ISBN:
9789400721296
Language:
English
Pages:
Online-Ressource (XII, 475 p. 120 illus, digital)
Series Statement:
New ICMI Study Series 15
Series Statement:
SpringerLink
Series Statement:
Bücher
Parallel Title:
Buchausg. u.d.T.
Keywords:
Mathematics
;
Education
;
Education
;
Mathematics
;
Mathematics—Study and teaching .
Abstract:
1. Aspects of proof in mathematics education: Gila Hanna and Michael de Villiers -- Part I: Proof and cognition -- 2. Cognitive development of proof: David Tall, Oleksiy Yevdokimov, Boris Koichu, Walter Whiteley, Margo Kondratieva, and Ying-Hao Cheng -- 3. Theorems as constructive visions: Giuseppe Longo -- Part II: Experimentation: Challenges and opportunities -- 4. Exploratory experimentation: Digitally-assisted discovery and proof: Jonathan M. Borwein -- 5. Experimental approaches to theoretical thinking: Artefacts and proofs -- Ferdinando Arzarello, Maria Giuseppina Bartolini Bussi, Allen Leung, Maria Alessandra Mariotti, and Ian Stevenson (With response by J. Borwein and J. Osborn) -- Part III: Historical and educational perspectives of proof -- 6. Why proof? A historian’s perspective: Judith V. Grabiner -- 7. Conceptions of proof – in research and in teaching: Richard Cabassut, AnnaMarie Conner, Filyet Asli Ersoz, Fulvia Furinghetti, Hans Niels Jahnke, and Francesca Morselli -- 8. Forms of proof and proving in the classroom: Tommy Dreyfus, Elena Nardi, and Roza Leikin -- 9. The need for proof and proving: mathematical and pedagogical perspectives: Orit Zaslavsky, Susan D. Nickerson, Andreas Stylianides, Ivy Kidron, and Greisy Winicki -- 10. Contemporary proofs for mathematics education: Frank Quinn -- Part IV: Proof in the school curriculum -- 11. Proof, Proving, and teacher-student interaction: Theories and contexts: Keith Jones and Patricio Herbst -- 12. From exploration to proof production: Feng-Jui Hsieh, Wang-Shian Horng, and Haw-Yaw Shy -- 13. Principles of task design for conjecturing and proving: Fou-Lai Lin, Kyeong-Hwa Lee, Kai-Lin Yang, Michal Tabach, and Gabriel Stylianides -- 14. Teachers’ professional learning of teaching proof and proving: 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: Viviane Durand-Guerrier, Paolo Boero, Nadia Douek, Susanna Epp, and Denis Tanguay -- 16. Examining the role of logic in teaching proof: Viviane Durand-Guerrier, Paolo Boero, Nadia Douek, Susanna Epp, and Denis Tanguay -- 17. Transitions and proof and proving at tertiary level: Annie Selden -- Part VI: Lessons from the Eastern cultural traditions -- 18. Using documents from ancient China to teach mathematical proof: Karine Chemla -- 19. Proof in the Western and Eastern traditions: Implications for mathematics education: Man Keung Siu -- Acknowledgements -- Appendix 1: Discussion Document -- Appendix 2: Conference Proceedings: Table of contents -- Author Index -- Subject Index.
Abstract:
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.
Description / Table of Contents:
Proof and Provingin Mathematics Education; Contents; Contributors; Chapter 1: Aspects of Proof in Mathematics Education; 1 ICMI Study 19; 2 Contents of the Volume; 3 Conclusion; Part1: Proof and Cognition; Chapter 2: Cognitive Development of Proof; 1 Introduction; 2 Perceptions of Proof; 2.1 What Is Proof for Mathematicians?; 2.2 What Is Proof for Growing Individuals?; 3 Theoretical Framework; 3.1 Theories of Cognitive Growth; 3.2 Crystalline Concepts; 3.3 A Global Framework for the Development of Mathematical Thinking; 4 The Development of Proof from Embodiment
Description / Table of Contents:
4.1 From Embodiment to Verbalisation4.2 From Embodiment and Verbalisation to Pictorial and Symbolic Representations; 4.3 From Embodiment, Verbalisation and Symbolism to Deduction; 5 Euclidean and Non-Euclidean Proof; 5.1 The Development of Euclidean Geometry; 5.2 The Beginnings of Spherical and Non-Euclidean Geometries; 6 Symbolic Proof in Arithmetic and Algebra; 6.1 The Increasing Sophistication of Proof in Arithmetic and Algebra; 6.2 Proof by Contradiction and the Development of Aesthetic Criteria; 7 Axiomatic Formal Proof; 7.1 Student Development of Formal Proof
Description / Table of Contents:
7.2 Structure Theorems and New Forms of Embodiment and Symbolism in Research Mathematics8 Summary; References*; Chapter 3: Theorems as Constructive Visions; 1 The Constructive Content of Euclid's Axioms; 2 From Axioms to Theorems; 3 On Intuition; 4 Little Gauss' Proof; 4.1 Arithmetic Induction and the Foundation of Mathematical Proof; 4.2 Prototype Proofs; 5 Induction vs. Well-Ordering in Concrete Incompleteness Theorems; 6 The Origin of Logic; 7 Conclusion; References; Part2: Experimentation: Challenges and Opportunities
Description / Table of Contents:
Chapter 4: Exploratory Experimentation: Digitally-Assisted Discovery and Proof1 Digitally-Assisted Discovery and Proof; 1.1 Exploratory Experimentation; 1.2 Digitally Mediated Mathematics; 1.3 Experimental Mathodology; 1.3.1 What Is Experimental Mathematics?; 1.4 Cognitive Challenges; 1.5 Paradigm Shifts; 2 Mathematical Examples; Example I: What Did the Computer Do?; Example II: What Is That Number?; Example III: From Discovery to Proof; Example IV: From Concrete to Abstract; Example V: A Dynamic Discovery and Partial Proof; Example VI: Knowledge Without Proof
Description / Table of Contents:
Example VII. A Mathematical Physics LimitExample VIII: Apéry's Formula; Example IX: When Is Easy Bad?; 3 Concluding Remarks; References; Chapter 5: Experimental Approaches to Theoretical Thinking: Artefacts and Proofs; 1 Introduction; 2 Part 1: From Straight-Edge and Compass to Dynamic Geometry Software; 2.1 Classical European Geometry; 2.2 The Modern Age in Europe; 2.3 Constructions with Straight-Edge and Compass in the Mathematics Classroom; 2.4 Constructions in a DGS; 2.5 DGS Constructions in the Classroom; 2.6 Experiments and Proofs with the Computer
Description / Table of Contents:
2.7 Implementation in Mathematics Classrooms
Note:
Description based upon print version of record
DOI:
10.1007/978-94-007-2129-6
URL:
Volltext
(kostenfrei)
URL:
Volltext
(lizenzpflichtig)
Permalink
