HOW DO STUDENTS ACQUIRE AN UNDERSTANDING OF LOGARITHMIC CONCEPTS? A dissertation submitted to the Kent State University Graduate School of Education, Health, and Human Services in partial fulfillment of the requirements for the degree of Doctor of Philosophy By Ellen Mulqueeny August 2012 © Copyright, 2012 by Ellen Mulqueeny All Rights Reserved ii A dissertation written by Ellen Mulqueeny B.C.E., Cleveland State University, 1979 M.S., Cleveland State University, 1993 Ph.D., Kent State University, 2012 Approved by _______________________, Co-director, Doctoral Dissertation Committee Michael Mikusa _______________________, Co-director, Doctoral Dissertation Committee Joanne Caniglia _______________________, Member, Doctoral Dissertation Committee Donald White Accepted by _______________________, Director, School of Teaching, Learning, Alexa Sandmann and Curriculum Studies _______________________, Dean, College and Graduate School of Daniel F. Mahony Education, Health and Human Services iii MULQUEENY, ELLEN, Ph.D., August 2012 Teaching, Leadership and Curriculum Studies HOW DO STUDENTS ACQUIRE AN UNDERSTANDING OF LOGARITHMIC CONCEPTS? (332 pp.) Co-Directors of Dissertation: Michael Mikusa, Ph.D. Joanne Caniglia, Ph.D. The use of logarithms, an important tool for calculus and beyond, has been reduced to symbol manipulation without understanding in most entry-level college algebra courses. The primary aim of this research, therefore, was to investigate college students’ understanding of logarithmic concepts through the use of a series of instructional tasks designed to observe what students do as they construct meaning. APOS Theory was used as a framework for analysis of growth. APOS theory is a useful theoretical framework for studying and explaining conceptual development. Closely linked to Piaget’s notions of reflective abstraction, it begins with the hypothesis that mathematical activity develops as students perform actions that become interiorized to form a process understanding of the concept, which eventually leads students to a heightened awareness or object understanding of the concept. Prior to any investigation, the researcher must provide an analysis of the concept development in terms of the essential components of this theory: actions, process, objects, and schemas. This is referred to as the genetic decomposition. The results of this study suggest a framework that a learner may use to construct meaning for logarithmic concepts. Using tasks aligned with the initial genetic decomposition, the researcher made revisions to the proposed genetic decomposition in the process of analyzing the data. The results indicated that historical accounts of the development of this concept might be useful to promote insightful learning. Based on this new set of data, iterations should continue to produce a better understanding of the student’s constructions. ACKNOWLEDGMENTS Many people have supported my efforts during this long process. I would like to acknowledge them and thank them for the many hours of their valuable time that they have given to me. My advisor and co-director, Dr. Michael Mikusa, has provided support since my arrival at Kent State University. He has offered his guidance and expertise, encouraging me to complete this journey. He challenged me to think as a researcher and not solely as a classroom instructor. This caused me to pause and forced me to reconceptualize my research interests several times in the course of this journey. Thanks also to Dr. Joanne Caniglia, my co-director, for her support and encouraging words along the way. I greatly appreciated both of you offering ideas for improving my study as well as your reading and guidance through the numerous drafts of this dissertation as it took its final form. Both of you have been a source of scholarly comfort for the past three years and I believe I am a better mathematics educator for this. I would also like to recognize Dr. Donald White for his mathematical insights and his critical analysis of this study. I do appreciate the opportunity you have given me to accomplish my goal of earning a doctorate. Such a long journey would not have been possible without the support of friends and colleagues along the way, Teresa Graham, Carol Phillips-Bey, Peggy Slavik, Jason Stone, Ieda Rodrigues, Kingsley Magpoc, and Wade Zwingler: thanks for just being there when I needed your support. iv TABLE OF CONTENTS ACKNOWLEDGMENTS ................................................................................................. iv LIST OF FIGURES ......................................................................................................... viii LIST OF TABLES ............................................................................................................. ix CHAPTER I. INTRODUCTION TO THE STUDY .........................................................................1 Background Information ............................................................................................4 Problem Statement ....................................................................................................10 Purpose of Research ..................................................................................................15 Research Questions ...................................................................................................16 Theoretical Basis for the Study .................................................................................17 A Constructivist Perspective ............................................................................17 Advanced Mathematical Thinking ...................................................................18 APOS Theory 21 The action construct ................................................................................22 The process construct ..............................................................................23 The object construct ................................................................................24 The schema construct. .............................................................................26 Instructional Implications ........................................................................................28 Goals of the Study .....................................................................................................30 Methodological Considerations ...............................................................................30 Summary of Chapter 1 ..............................................................................................32 Definition of Terms ..................................................................................................33 II. REVIEW OF RELATED LITERATURE ................................................................36 Introduction ..............................................................................................................36 Influence of Symbols in Algebraic Thinking ...........................................................37 Functions and Associated Learning Difficulties .......................................................46 Students’ Understanding of Exponents and Exponential Functions .........................51 Inverse Functions .....................................................................................................56 Historical Development of Logarithmic Concepts ...................................................59 Students’ Understanding of Logarithms ..........................................................62 Advanced Mathematical Thinking and APOS Theory .............................................69 Summary of Related Work ......................................................................................71 v III. QUALITATIVE METHODOLOGY ........................................................................75 Introduction ...............................................................................................................75 Qualitative Versus Qualitative Research Design ......................................................76 Qualitative Methodology ..........................................................................................81 Qualitative Research in Mathematics Education .............................................82 Teaching Experiments .....................................................................................83 Research Design........................................................................................................87 Research Site ............................................................................................................88 Sample.......................................................................................................................90 Procedure ..................................................................................................................92 The Role of APOS Theory ...............................................................................93 The genetic decomposition .....................................................................95 Defining the genetic decomposition of logarithms ........................98 Overview of instructional design .................................................103 Protocols ........................................................................................................106 Triangulation ..................................................................................................107 Inter-Rater Reliability .............................................................................................107