This book is for college and secondary school teachers who want to know how they can use the history of mathematics as a pedagogical tool to help their students construct their own knowledge of mathematics. Often, a historical development of a particular topic is the best way to present a mathematical topic, but teachers may not have the time to do the research needed to present the material. This book provides its readers with the historical ideas and insights which can be immediately applied in the classroom

The book is divided into two sections: the first on the use of history in secondary school mathematics, and the second on its use in university mathematics. So, teachers planning a discussion of logarithms, will find here the historical background of that idea along with suggestions for incorporating that history in the development of the idea in class. Teachers of abstract algebra will benefit by reading the three articles in the book dealing with aspects of that subject and considering their ideas for presenting groups, rings, and fields.

### Table of Contents

Preface

1. History of Mathematics Can Help Improve Instruction and Learning

2. The Role in the History of Mathematics of Algorithms and Analogies

3. Using Problems from the History of Mathematics in Classroom Instruction

4. Revisiting the History of Logarithms

5. Napier’s Logarithms Adapted for Today’s Classroom

6. Trigonometry Comes Out of the Shadows

7. Alluvial Deposits, Conic Sections, and Improper Glasses

8. An Historical Example of Mathematical Modelig: The Trajectory of a Cannonball

9. Concept of Function—Its History and Teaching

10. My Favorite Ways of Using History in Teaching Calculus

11. Improved Teaching of the Calculus Through the Use of Historical Materials

12. Euler and Heuristic Reasoning

13. Converging Concepts of Series: Learning from History

14. Historical Thoughts on Infinite Numbers

15. Historical Ideas in Teaching Linear Algebra

16. Wessel on Vectors

17. Who Needs Vectors

18. The Teaching of Abstract Algebra: An Historical Perspective

19. Toward the Definition of an Abstract Ring

20. In Hilbert’s Shadow: Notes Toward a Redefinition of Introductory Group Theory

21. An Episode in the History of Celestial Mechanics and Its Utility in the Teaching of Applied Mathematics

22. Mathematical Thinking and History of Mathematics

23. A Topics Course in Mathematics

Niels Henrik Abel (1802-1829) A Tribute

About the Authors

### About the Editors

**Frank Swetz** is Professor of Mathematics and Education at the Pennsylvania State University at Harrisburg. His research concerns societal impact on the learning and teaching of mathematics. These interests have led him into studies on the history of mathematics and ethnomathematics. His findings are incorporated into teaching strategies which humanize mathematics teaching. His most recent books are: *The Sea Island Mathematical Manual: Surveying and Mathematics in Ancient China* (1992) and *From Five Fingers to Infinity: A Journey through the History of Mathematics* (1993).

**Otto Bekken** is Professor of Mathematics at Adger College, Kristiansand, Norway. His major research interests focus on the life and work of Niels Henrik Abel. He is particularly active in the incorporation of the history of mathematics into teacher training programs. He has written materials for this purpose for use in Scandinavia and Peru. He served as the chief organizer and director of the Kristiansand Conference.

**John Fauvel** is Lecturer in Mathematics at the Open University, England, and chaired the Open University course, “Topics in the History of Mathematics.” He has been an editor of several books, including *Darwin to Einstein: Historical Studies on Science and Belief* (1980), *Conceptions of Inquiry* (1981), *The History of Mathematics: A Reader* (1987) and *Let Newton Be!* (1988). He is the chair of the International Study Group on the Relations between History and Pedagogy of Mathematics.

**Victor Katz** is Professor of Mathematics at the University of the District of Columbia in Washington, D.C. He has been interested in the history of mathematics for many years, and in particular, in ways in which it can be used in the mathematics classroom. He has published several papers on both history and on its use in teaching. His textbook *A History of Mathematics* (Harper Collins, 1993) contains many suggestions as to how the material can be used in teaching on both the secondary and the undergraduate levels.