ABSTRACT Title of Document: MATH AND MATH-IN-SCHOOL: CHANGES IN THE TREATMENT OF THE FUNCTION CONCEPT IN TWENTIETH CENTURY SECONDARY ALGEBRA TEXTBOOKS. Kyle M. Cochran, M.A., 2008 Directed By: Associate Professor Daniel Chazan, Department of Curriculum and Instruction. The exercises found in the chapters on function in five American textbooks (each taken from different twenty-year spans of the twentieth century) were analyzed using Mesa’s (2000) coding scheme. Problems were analyzed based on the context, the operations needed to solve the problem, the representations used, and the control structures (or checks) available to the students. This analysis allowed for the identification of trends across time. These trends were compared to trends in the concept of function in the mathematics discipline and trends in recommendations for mathematics education. This analysis was undertaken to address three basic research questions. First, is there evidence of change in the treatment of function in school algebra across time? Second, how do any changes that exist in the texts correlate with recommendations for mathematics education? And third, how do changes in the texts correlate with the developments of the function concept in the mathematics discipline?MATH AND MATH-IN-SCHOOL: CHANGES IN THE TREATMENT OF THE FUNCTION CONCEPT IN TWENTIETH CENTURY SECONDARY ALGEBRA TEXTBOOKS. By Kyle M. Cochran Thesis submitted to the Faculty of the Graduate School of the University of Maryland, College Park, in partial fulfillment of the requirements for the degree of Master of Arts 2008 Advisory Committee: Associate Professor Daniel Chazan, Chair Professor Emeritus James Fey Professor Denny Gulick© Copyright by Kyle M. Cochran 2008ii Acknowledgements I would like to thank my wife, Rachel, and my family for their support in the writing of this document, including editing advice, general comments, and the knowledge that I could call any of them at any time. I would also like to thank my advisor, Daniel Chazan, who helped me to see both unifying structure where it might not be obvious and to see complexity in the seemingly simple things. His help to me on this thesis and my master’s education has been invaluable. To the committee as a whole, Daniel Chazan, James Fey, and Denny Gulick, I would like to express thanks for their contributions to me and my education and their contributions to mathematics and mathematics education as a whole. Finally, I would like to thank all of you (you know who you are) who have helped me to be a better student, a better teacher, and a better person.iii Table of Contents Acknowledgements....................................................................................................... ii Table of Contents......................................................................................................... iii List of Tables ................................................................................................................ v List of Figures.............................................................................................................. vi Chapter 1: Rationale ..................................................................................................... 1 Where the Journey Began ......................................................................................... 1 Background............................................................................................................... 2 Research Questions................................................................................................... 4 Chapter 2: Theoretical and Analytic Frameworks........................................................ 8 Conception and The Coding of Data ........................................................................ 8 Recommendations and Reform............................................................................... 10 A Brief History of the Function Concept................................................................ 20 A Hypothetical Enterprise....................................................................................... 23 Chapter 3: Methodology ............................................................................................. 25 Textbook Selection ................................................................................................. 25 The Decision to use Textbooks........................................................................... 25 How the Texts were Chosen ............................................................................... 26 How the Sections were Chosen........................................................................... 30 The Coding System................................................................................................. 33 Examples of the Coding System......................................................................... 35 Modifications to the Coding System................................................................... 40 Chapter 4: Results....................................................................................................... 44 Treatment of Function in the Textbooks................................................................. 44 Wentworth’s Elementary Algebra (1906)........................................................... 45 Hart’s Progressive High School Algebra (1935)................................................ 46 Welchons and Krickenberger’s Algebra, Book One (1949) ............................... 48 Dolciani’s Modern Algebra: Structures and Methods, Book One (1962) .......... 50 UCSMP’s Algebra (1990)................................................................................... 52 Presentation of Results............................................................................................ 53 Definition of Function and Placement in the Textbook...................................... 54 Problems ............................................................................................................. 55 Operations........................................................................................................... 57 Representation Systems ...................................................................................... 59 Control Structures