Theorems in School: From History, Epistemology and Cognition to Classroom Practice

(New Directions in Mathematics and Science Education)

 P. Boero

Sense Publishers | 2007 | 335 páginas | pdf | 3 Mb

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During the last decade, a revaluation of proof and proving within mathematics curricula was recommended; great emphasis was put on the need of developing proof-related skills since the beginning of primary school. This book, addressing mathematics educators, teacher-trainers and teachers, is published as a contribution to the endeavour of renewing the teaching of proof (and theorems) on the basis of historical-epistemological, cognitive and didactical considerations. Authors come from eight countries and different research traditions: this fact offers a broad scientific and cultural perspective. In this book, the historical and epistemological dimensions are dealt with by authors who look at specific research results in the history and epistemology of mathematics with an eye to crucial issues related to educational choices. Two papers deal with the relationships between curriculum choices concerning proof (and the related implicit or explicit epistemological assumptions and historical traditions) in two different school systems, and the teaching and learning of proof there. The cognitive dimension is important in order to avoid that the didactical choices do not fit the needs and the potentialities of learners. Our choice was to firstly deal with the features of reasoning related to proof, mainly concerning the relationships between argumentation and proof. The second part of this book concentrates on some crucial cognitive and didactical aspects of the development of proof from the early approach in primary school, to high school and university. We will show how suitable didactical proposals within appropriate educational contexts can match the great (yet, underestimated!) young students' potentialities in approaching theorems and theories.

CONTENTS
Preface
The ongoing value of proof
Gila Hanna 3
Introduction
Theorems in school: An introduction
Paolo Boero 19
Part I: The historical and epistemological dimension
1 Origin of mathematical proof: History and epistemology
Gilbert Arsac 27
2 The proof in the 20th century: From Hilbert to automatic theorem proving
Ferdinando Arzarello 43
3 Students’ proof schemes revisited
Guershon Harel 65
Part II: Curricular choices, historical traditions and learning of proof: Two national case studies
4 Curriculum change and geometrical reasoning
Celia Hoyles and Lulu Healy 81
5 The tradition and role of proof in mathematics education in Hungary
Julianna Szendrei-Radnai and Judit Török 117
Part III: Argumentation and proof
6 Cognitive functioning and the understanding ofmathematical processes of proof
Raymond Duval 137
7 Some remarks about argumentation and proof
Nadia Douek 163
Part IV: Didactical aspects
8 Making possible the discussion of “impossible in mathematics”
Greisy Winicki-Landman 185
9 The development of proof making by students
Carolyn A. Maher, Ethel M. Muter and Regina D. Kiczek 197
10 Approaching and developing the culture of geometry theorems in school: A theoretical framework
Marolina Bartolini Bussi, Paolo Boero, Franca Ferri, Rossella Garuti and Maria Alessandra Mariotti 211
11 Construction problems in primary school: A case from the geometry of circle
Maria G. Bartolini Bussi, Mara Boni and Franca Ferri 219
12 Approaching theorems in grade VIII: Some mental processes underlying producing and proving
conjectures, and conditions suitable to enhance them
Paolo Boero, Rossella Garuti and Enrica Lemut 249
13 From dynamic exploration to “theory” and “theorems” (from 6th to 8th grades)
Laura Parenti, Maria Teresa Barberis, Massima
Pastorino and Paola Viglienzone 265
14 Geometrical proof: The mediation of a microworld
Maria Alessandra Mariotti 285
15 The transition to formal proof in geometry
Ferdinando Arzarello, Federica Olivero, Domingo Paola and Ornella Robutti 305