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The Fundamental Theorem of Calculus ISSN: 1402-1544 ISBN 978-91-86233-XX-X Se i listan och fyll i siffror där kryssen är DOCTORAL T H E SIS Anna Klisinska Department of Mathematics ISSN: 1402-1544 ISBN 978-91-86233-46-4 Theorem of Calculus: The Fundamental The Fundamental Luleå University of Technology 2009 Theorem of Calculus A case study into the didactic transposition of proof a case into study the didactic transposition of proof Anna Klisinska Luleå University of Technology The Fundamental Theorem of Calculus A case study into the didactic transposition of proof ANNA KLISIēSKA Tryck: Universitetstryckeriet, Luleå ISSN: 1402-1544 ISBN 978-91-86233-46-4 Luleå www.ltu.se To Marek Empty room Empty heart Since you've been gone I must move on iii iv ABSTRACT The relationship between academic mathematics as practiced by researchers at universities and classroom mathematics (the mathematical practices in classrooms in primary, lower and upper secondary education as well as in undergraduate university education) is a fundamental question in mathematics education. The focus of the study presented here is on how this relationship is seen from the perspective of mathematics education and by researching mathematicians, with a focus on proof. The Fundamental Theorem of Calculus (FTC) and its proof provide an illuminating but also curious example. The propositional content of the statements, which are connected to this name, varies. Consequently, also the proofs differ. The formulations of different versions of “the” FTC cannot be understood in isolation from its historical and institutional context. The study comprises a historical account of the invention of the FTC and its proof, including its appearance in calculus textbooks. Interviews with researching mathematicians from different sub-fields provide a picture of what meaning and relevance they attribute to the FTC. The outcomes of the historical account and the data about the mathematicians’ views are discussed from the perspective of the theory of didactic transposition. v vi ACKNOWLEDGEMENTS First of all, many warm thanks to my supervisor Professor Eva Jablonka for all excellent suggestions, guidance, and continued support in my work. Her generosity in sharing knowledge and encouragement has been priceless to me. Without her inspiration and guidance it would not have been possible for me to finish this work. Thank you from all my heart for making it happen. I am deeply indebted to Prof. Anna Sierpinska who was such an important source of inspiration for me. Her personality, expertise and judgement have helped me to see things clearly. I would like to express my gratitude to my assisting supervisor Prof. Christer Bergsten for his valuable knowledge and all his constructive ideas. My thanks also go to Prof. Lars-Erik Persson for unconditional support during my doctoral studies. Special thanks to Prof. Lech Maligranda for sharing ideas, thoughts and experiences with me. I wish to express my deep appreciation to the mathematicians who participated in these studies. Last but not least, I wish to express my tremendous gratitude to my children, Patryk and Piotr, for being a joy in my life. Thank you to all my dear friends for their moral support, help and understanding during the long and difficult period that it took to carry out and write this thesis. Christina, you are the best. vii viii CONTENTS ABSTRACT ...............................................................................................................................................V ACKNOWLEDGEMENTS...................................................................................................................VII CONTENTS............................................................................................................................................. IX 1. INTRODUCTION..................................................................................................................................1 BACKGROUND AND OUTLINE ................................................................................................................1 THE NOTION OF MATHEMATICAL PROOF ...............................................................................................2 2. CONCEPTUAL FRAMEWORK .........................................................................................................6 INTRODUCTION .....................................................................................................................................6 EPISTEMOLOGICAL OBSTACLES ............................................................................................................7 DIDACTIC TRANSPOSITION AND MATHEMATICAL PRAXEOLOGIES .......................................................12 Didactic transposition ............................................................................................................12 Mathematical praxeologies.....................................................................................................18 3. RESEARCH QUESTIONS AND METHODOLOGY ......................................................................22 GENERAL RESEARCH GOAL AND SPECIFIC RESEARCH QUESTIONS .......................................................22 RESEARCH STRATEGIES AND METHODS...............................................................................................23 Approaches for investigating the research questions .............................................................23 The historical study.................................................................................................................25 The case study: interviews with mathematicians ....................................................................25 4. FORMS AND FUNCTIONS OF PROOF IN MATHEMATICS AND IN MATHEMATICS EDUCATION: AN ISSUE OF DIDACTIC TRANSPOSITION..........................................................38 INTRODUCTION ...................................................................................................................................38 PROOF IN MATHEMATICS ....................................................................................................................38 Types of mathematical proof...................................................................................................38 Functions of proof in mathematics .........................................................................................41 PROOF IN MATHEMATICS EDUCATION .................................................................................................50 Functions of proof in mathematics education.........................................................................50 FORMS OF DIDACTIC TRANSPOSITIONS OF MATHEMATICAL PROOF .....................................................57 5. THE HISTORICAL DEVELOPMENT OF THE FTC AND ITS PROOF ....................................60 INTRODUCTION ...................................................................................................................................60 INVENTION OF THE CALCULUS AND INSTITUTIONALISATION OF THE FTC...........................................61 ix Predecessors...........................................................................................................................61 Prerequisites...........................................................................................................................62 The “FTC” and the calculus ..................................................................................................64 Arithmetisation of the calculus ...............................................................................................73 The FTC in 20th century mathematics....................................................................................76 NAME GIVING AND TEXTBOOKS ..........................................................................................................79 Textbooks................................................................................................................................79 Emergence of a shared name for the “FTC”..........................................................................86 DISCUSSION ........................................................................................................................................93 6. THE MATHEMATICIANS’ VIEWS: OUTCOMES OF THE INTERVIEWS.............................97 INTRODUCTION ...................................................................................................................................97 RESPONSES TO THE INTERVIEW QUESTIONS ........................................................................................97 DISCUSSION AND ANALYSIS..............................................................................................................116 What is the FTC, really?.......................................................................................................116 Significance and the problem that led to the FTC ................................................................118 Suggested didactic transpositions and their constraints.......................................................119 7. OUTCOMES AND CONCLUSIONS...............................................................................................121 RESEARCH QUESTION 1.....................................................................................................................121 Transformations of functions of proof in teaching institutions.............................................121 The
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