Over the years, the journals of the National Council of Teachers of Mathematics (NCTM) have published numerous articles on the history of mathematics and its use in teaching. These journals include *Teaching Children Mathematics, Mathematics Teaching in the Middle School,* and *Mathematics Teacher.* *Convergence* founding co-editor Frank Swetz arranged with NCTM to republish in *Convergence* (in pdf format) selected articles from *Mathematics Teacher,* beginning in 2015. One of the editor’s picks for this year is an article by David Bressoud on the origins of trigonometry and how the history of the subject suggests a new approach to trigonometry in our modern-day schools:

David M. Bressoud, "Historical Reflections on Teaching Trigonometry," *Mathematics Teacher,* Vol. 103, No. 8 (April 2010), pp. 578-582, plus three supplementary sections, "Hipparchus," "Euclid," and "Ptolemy." Reprinted with permission from *Mathematics Teacher, *© 2010 by the National Council of Teachers of Mathematics. All rights reserved.

Bressoud recently recalled how he came to write this article:

"Historical Reflections on Teaching Trigonometry" was based on a talk I gave at the 2009 NCTM Annual Meeting, "Lessons from the History of Mathematics." The title had been a place-holder as I decided what I wanted to talk about. Two things came together to convince me that I should talk about trigonometry. One was a visit to Marilyn Carlson and Pat Thompson at Arizona State University where I learned about their work and that of Kevin Moore, then a graduate student, on student difficulty with making the transition from trigonometric functions as ratios of sides of right triangles to a functional understanding of trigonometric functions. At the same time, I had just read Glen Van Brummelen's wonderful book, *The Mathematics of the Heavens and the Earth: The Early History of Trigonometry*, which showed that the functional approach, the sine and cosine as functions of arc length, is the historical approach. Al Goetz, then editor of* Mathematics Teacher,* was in the audience for my talk and convinced me to write it up for the magazine. My initial submission included the material from the three supplements in the article. I was not surprised when Al said that made it too long, so I carved them out as online supplements.

The article, "Historical Reflections on Teaching Trigonometry" (download it here), first reviews the invention of "circle trigonometry" by 2nd century BCE Greek astronomers who, given the length of an arc connecting two points on a circle, sought the length of the chord connecting the two points. By the end of the 5th century CE, Indian astronomers had focused instead on the half-chord, or *sine,* and this convention was adopted by later Arab and European astronomers. It was Euler who finally fixed the radius of the defining circle at 1. Bressoud admits that "triangle trigonometry," with ancient origins in shadow measurement, is important, too. However, he argues that, in order to best help students understand the trigonometric *functions,* teachers should focus first on the original functional relationship of circle trigonometry; namely, chord length as a function of arc length.

#### About the Author

David M. Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul, Minnesota. He has published extensively in history of mathematics and its use in teaching, including the MAA books *A Radical Approach to Real Analysis,* *A Radical Approach to Lebesgue's Theory of Integration,* and *Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture*. He served as MAA President during 2009 and 2010 and currently serves as Director of the Conference Board of the Mathematical Sciences (CBMS). - Janet Beery, Editor, *MAA Convergence*

#### About NCTM

The National Council of Teachers of Mathematics (NCTM) is the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development, and research. It publishes five journals, one for every grade band, as well as one on the latest research and another for teacher educators. With 80,000 members and more than 200 Affiliates, NCTM is the world’s largest organization dedicated to improving mathematics education in prekindergarten through grade 12. For more information on NCTM membership, visit http://www.nctm.org/membership.

#### Other *Mathematics Teacher* articles in *Convergence*

Patricia R. Allaire and Robert E. Bradley, “Geometric Approaches to Quadratic Equations from Other Times and Places,” *Mathematics Teacher,* Vol. 94, No. 4 (April 2001), pp. 308-313, 319.

Keith Devlin, "The Pascal-Fermat Correspondence: How Mathematics Is Really Done," *Mathematics Teacher,* Vol. 103, No. 8 (April 2010), pp. 578-582.

Peter N. Oliver, "Pierre Varignon and the Parallelogram Theorem," *Mathematics Teacher,* Vol. 94, No. 4 (April 2001), pp. 316-319.

Peter N. Oliver, "Consequences of the Varignon Parallelogram Theorem," *Mathematics Teacher,* Vol. 94, No. 5 (May 2001), pp. 406-408.

Shai Simonson, “The Mathematics of Levi ben Gershon,” *Mathematics Teacher,* Vol. 93, No. 8 (November 2000), pp. 659-663.