The Programme

There is international recognition of the importance of reasoning and proof in students’ learning of mathematics at all levels of education, and of the difficulties met by students and teachers in this area. Indeed, many students face difficulties with reasoning about mathematical ideas and constructing or understanding mathematical arguments that meet the standard of proof. Teachers also face difficulties with reasoning and proof, and existing curriculum materials tend to offer inadequate support for classroom work in this area. All of these paint a picture of reasoning and proof as important but difficult to teach and hard to learn. A rapidly expanding body of research has offered important insights into this area, but there are still many open questions for which theoretical and empirically based responses are sorely needed (for reviews of the literature in this area, see: Harel & Sowder, 2007; Mariotti, 2006; Stylianides, Bieda, & Morselli, 2016; Stylianides, Stylianides, & Weber, 2017).

TSG-18 offered during ICME-13 a forum for an overview of the state of the art, invited contributions from experts in the field (Viviane Durand-Guerrier, Gila Hanna, Eric Knuth, and Maria Alessandra Mariotti), presentation of high-quality research reports from members of the TSG organizing team and other TSG participants, and discussion of directions for future research. Associated with the TSG there were in total 21 regular presentations (8-page papers), 35 oral communications (4-page papers), and 12 posters.

The regular presentations (8-page papers) were organized around four themes as described below. Although several presentations (and associated papers) addressed issues that spanned several themes, practical considerations related to the organization of the TSG sessions during the conference necessitated a best-fit approach.

Theme 1: Epistemological issues related to proof and proving

The following presentations were offered under this theme:

  • Reflections on proof as explanation (Gila Hanna);

  • Working on proofs as contributing to conceptualization: The case of IR completeness prolegomena to a didactical study (Viviane Durand-Guerrier & Denis Tanguay);

  • Types of epistemological justifications (Guershon Harel);

  • Reasoning and proof in elementary teacher education: The key role of cultural analysis of the content (Paolo Boero, Giuseppina Fenaroli, & Elda Guala).

Theme 2: Classroom-based issues related to proof and proving

The following presentations were offered under this theme:

  • Constructing and validating a mathematical model: The teacher’s prompt (Maria Alessandra Mariotti & Manuel Goiuzueta);

  • Classroom-based interventions in the area of proof: Addressing key and persistent problems of students’ learning (Andreas J. Stylianides & Gabriel J. Stylianides);

  • Developing a curriculum for explorative proving in lower secondary school geometry (Mikio Miyazaki, Junichiro Nagata, Kimiho Chino, Taro Fujita, Daisuke Ichikawa, Shizumi Shimizu, & Yasuo Iwanaga);

  • Proof validation and modification by example generation: A classroom-based intervention in secondary school geometry (Kotaro Komatsu, Tomoyuki Ishikawa, & Akito Narazaki).

Theme 3: The teaching and learning of proof—issues and dilemmas

The following presentations were offered under this theme:

  • Teacher noticing of justifying in the elementary classroom (Mary Kathleen Melhuish & Eva Thanheiser);

  • How can a teacher support students in constructing a proof? (Bettina Bedemonte);

  • Reasoning-and-proving in school mathematics textbooks: A case study from Hong Kong (Kwong Cheong Wong & Rosamund Sutherland);

  • Irish teachers’ perceptions of reasoning-and-proving amidst a national educational reform (Jon D. Davis);

  • Identifying and using key ideas in proofs (Xiaoheng Yan, Gila Hanna, & John Mason);

  • Mathematical argumentation in pupils’ written dialogues (Silke Lekaus & Gjert-Anders Askevold);

  • What makes a good proof? Students evaluating and providing feedback on student-generated proofs (Tina Kathleen Rapke & Amanda Allan);

  • Use of examples of unsuccessful arguments to facilitate students’ reflection on their proving processes (Yosuke Tsujiyama & Koki Yui);

  • Allowance by experts for a break in “linearity” of deductive logic in the process of proving (Shiv Smith Karunakaran);

  • Systematic exploration of examples as proof: Analysis from four theoretical perspectives (Orly Buchbinder).

Theme 4: Issues related to the use of examples in proof and proving

The following presentations were offered under this theme:

  • The role of examples in proving related activities (Eric Knuth, Amy Ellis, & Orit Zaslavsky);

  • When is a generic argument a proof? (David A. Reid & Estela Aurora Vallejo Vargas);

  • How do pre-service teachers rate the conviction, verification and explanatory power of different kinds of proofs (Leander Kempen).