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*Routes of Learning: Highways, Pathways, and Byways in the History of Mathematics,* by Ivor Grattan-Guinness, 2009, 372 pp.; hardback, $75, ISBN 978-0-8018-9247–9; paperback, $35, ISBN 978-0-8018-9248-6; The Johns Hopkins University Press, 2715 N. Charles St., Baltimore MD 21218, www.press.jhu.edu.

The last three decades of the 20th century mark a renewed interest in the history of mathematics in the mathematical community. Motivated by an uncertainty as to the position of mathematics in society and the need to enrich school mathematics with more meaning, this revival has been fruitful. Ivor Grattan-Guinness was an active participant in this movement as a researcher, lecturer, editor, and writer. In particular, he emerged as a commentator and a critic on many of the trends he saw taking place. This book is a collection of some of his commentaries. It is Grattan-Guinness on Grattan-Guinness. The title indicates that the book’s contents should transport the reader between two points, be they perspectives, opinions, or planes of understanding. The paths of our journeys have been divided into three categories: “Highways” – where the subject is specifically the history of mathematics; “Pathways” – where the concern is mainly about teaching of the history of mathematics; and finally “Byways” – which consider, in the author’s opinion, two neglected aspects of the history of mathematics: numerology and the connection of mathematics to music. The journeys delineated can be thought-provoking, but are also frustrating to follow.

“Highways” consists of eight essays. The first of this collection, “The Mathematics of the Past,” calls attention to the need for distinguishing the “history of mathematics” from the “heritage of mathematics.” Where the first designation concerns relevant facts and motivation, the second focuses on resulting effects and knowledge, an interesting distinction that raises the worthwhile question, “What do mathematics historians write about?” The second essay, “Decline, then Recovery,” presents an excellent overview of twentieth century activities in the history of mathematics. The remaining essays in this section briefly consider the content and scope of historical coverage, the history of science journals, and the nature of scientific revolutions. In this coverage, Grattan-Guinness raises the issue of the relationship between the history of mathematics and the history of science in general, an important relationship that needs to be clarified and strengthened.

Part two, actually titled “Pathways in Mathematics Education,” examines the use of mathematics history in classroom teaching. Here the author appears as a theorist speculating on what could be done. Classroom and curriculum realities are not adequately considered. Approaches to teaching Euclid, number systems, the “Achilles Paradox” and calculus are discussed. The need for the existence of a history of the teaching of school mathematics is emphasized. Efforts of the National Council of Teachers of Mathematics (NCTM) and the Mathematical Association of America on this issue appear unknown to the author. NCTM published *A History of School Mathematics* in 2003 edited by George Stanic and Jeremy Kilpatrick. Indeed, throughout this volume, relevant contributions in North America to enhance the history of mathematics and its meanings and teaching are neglected.

Finally, “Byways in Mathematics and its Culture” calls attention to the historical existence of numerology in religion, specifically Christianity, and mathematical relationships with music. Numerology in religion is a rich topic for examination and discussion. Grattan-Guinness hints at this possibility but unfortunately does not transport the reader further. The included case study in mechanics, “Equilibrium in the Crucifixion Process” (pp. 279-282), is of questionable appeal. “Byways” results in dead ends and cul-de-sacs.

In surveying the history of mathematics and its modern trends and issues, Ivor Grattan-Guinness has omitted the impact of the Internet as a conveyor of information and resources on the subject. The MacTutor History of Mathematics archive, http://www-history.mcs.st-and.ac.uk/, and efforts such as *Convergence* would seem to warrant mention. In general, I found the reading of this book difficult. The writing style is erudite and the development of ideas limited. The journeys taken are often circuitous. The author is fond of analogies. On page 77 of this book he tells us that “The history of mathematics shows that since antiquity, mathematics has developed as a wide and ever widening rainbow of ideas and theories. But this rainbow has negative as well as positive aspects. It delights those who look at its many colors; but is also *distant* from us, and seemingly irrelevant to our cultural lives....” So too it might be said of *Routes of Learning*.

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Frank J. Swetz (Pennsylvania State University, Harrisburg), "Review of [i]Routes of Learning: Highways, Pathways, and Byways in the History of Mathematics[/i]," *Loci* (April 2010), DOI:10.4169/loci003481