Tools of American Mathematics Teaching: 1800-2000 (review)

In 1800, the US was a relatively simple agricultural economy that demanded little mathematical skill from its workers. Those involved in commerce needed to know nothing more than basic addition. The only individuals needing substantial mathematical knowledge were the few involved in surveying or navigation. In this volume, the three authors trace the development of mathematical knowledge by looking at the tools used for teaching mathematics, and in the process, they show how the country developed the skills that would be needed for the computing age that began in 1946.

This book deals with a surprising range of pedagogical tools, beginning with simple objects, such as textbooks, graph paper, and blackboards before eventually moving to machines, both mechanical and electronic, that have been used to convey mathematical concepts to students. Most of these objects will be familiar to potential readers, though a few individuals raised in the era of the graphing calculator will not know about protractors, abaci, ornomograms. The slide rule, once a staple of engineering education, has a central place in this book. The authors show how the modern slide rule developed in the 19th century, some three centuries after the invention of the underlying technology of logarithms.

Of the objects covered in the book, perhaps the most novel are the geometric models, the constructions of string and wood developed to explain solid geometry concepts. Many of these models were drawn from the collection of the National Museum of American History, where author Kidwell is curator of mathematical objects. Such models are regularly displayed in science museums but are rarely described in a way that explains the extraordinary insight of the most prominent of the model makers: Alexander Brill, Richard P. Baker, and Albert H. Wheeler. Brill's models could be extraordinarily beautiful and suggested the wireframe models that were common in computer graphics' early days.

The book's first section deals with tools for presenting mathematical concepts; the second, with tools of computation; the third,with measurement and representation; and the last,with machines for mathematical learning. There is little literature that covers this material or the development of mathematical pedagogy in the US.The best known,Patricia Cline Cohen's A Calculating People: The Spread of Numeracy in Early America (University of Chicago Press, 1982) is a fine treatment of the subject but does not deal with the objects and machinery discussed in this book.

The book includes a discussion of the modern machinery that has been used to teach mathematics. It explains how time-sharing systems and personal computers have been used to teach mathematics. It treats contributions by computer scientists such as John Kemeny, Seymour Papert, and Don Bitzer, Peter Braunfeld, and Wayne Lichtenberger. These last three were the developers of the PLATO system, an early effort at computer-mediated instruction. The book ends with a discussion of the tools that have become staples of college education: statistical systems and symbolic mathematics packages.

Many ideas in this book were familiar to the early computer designers. They knew how to work with slide rules, how to construct nomograms, how to express complicated geometric ideas in two or three dimensions. These ideas not only formed part of their intellectual background, they also shaped the way that these individuals tried to express the new concepts of the stored program computer. The files of many a computing pioneer are filled with diagrams and calculations that are described in this book. Reading Tools of American Mathematics Teaching reminds us that the computer was not created in a vacuum and that earlier designers had a common mathematical training imparted by tools that were 200 years in the making but are no longer familiar to us. In short, this book is an important contribution to the history of computation, not only helping us to understand how the US developed its mathematical expertise, but also showing how we moved to the basic categories of mathematics that have shaped digital computation. [End Page...