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Paul Lockhart
Harvard University Press
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Katherine Safford-Ramus
, on

This was a difficult book to review for several reasons, each related to categorization. We mathematicians love to see pattern, but Measurement defied pigeon-holing. First, the genre was elusive. The introductory remarks reach back to early philosophers and their struggle to separate form and being, reality versus perfection. This positions it in the realm of philosophy of mathematics. The author then discusses the nature of problem-solving. The questions “What is a problem?” and “How do we teach problem-solving?” have occupied those of us in mathematics education for decades. He argues for changing the way that we teach mathematics — stressing its beauty and structures. Such arguments place the book in the literature of mathematics education. Finally, the foci of the book are classic and analytic geometry. It could easily be classified as a textbook for use in a twelfth grade mathematics elective, a mathematics service course, college geometry, or in a capstone course for mathematics majors.

The second challenge was the identification of the intended readership. It was clear from the introductory sections that the author targeted amateur mathematicians, but I am not confident that every such reader would persevere without the assistance of a “guide on the side” to clarify points of confusion. The flyleaf of this book states that the author “succeeds in making complex ideas about the mathematics of shape and motion intuitive and graspable.” As such, it serves as a useful resource for those of us charged with the duty to perform that task on a daily basis in secondary and tertiary institutions of learning. One appropriate audience is cited above — mathematics majors in a capstone course. In fact, I am considering just such a role for the book when I teach our capstone course next year.

The third conundrum arose from the organization of the text. The book is not divided into chapters. It consists, instead, of two parts, “Size and Shape” and “Time and Space.” Each of these is subdivided into numbered but untitled sections which are themselves separated by questions that are then discussed. When reading forward (arguably the way one does read a text) this presents no problem, but it did cause a certain difficulty if you want to refer back to earlier material. Perhaps the biggest obstacle for the amateur reader, however, lay in the frequent conclusion of a section with a series of challenge questions. These were not then answered, leaving the reader in suspense wondering if their response was correct. When used as a textbook this device opens the door to individual research or group projects under the guidance of the instructor. However, I fear that the accumulation of too many loose ends might eventually discourage a layperson.

Having said all that, I found the book interesting to read and felt that it brought another perspective to the material that I teach. It is always enjoyable to find different ways to clarify mathematics concepts for students or to see your own words paraphrased, reinforcing your personal practice. I enthusiastically recommend it to fellow mathematics instructors and guardedly to the general reader. In a more numerate world, it would be a nice candidate for a mathematics book club, meeting weekly like a bible study, arguing the fine points of the questions probed by the author. 

Katherine Safford-Ramus is Professor of Mathematics at Saint Peter’s University in Jersey City, New Jersey.

The table of contents is not available.