracted by experiences in their undergraduate proof-based courses.

While there is much else that could interest mathematicians and mathematics education researchers, especially the fourth part that considers to what extent generic examples can be considered proofs, this review will give readers an idea of what can be found in this volume.

Annie Selden is Adjunct Professor of Mathematics at New Mexico State University and Professor Emerita of Mathematics from Tennessee Technological University. She regularly teaches graduate courses in mathematics and mathematics education. In 2002, she was the recipient of the Association for Women in Mathematics 12th Annual Louise Hay Award for Contributions to Mathematics Education. In 2003, she was elected a Fellow of the American Association for the Advancement of Science. She remains active in mathematics education research and curriculum development.