Mathematical education in the United States has long relied on a variety of ingenious devices to convey important ideas and give students valuable experience in the classroom.

Such historical objects, from protractors and linkages to geometric models and calculators, were the subject of a recent presentation at the MAA's Carriage House Conference Center. They are also the subject of a new book, *Tools of American Mathematics Teaching, 1800–2000* (Johns Hopkins University Press, 2008), by Peggy A. Kidwell, Amy Ackerberg-Hastings, and David L. Roberts.

U.S. educators, struggling to teach mathematics well, made one particular tool a mainstay in instruction: the protractor. Hundreds of years old, this object (along with the ruler and compass) became ubiquitous in U.S. schools. It played a key role in the formal treatment of geometry. Ackerberg-Hastings, who holds a doctorate in the history of technology and science from Iowa State University, summarized its historical relevance to 19th-century U.S. education.

Linkages, on the other hand, never caught on in the U.S. classroom. Inspired by James Watt's steam engine and developed to allow users to draw a straight line, linkages played a more important role in mechanical engineering than in mathematics education, said Roberts, who holds degrees in mathematics and the history of science. He offered brief histories of a slew of these relatively little-known tools—the slider-crank linkage, Peaucellier Inversor, and others—and portraits of the mathematicians who promoted them: George Halsted, Frank Morley, and Robert Yates.

Models of simple geometric objects, such as spheres, cones, and cubes, did capture U.S. educators' attention. Created by Josiah Holbrook in the 1830s, they were inspired by his notion that they were "good enough for the best and cheap enough for the poorest," which reflected a populist ideal of education.

More sophisticated geometric models, on the other hand, were imported by U.S. mathematicians who had received their doctorates abroad. Some sets of models, dating from 1893, included projections of polytopes, created by German mathematician Victor Schlegel and distributed by Ludwig Brill. Other examples included the hyperbolic paraboloid, the quintic scroll, and Baker's Gaussian Surface and Riemann Surface.

*Peggy A. Kidwell, Amy Ackerberg-Hastings, and David L. Roberts*

Kidwell, who is Curator of Mathematics at the Smithsonian's National Museum of American History, noted that today's popular mathematical models and tools will themselves eventually become objects of historical study as trends and fashions in math teaching ebb and flow.

The Smithsonian’s website features an online exhibition, curated by Kidwell, called “Slates, Slide Rules, and Software: Teaching Math in America.” Kidwell was also responsible for the Smithsonian exhibit “Mobilizing Minds: Teaching Math and Science in the Age of Sputnik.”—*H. Waldman*