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Historical Modules for the Teaching and Learning of Mathematics

Victor J. Katz and Karen Dee Michalowicz, editors
Mathematical Association of America
Publication Date: 
Electronic Book
[Reviewed by
John McCleary
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The goals for school mathematics education set recently by the National Council of Teachers of Mathematics (NCTM) set a premium on learning through multiple mathematical perspectives. One of these perspectives is historical. The modules collected on this CD-ROM are designed to provide this view for topics at the core of the standard mathematics curriculum from middle school through high school. The modules richly succeed in this goal but that is only a small part of a much more ambitious project. By providing copious and relevant activities, students and teachers are invited into first-hand experiences of the mathematics of the past, enriching their standard work, and obtaining valuable motivation for some important ideas. Ancient mathematicians didn't always find the most elegant or general solution to a problem; their approaches still contain insight, alternative ideas, and even just plain fun with mathematical ideas.

A typical example is the module on linear equations. After a tour of the development of algebraic notation in Egypt, China, India, and the Middle East, a review of proportions is presented as a discussion of problems from ancient texts — the Nine Chapters of the Mathematical Art, the Rhind Papyrus, Bhaskara's Lilavati, the Treviso Arithmetic and Euclid's Elements. The heart of the module treats linear equations in one unknown and introduces various methods of solution. The Egyptian method of false position is based on estimation and correction, allowing teachers to emphasize a core skill through ancient problems. The Chinese method of double false position uses two estimates whose difference lead to an exact solution. With the Islamic contribution of al-jabr we meet the familiar standard method, but the authors go on to a more general discussion through work of Colin MacLaurin and Leonard Euler. There are many excerpts from Euler's writings in the modules, giving teacher and student the opportunity to read original sources.

The modules provide ready-made problem sets, often consisting of problems from historical sources. But there are also many alternative student projects: there are skits (including an excerpt from Plato's Meno and a dialogue of Alfred Renyi), well-constructed worksheets (Student Pages), an internet treasure hunt, and hands-on projects like building a slide rule and paper-folding geometric constructions. Add to this a rich array projects in history that give students the opportunity to write narratives — on the development of ideas, on historical figures, or even on the interconnections between cultures around the world.

This collection was an ambitious project: there are 11 modules, written by 29 authors with assistance from 6 editors together with the two main editors. Many of the authors have positions in high schools and know their audience — the student pages are well-posed and the teacher's resources are well-crafted. Mathematics is much more than a technical and lifeless skill in calculation. These modules go a long way in bringing the rich life of mathematics to hand for teachers to use in the classroom. Anyone teaching elementary topics can find a tasty morsel to bring to class in these modules, be it a motivating story, an illuminating worksheet, or an enriching class project. I highly recommend this set of materials to every middle school, high school, and college teacher.

John McCleary ( is Professor of Mathematics at Vassar College.
The table of contents is not available.