Proof and Proving in Mathematics Education New ICMI Study Series VOLUME 15 Published under the auspices of the International Commission on Mathematical Instruction under the general editorship of Bill Barton, President Jaime Carvalho e Silva, Secretary-General For further volumes: http://www.springer.com/series/6351 Information on the ICMI Study programme and on the resulting publications can be obtained at the ICMI website http://www.mathunion.org/ICMI/ or by contacting the ICMI Secretary- General, whose email address is available on that website. Gila Hanna • Michael de Villiers Editors Proof and Proving in Mathematics Education The 19th ICMI Study Editors Gila Hanna Ontario Institute for Studies in Education (OISE) University of Toronto Toronto, ON, Canada gila.hanna@utoronto.ca Michael de Villiers Faculty of Education Edgewood Campus University of KwaZulu-Natal South Africa profmd@mweb.co.za ISSN 1387-6872 ISBN 978-94-007-2128-9 e-ISBN 978-94-007-2129-6 DOI 10.1007/978-94-007-2129-6 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2011940425 The images or other third party material in this book may be included in the book’s Creative Commons license, unless indicated otherwise in a credit line to the material or in the Correction Note appended to the book. For details on rights and licenses please read the Correction https://doi.org/ 10.1007/978-94-007- 2129-6_20. 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You do not have permission under this licence to share adapted material derived from this book or parts of it. v Contents 1 Aspects of Proof in Mathematics Education ......................................... 1 Gila Hanna and Michael de Villiers Part I Proof and Cognition 2 Cognitive Development of Proof ............................................................ 13 David Tall, Oleksiy Yevdokimov, Boris Koichu, Walter Whiteley, Margo Kondratieva, and Ying-Hao Cheng 3 Theorems as Constructive Visions ......................................................... 51 Giuseppe Longo Part II Experimentation: Challenges and Opportunities 4 Exploratory Experimentation: Digitally-Assisted Discovery and Proof ................................................................................ 69 Jonathan Michael Borwein 5 Experimental Approaches to Theoretical Thinking: Artefacts and Proofs ............................................................................... 97 Ferdinando Arzarello, Maria Giuseppina Bartolini Bussi, Allen Yuk Lun Leung, Maria Alessandra Mariotti, and Ian Stevenson Part III Historical and Educational Perspectives of Proof 6 Why Proof? A Historian’s Perspective .................................................. 147 Judith V. Grabiner 7 Conceptions of Proof – In Research and Teaching .............................. 169 Richard Cabassut, AnnaMarie Conner, Filyet Aslı İş çimen, Fulvia Furinghetti, Hans Niels Jahnke, and Francesca Morselli vi Contents 8 Forms of Proof and Proving in the Classroom ..................................... 191 Tommy Dreyfus, Elena Nardi, and Roza Leikin 9 The Need for Proof and Proving: Mathematical and Pedagogical Perspectives ................................................................. 215 Orit Zaslavsky, Susan D. Nickerson, Andreas J. Stylianides, Ivy Kidron, and Greisy Winicki-Landman 10 Contemporary Proofs for Mathematics Education ............................. 231 Frank Quinn Part IV Proof in the School Curriculum 11 Proof, Proving, and Teacher-Student Interaction: Theories and Contexts ............................................................................ 261 Keith Jones and Patricio Herbst 12 From Exploration to Proof Production ................................................. 279 Feng-Jui Hsieh, Wang-Shian Horng, and Haw-Yaw Shy 13 Principles of Task Design for Conjecturing and Proving .................... 305 Fou-Lai Lin, Kai-Lin Yang, Kyeong-Hwa Lee, Michal Tabach, and Gabriel Stylianides 14 Teachers’ Professional Learning of Teaching Proof and Proving ................................................................................... 327 Fou-Lai Lin, Kai-Lin Yang, Jane-Jane Lo, Pessia Tsamir, Dina Tirosh, and Gabriel Stylianides Part V Argumentation and Transition to Tertiary Level 15 Argumentation and Proof in the Mathematics Classroom ................. 349 Viviane Durand-Guerrier, Paolo Boero, Nadia Douek, Susanna S. Epp, and Denis Tanguay 16 Examining the Role of Logic in Teaching Proof ................................... 369 Viviane Durand-Guerrier, Paolo Boero, Nadia Douek, Susanna Epp, and Denis Tanguay 17 Transitions and Proof and Proving at Tertiary Level .......................... 391 Annie Selden Part VI Lessons from the Eastern Cultural Traditions 18 Using Documents from Ancient China to Teach Mathematical Proof ................................................................................ 423 Karine Chemla vii Contents 19 Proof in the Western and Eastern Traditions: Implications for Mathematics Education ............................................. 431 Man Keung Siu Appendix 1: ICMI Study 19: Proof and Proving in Mathematics Education: Discussion Document ............................... 443 Appendix 2: Conference Proceedings: Table of Contents ........................... 453 Author Index .................................................................................................... 461 Subject Index ................................................................................................... 471 Correction to: Proof and Proving in Mathematics Education ................... C1 ix Contributors Ferdinando Arzarello Department of Mathematics , University of Torino , Torino , Italy, ferdinando.arzarello@unito.it Maria Giuseppina Bartolini Bussi Department of Mathematics, University of Modena and Reggio Emilia (UNIMORE) , Modena , Italy, bartolini@unimore.it Paolo Boero Dipartimento di Matematica , Università di Genova , Genova, Italia, boero@dima.unige.it Jonathan Michael Borwein Centre for Computer-Assisted Research Mathematics and its Applications, CARMA, University of Newcastle, Callaghan, NSW 2308, Australia, jonathan.borwein@newcastle.edu.au Richard Cabassut LDAR Laboratoire de Didactique AndrØ Revuz , Paris 7 University, Paris , France. IUFM Institut Universitaire de Formation des Maîtres , Strasbourg University, Strasbourg , France, richard.cabassut@unistra.fr Karine Chemla CNRS, UniversitØ Paris Diderot, Sorbonne Paris CitØ, Research Unit SPHERE, team REHSEIS, UMR 7219, CNRS, F-75205 Paris, France, chemla@univ-paris-diderot.fr Ying-Hao Cheng Department of Mathematics , Taipei Municipal University of Education , Taipei , Taiwan, yinghao.cheng@msa.hinet.net AnnaMarie Conner Department of Mathematics & Science Education , University of Georgia, Athens, GA, USA, aconner@uga.edu Michael de Villiers School of Science, Mathematics & Technology Education , University of KwaZulu-Natal , Durban , South Africa, profmd@mweb.co.za Nadia Douek Institut Universitaire de Formation des Maîtres , UniversitØ de Nice , Nice , France, ndouek@wanadoo.fr x Contributors Tommy Dreyfus Department of Mathematics, Science and Technology Education, School of Education , Tel Aviv University, Tel-Aviv, Israel, tommyd@post.tau.ac.il Viviane Durand-Guerrier DØpartement de mathØmatiques , I3M, UMR 5149, UniversitØ Montpellier 2 , Montpellier, France, vdurand@math.univ-montp2.fr Susanna S. Epp Department of Mathematical Sciences , DePaul University, Chicago , IL , USA, sepp@depaul.edu Fulvia Furinghetti Dipartimento di Matematica , Università di Genova , Genova , Italy, furinghe@dima.unige.it Judith V. Grabiner Department of Mathematics, Pitzer College, Claremont, CA, USA, jgrabiner@pitzer.edu Gila Hanna Ontario Institute for Studies in Education , University of Toronto , Toronto , Canada, gila.hanna@utoronto.ca Patricio Herbst School of Education, University of Michigan, Ann Arbor, MI, USA, pgherbst@umich.edu Wang-Shian Horng Department of Mathematics , National Taiwan Normal University, Taipei , Taiwan, horng@math.ntnu.edu.tw Feng-Jui Hsieh Department of Mathematics , National Taiwan Normal University, Taipei , Taiwan, hsiehfj@math.ntnu.edu.tw Filyet Aslı İş çimen Department of Mathematics and Statistics , Kennesaw State University, Kennesaw, GA , USA, ersozas@yahoo.com Hans Niels Jahnke Fakultät für Mathematik , Universität Duisburg-Essen , Essen , Germany, njahnke@uni-due.de Keith Jones Mathematics and Science Education Research Centre, School of Education , University of Southampton , Highfield , Southampton , UK, d.k.jones@soton.ac.uk Ivy Kidron Applied Mathematics Department , Jerusalem College of Technology (JCT) , Jerusalem , Israel, ivy@jct.ac.il Boris Koichu Department of Education in Technology and Science , Technion – Israel Institute of Technology, Haifa , Israel, bkoichu@technion.ac.il Margo Kondratieva Faculty of Education and Department of Mathematics and Statistics , Memorial University of Newfoundland , St. John’s , Canada, margo. kon@gmail.com Kyeong-Hwa Lee Department of Mathematics Education , Seoul National University, Seoul , South Korea, khmath@snu.ac.kr Roza Leikin Department of Mathematics Education, Faculty of Education , University of Haifa , Haifa , Israel, rozal@construct.haifa.ac.il xi Contributors Allen Yuk Lun Leung Department of Education Studies , Hong Kong Baptist University, Kowloon Tong , Hong Kong, aylleung@hkbu.edu.hk Fou-Lai Lin Department of Mathematics , National Taiwan Normal University, Taipei , Taiwan, linfl@math.ntnu.edu.tw Jane-Jane Lo Department of Mathematics, Western Michigan University, Kalamazoo , MI , USA, jane-jane.lo@wmich.edu Giuseppe Longo CNRS – École Normale SupØrieure et CREA, École Polytechnique , Paris , France, longo@di.ens.fr Maria Alessandra Mariotti Department of Mathematics and Computer Science , University of Siena , Siena , Italy, mariotti21@unisi.it Francesca Morselli Dipartimento di Matematica , Università di Genova , Genoa , Italy, morselli@dima.unige.it Elena Nardi School of Education , University of East Anglia , Norwich , UK, e.nardi@uea.ac.uk Susan D. Nickerson Department of Mathematics and Statistics , San Diego State University, San Diego , CA , USA, snickers@sciences.sdsu.edu Judy-anne Osborn Centre for Computer Assisted Mathematics and Its Applications, School of Mathematical and Physical Sciences, University of Newcastle , Callaghan , NSW, Australia, Judy-anne.Osborn@anu.edu.au Frank Quinn Virginia Tech, Blacksburg, VA 24061, USA, quinn@math.vt.edu Annie Selden Department of Mathematical Sciences , New Mexico State University, Las Cruces , NM , USA, js9484@usit.net Haw-Yaw Shy Department and Graduate Institute of Mathematics , National Changhua University of Educatio, Changua, Taiwan, shy@cc.ncue.edu.tw Man Keung Siu Department of Mathematics , University of Hong Kong , Hong Kong SAR , China, mathsiu@hkucc.hku.hk Ian Stevenson Department of Education and Professional Studies , King’s College , London , UK, ian.stevenson@kcl.ac.uk Andreas J. Stylianides Faculty of Education , University of Cambridge , Cambridge , UK, as899@cam.ac.uk Gabriel Stylianides Department of Education , University of Oxford , Oxford , UK, gabriel.stylianides@education.ox.ac.uk Michal Tabach School of Education , Tel-Aviv University, Tel-Aviv, Israel, tabach.family@gmail.com David Tall Mathematics Education Research Centre , University of Warwick , Coventry, UK, david.tall@warwick.ac.uk xii Contributors Denis Tanguay DØpartement de mathØmatiques , UniversitØ du QuØbec à MontrØal (UQAM) , Montreal , QC , Canada, tanguay.denis@uqam.ca Dina Tirosh School of Education , Tel-Aviv University, Tel-Aviv, Israel, dina@post.tau.ac.il Pessia Tsamir School of Education , Tel-Aviv University, Tel-Aviv, Israel, pessia@post.tau.ac.il Greisy Winicki-Landman Department of Mathematics and Statistics , California State Polytechnic University, Pomona , CA , USA, greisyw@csupomona.edu Walter Whiteley Department of Mathematics and Statistics , York University, Toronto , Canada, whiteley@mathstat.yorku.ca Kai-Lin Yang Department of Mathematics , National Taiwan Normal University, Taipei , Taiwan, kailinyang3@yahoo.com.tw Oleksiy Yevdokimov Department of Mathematics & Computing , University of Southern Queensland , Toowoomba , Australia, oleksiy.yevdokimov@usq.edu.au Orit Zaslavsky Department of Teaching and Learning, New York University, New York, NY, USA Department of Education in Technology and Science, Technion – Israel, Institute of Technology, Haifa, Israel, oritrath@gmail.com and licenses please read the Correction https://doi.org/10.1007/978-94-007-2129-6_20 This book has been made open access under a CC BY-NC-ND 4.0 license. For details on rights 1 This volume, Proof and proving in mathematics education , is a Study Volume sponsored by the International Commission on Mathematical Instruction (ICMI). ICMI Studies explore specific topics of interest to mathematics educators; they aim at identifying and analysing central issues in the teaching and learning of these topics. To this end, the ICMI convenes a Study Conference on chosen topics: A group of scholars from the conference then prepares a Study Volume that reports on the outcomes of the conference. The present Study Volume examines several theoretical and practical notions about why and how mathematics educators should approach the teaching and learning of proof and proving. The authors of the chapters here are presenting major themes and subthemes that arose from the presentations and discussions at the 19th ICMI Study Conference. 1 ICMI Study 19 The 19th ICMI Study, intended to examine issues of proof and proving in mathematics education, was officially launched in 2007 with the selection of Gila Hanna and Michael de Villiers as Co-Chairs. In consultation with them, the ICMI Executive invited eight additional experts in the field of proof in mathematics education to G. Hanna ( * ) Ontario Institute for Studies in Education , University of Toronto , Toronto , Canada e-mail: gila.hanna@utoronto.ca M. de Villiers School of Science, Mathematics & Technology Education , University of KwaZulu-Natal , Durban , South Africa e-mail: profmd@mweb.co.za Chapter 1 Aspects of Proof in Mathematics Education Gila Hanna and Michael de Villiers © The Author(s) 2021 G. Hanna and M. de Villiers (eds.), Proof and Proving in Mathematics Education , New ICMI Study Series, https://doi.org/10.1007/978-94-007-2129-6_1 and licenses please read the Correction https://doi.org/10.1007/978-94-007-2129-6_20 This book has been made open access under a CC BY-NC-ND 4.0 license. For details on rights 2 G. Hanna and M. de Villiers serve on an International Program Committee (IPC). The Co-Chairs prepared a draft Discussion Document, circulated it to the entire IPC, and then revised it in light of the IPC members’ input. At its first meeting (Essen, Germany, November 2007), the IPC settled on the themes of the Study and finalised the Discussion Document, which was later published in the Bulletin of the International Commission on Mathematical Instruction as well as in a number of mathematics education jour- nals (see Appendix 1). Clearly, we could not include in a single ICMI Study all the themes germane to the teaching of proof. Thus the IPC originally selected seven themes that it judged to be most relevant to mathematics education and within the IPC mem- bers’ realm of expertise. The Discussion Document called for contributions that would address these themes and contained a list of criteria by which the contributions would be assessed. At its second meeting (Sèvres, France, November 2008), the IPC selected the contributions that had been recommended by reviewers after a strict refereeing process and that were also most closely related to the conference themes. (Unfortunately a few excellent submissions had to be excluded because they treated themes beyond the conference’s scope.) The IPC then drew up an invitation list of about 120 contributors. Taking into account the submissions that had been accepted, the IPC developed a programme that included only six of the original seven themes. Each of these themes was the focus of a Working Group (WG) that met throughout the Study Conference and whose major aim was to prepare one or more chapters for this book. WG1: Cognitive Development of Proof , co-chaired by David Tall and Oleksiy Yevdokimov, focused on the characteristics of the cognitive development of proof at various school levels, with a view to building an overall picture of the cognitive development of proof. WG2: Argumentation , chaired by Viviane Durand-Guerrier, focused on the relation- ship between proof and argumentation from the perspective of opposing qualities such as formal vs. informal, form vs. content, syntax vs. semantics, truth vs. valid- ity, mathematical logic vs. common sense, formal proof vs. heuristics, and continuity vs. discontinuity. WG3: Dynamic Geometry Software/Experimentation , chaired by Ferdinando Arzarello, focused on the ways in which mathematical investigations using advanced technology and different semiotic resources relate to the formal aspects of mathe- matical discourse and to the production of proofs. WG4: Proof in the School Curriculum, Knowledge for Teaching Proof, and the Transition from Elementary to Secondary , chaired by Fou-Lai Lin, focused on the knowledge that teachers need to teach proof effectively and on how proving activities should be designed to best foster successful instruction about proof and proving. WG5: The Nature of Proof for the Classroom , co-chaired by Tommy Dreyfus, Hans Niels Jahnke, and Wann-Sheng Horng, examined aspects of the teaching of proof 3 1 Aspects of Proof in Mathematics Education from the primary through the tertiary level. It addressed questions about the form, status, and role that proof must assume at each level to ensure success in generating mathematical understanding. WG6: Proof at the Tertiary Level , chaired by Annie Selden, explored all aspects of the teaching and learning of proof and proving at the tertiary level, including the transition from secondary school to university and the transition from undergradu- ate to graduate work in mathematics. Complementary to the Working Groups, the IPC broadened the Study’s scope by inviting four distinguished scholars to deliver plenary talks on topics related to proof in mathematics, but not necessarily intimately connected to mathematics education. In their talks, Giuseppe Longo, Jonathan Borwein, Judith Grabiner and Frank Quinn examined proof from the four perspectives of epistemology, experimental mathematics, the history of mathematics, and mathematics itself. The IPC also invited a panel of eminent experts, Karine Chemla, Wann-Sheng Horng and Man Keung Siu to discuss proof as perceived in ancient Chinese mathematics writing. The ICMI Study Conference itself took place at the National Taiwan Normal University in Taipei, Taiwan, from May 10 to May 15, 2009 (see Appendix 2). 2 Contents of the Volume A common view of mathematical proof sees it as no more than an unbroken sequence of steps that establish a necessary conclusion, in which every step is an application of truth-preserving rules of logic. In other words, proof is often seen as synonymous with formal derivation. This Study Volume treats proof in a broader sense, recognis- ing that a narrow view of proof neither reflects mathematical practice nor offers the greatest opportunities for promoting mathematical understanding. In mathematical practice, in fact, a proof is often a series of ideas and insights rather a sequence of formal steps. Mathematicians routinely publish proofs that contain gaps, relying on the expert reader to fill them in. Many published proofs are informal arguments, in effect, but are still considered rigorous enough to be accepted by mathematicians. This Volume examines aspects of proof that include, but are not limited to, explo- rations, explanations, justification of conjectures and definitions, empirical reasoning, diagrammatic reasoning, and heuristic devices. The chapter authors, whilst by and large accepting the common view of proof, do diverge on the importance they attach to various aspects of proof and particularly on the degree to which they judge formal derivation as necessary or useful in promoting an understanding of mathematics and mathematical reasoning. The remainder of the Volume is divided into six parts. These are arranged according to major themes that arose from the conference as a whole, rather than by working groups. 4 G. Hanna and M. de Villiers Part I: Proof and Cognition In Chap. 2, “Cognitive development of proof” David Tall, Oleksiy Yevdokimov, Boris Koichu, Walter Whiteley, Margo Kondratieva and Ying-Hao Cheng examine the development of proof from the child to the adult learner and on to the mature research mathematician. The authors first consider various existing theories and viewpoints relating to proof and proving from education research, brain research, cognitive science, psychology, semiotics, and more, and then go on to offer their own theory of “the broad maturation of proof structures”. Their resulting framework for the broad maturation of proof structures consists of six developmental stages which they illustrate with an interesting array of well-chosen examples. They also appropriately elaborate on the novel notion of a “crystalline concept” which they defi ne as “a concept that has an internal structure of constrained relationships that cause it to have necessary properties as part of its context.” In his plenary chapter “Theorems as constructive visions” Giuseppe Longo describes mathematics and proofs as conceptual constructions that, though sup- ported by language and logic, originate in the real activities of humans in space and time. He points out in particular the crucial role of cognitive principles such as symmetry and order in attaining mathematical knowledge and understanding proof, citing several examples to show that in constructing a proof the notions of symmetry and order derived from actual experience are no less essential than logi- cal inference. He concludes that “Mathematics is the result of an open-ended ‘game’ between humans and the world in space and time; that is, it results from the inter-subjective construction of knowledge made in language and logic, along a passage through the world, which canalises our praxes as well as our endeavour towards knowledge.” Part II: Experimentation: Challenges and Opportunities Mathematical researcher Jonathan Borwein, in his plenary chapter “Exploratory experimentation: Digitally-assisted discovery and proof” argues that current com- puting technologies offer revolutionary new scaffolding both to enhance mathemat- ical reasoning and to restrain mathematical error. He shares Pólya’s view that intuition, enhanced by experimentation, mostly precedes deductive reasoning. He then gives and discusses some illustrative examples, which clearly show that the bound- aries between mathematics and the natural sciences, and between inductive and deductive reasoning, are blurred and getting more blurred. Borwein points out that the mathematical community faces a great challenge to re-evaluate the role of proof in light of the power of computer systems, the sophis- tication of mathematical computing packages, and the growing capacity to data- mine on the Internet. As the prospects for inductive mathematics blossom, the need to ensure that the role of proof is properly founded remains undiminished. 5 1 Aspects of Proof in Mathematics Education The chapter “Experimental approaches to theoretical thinking: Artefacts and proofs” by Ferdinando Arzarello, Mariolina Bartolini Bussi, Andy Leung, Maria Alessandra Mariotti, and Ian Stevenson examines the dynamic tension between the empirical and the theoretical aspects of mathematics, especially in relation to the role of technological artefacts in both. It does so against the background of offering teach- ers a comprehensive framework for pursuing the learning of proof in the classroom. The authors discuss and analyse their subject from different linked perspectives: historical, epistemological, didactical and pedagogical. They first present examples of the historical continuity of experimental mathematics from straight-edge and compass construction to the modern use of different dynamic mathematics soft- ware. They draw these examples from a few different cultures and epochs in which instruments have played a crucial role in generating mathematical concepts, theo- rems and proofs. Second, the authors analyse some didactical episodes from the classroom, where the use of instruments in proving activities makes the aforementioned dynamic ten- sion explicit. Specifically, they examine how this tension regulates students’ cogni- tive processes in solving mathematical problems, first making explorations with technological tools, then formulating suitable conjectures and finally proving them. The chapter is followed by a commentary “Response to Experimental approaches to theoretical thinking: Artefacts and proofs” by Jonathan Borwein and Judy- anne Osborn. Part III: Historical and Educational Perceptions of Proof In her plenary address “Why proof? A historian’s perspective,” historian of mathemat- ics Judith Grabiner traces some of the main aspects of the history of mathematical proof in the Western tradition. She first addresses the birth of logical proof in Greek geometry and why the Greeks moved beyond visualisation to purely logical proof. Then she looks at the use of visual demonstration in Western mathematics since the Greeks, and proceeds to discuss two characteristics of more modern mathematics, abstraction and symbolism, and their power. There follows a discussion of how and why standards of proof change, noting in particular the influence of ideas from phi- losophy. Finally, the author discusses how proof in mathematics interacts with the ‘real world’, arguing that proof did not develop in a cultural or intellectual vacuum. In the chapter “Conceptions of proof – In research and in teaching”, Richard Cabassut, AnnaMarie Conner, Filyet Aslı İş çimen, Fulvia Furinghetti, Hans Niels Jahnke and Francesca Morselli describe mathematicians’ conceptualisations of proof and contrast them with those of mathematics educators. The authors argue that practising mathematicians do not rely on any specific formal definition of proof but they do seem to know what a proof is. On the other hand, mathematics educa- tors’ conceptions of proof derive from the need to teach students to construct proper proofs and to recognise the subtle differences between argumentation and mathematical proof. The authors then discuss the ideas of “genetic”, “pragmatic,” 6 G. Hanna and M. de Villiers and “conceptual” proofs. They next examine in detail some epistemological and pedagogical beliefs about the nature and role of proof in mathematics, about the role of proof in school mathematics, about difficulties in proving, about how proof should be taught in school, and about the self as mathematical thinker in the context of proof. The authors conclude by discussing “metaknowledge about proof”, its importance and its role in the mathematics curriculum. Tommy Dreyfus, Elena Nardi and Roza Leikin review diverse forms of proofs in their chapter “Forms of proof and proving in the classroom”. Relying on many empirical studies presented at the ICMI 19 Conference and on published empirical research papers, they describe a variety of proofs (e.g., by visual, verbal, and dynamic representations) and an array of mathematical arguments (from example- based, deductive and inductive to generic and general). They discuss different degrees of rigour, where and how these are used, and the contexts in which they appear. The authors also report on students’ and teachers’ beliefs about various aspects of proof and proving. They discuss the pedagogical importance of multiple- proof tasks and of taking into account the mathematical, pedagogical, and cognitive structures related to the effective teaching of proof and proving. They conclude with a plea for additional empirical research, longitudinal studies, and investigations on the long-term effects of the different approaches to proof. In the chapter “The need for proof and proving: mathematical and pedagogical perspectives”, Orit Zaslavsky, Susan D. Nickerson, Andreas Stylianides, Ivy Kidron and Greisy Winicki-Landman explore three main questions: Why teach proof? What are (or may be) learners’ needs for proof? How can teachers facilitate the need for proof? First, they discuss the connection between different functions of proof in math- ematics and the needs those evoke for teaching proof. They briefly explore the epis- temology of proof in the history of mathematics in order to illuminate the needs that propelled the discipline’s development. Second, the authors take a learner’s per- spective on the need to prove, and examine categories of intellectual need that may drive learners to prove (i.e., needs for certitude, for understanding, for quantifica- tion, for communication, and for structure and connection). Finally, the authors address pedagogical issues involved in teachers’ attempts to facilitate learners’ need to prove; uncertainty, cognitive conflict or the need for explanation or organised structure may help drive learners to prove. In his plenary “Contemporary proofs for mathematics education”, Frank Quinn argues that the proofs encountered in mathematical practice provide a very high level of reliability, because the proof process creates a record sufficiently detailed to allow easy detection and repair of errors. He therefore recommends the introduction of two key ideas, “potential proof” and “formal potential proof,” into school mathematics and undergraduate mathematics education. The first entails asking students to show their work – that is, to provide a detailed record of their solution so that it can be checked for errors. The second means asking students to supply explicit explanations to justify their work. Citing several examples, Quinn argues that students should thereby learn that a train of reasoning leading to a correct conclusion does not count as a proof unless it is a potential proof that has been found to be error-free. 7 1 Aspects of Proof in Mathematics Education Part IV: Proof in the School Curriculum Keith Jones and Patricio Herbst, in their chapter “Proof, proving, and teacher- student interaction: Theories and contexts”, seek to identify theoretical frameworks that would help understand the teacher’s role in proof education. They focus on three theories that might shed light on teacher-student interaction in teaching of proof across diverse contexts. They first discuss the theory of socio-mathematical norms, characterised by inquiry-based mathematics classrooms and the use of class- room interactions to arrive at shared norms of mathematical practice. Second comes the theory of teaching with variation, in which the teacher uses two types of varia- tions: conceptual variation (highlighting a new concept by contrasting inadmissible examples), and procedural variation (refocusing the learner’s attention from a con- crete problem to its symbolic representation). Third, the authors examine the theory of instructional exchanges that, borrowing from Brousseau’s notion a “didactical contract” presumes that teacher and students are mutually responsible for whatever learning takes place in the classroom. Feng-Jui Hsieh, Wang-Shian Horng and Haw-Yaw Shy, in “From exploration to proof production”, explain how exploration, especially hands-on exploration, is introduced and integrated into the teaching of proof in Taiwan. They describe a conceptual model for the relationship between exploration, problem solving, prov- ing and proof, and illustrate it with two exploratory teaching experiments. The authors distinguish two different positions in regard to “exploration” as a learning and conceptualising activity. The first position views exploration as a men- tal process, the second, as an activity that involves manipulating and interacting with external environments (e.g., hands-on or dynamic computer software environ- ments). Exploration generally provides learners with valuable opportunities to con- struct mathematics objects, transform figures, probe in multiple directions, perceive divergent visual information, and receive immediate feedback on their actions. The authors also give two extracts from a Taiwanese textbook, which demonstrate the integration of exploration in proving. Last, they provide a useful but tentative com- parison of dynamic computer and hands-on explorations, and summarise some of the positive and negative issues raised by integrating exploration, as well as suggest- ing areas for future research. Fou-Lai Lin, Kyeong-Hwa Lee, Kai-Lin Yang, Michal Tabach and Gabriel Stylianides develop some principles for designing tasks that teach conjecturing and proving in the chapter “Principles of task design for conjecturing and proving”. They extract some first principles from design research and the literature for design- ing tasks for mathematics learning generally. They also briefly reflect on a few his- torical examples, such as Fermat and Poincaré’s conjectures, within the context of Lakatos’ model. They discuss the strategy of promoting ‘what-if-not’ questions, which encourage students to conjecture the consequence of some change in a statement’s premise or conclusion or to explore the transformation and application of algorithms and for- mulae in other areas. They also explore students’ attainment of conviction and