University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 8-2011 Preservice Secondary School Mathematics Teachers' Current Notions of Proof in Euclidean Geometry Michael Ratliff [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Curriculum and Instruction Commons Recommended Citation Ratliff, Michael, "Preservice Secondary School Mathematics Teachers' Current Notions of Proof in Euclidean Geometry. " PhD diss., University of Tennessee, 2011. https://trace.tennessee.edu/utk_graddiss/1118 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by Michael Ratliff entitled "Preservice Secondary School Mathematics Teachers' Current Notions of Proof in Euclidean Geometry." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Teacher Education. P. Mark Taylor, Major Professor We have read this dissertation and recommend its acceptance: Vena M. Long, JoAnn Cady, Jerzy Dydak Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) To the Graduate Council: I am submitting herewith a dissertation written by Michael L. Ratliff entitled “Preservice Secondary School Mathematics Teachers‟ Current Notions of Proof in Euclidean Geometry.” I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Education. Dr. P. Mark Taylor, Major Professor We have read this dissertation and recommend its acceptance: Dr. Vena M. Long Dr. JoAnn Cady Dr. Jerzy Dydak Accepted for the Council: Dr. Carolyn R. Hodges, Vice Provost and Dean of the Graduate School (Original signatures are on file with official student records.) Preservice Secondary School Mathematics Teachers‟ Current Notions of Proof in Euclidean Geometry A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville Michael L. Ratliff August 2011 Copyright © by Michael L. Ratliff All rights reserved. ii Acknowledgements There are many people who are responsible for my education, and I express gratitude to all of them. I could not have completed this degree without the instruction, guidance, encouragement, and support provided by these people. There are a few from this group that I want to acknowledge who are most responsible for the completion of this degree. I want to thank the ACCLAIM Management Team for their belief in me and support, especially Dr. Vena Long, Dr. Bill Bush, Dr. Mike Mays, Dr. Carl Lee, and Dr. Robert Mayes. This opportunity has forever changed my professional life. Also, I want to thank my dissertation committee: Dr. P. Mark Taylor (chair), Dr. Vena Long, Dr. JoAnn Cady, and Dr. Jerzy Dydak. I appreciate the guidance and also patience that each of you demonstrated as I completed this work. In addition, I want to thank a few people from my home institution, Lindsey Wilson College: Dr. Bill Luckey, Dr. William Julian, Dr. Bettie Starr, and my Division colleagues. I could not have finished without the support and encouragement that all of you provided. I‟m thankful for the many life-long friendships that I now have with those from my cohort and the other two cohorts. I hope that all of you achieve your professional and personal goals in life, and I thank all of you for the support and encouragement offered along our doctoral paths. I want to thank my family for their support and encouragement. And, I want to thank my very good friend, Faylene, for your support and encouragement this past year. Lastly, this dissertation is dedicated to my parents, William S. and Katherine L. Ratliff. Thank you for everything (and I think I‟m finished with school now). iii Abstract Much research has been conducted in the past 25 years related to the teaching and learning of proof in Euclidean geometry. However, very little research has been done focused on preservice secondary school mathematics teachers‟ notions of proof in Euclidean geometry. Thus, this qualitative study was exploratory in nature, consisting of four case studies focused on identifying preservice secondary school mathematics teachers‟ current notions of proof in Euclidean geometry, a starting point for improving the teaching and learning of proof in Euclidean geometry. The unit of analysis (i.e., participant) in each case study was a preservice mathematics teacher. The case studies were parallel as each participant was presented with the same Euclidean geometry content in independent interview sessions. The content consisted of six Euclidean geometry statements and a Euclidean geometry problem appropriate for a secondary school Euclidean geometry course. For five of the six Euclidean geometry statements, three justifications for each statement were presented for discussion. For the sixth Euclidean geometry statement and the Euclidean geometry problem, participants constructed justifications for discussion. A case record for each case study was constructed from an analysis of data generated from interview sessions, including anecdotal notes from the playback of the recorded interviews, the review of the interview transcripts, document analyses of both previous geometry course documents and any documents generated by participants via assigned Euclidean geometry tasks, and participant emails. After the four case records were completed, a cross-case analysis was conducted to identify themes that traverse the individual cases. iv From the analyses, participants‟ current notions of proof in Euclidean geometry were somewhat diverse, yet suggested that an integration of justifications consisting of empirical and deductive evidence for Euclidean geometry statements could improve both the teaching and learning of Euclidean geometry. v Table of Contents Chapter 1 : INTRODUCTION........................................................................................................ 1 Background of the Study ............................................................................................................ 1 United States Schools, Geometry and Proof ........................................................................... 1 Learned Societies, Organizations, Committees and Proof.................................................... 12 The Researcher‟s Interest and Motivation ............................................................................ 14 Statement of Problem ................................................................................................................ 15 The Research Questions ............................................................................................................ 16 Delimitations and Limitations ................................................................................................... 17 Chapter 2 : REVIEW OF LITERATURE .................................................................................... 19 Defining Proof .......................................................................................................................... 19 Proof Schemes .......................................................................................................................... 21 External Conviction Proof Schemes ..................................................................................... 22 Empirical Proof Schemes ...................................................................................................... 23 Deductive Proof Schemes ..................................................................................................... 23 Functions of Proof..................................................................................................................... 23 Pedagogical Approaches ........................................................................................................... 26 Recommendations and Standards ............................................................................................. 27 Proof and Dynamic Geometry Software ................................................................................... 28 Discussion ................................................................................................................................. 31 Chapter 3 : RESEARCH DESIGN ............................................................................................... 33 Rationale for Case Study .......................................................................................................... 33 Participants ................................................................................................................................ 34 Data Collection ......................................................................................................................... 34 Phase One.............................................................................................................................