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  • © 2012

Proof and Proving in Mathematics Education

The 19th ICMI Study

  • First comprehensive book covering a wide range of aspects related to the teaching and learning of proof

  • Provides illustrative examples of proof practices and activities in different contexts and levels

  • Extensive survey of available research findings, and indicating directions for future research

Part of the book series: New ICMI Study Series (NISS, volume 15)

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Softcover Book USD 49.99
Price excludes VAT (USA)
Hardcover Book USD 59.99
Price excludes VAT (USA)

Table of contents (20 chapters)

  1. Front Matter

    Pages i-xii
  2. Aspects of Proof in Mathematics Education

    • Gila Hanna, Michael de Villiers
    Pages 1-10
  3. Proof and Cognition

    1. Front Matter

      Pages 11-11
    2. Cognitive Development of Proof

      • David Tall, Oleksiy Yevdokimov, Boris Koichu, Walter Whiteley, Margo Kondratieva, Ying-Hao Cheng
      Pages 13-49
    3. Theorems as Constructive Visions

      • Giuseppe Longo
      Pages 51-66
  4. Experimentation: Challenges and Opportunities

    1. Front Matter

      Pages 67-67
    2. Experimental Approaches to Theoretical Thinking: Artefacts and Proofs

      • Ferdinando Arzarello, Maria Giuseppina Bartolini Bussi, Allen Yuk Lun Leung, Maria Alessandra Mariotti, Ian Stevenson
      Pages 97-143
  5. Historical and Educational Perspectives of Proof

    1. Front Matter

      Pages 145-145
    2. Why Proof? A Historian’s Perspective

      • Judith V. Grabiner
      Pages 147-167
    3. Conceptions of Proof – In Research and Teaching

      • Richard Cabassut, AnnaMarie Conner, Filyet Aslı İşçimen, Fulvia Furinghetti, Hans Niels Jahnke, Francesca Morselli
      Pages 169-190
    4. Forms of Proof and Proving in the Classroom

      • Tommy Dreyfus, Elena Nardi, Roza Leikin
      Pages 191-213
    5. The Need for Proof and Proving: Mathematical and Pedagogical Perspectives

      • Orit Zaslavsky, Susan D. Nickerson, Andreas J. Stylianides, Ivy Kidron, Greisy Winicki-Landman
      Pages 215-229
  6. Proof in the School Curriculum

    1. Front Matter

      Pages 259-259
    2. From Exploration to Proof Production

      • Feng-Jui Hsieh, Wang-Shian Horng, Haw-Yaw Shy
      Pages 279-303
    3. Principles of Task Design for Conjecturing and Proving

      • Fou-Lai Lin, Kai-Lin Yang, Kyeong-Hwa Lee, Michal Tabach, Gabriel Stylianides
      Pages 305-325
    4. Teachers’ Professional Learning of Teaching Proof and Proving

      • Fou-Lai Lin, Kai-Lin Yang, Jane-Jane Lo, Pessia Tsamir, Dina Tirosh, Gabriel Stylianides
      Pages 327-346
  7. Argumentation and Transition to Tertiary Level

    1. Front Matter

      Pages 347-347

About this book


One of the most significant tasks facing mathematics educators is to understand the role of mathematical reasoning and proving in mathematics teaching, so that its presence in instruction can be enhanced. This challenge has been given even greater importance by the assignment to proof of a more prominent place in the mathematics curriculum at all levels.

Along with this renewed emphasis, there has been an upsurge in research on the teaching and learning of proof at all grade levels, leading to a re-examination of the role of proof in the curriculum and of its relation to other forms of explanation, illustration and justification.

This book, resulting from the 19th ICMI Study, brings together a variety of viewpoints on issues such as:

  • The potential role of reasoning and proof in deepening mathematical understanding in the classroom as it does in mathematical practice.

  • The developmental nature of mathematical reasoning and proof in teaching and learning from the earliest grades.
  • The development of suitable curriculum materials and teacher education programs to support the teaching of proof and proving.

The book considers proof and proving as complex but foundational in mathematics. Through the systematic examination of recent research this volume offers new ideas aimed at enhancing the place of proof and proving in our classrooms.


  • ICMI 19
  • mathematical reasoning
  • mathematics teaching
  • proof
  • proving
  • Open access

Editors and Affiliations

  • Ontario Institute for Studies in Education (OISE), University of Toronto, Toronto, Canada

    Gila Hanna

  • Faculty of Education, University of KwaZulu-Natal, Ashwood, South Africa

    Michael de Villiers

About the editors

Gila Hanna is Professor Emeritus at the University of Toronto. She taught graduate courses in mathematics education and in measurement and evaluation; she also led a large number of research projects and supervised several Ph.D. and M.A. theses in mathematics education. Her primary research interests are the role of proof and truth in mathematics and mathematics education, and gender issues in mathematics teaching and learning. She has been Convenor of the International Organisation of Women and Mathematics Education (1988-1992), Vice-Chair of the Canadian Mathematics Education Study Group (1986-1990), and has served on several Social Sciences and Humanities Research Council of Canada (SSHRC) adjudication committees and on the scientific program committees of a number of international conferences on mathematics education. From 1989 to 2000 she was Co-Editor of the international journal Educational Studies in Mathematics and is now one of its Advisory Editors. From 2000 to 2007 she was Co-Founder and Co-Editor of the Canadian Journal of Science, Mathematics and Technology Education. She served as Co-Chair of ICMI Study 19 on proof and proving in mathematics education. She has published extensively on proof and other aspects of mathematics education, and has delivered lectures at several universities as well as at numerous international conferences on mathematics education. In 2003 she was appointed “Fields Institute Fellow”, a lifetime appointment conferred on certain individuals who have made outstanding contributions to the Fields Institute for Research in Mathematical Sciences, its programs, and to the Canadian mathematical community.

After teaching mathematics and science at high school for a few years, Michael de Villiers, received the Science Teacher of the Year award in South Africa in 1983. Later that same year he was appointed as a researcher at the Research Unit for Mathematics Education of the University of Stellenbosch, South Africa. He has been at the University of Durban-Westville since 1991 (Univ. of KwaZulu-Natal since 2004) where he teaches both under-graduate and graduate courses in mathematics & mathematics education. He has published 7 books and over 170 articles peer reviewed articles in mathematics and mathematics education, many of which have been in international journals. Two special points in a triangle have been named after him. 

Between 1988-1997, he was editor of Pythagoras, the official research journal of the Association for Mathematics Education of South Africa (AMESA). He is a regular speaker at local and international conferences on mathematics and mathematics education. His main research interests are Geometry, Proof, Applications and Modelling, and the History and Philosophy of Mathematics. Since 1997, he has also been vice-chair of the SA Mathematics Olympiad Committee which is responsible for setting and organizing the nationwide South African Mathematics Olympiad, seeing a growth of participation from less than 10000 to over 60000 currently.

Bibliographic Information

  • Book Title: Proof and Proving in Mathematics Education

  • Book Subtitle: The 19th ICMI Study

  • Editors: Gila Hanna, Michael de Villiers

  • Series Title: New ICMI Study Series

  • DOI:

  • Publisher: Springer Dordrecht

  • eBook Packages: Humanities, Social Sciences and Law, Education (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s) 2012

  • Hardcover ISBN: 978-94-007-2128-9Published: 17 February 2012

  • Softcover ISBN: 978-94-017-8066-7Published: 16 April 2014

  • eBook ISBN: 978-94-007-2129-6Published: 14 June 2012

  • Series ISSN: 1387-6872

  • Series E-ISSN: 2215-1745

  • Edition Number: 1

  • Number of Pages: XII, 475

  • Number of Illustrations: 120 b/w illustrations

  • Topics: Mathematics Education

Buying options

Softcover Book USD 49.99
Price excludes VAT (USA)
Hardcover Book USD 59.99
Price excludes VAT (USA)