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This book explores new trends and developments in mathematics education research related to proof and proving, the implications of these trends and developments for theory and practice, and directions for future research. With contributions from researchers working in twelve different countries, the book brings also an international perspective to the discussion and debate of the state of the art in this important area.

The book is organized around the following four themes, which reflect the breadth of issues addressed in the book:

- Theme 1: Epistemological issues related to proof and proving;
- Theme 2: Classroom-based issues related to proof and proving;
- Theme 3: Cognitive and curricular issues related to proof and proving; and
- Theme 4: Issues related to the use of examples in proof and proving.

Under each theme there are four main chapters and a concluding chapter offering a commentary on the theme overall.

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Preface

Andreas J. Stylianides ; Guershon Harel

as899@cam.ac.uk

THEME 1: EPISTEMOLOGICAL ISSUES RELATED TO PROOF AND PROVING

Chapter 1. Reflections on proof as explanation

Gila Hanna - gila.hanna@utoronto.ca

Chapter 2. Working on proofs as contributing to conceptualization - The case of IR completeness

Viviane Durand-Guerrier ; Denis Tanguay

viviane.durand-guerrier@umontpellier.fr

Chapter 3. Types of epistemological justifications, with particular reference to complex numbers

Guershon Harel

harel@math.ucsd.edu

Chapter 4. Mathematical argumentation in elementary teacher education: The key role of the cultural analysis of the content

Paolo Boero ; Giuseppina Fenaroli; Elda Guala

boero@dima.unige.it

Chapter 5. Toward an evolving theory of mathematical practice informing pedagogy: What standards for this research paradigm should we adopt?

Keith Weber ; Paul Dawkins

keith.weber@gse.rutgers.edu

THEME 2: CLASSROOM-BASED ISSUES RELATED TO PROOF AND PROVING

Chapter 6. Constructing and validating the solution to a mathematical problem: The teacher´s prompt

Maria Alessandra Mariotti ; Manuel Goizueta

mariotti21@unisi.it

Chapter 7. Addressing key and persistent problems of students´ learning: The case of proof

Andreas J. Stylianides ; Gabriel J. Stylianides

as899@cam.ac.uk

Chapter 8. How can a teacher support students in constructing a proof?

Bettina Pedemonte

bettina.pedemonte@sjsu.edu

Chapter 9. Proof validation and modification by example generation: A classroom-based intervention in secondary school geometry

Kotaro Komatsu ; Tomoyuki Ishikawa; Akito Narazaki

kkomatsu@shinshu-u.ac.jp

Chapter 10. Classroom-based issues related to proofs and proving

Ruhama Even

ruhama.even@weizmann.ac.il

THEME 3: COGNITIVE AND CURRICULAR ISSUES RELATED TO PROOF AND PROVING

Chapter 11. Mathematical argumentation in pupils´ written dialogues

Gjert-Anders Askevold; Silke Lekaus

slek@hib.no

Chapter 12. The need for "linearity" of deductive logic: An examination of expert and novice proving processes

Shiv Smith Karunakaran

karunak3@msu.edu

Chapter 13. Reasoning-and-proving in algebra in school mathematics textbooks in Hong Kong

Kwong-Cheong Wong ; Rosamund Sutherland

wongkwongcheong@gmail.com

Chapter 14. Irish teachers´ perceptions of reasoning-and-proving amidst a national educational reform

Jon D. Davis

jon.davis@wmich.edu

Chapter 15. About the teaching and learning of proof and proving: Cognitive issues, curricular issues and beyond

Lianghuo Fan ; Keith Jones

l.fan@southampton.ac.uk

THEME 4: ISSUES RELATED TO THE USE OF EXAMPLES IN PROOF AND PROVING

Chapter 16. How do pre-service teachers rate the conviction, verification and explanatory power of different kinds of proofs?

Leander Kempen

kempen@khdm.de

Chapter 17. When is a generic argument a proof?

David Reid ; Estela Vallejo Vargas

dreid@math.uni-bremen.de

Chapter 18. Systematic exploration of examples as proof: Analysis with four theoretical frameworks

Orly Buchbinder

orly.buchbinder@unh.edu

Chapter 19. Using examples of unsuccessful arguments to facilitate students´ reflection on their processes of proving

Yosuke Tsujiyama ; Koki Yui

tsujiyama@chiba-u.jp

Chapter 20. Genericity, conviction, and conventions: Examples that prove and examples that don´t prove

Orit Zaslavsky

orit.zaslavsky@nyu.edu

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