Copyright by Tesfayohannes Kiflemariam Haile 2008 The Dissertation Committee for Tesfayohannes Kiflemariam Haile Certifies that this is the approved version of the following dissertation: A Study on the Use of History in Middle School Mathematics: The Case of Connected Mathematics Curriculum Committee: Philip Uri Treisman, Supervisor O.L. Davis, Jr., Co-Supervisor Sherry Field Norvell Northcutt Anthony Petrosino A Study on the Use of History in Middle School Mathematics: The Case of Connected Mathematics Curriculum by Tesfayohannes Kiflemariam Haile, BSc., MSc. Dissertation Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy The University of Texas at Austin December 2008 Dedication To my brother Amanuel Kiflemariam Haile Acknowledgements I feel that acknowledgement in writing does not do justice to conveying the deep gratitude I have for a number of individuals and organizations that supported me in carrying out this study. But, I will try anyway. First, I would like to express my gratitude to my parents, who did their best early on to instill in me the values of discipline, perseverance, and hard work—factors that are extremely important for success in graduate study. I hear your whisper coming from afar, thousands of miles away, “e’ze m’hrokha aiwda’en du yu?” I understand why you ask if my formal education has any ending. I hope you will be prouder when I accomplish my goals. I am also grateful to my siblings for their love and support. My deep gratitude goes to the Fulbright scholarship program for sponsoring part of my graduate study. I am very privileged to have received the support. This program conveyed to me America’s desire to connect with people around the globe across cultural and educational avenues. I would like to express my heartfelt appreciation to the Graduate Office and International Office at the University of Texas at Austin for the financial support and good services I received. Special thanks to Dr. Darlene Grant from the Graduate Office. I extend my gratitude to all members of my dissertation committee for their support and general guidance. The comments provided by each member during the proposal defense were constructive and very helpful in reflecting and reshaping the study. Dr Davis, Jr., Dr. Field, and Dr. Northcutt provided comments that were helpful in rethinking my approach to the study; Dr. Petrosino offered specific suggestions on the v literature and raised a critical question on the significance of the study. Dr. Triesman’s comments regarding the thesis and his suggestion regarding study participants (particularly teachers) were useful. Dr.Treisman helped find potential teacher interviewees and secured funds (through the Dena Center) for teacher interviews. Thanks to all my dissertation study participants without whom this study would have been harder to realize. Special thanks should also go to the Austin Independent School District research and evaluation section for allowing me to conduct interviews in the district schools. I also extend my thanks to Pearson Education for granting me permission to use pages from the Connected Mathematics books for appendices. Last but not least, I would like to forward my thanks to my graduate student friend Willy Rotich for reading the manuscript and giving helpful comments. I would also like to extend my appreciation to Ashley Schmiedekamp, Ashwini Salpekar, and Melissa Campos-Hernandez for editorial help. vi A Study on the Use of History in Middle School Mathematics: The Case of Connected Mathematics Curriculum Publication No._____________ Tesfayohannes Kiflemariam Haile, PhD The University of Texas at Austin, 2008 Supervisors: Philip Uri Treisman and O.L. Davis, Jr. This dissertation explores the use of history of mathematics in middle school mathematics. A rationale for the importance of the incorporation of historical dimensions (HD) of mathematics is provided through a review of the literature. The literature covers pedagogical, philosophical, psychological, and social issues and provides arguments for the use of history. The central argument is that history can help reveal significant aspects regarding the origins and evolutions of ideas that provide contexts for understanding the mathematical ideas. History can be used as a means to reflect on significant aspects— errors, contractions, challenges, breakthroughs, and changes—of mathematical developments. Noting recent NCTM (2000) calls for school math to include so-called process standards, I contend that incorporating the history of mathematics can be considered as part of this standard. This study examines how HD is addressed in a contemporary mathematics curriculum. Specifically, the study examines the Connected Mathematics Project (CMP) as a case. This curriculum has some historical references which triggered further exploration on how seriously the historical aspects are vii incorporated. The analysis and discussion focus on four CMP units and interviews with three curriculum experts, eight teachers, and 11 middle school students. The analysis of textbooks and interviews with the experts explore the nature and purpose of historical references in the curriculum. The interviews with teachers and students focus on their perspectives on the importance of HD in learning mathematics. This study examines specifically historical incorporations of the concepts of fractions, negative numbers, the Pythagorean Theorem, and irrational numbers. The analysis reveals that CMP exhibits some level of historical awareness, but the incorporation of HD was not systematically or seriously considered in the development of the curriculum. The interviews suggest that the teachers did not seriously use the limited historical aspects available in the textbooks. The experts’ and teachers’ interviews suggest skepticism about the relevance of HD for middle school mathematics. The teachers’ accounts indicate that students are most interested in topics that are related to their experience and to future applications. The students’ accounts do not fully support the teachers’ assessment of students’ interest in history. I contend that incorporating HD can complement instruction in ways that relate to students’ experiences and to applications besides adding an inquiry dimension to instruction. viii Table of Contents List of Tables ....................................................................................................... xiii List of Figures ........................................................................................................xv Chapter One: Introduction .......................................................................................1 Chapter Two: Review of the Literature ...................................................................7 Rationale for Incorporation of Historical Dimension into School Mathematics ........................................................................................7 Pedagogical Grounds .............................................................................7 Philosophical Grounds .........................................................................13 Cognitive (psychological) Grounds .....................................................19 Social/Cultural Grounds.......................................................................26 Ways of Incorporating History of Mathematics in School Mathematics .....30 Conceptual Challenges and History of Selected Concepts ...........................34 Conceptual Challenges Associated with Selected Concepts ...............34 Fractions ......................................................................................34 Negative Numbers ......................................................................37 Pythagorean Theorem .................................................................40 Irrational Numbers ......................................................................42 Historical Reviews of the Selected Topics ..........................................46 Fractions ......................................................................................46 Negative Numbers ......................................................................55 The Pythagorean Theorem ..........................................................61 Irrational Numbers ......................................................................67 Chapter Three: Research Methodology .................................................................74 Research Positionality ...................................................................................74 Conceptual Framework .................................................................................76 Methods of Data Gathering and Analysis .....................................................78 Content Analysis ..................................................................................78 ix Interviews .............................................................................................80 The Setting ....................................................................................................85 Chapter Four ..........................................................................................................86 Incorporation of HD in the Curriculum: ................................................................86 The Case of the Connected Mathematics Project ..................................................86 Analysis