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ASSUMPTION DIGGING in EUCLIDEAN GEOMETRY By Assumption Digging in Euclidean Geometry Item Type text; Electronic Thesis Authors Kaushish, Madhav Publisher The University of Arizona. Rights Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction, presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 24/09/2021 07:09:01 Link to Item http://hdl.handle.net/10150/636511 ASSUMPTION DIGGING IN EUCLIDEAN GEOMETRY by Madhav Kaushish ____________________________ Copyright © Madhav Kaushish 2019 A Thesis Submitted to the Faculty of the DEPARTMENT OF MATHEMATICS In Partial Fulfillment of the Requirements For the Degree of MASTER OF SCIENCE In the Graduate College THE UNIVERSITY OF ARIZONA 2019 Madhav Kaushish 3 ACKNOWLEDGEMENTS I would like to thank the following sets of people, who are to blame for the existence of this thesis (I bear minimal responsibility): 1. My advisor Rebecca McGraw, as well the other two members of my thesis committee, Jennifer Eli and Aditya Adireja, for pushing me outside of my comfort zone and making my thesis far better. 2. K.P. Mohanan, Tara Mohanan, and Jayasree Subramanian, and other members of ThinQ, for working with me to come up with the ideas in this thesis. Also, for completely destroying my bad ideas and making my other ideas far better. 3. David Taylor and Eric Elert, my fellow graduate students, for being my first sounding boards for ideas, whether they were interested in hearing them or not! Also, for making it explicit when my ideas were terrible. 4. My family and friends back home for still liking me (hopefully?) despite me not picking up any of their phone calls or responding to their texts! 5. Arnav and Dhody for willing to argue/fight with me ever since I remember, and being kind enough to lose each time. 6. My parents, Gauri and Uday, and my grandparents, for the extreme, and often un-called for, levels of concern about my life. 7. Oliver Schirokauer, Jim Walsh, Bob Bosch, Jack Calcut, Sandhya Saini, and Rajyashree Sood, my math teachers at Oberlin and Vasant Valley School, for my interest in mathematics. 8. David Glickenstein, Klaus Lux, and my other math professors at The University of Arizona, for giving me insight into the nature of mathematics. Assumption Digging in Euclidean Geometry 4 9. Bertrand Russell, Noam Chomsky, David Hilbert, Albert Einstein, John Dewey, Charles Darwin, Richard Feynman, Daniel Kahneman, Ian Stewart, David Tall, and George Polya, amongst others, whose ideas have been largely unacknowledged in the body of this thesis, but form the basis of the way I think about education. Madhav Kaushish 5 TABLE OF CONTENTS ABSTRACT ............................................................................................................................................................ 8 INTRODUCTION ................................................................................................................................................... 9 TYPES OF THEORY BUILDING ......................................................................................................................................... 10 Theory Building Inspired by the Real World. ..................................................................................................... 11 Generalizing Existing Theories. ......................................................................................................................... 12 Changing an Existing Theory. ............................................................................................................................ 13 Putting Theories on Firmer Footing. .................................................................................................................. 13 ASSUMPTION DIGGING ................................................................................................................................................ 14 An Analogy. ....................................................................................................................................................... 14 Unpacking Assumption Digging. ....................................................................................................................... 15 Operationalizing Assumption Digging ............................................................................................................... 19 Classroom Norms for Assumption Digging. ...................................................................................................... 20 VALUE OF ASSUMPTION DIGGING .................................................................................................................................. 21 EUCLIDEAN GEOMETRY ................................................................................................................................................ 22 LITERATURE REVIEW .......................................................................................................................................... 23 ASSUMPTION DIGGING ................................................................................................................................................ 23 Assumption Digging and RME. .......................................................................................................................... 23 Theory Building. ................................................................................................................................................ 24 Understanding Related to Assumption Digging. ............................................................................................... 25 EUCLIDEAN GEOMETRY ................................................................................................................................................ 29 Van Hiele Levels. ............................................................................................................................................... 30 RESEARCH QUESTIONS ................................................................................................................................................. 31 Assumption Digging in Euclidean Geometry 6 METHODOLOGY ................................................................................................................................................. 32 INSTRUCTIONAL SETTING AND STUDENT BACKGROUND ...................................................................................................... 32 The Workshops. ................................................................................................................................................. 32 The Facilitators. ................................................................................................................................................. 33 Education in India ............................................................................................................................................. 33 Student Background. ......................................................................................................................................... 35 Instructional Setting. ......................................................................................................................................... 36 Prior Geometry Experience. ............................................................................................................................... 37 The Mathematics. ............................................................................................................................................. 37 DATA COLLECTION ...................................................................................................................................................... 38 DATA ANALYSIS METHOD ............................................................................................................................................. 39 RESULTS ............................................................................................................................................................. 41 PRELIMINARIES ........................................................................................................................................................... 41 SESSION 1 – GANGA ................................................................................................................................................... 41 Ganga: Altitude of Triangle (AD2). .................................................................................................................... 42 Ganga: Parallel Lines (AD1 & AD2). .................................................................................................................. 43 Ganga: Equidistant (AD1 & AD2). ..................................................................................................................... 44 Ganga: Breaking a Parallelogram into Triangles (AD1). ................................................................................... 47 Ganga: Making Parallelograms from Triangles (AD1). ..................................................................................... 49 Ganga: Student finds Flaw in reasoning (AD1 & AD3). ..................................................................................... 50 Ganga: Fixing the argument (AD1 & AD2). ......................................................................................................
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