In the early 1990s, geometry in Russian schools was taught using the textbooks that had won the nationwide competition. But already by the mid-1990s new geometry textbooks began to appear. If in the USSR the commission to write a textbook came from the government, then now it became possible to create a new textbook based on the private initiative of some specialist or some organization that wished to finance such work. If in the USSR only one publishing house,Prosveschenie, published instructional materials for schools, then in the 1990s dozens of publishing houses appeared in Russia that were involved in publishing instructional materials and textbooks that
competed in a market. Among the new publishing houses wereDrofa, MIROS(which combined publishing activities with scientific research), Mnemozina, andSpetsial’naya literatura. In the late 1990s, the Russian Ministry of Education, together with the National Training Foun- dation (NTF), conducted several competitions for “New Generation Textbooks” for different grades (1st–11th) and in different subjects.
These competitions touched on the teaching of geometry as well, and certain new textbooks were prepared for them and subsequently published with proper support. Below, we will describe briefly some of the books that appeared during those years. We remind the reader that because in this chapter we are focusing on so-called ordinary schools, we will not discuss textbooks for schools with an advanced course of study in mathematics or schools specializing in the humanities.
5.4.1 I. F. Sharygin’s textbooks
I. F. Sharygin (1938–2004), a graduate of Moscow State University, accomplished much for geometry education in Russia. From his pen came several problem books which contained diverse and difficult problems (Sharygin, 1982, 1984) and were a crucial source for gifted students interested in geometry. At the same time, in collaboration with L. N. Erganzhieva, he published the book Visual Geometry (Sharygin and Erganzhieva, 1995), which, along with the sections on geometry that he wrote for the textbook edited by him and G. V. Dorofeev (Dorofeev and Sharygin, 2002), did much for the geometric development of fifth and sixth graders. Recalling Albert Schweitzer’s words concerning reverence for life, one can say that what was characteristic of Sharygin was reverence for geometry — a sense of awe based on a profound knowledge of the properties of geometric figures and, one might say, a personal relationship with these figures.
One of the authors of this chapter (A. Karp) remembers how, during one conference, Sharygin with sincere pain spoke about the fact that although school geometry was based on the triangle and the circle, in the geometry course the triangle receives much attention but the circle is forgotten and pushed into the background. Sharygin tried to convey this awe for geometry not only in the books cited above but
also in his textbooks (Sharygin, 1997, 1999). The annotations to them state:
The new textbook in geometry for ordinary schools embodies the author’s visual–empirical conception of a school course in geom- etry. This is expressed first and foremost in the rejection of the axiomatic approach. Axioms, of course, are present, but they are not foregrounded. Greater attention, by comparison with traditional textbooks, is devoted to techniques for solving geometric problems.
(Sharygin, 1997, p. 2)
Addressing the students, Sharygin writes:
Far from all students feel a great love for mathematics. Some are not too good at carrying out arithmetic operations, have a poor grasp of percentages, and in general have reached the conclusion that they have no mathematical abilities. I have good news for them: geometry is not exactly mathematics. At least, it’s not the mathematics with which you have had to deal up to now. Geometry is a subject for those who like to daydream, draw, and look at pictures, those who know how to observe, notice, and draw conclusions. (Sharygin, 1997, pp. 3–4)
Sharygin’s textbooks are full of illustrations, including the works of M. C. Escher, Victor Vasarely, and Anatoly Fomenko. The mathemati- cal content of his textbooks, however, is quite traditional. In his posthu- mously published article “Do Twenty-First Century Schools Need Geometry?” Shargyin (2004) identified three basic types of courses that taught anti-geometry (false geometry and pseudogeometry). The first type is built on a formal–logical (axiomatic) foundation; the second type is the practical–applied course with a narrowly pragmatic profile;
and about the third type he wrote: “And yet I am convinced that the coordinate method (along with trigonometry) constitutes one of the most effective means for ruining geometry, and even for destroying geometry” (p. 75). Nonetheless, both axioms (basic properties) and trigonometry with coordinates are to a certain degree present in his textbooks as well.
Sharygin introduced into his textbooks sections that were usually not included in textbooks for ordinary schools (he did, however, mark
them with an asterisk to indicate that they are optional and not part of the mandatory program). For example, the textbook for grades 10 and 11 discusses the Schwarz boot, and in its chapter on regular polyhedra half of the sections are optional, including a section explaining that the number of regular polyhedra is finite.
5.4.2 The textbooks of I. M. Smirnova and V. A. Smirnov
In contrast to Sharygin, the authors of these textbooks (Smirnova and Smirnov, 2001a, 2001b), professors at Moscow Pedagogical University, emphasize their adherence to the axiomatic approach.
A note in their textbookGeometry 7states:
The textbook is based on the axiomatic approach to structuring a course in geometry and corresponds to the mathematics program in ordinary schools. In addition to classical plane geometry, topics in spatial geometry, contemporary geometry, and popular-scientific geometry have been included in it as supplementary material.
(Smirnova and Smirnov, 2001a, p. 2)
The content of the course is wholly traditional (in particular, the authors once again return to Kiselev’s approach to the defining similarity, presenting a theorem about the proportionality of segments cut off by parallel straight lines from the sides of an angle, a theorem that is effectively unprovable in school). This textbook contains fewer problems than, for example, the one by Atanasyanet al.
The authors strive to make their geometry textbooks interesting and entertaining. The following words are printed on the covers of the textbooks: “Geometry is not hard. Geometry is beautiful.” These textbooks probably contain even more optional sections than the one by Sharygin (1999). Thus, at the end of seventh grade, six optional sections are given: “Parabola,” “Ellipse,” “Hyperbola,” “Graphs,”
“Euler’s Theorem,” and “The Four-Color Problem.” The textbook Geometry 10–11 includes such sections as “Semiregular Polyhedra,”
“Star Polyhedra,” “Crystals — Nature’s Polyhedra,” “The Orientation of Space,” “The Moebius Strip,” and “Polyhedra in Optimization Problems.”
5.4.3 The textbooks of A. L. Werner and his coauthors
After Alexandrov’s death, his coauthors and collaborators prepared several new textbooks. Adhering to the same principles on which the earlier textbooks had been based, the authors attempted to address certain critical new problems.
One of them consisted in the need to fill out the course in plane geometry with elements of three-dimensional geometry. We have already pointed out that without this, the spatial imagination of students who are immersed for three years in the world of plane geometry grows weaker (or atrophies altogether). The problem of overcoming students’spatial blindness is well known to teachers who are beginning to teach a course in three-dimensional geometry. Fur- thermore, since a complete (11-year) secondary education once again became nonmandatory, it was deemed necessary to provide students with some rudimentary knowledge of three-dimensional geometry in basic schools (the nine-year program).
The authors of existing textbooks often merely supplemented their plane geometry textbooks with one last chapter, which presented the rudiments of three-dimensional geometry. This in no way solved the problem of developing students’ spatial imaginations: as before, they were immersed for three years in the world of plane geometry. There- fore, during the very first year of competitions, the NTF announced a competition for a new textbook,Geometry 7, and during the second year, for a second textbook,Geometry 8–9, in which a systematic course in plane geometry would be supplemented with elements of three- dimensional geometry, presented in a visual–intuitive fashion. Both competitions were won by textbooks written by a working group that included A. L. Werner, V. I. Ryzhik, and T. G. Khodot, an associate professor at Herzen University’s geometry department (Werneret al., 1999, 2001a, 2001b).
The elements of three-dimensional geometry in these textbooks were presented along with analogous topics in plane geometry: per- pendiculars in a plane were accompanied by perpendiculars in space, parallels in a plane were accompanied by parallels in space, the circle and the disk were accompanied by the sphere and the ball, and so on. Each
textbook placed special emphasis on the main theme of each course: in grade 7, the geometry of constructions; in grade 8, the geometry of computations; in grade 9, the ideas and methods of post-Euclidean geometry — vectors, coordinates, and transformations.
The same main themes were followed in another series of textbooks prepared by Werner and Ryzhik on the basis of textbooks prepared under the supervision of Alexandrov as part of a project to create the so-calledAcademic School Textbook(the heads of the project were the academician V. V. Kozlov, vice president of the Russian Academy of Sciences; the academician N. D. Nikandrov, president of the Russian Academy of Education; and A. M. Kondakov, general director of the Prosveschenie publishing house and corresponding member of the Russian Academy of Education).
Among the distinctive features of this series of textbooks (Alexan- drov et al., 2008, 2009, 2010) were their sections on logic and set theory, as well as their heightened attention to the history of geometry.
The textbooks placed considerable emphasis on issues pertaining to the language of geometry, providing translations of geometric terms accompanied by lists of words with the same roots. They contained numerous illustrations showing various architectural constructions (“frozen geometry”) and discussed symmetry and its role in connection with this, and so on.
Ryzhik broke down the problems in the book into sections whose titles indicated to teachers and students the main form of activity involved in solving them. Among these titles were the following:
• Analyzing solutions. Students are not only given completed proofs, which are part of the theoretical course, but also shown how these proofs are found.
• Supplementing theory.Students are given theoretical propositions that do not belong to the main theme of the course, but are useful for solving other problems. Students can refer to them along with the theoretical propositions that belong to the main theme of the course.
• Looking. Students are taught to interpret information presented in visual form, and students’ spatial (two- and three-dimensional) imaginations are developed.
• Drawing. Students develop their spatial thinking skills.
• Representing. The problems in this category may be solved using only visual representations, without boring theoretical explanations.
• Working with formulas. Important problems that link the courses in geometry and algebra.
• Planning.Designing an algorithm that leads to the solution of a problem.
• Finding the value. Ordinary classroom computation problems.
• Proving. Problems involving proofs.
• Investigating. Problems whose conditions or possible results may contain some uncertainty, incompleteness, and ambiguity.
• Constructing. Construction problems.
• Applying geometry. Problems from outside mathematics that must be translated into mathematical language.
6 Concerning Some Problems with the Course in Geometry in Russia in Recent Decades
Pondering the development of and changes in the course in geometry in Russia over the past half-century (a short description of which has been given above), one cannot help noticing several basic problems around which discussions have revolved. It must be acknowledged that the existence of these discussions in itself shows that these problems are difficult and that no simple solutions to them can be expected. They can, however, be examined in greater detail, as we will attempt to do below.