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 ManKeung 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.
WannSheng 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 socalled 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 nonEuclidean 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."
"The Use of History in Teacher Training" is the fourth part of the book. Ian Isaacs, V. Mohan Ram and Ann Richards remind us how important it is to the future of mathematics that elementary school teachers encourage, or at least not discourage, young mathematics students. They give specific examples of how they use history "to modify the belief systems and perceptions of these preservice teachers regarding the nature of mathematics and the purpose of school mathematics."
Maxim Bruckheimer and Abraham Arcavi remind us that we can't teach mathematics using history unless we have a repertoire of facts from the history of mathematics. They give us some anecdotes, including the thrilling and tragic story of Feuerbach wielding a sword and threatening to behead students who could not solve problems in class. They also challenge us to use original sources and share one of their worksheets based on the original works of Viète.
Katz put almost half of the essays in this collection into Part V, "The History of Mathematics." This section reflects the fact, of which Bruckheimer and Arcavi reminded us, that incorporating history into a mathematics course requires a knowledge of history as well as of mathematics.
In the first essay of the section, Eleanor Robson uses Mesopotamian mathematics to contend that writing arose first to record mathematics. This makes mathematics a function of civilization that predates even writing. She also emphasizes the importance of context when viewing mathematical artifacts, and emphasizes how mathematics is the product of the society from which it arises.
George Heine gives us a delightful example attributed to the Persian scholar ibn Sina (9801037), but in the spirit of the Ancient Greek Nichomachus or the more modern Conway and Guy in their Book of Numbers. His example works for any square array of consecutive odd numbers, but Heine gives us the following case of a 5x5 array:
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The highlighted entries sum, respectively, to 5 and 5^{2}=25.
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The highlighted entries sum to 5^{3}=125 and 5^{4}=625. In the third array, the sum on the opposite diagonal is also 125.
Torkil Heiede, in "The History of NonEuclidean Geometry" traces how attempts to prove Euclid's Fifth Postulate grew into the geometrical revolution of Bolyai and Lobachevsky. He also gives a remarkable list of eight concise statements equivalent to Euclid's Fifth Postulate, all much simpler to state and some easier to believe than Euclid's version.
Some authors submit to an occasional temptation to change what is taught so that it can be more easily taught from an historical perspective. Others are sometimes a bit optimistic about when history adds understanding to the presentation of a topic. Early techniques in linear algebra, for example, are so burdened with nowobsolete notation that no presentation can be both understandable to the students and historically faithful.
Overall, this collection of essays goes well beyond the promise of its title. It presents a broad spectrum of ideas about how to use history in teaching, from things as basic as particular classroom activities to concepts as profound as different ways to consider the nature of mathematics. The perspective certainly is international. There are contributions from every continent, and only three of the 31 contributors are from the United States, matching the contributions from Israel, Italy and Portugal and one fewer than the number from France.
Beyond its title, though, this collection of essays captures, in a way that ordinary textbooks on the subject do not, some of the ways that the beauty and vitality of mathematics grows from its roots in history. In his essay, Heiede also asks "But does it matter if a teacher does not know about nonEuclidean geometry[?]" This wonderful book will convince you that it does matter. If we are to keep mathematics out of the museum, somewhere between the mastodon bones and the mummy, then we should keep it connected to its roots.
If only it had a better title...
Ed Sandifer (sandifer@wcsu.ctstateu.edu) is a professor of mathematics at Western Connecticut State University and an enthusiastic fan of Leonhard Euler.