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It is well-known that using clichés is considered poor form in writing. However, there is often a fine, essentially nonexistent line between a cliché and an old truism with a great deal of wisdom. The title of this book is one form of what is perhaps the best of all the words of wisdom, namely to be the best, you must learn from the best. No one knew this better than Niels Henrik Abel, Norway's greatest mathematician. He wrote the following in the margin of one of his notebooks, «It appears to me that if one wants to make progress in mathematics one should study the masters». This book is a publication of some of the papers presented at an international conference on the History of Mathematics held in Kristiansand, Norway in 1988. It is fitting that Abel lived in that area for some time. Reading about the actions of the masters is always refreshing and helps to improve your self-esteem. To know that even the great ones struggled and made colossal errors reminds us that mathematical progress is not linear, but extremely chaotic. If a chart could be made of the development of mathematics, it would exhibit a gross upward movement. However, if one was to perform an expansion transformation, the local behavior would resemble Brownian motion. It is also sad to be informed about some of the spiteful actions that even geniuses are capable of. The range of topics covered in this collection of papers is wide and includes some of the applied mathematical motivations in the development of new areas of mathematics. It is reasonable to argue that most of the development of mathematics throughout history originated in "simple" problems that had to be solved. Problems from the simplification of calculations to the trajectories of cannonballs to a set of bridges in the old city of Konigsberg all served as the impetus that led to the creation of new mathematics. Many of the papers also present problems that can be used in college classes. It is good for us all to occasionally revisit the historical origins of the topics that we present and re-present in class after class. Looking at it from the perspective of those who created it is sometimes the best way to get new insights into the material, and many such items are found in this book.
Published in Journal of Recreational Mathematics, reprinted with permission.
Preface
History in School Mathematics
History of Mathematics Can Help Improve Instruction and Learning by Shmuel Avital
The Role in the History of Mathematics of Algorithms and Analogies by Phillip S. Jones
Using Problems from the Histoiy of Mathematics in Classroom Instruction by Frank J.Swetz
Revisiting the History of Logarithms by John Fauvel
Napier’s Logarithms Adapted for Today’s Classroom by Victor J. Katz
Trigonometry Comes Out of the Shadows by Frank J. Swetz
Alluvial Deposits, Conic Sections, and Improper Glasses, or History of Mathematics Applied in the Classroom by Jan A. van Maanen
An Historical Example of Mathematical Modeling: The Trajectory of a Cannonball by Frank J. Swetz
History in Higher Mathematics
Concept of Function—Its History and Teaching by Man-Keung Siu
My Favorite Ways of Using History in Teaching Calculus by V. Frederick Rickey
Improved Teaching of the Calculus Through the Use of Historical Materials by Michel Helfgott
Euler and Heuristic Reasoning by Man-Keung Siu
Converging Concepts of Series: Learning from History by Joel P. Lehmann
Historical Thoughts on Infinite Numbers by Lars Mejlbo
Historical Ideas in Teaching Linear Algebra by Victor J, Katz
Wessel on Vectors by Otto B. Bekken
Who Needs Vectors? by Karen Reich
The Teaching of Abstract Algebra: An Historical Perspective by Israel Kleiner
Toward the Definition of an Abstract Ring by David M. Burton and Donovan H. Van Osdol
In Hilbert’s Shadow: Notes Toward a Redefinition of Introductory Group Theory by Anthony D. Gardiner
An Episode in the History of Celestial Mechanics and Its Utility in the Teaching of Applied Mathematics by Eric J. Alton
Mathematical Thinking and History of Mathematics by Man-Keung Siu
A Topics Course in Mathematics by Abe Shenitzcer
Niels Henrik Abel (1802-1829): A Tribute
About the Authors