45-1504aQA4532006-924363 CIP
Science & Technology \ Mathematics
Kiselev, A.P.  Kiselev's geometry: book I. planimetry, tr. by Alexander Givental.  Sumizdat, 2006.  240p bibl index ISBN 0-9779852-0-2, $39.95; ISBN 9780977985203, $39.95. Reviewed in 2007nov CHOICE.
This title has been reviewed jointly with "Extending the Frontiers of Mathematics," by Edward B. Burger.
Beware the code words "quantitative reasoning" often bandied about by the secret enemies of geometry who would suppress that subject like some sort of pagan religion or foolish luxury. In fact, most high schools, for various reasons, bypass rigorous geometry for weak calculus; as a result, many college students now hit a brick wall trying to follow any precise logical argument, or worse, construct one. Plato warned us! In the book by Kiselev, translator Givental, himself a very distinguished mathematician, aims to rescue geometry for our time by bringing into English a classic of Russian pedagogy, a book with a track record extending back more than a century. Indeed, under the Soviets, an era of prodigious mathematical achievement, Kiselev's book actually attained the status of stable, meaning it was the entire nation's official book for classroom use, and it held that status for decades, remaining popular still. SUMIZDAT (the name evidently a portmanteau of sum and samizdat) calls itself a publisher promoting nonsense-free mathematics and science curricula. Givental's excellent and concise Linear Algebra and Differential Equations (2001), albeit published by the AMS, also nicely fits this category. Certainly library shelves must make a place for Kiselev's classic!

Because we don't teach geometry either when, or as deeply as we should, we do damage control instead. We offer "transition to abstraction" courses where students will first meet mathematical proof, but detached from any consistent content. Though such courses sometimes do other things well, they rarely achieve their stated aim--sound preparation for subjects such as abstract algebra, real analysis, and topology. Teachers of these advanced courses generally find that they must redo the principles of proof all over from scratch. Mathematical proof, isolated from substantial content, strikes most students only as a quaint conventionalized genre, like fugue or villanelle, with productions conforming to received rules that either win praise from the teacher or not, for mysterious reasons. This book by Burger (Williams College) will serve such courses better than many competitors. Though sufficiently accessible to reach the weakest students, he nevertheless avoids the trap of exercises that request purely hygienic proofs--formal verifications of statements no sentient person could doubt. Indeed, learning to value proofs should stand prior to learning to construct them, and already Burger's first chapter on puzzles sets the tone right, making it clear that mathematics holds surprises for us even in the simplest situations. Summing Up: Both books: Recommended. Upper-division undergraduates through faculty.

 -- D. V. Feldman, University of New Hampshire