Illinois State University ISU ReD: Research and eData Theses and Dissertations 5-26-2016 An Intervention Study Aimed At Enhancing Seventh-Grade Students’ Development Of The Concept Of A Variable Sinan Kanbir Illinois State University, [email protected] Follow this and additional works at: https://ir.library.illinoisstate.edu/etd Part of the Elementary and Middle and Secondary Education Administration Commons, Junior High, Intermediate, Middle School Education and Teaching Commons, and the Science and Mathematics Education Commons Recommended Citation Kanbir, Sinan, "An Intervention Study Aimed At Enhancing Seventh-Grade Students’ Development Of The Concept Of A Variable" (2016). Theses and Dissertations. 575. https://ir.library.illinoisstate.edu/etd/575 This Dissertation is brought to you for free and open access by ISU ReD: Research and eData. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of ISU ReD: Research and eData. For more information, please contact [email protected]. AN INTERVENTION STUDY AIMED AT ENHANCING SEVENTH-GRADE STUDENTS’ DEVELOPMENT OF THE CONCEPT OF A VARIABLE Sinan Kanbir 311 Pages Thirty-two seventh-grade students attending a public middle school in a rural district in a midwestern state of the United States of America participated in a mixed- method intervention study aimed at enhancing student understandings of the algebraic concept of a variable, especially in relation to structure and modeling. Participating students were randomly allocated to two class groups, with one class initially being taught by Mr. X and the other by Mr. Y, who had both participated in professional development sessions led by two senior mathematics educators. Mr. X initially led one of the class groups in seven sessions on algebraic structure and Mr. Y led the other class group in seven sessions on modeling using algebra. Then, halfway through the intervention, the class groups swapped, with the teachers repeating the workshops with their new classes. Pre-teaching, mid-intervention, post-teaching, and retention assessment of learning occurred over a period of 24 weeks, with data gathered from pencil-and-paper tests and from 56 one-to-one interviews at various stages of the study. The theoretical bases for the study derived from the writings of two main scholars—first, the semiotic theory of Charles Sanders Peirce; and second the theory of apperception put forward by Johann Friedrich Herbart. Six research questions were identified, and answered after all data had been collected and analyzed. Most of the participating students showed large gains in their understandings of algebraic structure and modeling, effect sizes of the intervention were large, and concept images of a variable were substantially modified. KEYWORDS: Algebra education, Charles Sanders Peirce, Common core algebra, Concept image, Johann Friedrich Herbart, Modeling in middle-school mathematics AN INTERVENTION STUDY AIMED AT ENHANCING SEVENTH-GRADE STUDENTS’ DEVELOPMENT OF THE CONCEPT OF A VARIABLE SINAN KANBIR A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Department of Mathematics ILLINOIS STATE UNIVERSITY 2016 © 2016 Sinan Kanbir AN INTERVENTION STUDY AIMED AT ENHANCING SEVENTH-GRADE STUDENTS’ DEVELOPMENT OF THE CONCEPT OF A VARIABLE SINAN KANBIR COMMITTEE MEMBERS: McKenzie A. Clements, Co-Chair Nerida F. Ellerton, Co-Chair David Barker John K. Rugutt ACKNOWLEDGMENTS There are many people who have helped me complete this dissertation in a timely way. I would like to thank my parents, and my brothers and sisters who have always supported me, throughout my studies. I have been very fortunate to be able to have had helpful and knowledgable committee members. First of all, I would like to express my gratitude to my co-chairs, Dr. Ken Clements and Dr. Nerida Ellerton. I started my mathematics education research journey with them and have valued their astute guidance and encouragement. They have been with me throughout my journey. I want them to know how grateful I am for the interest, time, expertise, and genuine friendship which they have offered, at all times. Despite their busy schedules, they were always there for me when I needed them most. Dr. David Barker contributed much with his great knowledge of the topic under consideration, and Dr. John Rugutt was of great assistance on statistical and design issues. Of course, the research described in this dissertation would not have been possible if I had not been able to find a school where the teaching intervention and data collection could happen. It is a pity that I am not free to name the school, its Principal, the two participating teachers, and the 32 participating seventh-grade students on this page, for they were absolutely wonderful throughout the study. There was no time that I felt that I was receiving less than 100 percent cooperation from them. I would also like to thank the Department of Mathematics at Illinois State University for everything I have learned during my time as a doctoral student in mathematics i education at the University. I look forward to being able to develop many of the skills that I have learned in future years. S. K. ii CONTENTS Page ACKNOWLEDGMENTS i CONTENTS iii TABLES viii FIGURES x CHAPTER I. IDENTIFYING THE PROBLEM, AND PLANNING FOR THE INVESTIGATION 1 Approaches to Introducing Algebra: Structure and Modeling 2 Overview of the Current Study 3 The Concepts of Algebraic Structure, and Modeling, as Presented in the Common Core State Standards for Mathematics 5 Structural Thinking in Algebra, Across the Grades 5 Algebraic Structure as it Appears in the Common Core State Standards for Mathematics 7 The Associative Properties for Addition and Multiplication, and the Distributive Property, in Common Core Mathematics 8 Modeling Across the Grades in CCSSM Algebra 13 Summary of Research on Learning and Teaching Algebra 18 Structural and Functional Approaches as Presented in the Mathematics Textbook Used by Participating Students 19 Textbook Treatment of the Associative Property for Addition 20 Textbook Treatment of the Associative Property for Multiplication 21 Textbook Treatment of the Distributive Property 22 iii Textbook Treatment of Modeling and Related Problem Solving 22 The Problem as it Appeared to be at the Beginning of the Study 24 II. LITERATURE REVIEW 29 The Design Research Foundation for the Study 32 Theoretical Bases for the Study 35 Signifiers, Objects and Interpretants—and Charles Sanders Peirce 37 Semiotic Aspects of Structural Properties for Rational Numbers 39 Semiotic Aspects of the Functional-Thinking Approach to Early Algebra 51 Radford’s Comments on “Objectification” for Middle- School Algebra 55 Some Scholars’ Preference for the Functional-Thinking Approach 58 Signs, Symbols, Representations, Cognitive Structures and Individual Learners 60 Literature Which Helped Frame the Design of the Teaching Intervention 68 Research Questions 73 III. RESEARCH DESIGN AND METHODOLOGY 75 Design Research 75 Setting up the Intervention 78 Two Pilot Studies 78 Preparing for the Current Study 83 Defining Desirable Learning Outcomes 85 Professional Development in a Theoretical Context 86 The Research Team for the Current Study 86 Model Lessons 87 Intervention Setting and Theoretical Base 89 The Teaching Approach 89 Procedures 91 iv Intervention Fidelity and Implementation 93 Instrumentation 94 The Main Test Instrument: The Algebra Test 95 The Interview Protocol 99 Research Hypotheses, and Issues Related to the Quantitative Analyses 102 Comparisons of Scores on Tests Administered at the Same Time 104 Comparisons Between Gain Scores at Different Times 109 Effect Sizes 113 Post-Hoc Statistical Analyses 116 Issues Related to the Qualitative Analyses 117 Semiotic Aspects of the Qualitative Analyses 117 The Qualitative Analyses and Cognitive Structure 118 Concluding Comments 120 IV. QUANTITATIVE ANALYSES OF DATA 122 Overview of Quantitative Data 123 Analyses of Pre-Teaching Data 126 Pre-Teaching Group Differences with Respect to Structure 127 Pre-Teaching Group Differences with Respect to Modeling 127 Mid-Intervention Group Differences with Respect to Structure 128 Mid-Intervention Group Differences with Respect to Modeling 129 Post-Teaching Group Differences with Respect to Structure 130 Post-Teaching Group Differences with Respect to Modeling 130 Retention Group Differences with Respect to Structure 131 Retention Group Differences with Respect to Modeling 132 Comparisons Between Mean Gain Scores at Different Times 132 First Mean-Gain Comparison (Structure, Mid-Intervention Versus Pre-Teaching) 133 v Second Mean-Gain Comparison (Modeling, Mid-Intervention Versus Pre-Teaching) 134 Third Mean-Gain Comparison (Structure, Post-Teaching Versus Pre-Teaching) 135 Fourth Mean-Gain Comparison (Modeling, Post-Teaching Versus Pre-Teaching) 136 Fifth Mean-Gain Comparison (Structure, Retention Versus Pre-Teaching) 136 Sixth Mean-Gain Comparison (Modeling, Retention Versus Pre-Teaching) 137 Seventh Mean-Gain Comparison (Structure, Retention Versus Post-Teaching) 138 Eighth Mean-Gain Comparison (Modeling, Retention Versus Post-Teaching) 139 Calculation of Effect Sizes 140 Summary and Concluding Comments on the Quantitative Analyses 143 V. QUALITATIVE ANALYSES OF DATA 145 Introduction: Intended and Attained Common Core Curricula 145 Peirce’s Triadic Semiotic Position and Herbart’s Theory of Apperception 148 Analyses of Qualitative Data Generated by the “Structure” Intervention 151 Analyzing