This book is about research on the teaching and learning of proof in mathematics over the last fifty years. The book comprises nine chapters, divided into three parts. The first part presents the theoretical framework for the research and situates it within the state of the art from a multidisciplinary perspective. The second part details a classical case study of students’ dynamics of proof and refutation, resulting in a typology of proofs that sheds light on the links between proving and knowing. Finally, the third part proposes entering the classroom with an experimental approach, in support of a reflection on the challenges of implementing and managing situations for the teaching and learning of proof in mathematics at the compulsory school level. The book balances theoretical discussions of interest to researchers and mathematics educators. The detailed case studies should serve as discussion generators for teachers involved in teaching proof and proving.

Since 2012, Dr. Nicolas Balacheff is Directeur de recherche CNRS émérite (senior scientist emeritus at the French National Center for Scientific Research – CNRS). He is attached to the computer science lab LIG (Laboratoire d’informatique de Grenoble), member of the team Models and Technologies for Human Learning he created in the early 2000. Since 1975, Nicolas research domain is mathematics education with a focus on the learning of mathematical proof, and issues related to the modeling of students’ conceptions. This research developed within the framework of the Theory of Didactical Situations. His work on the categorization of students’ proof contributed to frame research in this area.   He extended this research to the learning and teaching of proof in a technology enhanced learning environment, in the middle of the 80s, as an early collaborator of the project Cabri‐géomètre. It is in relation with this multidisciplinary work with computer scientists that he proposed the cK¢ model of students’ conceptions fulfilling two constraints: being epistemologically valid and computationally tractable. One component of this model, the “control structure”, is instrumental to bridging knowing and proving. It is the focus of his current research. Co‐founder in 1980 of the French journal Recherches en Didactique des Mathématiques, he participated actively to the development of research in Didactique des Mathématiques, and directed the translation of Brousseau theoretical foundation (Theory of didactical situations, 1997, Kluwer). Nicolas served as President of PME (1988‐1990), member of the NATO panel of the Special program on Advanced Educational Technology" (1990‐1993). Founder and scientific manager of Kaleidoscope, the European Network of Excellence on technology enhanced learning (2004‐2007). Nicolas achieved the graduation in mathematics, then in computer science at the University of Grenoble. He got his PhD in 1978 on the classification of students reasoning and his Doctorat ès‐Science in 1988 on the learning of mathematical proof. He got the position of CNRS senior scientist in 1988, attached to labs in Lyon then in Grenoble since 1992. He directed the Leibniz Laboratory (computer science, cognitive science and discrete mathematics) from 2000 to 2006.