Chapter Two Recovering Mathematics
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UNIVERSITY OF CALGARY Recovering Mathematics: An Ontological Turn Toward Understanding the Teaching of Mathematics with Children by Mary Margaret Stordy A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY GRADUATE DIVISION OF EDUCATIONAL RESEARCH CALGARY, ALBERTA SEPTEMBER, 2009 © Mary Margaret Stordy 2009 Library and Archives Bibliotheque et 1*1 Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington OttawaONK1A0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-54484-6 Our file Notre reference ISBN: 978-0-494-54484-6 NOTICE: AVIS: The author has granted a non L'auteur a accorde une licence non exclusive exclusive license allowing Library and permettant a la Bibliotheque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par Nnternet, preter, telecommunication or on the Internet, distribuer et vendre des theses partout dans le loan, distribute and sell theses monde, a des fins commerciales ou autres, sur worldwide, for commercial or non support microforme, papier, electronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in this et des droits moraux qui protege cette these. Ni thesis. Neither the thesis nor la these ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent etre imprimes ou autrement printed or otherwise reproduced reproduits sans son autorisation. without the author's permission. In compliance with the Canadian Conformement a la loi canadienne sur la Privacy Act some supporting forms protection de la vie privee, quelques may have been removed from this formulaires secondaires ont ete enleves de thesis. cette these. While these forms may be included Bien que ces formulaires aient inclus dans in the document page count, their la pagination, il n'y aura aucun contenu removal does not represent any loss manquant. of content from the thesis. •+• Canada ABSTRACT Recovering Mathematics: An Ontological Turn Toward Coming to Understand the Teaching of Mathematics with Children is an interpretive study into the teaching of mathematics to children. Drawing from Gadamer's (1989) ontological hermeneutics, this research examines lived experience through narrative pedagogic events to explore the idea of recovering mathematics as a living human enterprise for children and teachers in schools. Attending to Caputo's (1987) idea of returning to the original difficulty, this text attempts to keep the difficulty of teaching alive while resisting the metaphysics of presence. In Truth and Method (1989), Gadamer claims that the happening of events is essential for understanding. This philosophical inquiry embraces Gadamer's idea of "the fecundity of the individual case" as a way to explore and make meaning from lived experiences with children, teachers, and primary / elementary pre-service teachers. Recovering Mathematics hermeneutically considers the relationship of mathematics to teaching in terms of the past and the present, the particular and the general, the philosophical and the practical, the part and the whole. It is a philosophical exploration into what might be possible when it comes to teaching mathematics to children when the world, which includes the living world of mathematics, is allowed entry. a ACKNOWLEDGEMENTS It is with gratitude that I wish to thank the members of my supervisory committee - Dr. Jo Towers, Dr. David Jardine, and my advisor, Dr. Jim Paul. Besides all the incredible support they have given me as a doctoral student, David introduced me to Gadamer's writings, Jo introduced me to the Canadian Mathematics Education Study Group, and Jim wisely convinced me that I needed to conceptualize my work in the first place. All three of them have opened up my academic world. I wish to thank them for their patience, their time, and their thoughtful questions and comments about this work. I wish to thank Dr. Nancy Dudley and Dr. Florence Glanfield for contributing their time and energy as gracious outside examiners. Their careful reading of this text and thoughtful questions have helped me see even more possibilities. I could never have undertaken this work without the support and love of my family. I wish to acknowledge my parents, Marion and Gerald Stordy, for their unwavering belief in me and for fostering in me an intense curiosity about the world. Their sense of responsibility to others has impacted me greatly, and I can only hope I will one day be as generous and as giving as they are. I also wish to thank my four siblings, Larry, Allan, Cheryl, and Paul, for their encouragement and support and for keeping my feet firmly on the ground, as only older siblings can. They are incredible people and I am proud to call them my brothers and sister. To my dear friends Dr. Alexandra Fidyk, Dr. Beverly Mathison, and Dr. Laurie Bowers, I am so grateful. Your ability to take phone calls at the strangest hour, your senses of humor, your lightness, and your wisdom have been such gifts. iii To my former colleagues and students at 'that Western Canadian school,' I am forever indebted. Directly influencing this research, I wish to acknowledge Dr. Sharon Friesen who has been my mentor and friend since the first day I witnessed Pat and her with children, and my former teaching partners and dear friends, Sherri Rinkel MacKay and Jeff Stockton, for their gifts of wisdom and laughter. I am deeply grateful. To my Newfoundland friends, my Calgary friends, my Prince Edward Island friends, and my friends from other pockets of this earth, thank you, all, for your patience and support. I hope you know how much you have helped me grow and become who I am. With love and gratitude, I wish to thank Jody Doyle for believing in me, particularly as I have proceeded through the final stages of this work. I will be forever grateful for his compassion, his generosity, and his understanding. Finally, I must turn and acknowledge my beloved furry family - Sophie, Maggie, Austin, and Katy - who have been beside me every step of the way. It is unbelievable what these two cats and two golden retrievers have done for my soul. I can only hope, one day, to become the human they think I am already. IV DEDICATION This work is dedicated to the memories of two amazing scholars, Dr. Patricia Clifford and Dr. Joe Kincheloe. Thank you, my two dear friends, for being in my corner, and helping me to see what is possible. TABLE OF CONTENTS ABSTRACT ii ACKNOWLEDGEMENTS Hi DEDICATION v TABLE OF CONTENTS vi CHAPTER ONE 1 BEING LED ON 1 Out in the Hallway 1 The Post Office 6 Keeping up with the Children: Sketches from the Archives 10 Feels Like Home 13 Possibilities 15 Being Led On 16 Blindsided 18 Opening Up 21 CHAPTER TWO 25 RECOVERING MATHEMATICS 25 CHAPTER THREE 41 HERMENEUTICS: AN ONTOLOGICAL TURN 41 Introduction 41 Hermeneutical Tracings 42 From Aristotle to Husserl 42 Husserl and Intentionality 44 Heidegger's Ontological Shift 44 Gadamer's Ontology of Understanding 45 Choosing the Method: Did I Have a Choice? 46 I am in the World 48 Dialogical Nature of Hermeneutics 50 Mathematics Requires a World 50 Memory 52 Truth 52 Understanding Through Conversation 54 Horizons 54 Conversation with Mathematics 56 Understanding the Part and the Whole 57 Play and Pedagogy 57 Possibilities 59 CHAPTER FOUR 60 ANOTHER WAY OF BEING WITH CHILDREN AND MATHEMATICS 60 The Question is Ontological 60 Frozen Future 63 SECTION ONE: GRADE ONE 66 vi If Five is the Answer, What Could the Question Be? 66 Learning With and From the Parents 71 The Seven-ness of Things 74 The Conversation Must Continue 79 SECTION TWO: GRADE TWO 80 Things are Not Always as They Seem: Entering the Classroom 80 A Foreign Land 82 Shifting Locations: Shifting Perspectives 85 The Gathering Area: Two Steps Forward and One Step Back 88 Castle Remnants: Keeping the Conversation Going 89 Exploring the Martian Landscape: Creating New Terrain 92 The Notebook: Working Alongside Eratosthenes 97 Craft Sale: Empowering the Children 103 Walking Beside my Mentor: Counting Grains of Sand 109 Challenging the Problem with Attentiveness 124 Alien Territory: Treading Rough Ground 136 CHAPTER FIVE 143 LIVING WITH THE COVER VERSION OF MATHEMATICS 143 Keeping the Difficulty Alive 143 Why Didn't Anyone Tell Me? 150 Mathematics Autobiographies: Living with the Cover Version 156 CHAPTER SIX 167 RESPONSIBILITY TO RECOVER 167 You Can't Get There From Here: Returning Home 167 We Tell our Stories to Find Out What They Mean 168 A Final Look Back 172 Personal Journal Entry, Fall 2006 172 Letter Unsent: A Tribute 173 ENDNOTES 176 REFERENCES 177 APPENDIX A 186 vii 1 CHAPTER ONE BEING LED ON Out in the Hallway "I didn't know it was going to be so hard," the young child said, standing outside of the classroom. Her arms were crossed, her brow furrowed and her bottom lip jutted out. Her dark eyes met mine. Then she looked away. "What?" I asked, not sure where we were headed. "School. If I had of known it was going to be like this, I never would have come out." I bent down and squatted beside her, carefully adjusting my skirt in the process. I placed my hand gently on her back and tried to meet her eyes. "Come out? You mean come to school?" "No! I never would have come out of my mom if I knew it was going to be so hard!" Before me, stood a six-year-old girl whom I was just getting to know.