Let History into the Mathematics Classroom

For years now I have believed that substantive mathematics classes benefit from an injection of history into the course content. I wouldn’t dream of teaching my upper-level Euclidean geometry course without mentioning the “Greek miracle”, Euclid and The Elements, and in the follow-up semester on non-Euclidean geometry, history is discussed extensively — the students find it fascinating (as do I) that after hundreds and hundreds of years of attempting to prove the Euclidean parallel postulate from the other axioms of Euclidean geometry, mathematicians gradually came to realize that it couldn’t be done, and that non-Euclidean geometries were as logically consistent as is Euclidean geometry.

Likewise, I like to begin a course in abstract algebra with a very quick historical overview of the subject. (See, for example, the first chapter of A Guide to Groups, Rings and Fields by Gouvêa), Just talking about how mathematicians came to realize that various systems that they were dealing with had certain features in common, and that isolating those features and studying structures that satisfied them was a useful thing to do. makes subsequent definitions like “group” and “ring” somewhat more palatable and natural to the students.

I suppose some people may disagree (classroom time is limited, after all, and time spent discussing history is time that could be spent discussing other things), but the authors of the book under review (there are a lot of them, each contributing individual chapters) obviously share my view, although the “classrooms” that they speak of in the title are for 11–18 year old students (in France). The book is divided into ten chapters, which can be divided into four basic groups. The first three chapters are based on geometry; the next three have as a unifying theme number systems and arithmetic; the next two involve drawing and approximating curves; the last two are on probability.

The chapters are generally written in the first person and describe the authors’ attempts to introduce historical discussions into their classrooms. The sources used by the authors are described and snippets from them are reproduced. The authors also go beyond the history by actually talking about the pedagogical issues involved in presenting this material to their students. Specific classroom questions and projects, and the students’ reactions to them, are described, so that this book is as much about education as it is about history.

While some of the history presented here is fairly standard (chapter 9, for example, is a nice discussion of basic probability and Leibniz’s text on the subject), some of it is a little bit “off the beaten track”. This is certainly not a criticism — I wound up learning a fair amount of new things myself. Despite having twice taught a one-semester course in the history of mathematics, for example, I had never heard of the “congruence machine” perfected by the Carissan brothers in the early 1900s. (I don’t feel too bad about this, though, since Katz’s book on the history of mathematics doesn’t mention them either.) And people who have always associated Fibonacci with his Liber Abaci, will find in chapter 2 a discussion of another of his works, Practica Geometriae, in connection with the general problem of dividing a triangle into two parts of equal area.

This isn’t a textbook; there are, for example, no exercises. However, there is a lot of material here that might be of interest to people who are interested in history and/or pedagogy. It’s certainly a book that is worth a look, even by people who teach older students in college. This is apparently one of the early volumes in a new Springer series called History of Mathematics Education; I intend to keep an eye out for new volumes in the series as they are published.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.