Generalization: Developing Mathematical Practices in Elementary School Item Type text; Electronic Dissertation Authors Dumitraşcu, Gabriela Georgeta 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 or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 06/10/2021 17:28:14 Link to Item http://hdl.handle.net/10150/556959 GENERALIZATION: DEVELOPING MATHEMATICAL PRACTICES IN ELEMENTARY SCHOOL by Gabriela Georgeta Dumitrașcu ________________________________________________ Copyright © Gabriela Georgeta Dumitrașcu 2015 A Dissertation Submitted to the Faculty of the DEPARTMENT OF TEACHING, LEARNING, AND SOCIOCULTURAL STUDIES In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY WITH A MAJOR IN TEACHING AND TEACHER EDUCATION In the Graduate College THE UNIVERSITY OF ARIZONA 2015 2 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE As members of the Dissertation Committee, we certify that we have read the dissertation prepared by Gabriela Georgeta Dumitrașcu, titled Generalization: Developing Mathematical Practices in Elementary School, and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy. ________________________________________________Date: 3/30/2015 Dissertation Director: Erin Turner ________________________________________________Date: 3/30/2015 Dissertation Director: Marta Civil _______________________________________________________________________ Date: 3/30/2015 Walter Doyle _______________________________________________________________________ Date: 4/15/2015 Barbara Dougherty Final approval and acceptance of this dissertation is contingent upon the candidate’s submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. ________________________________________________ Date: 3/30/2015 Dissertation Director: Erin Turner ________________________________________________ Date: 3/30/2015 Dissertation Director: Marta Civil 3 STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of the requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that an accurate acknowledgement of the source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the copyright holder. SIGNED: Gabriela Georgeta Dumitrașcu 4 TABLE OF CONTENTS LIST OF TABLES .................................................................................................................... 6 LIST OF FIGURES .................................................................................................................. 7 ABSTRACT .............................................................................................................................. 9 CHAPTER 1 — INTRODUCTION ....................................................................................... 11 The Process of Generalization in Mathematics ...................................................................... 11 Insights from Educational Psychology ................................................................................... 17 The Research Problem ............................................................................................................ 19 The Research Questions .......................................................................................................... 21 CHAPTER 2 — LITERATURE REVIEW ............................................................................. 24 Theoretical Framework ........................................................................................................... 24 The Concept of Generalization ............................................................................... 24 The Concept of Generalization in Mathematics ..................................................... 29 Research on Content Analysis of Textbooks ........................................................................... 41 Textbooks as Blueprint for Classroom Instruction ................................................. 41 The Importance of Research on Content Analysis of Mathematics Textbooks ...... 43 CHAPTER 3 — METHODS .................................................................................................. 48 Overview ................................................................................................................................. 48 Study Design ........................................................................................................................... 48 The Qualitative Research Paradigm ........................................................................ 48 Research Design ...................................................................................................... 50 Data Collection Strategies ....................................................................................... 51 Data Analysis Procedures ....................................................................................... 51 The first round of analysis. ................................................................................. 51 The second round of analysis. ............................................................................. 53 The third round of analysis. ................................................................................ 54 Analytical Criteria ................................................................................................... 55 A general theoretical framework. ........................................................................ 56 The Structure of the Coding Procedure. ............................................................. 61 Applying the theoretical framework. .................................................................. 63 Reliability of the coding system. ........................................................................ 70 Reporting the Findings ............................................................................................ 71 CHAPTER 4 — FINDINGS ................................................................................................... 72 Findings: Part I ........................................................................................................................ 72 5 Investigations in Number, Data, and Space ............................................................ 73 Description. ......................................................................................................... 73 Analysis............................................................................................................... 74 Summary of Findings for Investigations Curriculum ......................................... 94 Everyday Mathematics ............................................................................................ 95 Description. ......................................................................................................... 95 Analysis............................................................................................................... 96 Summary of Findings for Everyday Mathematics ............................................ 106 Math Trailblazers .................................................................................................. 107 Description. ....................................................................................................... 107 Analysis............................................................................................................. 108 Summary of findings for Math Trailblazers ...................................................... 114 Comparison and Contrast ...................................................................................... 114 Findings: Part II .................................................................................................................... 120 Description of Arithmetic Operations in Everyday Mathematics Textbooks ....... 122 Analysis................................................................................................................. 123 Section A. Findings for Core Aspect A – Strand 1. ........................................... 125 Section B. Findings for Core Aspect B – Strand 1 ........................................... 148 Summary of Findings ............................................................................................ 153 CHAPTER 5 — DISCUSSION ............................................................................................ 155 Discussion of Findings .......................................................................................................... 155 Discussion of the Findings for the First Research Question ................................. 155 Discussion of the Findings for the Second Research Question ............................ 163 Implications..........................................................................................................................