This book is a collection of articles by international specialists in the history of mathematics and its use in teaching, based on presentations given at an international conference in 1996. Although the articles vary in technical or educational level and in the level of generality, they show how and why an understanding of the history of mathematics is necessary for informed teaching of various subjects in the mathematics curriculum, bot at secondary and at university levels. May of the articles can serve teachers directly as the basis of classroom lessons, while others will give teachers plenty to think about in designing courses or entire curricula. For example, there are articles dealing with the teaching of geometry and quadratic equations to high school students, of linear algebra, combinatorics, and geometry to university students, and of the notion of pi at various levels. But there is also an article showing how to use historical problems in various courses and one dealing with mathematical anomalies and their classroom use.

Although the primary focus or subject of the book is the teaching of mathematics through its history, some of the articles deal more directly with topics in the history of mathematics not usually found in textbooks. These articles will give teachers valuable background. They include one on the background of Mesopotamian mathematics by one of the world’s experts in this field, one on the development of mathematics in Latin America by a mathematician who has done much primary research in this little known field, and another on the development of mathematics in Portugal, a country whose mathematical accomplishments are little known. Finally, an article on the reasons for studying mathematics in Medieval Islam will give all teachers food for thought when they discuss similar questions, while a short play which covers the work of Augustus DeMorgan will help teachers provide an outlet for their class thespians.

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Table of Contents

Preface

Part I: General Ideas on the Use of History in Teaching

Part II: Historical Ideas and their Relationship to Pedagogy

Part III: Teaching a Particular Subject Using History

Part IV: The Use of History in Teacher Training

Part V: The History of Mathematics

Notes on Contributors

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About the Editor

**Victor J. Katz** was born in Philadelphia. He received his Ph.D. in mathematics from Brandeis University in 1968. He has long been interested in the history of mathematics and, in particular, its use in teaching. His well-regarded textbook, *A History of Mathematics: An Introduction*, is now in its second edition. Its first edition won the Watson Davis prize of the History of Science Society in 1995, a prize awarded annually to the best book on the history of science aimed at undergraduates. He has published numerous articles on the history of mathematics and its use in teaching. He has also directed two NSF-sponsored projects which helped college teachers learn the history of mathematics and how to use it in teaching, and involved high school teachers in writing materials using history in the teaching of various topics in the high school curriculum.

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MAA Review

Don't be turned off by the title. Victor Katz has gathered a diverse and fascinating selection of 26 essays on the history of mathematics and on ways to use it to teach mathematics, just like it says in the title. The title, though, does not capture the enthusiasm of the various authors, the depth and breadth of their topics, or their conviction that understanding and using history can enrich and improve the ways we teach mathematics.

Katz has divided the essays into five groups, proceeding from the more pedagogical in Part I to the more historical in Part 5. The first four parts consist of three to five essays each, and the fifth part consists of eleven.

The three essays in "Part I: General Ideas on the Use of History in Teaching" lay a foundation and motivation for the incorporation of history. Siu Man-Keung opens the work with some ways to include history without sacrificing mathematical content. Frank Swetz follows with an account of mathematical education from Mesopotamia through China to the Italian Renaissance.

Wann-Sheng Horng contributes "Euclid versus Liu Hui: A Pedagogical Reflection" to "Part II: Historical Ideas and their Relationship to Pedagogy." He gives a provocative comparison between the structural approach to mathematics that the Greeks used to the more operational approach of the Chinese, with special emphasis on Euclid's and Liu Hui's descriptions of the so-called Euclidean algorithm.

The third part of the book turns to "Teaching a Particular Subject Using History." Janet Heine Barnett shows how mathematical anomalies such as incommensurables, infinity and non-Euclidean geometries open mathematical minds and "prepare new intuitions." Evelyne Barbin gives a delightful account of how the meaning of "obvious" has evolved. For example, geometric proofs of proportionality may be beautiful or tedious, depending on your aesthetic, but those same theorems proved symbolically become obvious "in the sort of 'blind' way that algebraic calculations allow.” Continued...