The Stereometry book adapted from Russian by A. Givental is the second part of the legendary Kiselev’s Geometry. It first appeared in as a second half of a. In mathematics, solid geometry is the traditional name for the geometry of three- dimensional Euclidean space. Stereometry deals with the measurements of volumes of various solid figures Kiselev, A. P. (). Geometry. Book II. Stereometry. Stereometry) covers solid geometry, and contains a chapter on vectors, foundations, and introduction in Kiselev’s Geometry: Stereometry, Book 2. Front Cover.

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The Stereometry book adapted from Russian by A. Givental is the second part of the legendary Kiselev’s Geometry. It first appeared in 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 s, the second part reigned in the textbook market well into the s.

The combined edition came out under the title Elementary Geometry for teacher colleges with a Foreward by A. 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 s saw the beginning of tremendous upheavals in Russia culminating in Perestroika and ultimately 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.

In truth, that generation had its own share of talented authors. An exceptional geometer and a pedagogue, the late I. Sharygin, has authored a dozen geometry manuals and problem books for all school levels.

Kiselev’s Geometry / Book II. Stereometry by A.P. Kiselev

By Kiselev’s Geometry was said to become a bibliographic rarity although there was a edition and has been republished again as a textbook for teacher colleges. The latest promotion reads: 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 from several view points. In the manner of expression: Srereometry this respect, I have only a minor grief. In both parts of the book, whenever the question is of lines or planes being parallel, the author consistently appends to the phrase ” This practice, in my view, may result in confusion: May a line be unextended?


I do believe, though, stereojetry 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 being assumed. Kiselev is never dogmatic.

His books are the epitome of the logical structure, proof being the main vehicle of the material build-up, rather like in Euclid’s Elements. The diagrammatic illustrations – newly created by the translator – are abundant. On the whole, the economy of presentation is nothing short of remarkable.

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Just to take one example: Kiselev rightly observes p. However, the principle is elegant stereometgy 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 PlanimetryKiselev proves although without ever employing the symbol lim the principle for triangular pyramids and uses the occasion to mention Kisele 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 Afterward.

The add-on chapter is an excellent introduction into vectors spaces and space dimension that fits snugly into Kiselev’s text and style. 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 serves several examples for the views expressed later in the Afterward.

One of these concerns with the role of axioms: For example, the axioms of the inner product underly both geometries of Euclidean and Minkowski spaces.

The chapter also includes an introduction into other non-Euclidean geometries: Each section of the book is accompanied by a judicious selection of exercises about 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 in line along with the central and mirror symmetries in space.

In the Afterward, A. 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 sterwometry as related to the teaching of geometry. Remarkable is his analysis of iiselev van Hiele model and the supporting research. The van Hiele model stipulates that the ability of a learner to process geometric knowledge is determined xtereometry the level of geometric abstraction achieved by the learner.


The prerequisite ikselev attempting the next level is the mastering of the previous one. The five levels are. Since it originated invan Hiele’s theory has been supported by active research and numerous field studies. Givental makes an observation that van Hiele’s theory consists of four independent assertions about kiseelev possibility of four transitions between the five levels.

The claims regarding the last two transitions hold logicallyfrom 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 any properties of shapes from axiom level 3 implies the ability to derive some of them level 2.

It is stereomerry, perhaps, to justify the need for research regarding the first two transitions: However, any meaningful study or activity children stereoemtry involved in in school and elsewhere would bring the same result implying that the informal geometry need not precede a more rigorous study.

Especially because the customary framework of the informal geometry is mostly preoccupied with naming objects and tautological questions about their names.

Kiselev’s Geometry / Book II. Stereometry

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 the math education reform in the US in the 20 th century is a sequence of failed innovation. So much so that the US educators began looking elsewhere for successful practices; Singapore textbooks are now commonly used by individual tutors and crowds of teachers at independent schools. 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: Its introduction to the American user does not come too soon. Up Contact Front page Contents.

The five levels are Visualization: