Kiselev's Geometry, Book II: Stereometry

Adapted from Russian by A. Givental, Stereometry is the second part of the legendary Kiselev's Geometry. It first appeared in 1892 as a second half of a single textbook and, for a long time, the two co-existed between the same covers. Indeed, the idea of a plane was introduced on page 1 while the last chapter of the book (that followed the stereometry part) was devoted to the geometric constructions in two dimensions. Kiselev's Geometry has demonstrated an unusual staying power, being in an uninterrupted circulation for a good part of a century. (For the historic outline, see the review of the first part.) As a matter of fact, the first part of the book met with stiffer competition so that, while its rule was weakened in the 1960s, the second part reigned in the textbook market well into the 1970s.

The combined 1980 edition came out under the title Elementary Geometry for Teacher Colleges with a foreword by A. N. Tikhonov who observed, albeit with some reservations, that the pedagogical mastery with which the book was written, the simplicity and consistency of the exposition, kept the book from becoming obsolete.

The 1980s saw the beginning of tremendous upheavals in Russia culminating in Perestroika and ultimately the disintegration of the USSR. The educational system became decentralized, market liberalization led to the creation and spread of private schools, each in a position to choose and even publish its own texts. That generation also had its own share of talented authors. An exceptional geometer and a pedagogue, the late I. F. Sharygin, authored a dozen geometry manuals and problem books for all school levels. By 2004 Kiselev's Geometry was said to become a bibliographic rarity (although there was a 1998 edition) and has been republished again as a textbook for teacher colleges. The latest promotional material reads: "Further improvement in teaching of mathematics is impossible unless the teachers become acquainted with the former staples of math education."

The same argument applies to the English speaking market. A good teacher should have deep understanding of the subject matter that comes from being acquainted with multiple pedagogical views and approaches. Kiselev's Geometry is an invaluable source of inspiration for teachers of geometry.

Kiselev's brevity is notorious. The sentences have been polished through many revisions and by assiduously heeding advice of generations of reviewers, teachers and students. In this respect, I have only a minor grievance. In both parts of the book, whenever the question is of lines or planes being parallel, the author consistently appends to the phrase "…do not intersect each other" the redundant expression "no matter how far they are extended," probably trying to appeal to the power of visual perception. This practice, in my view, may result in confusion: is it possible not to extend "however far" a straight line (as opposed to a line segment)? May a line be unextended? (I do believe, though, that the usage is not a slip of the tongue, but is likely to have a pedagogical reason suppoted by experience.)

The pursuit of brevity is also manifest in a superb balance between what is actually proved in the text, what is left to be proved by the reader, and what is assumed. Kiselev is never dogmatic. His books are the epitome of the logical structure, proof being the main vehicle for building up the material, rather like in Euclid's Elements. The diagrams — newly created by the translator — are abundant. On the whole, the economy of presentation is nothing short of remarkable.

Just to take one example: Cavalieri's Principle. Kiselev rightly observes (p. 45) that to justify it one needs methods that would go beyond elementary mathematics. However, the principle is elegant and useful and mentioning it supplies an engaging historical background along with a viable demonstration of the continuity of the evolution of mathematics over time. So, based on the theory of limits developed in the first part (Planimetry), Kiselev proves (although without ever employing the symbol lim) the principle for triangular pyramids and uses the occasion to mention Hilbert's third problem. Later on, he applies the principle to determine the volume of a ball (but now without proof) and makes use of the opportunity to mention the work of Archimedes and of the recovery of his tomb by Cicero.

The book is comprised of three original chapters (Lines and Planes, Polyhedra, Round Solids) and one (Vectors and Foundations) added by the translator who also wrote an afterword. The add-on chapter is an excellent introduction to vectors spaces and space dimension that fits snugly into Kiselev's text and style. The 50+ page chapter (one third of the book, actually) develops a solid foundation for Kiselev's Geometry and forms an extra link between the Planimetry and Stereometry parts. For example, this is where we meet the notions of material point and barycenter which lead to the plane theorems of Ceva and Menelaus. The chapter also provides several examples for the views expressed later in the afterword. One of these concerns with role of axioms: nowadays axioms are used for unification purposes, to study similar articles at once. For example, the axioms of the inner product underly the geometry of both Euclidean and Minkowski spaces. The chapter also includes an introduction into other non-Euclidean geometries: spherical and hyperbolic.

Each section of the book is accompanied by a judicious selection of exercises (about 250 of which have been added in the translation.) Many problems are solved in the body of the book, but the exercises come without solutions. The chapters on space symmetries and regular polyhedra have been expanded by the translator; this is the only stereometry book I am aware of which discusses the symmetry about a line along with the central and mirror symmetries in space.

In the afterword, Givental offers his thoughts on the changing role of axiomatics (with a reference to Chapter 4) in modern mathematics and the contemporary ideology of math education as related to the teaching of geometry. His analysis of the van Hiele model and the supporting research is remarkable. The van Hiele model stipulates that the ability of a learner to process geometric knowledge is determined by the level of geometric abstraction achieved by the learner. The prerequisite for attempting the next level is the mastering of the previous one. The five levels are

0. Visualization: student identifies shapes.
1. Analysis: student attributes properties to shapes.
2. Abstraction: student derives relationships between the properties of shapes.
3. Deduction: student develops an appreciation of the logical structure that tracks the properties of shapes to axioms.
4. Rigor: student is able to handle any axiomatic approach without relying on intuition.

Since it originated in 1957, van Hiele's theory has been supported by research and numerous field studies. Givental observes that van Hiele's theory consists of four independent assertions about the possibility of four transitions between the five levels. The claims regarding the last two transitions hold logically, from the definition, simply because many is more than one. The ability to handle an axiomatic approach in general (level 4) implies the ability to handle one of them (level 3). Likewise, the ability to derive all properties of shapes from axioms (level 3) implies the ability to derive some of them (level 2).

It is possible, perhaps, to justify the need for research regarding the first two transitions: what the theory claims is that the ability to abstract can be achieved only after the two preliminary stages. Giventhal argues, however, that any meaningful study or activity in which children get involved in school and elsewhere would bring the same result, implying that the informal geometry need not precede a more rigorous study. This conclusion is reinforced by the fact that the customary approach to informal geometry is mostly preoccupied with naming objects and tautological questions about their names. Givental gives several convincing examples to illustrate this point.

I wish to end the review with a general remark. Although we owe the English edition of Kiselev's Geometry to the private initiative of a single individual, its appearance should be considered in a broader context of the changes in school programs that are shaking the US math education establishment once more.

However good or theoretically justified a particular reform might be, its failure is practically a foregone conclusion if forced en masse on the unprepared population of students and teachers. The history of math education reform in the US in the 20^th century is a sequence of failed innovations. So much so that US educators have begun to look elsewhere for successful practices. Singapore textbooks are now commonly used by individual tutors and crowds of teachers at independent schools. Lessons in Geometry by Jacques Hadamard is in preparation by the Education Development Center and is due in December 2008. Kiselev's Geometry was one of the cornerstones of the Soviet school of math education — admittedly one of the best in the world. For generations, it influenced geometry teaching in the Eastern Europe and China. Its appearance in the US should be embraced by every single teacher and teacher college: the text worked well for generations of Soviet boys and girls and their teachers. Its introduction to the American user does not come too soon.

Note that while the book's list price is $30, it can be had directly at sumizdat for $20. The two books go for a sale price of $45.

Alex Bogomolny is a business and educational software developer who lives with his wife and a little son in East Brunswick, NJ. His popular web site Interactive Mathematics Miscellany and Puzzles is set to welcome its 30,000,000^th visitor.