45-1504a | QA453 | 2006-924363 CIP |

Science & Technology \ Mathematics | ||

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