A History of Trigonometry Education in the United States: 1776-1900
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A History of Trigonometry Education in the United States: 1776-1900 Jenna Van Sickle Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy under the Executive Committee of the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2011 ©2011 Jenna Van Sickle All Rights Reserved ABSTRACT A History of Trigonometry Education in the United States: 1776-1900 Jenna Van Sickle This dissertation traces the history of the teaching of elementary trigonometry in United States colleges and universities from 1776 to 1900. This study analyzes textbooks from the eighteenth and nineteenth centuries, reviews in contemporary periodicals, course catalogs, and secondary sources. Elementary trigonometry was a topic of study in colleges throughout this time period, but the way in which trigonometry was taught and defined changed drastically, as did the scope and focus of the subject. Because of advances in analytic trigonometry by Leonhard Euler and others in the seventeenth and eighteenth centuries, the trigonometric functions came to be defined as ratios, rather than as line segments. This change came to elementary trigonometry textbooks beginning in antebellum America and the ratios came to define trigonometric functions in elementary trigonometry textbooks by the end of the nineteenth century. During this time period, elementary trigonometry textbooks grew to have a much more comprehensive treatment of the subject and considered trigonometric functions in many different ways. In the late eighteenth century, trigonometry was taught as a topic in a larger mathematics course and was used only to solve triangles for applications in surveying and navigation. Textbooks contained few pedagogical tools and only the most basic of trigonometric formulas. By the end of the nineteenth century, trigonometry was taught as its own course that covered the topic extensively with many applications to real life. Textbooks were full of pedagogical tools. The path that the teaching of trigonometry took through the late eighteenth and nineteenth centuries did not always move in a linear fashion. Sometimes trigonometry education stayed the same for a long time and then was suddenly changed, but other times changes happened more gradually. There were many international influences, and there were many influential Americans and influential American institutions that changed the course of trigonometry instruction in this country. This dissertation follows the path of those changes from 1776 to 1900. After 1900, trigonometry instruction became a topic of secondary education rather than higher education. Table of Contents Chapter I: Introduction…………………………………………………………….. 1 Need for the Study………………………………………………................ 1 Purpose of the Study……………………………………………................. 3 Procedures of the Study…………………………………………................ 4 Chapter II: Literature Review……………………………………………………… 7 The History of Mathematics Education…………………………………… 10 References that Directly Inform this Study………………………………. 10 Allen‘s Teaching of Trigonometry………………………………………... 22 Mathematics Education References for Methodology……………………. 23 Overview of the History of Trigonometry………………………………… 32 ―Historical Reflections on the Teaching of Trigonometry‖………………. 40 Resources on the History of Trigonometry……………………………….. 43 The History of Higher Education…………………………………………. 46 Summary and Synthesis…………………………………………………… 55 From the ―Line System‖ to the ―Ratio System‖…………………………... 56 Textbook Adoption in Colleges and Universities………………................. 57 The French and European Influences……………………………………... 58 Chapter III: Methodology…………………………………………………………. 59 Methods for Historical Research………………………………………….. 59 Methods for History of Mathematics Education…………………………. 61 Methods for Textbook Analysis…………………………………………... 65 Summary…………………………………………………………………... 67 Chapter IV: The Early Establishment of Trigonometry Education………………... 68 A Note on Terminology…………………………………………………… 69 On the Construction of Trigonometric Tables…………………………….. 71 Eighteenth and Early Nineteenth Century Teaching of Trigonometry……. 72 English Textbooks………………………………………………………… 73 Scottish Textbooks………………………………………………………... 84 American Textbooks………………………………………………………. 87 Trends in Textbooks—English, Scottish, and American…………………. 92 Summary…………………………………………………………………... 100 i Chapter V: Antebellum Trigonometry Education…………………………………. 103 A Note on Terminology…………………………………………………… 104 Early to Mid-Nineteenth Century Teaching of Trigonometry…………….. 105 French Textbooks…………………………………………………………. 106 Translations and Adaptations of French Textbooks………………………. 108 Liberal Translations and Adaptations of French Textbooks……………… 116 American Textbooks………………………………………………………. 118 New Pedagogy Takes Shape………………………………………………. 128 Changes in Structure that Reflect Changes in Mathematical Thinking…… 130 Debate on the Ratio System Versus the Line System…………………….. 131 On the Field of Analytic Trigonometry…………………………………… 140 Summary…………………………………………………………………... 145 Chapter VI: Turning Trigonometry on its Head…………………………………… 146 A Note on Terminology…………………………………………………… 146 Late Nineteenth Century Teaching of Trigonometry……………………... 148 American Textbooks……………………………………………………… 150 Final Exams in Trigonometry……………………………………………... 168 German Influence in Mathematics Education…………………………….. 172 Trigonometric Definitions Change Radically……………………………... 173 Pedagogy Develops Further……………………………………………….. 176 Changes in the Content of Trigonometry…………………………………. 180 Summary…………………………………………………………………... 187 Chapter VII: Conclusion and Recommendations………………………………….. 188 Research Questions Answered……………………………………………. 189 A Consideration of the Reasons for the Changes…………………………. 196 Limitations of the Study…………………………………………………... 197 Recommendations for Further Study……………………………………… 199 Final Remarks……………………………………………………………... 200 References………………………………………………………………………….. 202 Appendix A: Late Nineteenth Century Textbooks………………………………… 208 ii List of Charts, Graphs, and Illustrations Figure 2.1……...........12 Figure 2.2…………...12 Figure 2.3……………35 Figure 2.4……………39 Figure 3.1……………64 Figure 3.2……………65 Figure 4.1……………69 Figure 4.2……………70 Figure 4.3……………73 Figure 4.4…………….87 Figure 4.5…………….93 Figure 5.1……………134 Figure 6.1……………150 Figure 6.2……………155 Figure 6.3……………156 Figure 6.4……………160 Figure 6.5……………163 Figure 6.6……………182 Figure A.1…………...207 iii Acknowledgements I would like to thank my adviser, Dr. Alexander P. Karp, for his continued guidance and for never letting me settle for anything less than the best. I would also like to thank Dr. J. Philip Smith for mentoring and encouraging me along the way, and for reading drafts of my chapters and responding with wonderfully helpful comments. I would also like to thank the members of my committee, Dr. Bruce Vogeli, Dr. Erica Walker, Dr. Patrick Gallagher, and Dr. Felicia Moore Mensah for taking time out of their busy lives to read this dissertation and participate in my oral defense. Thank you to all the teachers throughout my life, who have taught me, guided me, encouraged me, and believed in me. I have had so many such teachers that I cannot possibly thank them all individually. Without all of you, I would not be here. I would like to express my deepest thanks to all of my family, for their absolutely unwavering support. My parents, Susan and Richard Cain, have always believed in me, loved me, and supported me in every way possible. My husband, Jacob, and my sons, Michael and John, have sacrificed everything for me, and it is to them that I have dedicated this dissertation. I am grateful every day for their unconditional love and constant companionship. Thank you to my Lord, God, and Savior Jesus Christ, for every good and perfect gift is from above. iv To Jacob, Michael, and John v 1 CHAPTER I: Introduction Need for the Study The history of mathematics education is in the focus of research today, both internationally and within the United States. The most notable work on the history of mathematics education in the United States is a two-volume History of School Mathematics, edited by George M. A. Stanic and Jeremy Kilpatrick, published by the National Council of Teachers of Mathematics (2003). The history of mathematics education was the topic of a study group at the International Congress on Mathematics Education in Copenhagen in 2004, and based on the experience of this study group the International Journal for the History of Mathematics Education was created. In recent years, important contributions to the history of mathematics education in the United States have been made by many researchers. These contributors include Amy Ackerberg- Hastings (2000) Leo Corry (2007),, Eileen Frances Donoghue (2001, 2008), Gloriana Gonzalez and Patricio G. Herbst (2006), Dustin Jones (2008), Jeremy Kilpatrick (1994, 2003), Sharon L. Senk and Denisse R. Thompson (2003), George M. A. Stanic (2003), etc. Understanding this history has implications for how mathematics is taught today (Cajori, 1890; Coxford and Jones, 1970; Butts, 1971). Furthermore, ―we, as mathematics 2 educators, need a record of our history not simply to serve as a partial guide to present actions but to better understand who we are‖ (Stanic and Kilpatrick, 2003, p. 13). Currently, there is concern about the state of trigonometry instruction. Current studies on trigonometry instruction include Fi, 2003; McMullin, 2003; Weber, 2005; Thompson, 2007; Bressoud, 2010, etc. A better understanding of the history of trigonometry education will aid researchers