University of Calgary PRISM: University of Calgary's Digital Repository Graduate Studies The Vault: Electronic Theses and Dissertations 2019-07-05 Cultivating the Growth of Mathematical Images Plosz, Jennifer A. Plosz, J. A. (2019). Cultivating the Growth of Mathematical Images (Unpublished master's thesis). University of Calgary, Calgary, AB. http://hdl.handle.net/1880/110606 master thesis University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca UNIVERSITY OF CALGARY Cultivating the Growth of Mathematical Images by Jennifer A. Plosz A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE GRADUATE PROGRAM IN EDUCATIONAL RESEARCH CALGARY, ALBERTA JULY, 2019 © Jennifer A. Plosz 2019 ii Abstract In this document, I present some of the findings of the study, Cultivating the Growth of Mathematical Images, in which I explore the role that spatial reasoning plays in the growth of mathematical images. This study involved Grade Five students labelled with learning disabilities, and their teacher. This study was a micro analysis of a participatory action research study, as it looked at the beginning stages of exploration with students and the impact that a more spatial approach to fractions might have on their growth. Information was gathered from students’ psycho educational assessment, an informal assessment of spatial reasoning ability, and a pre- assessment task that looked at their understanding of basic number and fractions. There were many layers of complexity surrounding each student’s psycho educational profile, their performance in the classroom, and the pre-assessment task offered to the students. Certain aspects of these experiences seemed consistent with each other, others contradictory, and still others contained much variability, such as in the area of working memory. During the second week of the study, video data were collected while students engaged with a task that created an interplay between visualizing and building fractions. In analysis of the data, close attention was paid to what sort of offerings the students were given such as signitive (written and oral), imaginative (visualizing), and perceptual (sensory), which are somewhat modified from Husserl (1970). During this task, a pattern of growth began to emerge which is discussed and connected to the Pirie-Kieren (1994) Dynamical Theory for the Growth of Mathematical Understanding. Generally, with the introduction of the signitive only, there was No appearance of movement. Then as the student began either producing or receiving perceptual experiences they progressed into the Image Making phase. The continued engagement with perceptual experiences appeared to create the beginnings of the imaginative, Image Making iii emergent Image Having phase, and some students were even able to reach the point of being predominately in the Image Having phase. Each participant went through the various phases at varying speeds. Within these various phases there was found to be much complexity in terms of contributing factors towards growth. The fact that some participants built more than others and therefore had more perceptual offerings seemed a strong contributing factor but other aspects such as their own personal commitment to sense-making as they built, their social interactions, and their own self-belief seemed to also impact growth. Keywords: visualization, images, embodied cognition, learning disabilities, dyslexia, mathematics education PREFACE iv Preface This thesis is original, unpublished, independent work by the author, J. A. Plosz The experiments reported in Chapters 5-10 were covered by Ethics Certificate number REB16-1205, issued by the University of Calgary Conjoint Health Ethics Board for the project “Cultivating the Growth of Mathematical Images” on January 18, 2017. ACKNOWLEDGEMENTS v Acknowledgments I want to express my deep appreciation to the children who offered so willingly and enthusiastically to be part of my study even when school has caused them so much scarring; to the philosophers, researchers and theorists whose shoulders I have stood upon to try to make sense of all this complexity; to my supervisor, Jo Towers, whose patience and gentle guidance has allowed me to stretch my own understandings and move in unexpected directions as a learner; and finally, to my family, to my father who was often my sounding board as I would often need to verbally organize my thoughts; to my kids, Roman and Kelly, who patiently understood as mom would once again be spending the day at the library; and last but definitely not least, my husband, who supported and encouraged me every step of the way, who took on much more of the domestic responsibility in support of my endeavor. Thank you to all of these people who have made this document a reality. DEDICATION Dedication To my fellow dyslexics, may our future path be smoother. TABLE OF CONTENTS vii Table of Contents Abstract ............................................................................................................................... ii Preface................................................................................................................................ iv Acknowledgments............................................................................................................... v Dedication .......................................................................................................................... vi Table of Contents .............................................................................................................. vii Figures............................................................................................................................... xii Tables ............................................................................................................................... xiv Entering into the topic ......................................................................................................... 1 Chapter One: Three horizons that surround this topic ........................................................ 5 1.1 Horizon of mathematics ............................................................................................ 5 1.1.1 Plato and Aristotle .............................................................................................. 6 1.1.2 Descartes and Kant ............................................................................................ 7 1.1.3 Frege and Brouwer ............................................................................................. 8 1.1.4 Hilbert and Gödel ............................................................................................... 9 1.1.5 Modern times ....................................................................................................11 1.2 Horizon of images ................................................................................................... 13 1.2.1 The philosophical tradition .............................................................................. 13 1.2.2 Age of experimental psychology ..................................................................... 14 1.2.3 Cognitive science ............................................................................................. 15 1.3 Horizon of dyslexia. ................................................................................................ 19 1.3.1 The reading brain. ............................................................................................ 20 1.3.2 Oral to written culture. ..................................................................................... 21 1.3.4 Birth of dyslexia as a disorder. ......................................................................... 26 Chapter Two: The Phenomenon ........................................................................................ 31 2.1 Perception ............................................................................................................... 33 2.2 Images: the work of art ........................................................................................... 36 2.2.1 Why not representation? .................................................................................. 38 2.2.2 Why not mental? .............................................................................................. 40 2.2.3 Why image? ..................................................................................................... 41 2.3 Cultivating growth in mathematical images ........................................................... 44 TABLE OF CONTENTS viii 2.3.1 Nature of images. ............................................................................................. 46 2.3.2 Pirie-Kieren Dynamical Theory. ...................................................................... 47 2.3.3 Husserlian distinctions. .................................................................................... 49 2.4 Elements of thought: images, language, and gesture .............................................. 55 2.4.1 A space. ...........................................................................................................