The Evolution of College Algebra- Competencies and Themes of a Qu

Western Kentucky University TopSCHOLAR® Dissertations Graduate School Fall 2017 The Evolution of College Algebra: Competencies and Themes of a Quantitative Reasoning Course at the University Of Kentucky Scott Taylor Western Kentucky University, scott.taylor892@topper.wku.edu Follow this and additional works at: https://digitalcommons.wku.edu/diss Part of the Higher Education Commons, History of Science, Technology, and Medicine Commons, Other History Commons, and the Science and Mathematics Education Commons Recommended Citation Taylor, Scott, "The Evolution of College Algebra: Competencies and Themes of a Quantitative Reasoning Course at the University Of Kentucky" (2017) Dissertations Paper 139 https://digitalcommons.wku.edu/diss/139 This Dissertation is brought to you for free and open access by TopSCHOLAR® It has been accepted for inclusion in Dissertations by an authorized administrator of TopSCHOLAR® For more information, please contact topscholar@wku.edu THE EVOLUTION OF COLLEGE ALGEBRA: COMPETENCIES AND THEMES OF A QUANTITATIVE REASONING COURSE AT THE UNIVERSITY OF KENTUCKY A Dissertation Presented to The Faculty of the Educational Leadership Doctoral Program Western Kentucky University Bowling Green, Kentucky In Partial Fulfillment Of the Requirements for the Degree Doctor of Education By Scott Taylor December 2017 CONTENTS LIST OF FIGURES vi LIST OF TABLES vii CHAPTER I: STATEMENT OF THE PROBLEM .1 Introduction College Algebra .2 Quantitative Reasoning College Algebra as a Quantitative Reasoning Course .9 Historical Influences 10 UK and KCTCS 13 Purpose and Central Research Questions 14 Empirical Research Questions 15 Chapter I summary 16 CHAPTER II: REVIEW OF LITERATURE .17 College Algebra 19 College Algebra in Kentucky 20 Commonalities 22 Disparities 23 Instrument variation and the myth of college readiness 24 History of Educational Reform 31 Mathematics and Early American Colleges 31 The Mathematical Association of America (MAA) 33 MAA and QR 42 iii Quantitative Reasoning Requirement 44 Current administrative policies 44 Institutional missions & philosophies of QR 46 Government, Politics, and War 49 WWII/GI Bill 49 The Space Race—an essential STEM race .50 National education reform 52 Effects of Economics and Funding 53 Chapter II summary 55 CHAPTER III: METHODOLOGY .57 Research Design 57 Role of the Researcher 59 Trustworthiness 59 Denial of the one-to-one function 60 Rejecting CA as the default QR 61 Other values 62 Sources of Data 64 College catalogs 65 Course syllabi 66 Mathematics textbooks 67 Other documents 69 iv Overview of Instrumentation 69 Procedures/Data Collection 70 Analysis Plan 70 Delimitations and Limitations of this Study 72 Chapter III summary 74 CHAPTER IV: FINDINGS .76 Common Topics—Textbooks Once Used in CA 77 Common topics—functions 77 Common topics—polynomial functions 87 Common topics—rational functions 94 Common topics—exponential functions 98 Common topics—logarithmic functions 104 Common Topics—Relating RQ1 with Textbooks .111 Common Topics—Course Descriptions from Catalogs 112 Common topics—Relating RQ1 with Course Descriptions 121 Summary of RQ1—transition to RQ2 123 Internal Forces—Documents from the UK Archives and the Math Website 123 Internal forces—examinations 127 Internal forces—syllabi 128 Internal Forces—Relating RQ2 with archival and website documents .131 Summary of RQ2—transition to RQ3 131 QR Evolution—Documents from the Self-Study 132 v QR Evolution—Relating RQ3 with Self-Study Documents 136 Chapter IV summary 136 CHAPTER V: CONCLUSIONS 137 Summaries on RQ1 137 Summaries on RQ2 141 Summaries on RQ3 143 Significance to Educational Leadership 145 Performance-based funding 145 Pathways and meta-majors 146 Liberal arts philosophy and academic integrity 147 Suggestions for Further Research 148 Conclusions 149 REFERENCES .151 APPENDIX A: Data References 176 APPENDIX B: Catalogs 181 APPENDIX C: Catalog Notes 200 APPENDIX D: First-Round Coding on Textbooks 202 APPENDIX E: Examinations 227 APPENDIX F: Sample HCC Syllabus 229 vi LIST OF FIGURES Figure Excerpt from Maura Corley’s CA fall syllabus 66 Figure Definition of a function from ABN 79 Figure Evaluating functions as Example from ABN 79 Figure Revised definition of a function from ABN 80 Figure Bold bullets were added in the third edition of ABN in Example 81 Figure Flow of material from definitions to Example to Example 82 Figure Example 1, part a, from the first edition of the MLS text 86 Figure Definition of a polynomial function from MLS 91 Figure Definition of a rational function from the ABN textbook 95 Figure 10 Definition of a rational function from MLS 97 Figure 11 Definition of an exponential function from ABN 100 Figure 12 The graph of the same exponential function with different domains 102 Figure 13 A four-part theorem regarding properties of exponential expressions 102 Figure 14 The MLS definition of an exponential function 102 Figure 15 Definition of a logarithm from the first edition of the ABN text 105 Figure 16 Properties of logarithms from the first edition of the ABN text 106 Figure 17 The definition of a logarithm from the first edition of the MLS text 108 Figure 18 The definition of a logarithm from the third edition of the MLS text 108 Figure 19 The definition of a logarithm from the fourth edition of the MLS text 109 Figure 20 Five properties of logarithms as given in the first edition of ABN 110 Figure 21 Screenshot of the UK catalog from 1865 112 Figure 22 Screenshot of the UK catalog from 1892 115 vii Figure 23 The first mention of college algebra in the 1908-1909 catalog 116 Figure 24 The return of specific topics in CA from the 1913-1914 catalog 117 Figure 25 The removal of functions in the 1918-1919 catalog 117 Figure 26 The 1921-1922 catalog returned to a limited information format 118 Figure 27 The 1931-1932 catalog returned to a limited information format 118 Figure 28 The 1940-1941 catalog included specific topics 118 Figure 29 The 1950-1951 catalog excluded specific topics 119 Figure 30 The wording from the 1976-1977 catalog was mostly unchanged 120 Figure 31 Another professor White in the 1908 edition of The Kentuckian 124 Figure 32 The 1909 edition of The Kentuckian described J.G White 124 Figure 33 The 1910 The Kentuckian described CA 125 viii LIST OF TABLES Table Topics covered in CA identified by era and starting year ……………122 ix  Then defining Euler’s number using compound interest  EX5 is exponential growth/decay  EX6 is radioactive decay word problem 2nd Ed MLS 𝑥  EX2 changed to graph 𝑓(𝑥) = 2𝑥 and graph 𝑓(𝑥) = (2)  Figures added to show different bases  EX3 graph 𝑓(𝑥) = 2−𝑥  EX4 fractional base  EX5 solving exponential equation for base  Compound interest formula given in box  EX6 compound interest word problem and Euler’s number  Euler’s number given to nine places  EX7 population growth problem 3rd Ed MLS  Introduction added to section  Opening conversation about doubling pennies  Definition of exponential given in words  Repeat concept from chapter about 𝑎𝑚 for rational values  Conversation about 2√3 and how √3 ≈ 1.7, ≈ 1.73, ≈ 1.732  Three graphs of different domains given  Four-part theorem given  EX1 is solving exponential equation for an exponent 219  EX2 is solving an exponential equation for the base  “Caution” added for extraneous solutions  Definition of exponential given in box  EX3 is evaluate  EX4 is graph 𝑓(𝑥) = 2𝑥 and graph 𝑓(𝑥) = (2)  Box of properties of the graph of an exponential, including (0,1) is a point; 𝑥 if a>1, f(x) increases, and if 0