The vicissitudes of the struggle against Kolmogorov’s reforms and subsequent events are described in A. M. Abramov’s chapter in this two-volume set. The campaign that unfolded at the time went beyond the bounds of a debate about methodology and became politicized, giving rise to a situation in which even observations that were fundamentally correct were exaggerated to the point of becoming nonsensical. Thus, the need to give up set theory, which was allegedly the cause of all difficulties and unsuitable for Soviet children in general, became one of the campaign’s slogans. It was decided that the word
“set” would not be used. The plan developed by I. M. Vinogradov’s committee contained an explicit proposal “not to use set theory as the basis for the teaching of mathematics in secondary schools” (Proekt,
1979, pp. 7–12). A. V. Pogorelov’s textbook, which had been written by this time, turned out to be “good” because the rejected word was not used in it. In his article “On the Concept of the Set in the Course in Geometry,” A. D. Alexandrov (1984a) showed that the content of this textbook, which had been recommended by Vinogradov’s committee, was in fact based on set theory. The same fact had been pointed out even earlier by Kolmogorov (1983), in his memo “On A. V. Pogorelov’s Teaching ManualGeometry 6–10”:
The very first page of the textbook states: “We conceive of every geometric figure as being composed of points.” It is difficult to understand this sentence except as an assertion that every figure is a set of points. However, the actual word “set” is not used anywhere in the textbook. (p. 45)
But, by this time, not using unapproved words was precisely what was important.
A. M. Abramov describes, in his detailed account, how the decision was made to conduct a nationwide competition for mathematics textbooks, in which all of the principal authors’ groups took part, and how the defeat of Kolmogorov gave other working groups a chance to make their own proposals heard. In addition to Pogorelov’s textbook, which has already been cited, we should also mention the textbook written under the supervision of the academician A. N. Tikhonov by L. S. Atanasyan (1921–1998), chair of the geometry department at the Moscow State Pedagogical Institute; Professor E. G. Poznyak (1923–
1993) of the mathematics division of the Moscow State University’s physics department; Poznyak’s colleagues V. F. Butuzov and S. B.
Kadomtsev; and the well-known mathematics educator I. I. Yudina, who joined them later. Another working group, formed under the supervision of the academician A. D. Alexandrov, included one of the authors of this chapter, A. L. Werner, who was then chair of the geometry department at the Leningrad (now St. Petersburg) Pedagogical Institute, and V. I. Ryzhik, a well-known teacher.
These three textbooks, which went on to become probably the most popular, won the competition. The manuscripts of the textbooks written by Atanasyan and his colleagues won first place in both
competitions (“Geometry 7–9” and “Geometry 10–11”). Second place in the competition “Geometry 7–9” was won by Pogorelov’s textbook, while in the competition “Geometry 10–11” Pogorelov’s textbook shared second and third place with the manuscript presented by the Kiev authors G. P. Bevz, V. G. Bevz, and N. G. Vladimirova. The manuscript of the textbook “Geometry 7–9” by Alexandrov, Werner, and Ryzhik came in third, and their “Geometry 10–11” fourth. Below, we describe these textbooks’ approaches in greater detail.
5.3.1 A. V. Pogorelov’s geometry textbook
Long before the nationwide competition, A. V. Pogorelov, an academi- cian and well-known geometer, published a book in elementary geom- etry (1974), which became the foundation for his school textbook.
Therefore, we will begin with his textbook (Pogorelov’s textbook was reissued many times; see, for example, Pogorelov, 2004a, 2004b.) The competition committee characterized his work as follows: “The manuscripts of the textbooks are characterized by a high level of rigor in the presentation of the theoretical material, brevity and precision of language, and the use of an axiomatic foundation in the construction of the course” (Konkurs, 1988, p. 49).
What Kolmogorov had been preparing to do (but did not do), Pogorelov did: at the very beginning of the course, he named the basic geometric figures — point and straight line — and presented a complete system of axioms for this course, which he described as the fundamental properties of the basic geometric figures. After this, precisely and methodically, Pogorelov presented definitions and proved subsequent propositions. The course is unified, self-contained, and similar to a course in the foundations of geometry.
Pogorelov’s geometry textbook is structured as an outline. It is divided into sections which are broken down into clauses. The theoretical text in each section is followed first by test questions and then by problems. People who worked with Pogorelov told the authors of this chapter that he always strove to shorten the text of his textbook and would repeat: “If you see that a sentence can be crossed out, then cross it out!” Pogorelov assumed that teachers by themselves would
add the necessary words in class, in accordance with their pedagogical approach.
The hand of an outstanding geometer can be seen in many of the proofs and in how the presentation of the topics is structured.
Nonetheless, as the textbook was put into use, critical observations arose. Let us return, for example, to the theorem about the intersection of the diagonals of a parallelogram, discussed above. Kiselev’s tacit introduction of a point O at the intersection of the diagonals was unacceptable for Pogorelov’s course, which was far more rigorous than Kiselev’s: indeed, it does not follow from anything that a parallelo- gram’s diagonals intersect at all. Kolmogorov’s proof, examined above, showed this, but it relied on transformations, which was unacceptable for Pogorelov’s course. The way out of this predicament was found, first, by proving on the basis of the congruence of the triangles (which was once again referred to as “equality”) that if the diagonals of a quadrilateral intersect and their point of intersection divides them in half, then this quadrilateral is a parallelogram. As for the theorem that the diagonals of a parallelogram intersect and are divided in half by their point of intersection, it was proven as follows (Fig. 2):
In the parallelogramABCD, consider the midpointOof the diagonal BD, draw the segmentAO, and extend it to a pointC1, such that the length ofAO equals the length of OC1. The quadrilateralABC1D turns out to be a parallelogram in accordance with a theorem proven earlier. From this it follows that the straight line DC←→1 is parallel to the straight line ←→AB; therefore,DC←→1 coincides with the straight lineDC←→ (since, given a point and a straight line, there is only one straight line that passes through the point and is parallel to the given
Fig. 2.
straight line). In an analogous manner, it is proven that the straight lineBC←→1coincides with the straight line←→BC as well. As a result, we find that the points C1 and C are identical, which means that the parallelogramsABCDandABC1Dare identical, and therefore in the given parallelogramABCDthe diagonals intersect and their point of intersection divides them in half.
For the students, this proof was decidedly not simple. In some cases, the efforts made to formulate a precise proof of the fact that some pair of straight lines intersected — a fact that was visually obvious to the students — completely overshadowed the substantive part of the theorem in the students’ eyes. The brevity of the textbook, which was meant to offer teachers an opportunity to make their own contributions, often simply made the lessons bare and dull if the teachers had nothing to add.
Pogorelov’s textbook has remained in use to this day, although it appears to be substantially less widespread now than in the 1980s.
5.3.2 The geometry textbooks of L. S. Atanasyan and his coauthors
L. S. Atanasyan and his coauthors began working on their textbooks over 30 years ago, in late 1970s, and today these textbooks are the most popular in Russia (Atanasyan et al., 2004, 2006). In its day, the nationwide competition committee characterized them as follows:
“The manuscripts are distinguished by the fact that the presentation of the material in them is accessible, by the fact that they are oriented toward students studying the material on their own, and by their explicit practical orientation” (Konkurs, 1988, p. 49).
In a conversation with one of the authors of this chapter (A. Werner), E. G. Poznyak himself said that the aim of their working group was to develop a simple textbook in the spirit of Kiselev’s.
Indeed, Atanasyan and his coauthors returned to the reliable path of Euclid (and Kiselev), which has stood the test of millennia. For example, they call two geometric figures “equal” if they coincide when superimposed on each other. This is exactly what we find in Kiselev and almost exactly what we find in Euclid. The proofs of the congruence of
triangles, and of much else, are what they are in Kiselev and even what they are in Euclid. The theorem about the intersection point of two diagonals, which we examined above as an example, was once again reunited with its old proof from Kiselev’s textbook, which relied on the congruence of triangles and never raised the question of whether the diagonals intersected at all. All of this was known and familiar to teachers, to students, and to students’ parents.
Certain innovations appeared in the discussion of similarity. Kiselev, following French models (Barbin, 2009), had departed from the Euclidean principle of using areas to prove theorems about relations between the lengths of segments. As a consequence, the theorems on which the basic propositions about similar triangles relied turned out to be very difficult: Pogorelov had made one such theorem a required part of his course (in his formulation, it read as follows:the cosine of an angle depends only on the angle’s degree measure), but in practice it turned out that students did not understand it. The textbook of Atanasyanet al.(just like the textbook of Alexandrov and his coauthors, which will be discussed below) returns to the spirit, if not the letter, of Euclid’s approach, using areas to prove theorems about similarity.
This noticeably simplified the course, not to mention the fact that introducing the concept of area early on made discussions of many geometric ideas and problems more accessible earlier than they had been previously.
The chapters devoted to post-Euclidean geometry are arguably more open to criticism. For example, according to what we have observed, the concluding chapter of the course in plane geometry,
“Rigid Motion,” is almost never studied in school in practice (the key section concerning the relationship between the concept of rigid motion, introduced in this chapter, and the concept of congruence, examined earlier, is marked with an asterisk, which denotes that material in the section is optional). Moreover, the idea of discussing transformations of the plane after all else in the course has been covered might itself give rise to objections.
On the other hand, the range of problems offered in the textbook of Atanasyan and his coauthors is rich and convenient for teachers.
These problems, along with good methodological supporting materials
(teachers’ manuals), appear to have been one of the most important reasons for the success of this textbook. Every section is accompanied by problems. Often, two similar problems are given in a row: one of them is solved by the teacher in class, and the other is assigned as homework. Each chapter also contains additional problems, and at the end of each class there is a set of more difficult problems.
Questions for review follow each chapter. In addition to problems, practical assignments accompany some sections, when appropriate.
Answers to problems and hints for some solutions appear at the end of the textbook.
5.3.3 The textbooks of A. D. Alexandrov and his coauthors
The manuscript of the geometry textbook for grades 7–9 by A. D.
Alexandrov and his coauthors was characterized by the nationwide competition committee as follows: “It is distinguished by its untradi- tional treatment of a number of topics, by the liveliness and readability of its language, by the overall orientation of its exercises toward students’ development” (Konkurs, 1988, p. 49).
Indeed, if the traditional view was that a geometry textbook should be laconic and dry, then the authors of this textbook (Alexandrovet al., 1983, 1992, 1992, 2006), and above all Alexandrov himself, strove to speak to the teacher and the students in a completely different language, not only explaining various propositions to them but also discussing their content and meaning. Below, for example, is a brief excerpt from the section of the textbook in which Alexandrov explains the meaning of the Pythagorean theorem:
The Pythagorean theorem is also remarkable because in itself it is not at all obvious. If you look closely, for example, at an isosceles triangle with an added median, then you will be able to see directly all of the properties that are formulated in the theorem that deals with it. But no matter how long you look at a right triangle, you will never see that its sides stand in this simple relation to one another:
a2 +b2 = c2. Yet this relation, as a relation between the areas corresponding to the sides, becomes obvious from the construction
Fig. 3.
depicted in Fig. 3. This is what the best style of mathematics consists of: taking something that is not obvious and making it obvious by means of a clever construction, technique, or argument. (Alexandrov et al., 1992, p. 139)
Alexandrov formulated several principles for teaching geometry (following Werner, 2002, p. 166):
• Since one of the aspects of geometry is its rigorous logical character, and since the students of grades 7–11 are already capable of grasping this logical character, the course in geometry must be presented sufficiently rigorously, without logical gaps in thebasic structureof the course.
• Since the second basic aspect of geometry is its visual character, in the teaching of geometry every element of the course should be initially presented in the most simple and visually intuitive way, using that which may be drawn on the blackboard, demonstrated on models, on real objects, as far as possible.
• Further, despite its high degree of abstraction, geometry arose from practical applications and is put to practical uses. Therefore, the teaching of geometry must unquestionably connect it with
real objects, with other disciplines, with art, architecture, and so on.
• A textbook aimed at ordinary secondary schools must not contain in its basic part anything that is extraneous, of secondary importance, or of little significance in the main body of the text.
• But since the abilities and interests of the students are quite varied, such a textbook must contain supplementary material, aimed at students who are stronger and have a greater interest in mathematics.
• Geometry must be presented geometrically. It contains its own methodology: the direct geometric methodology of grasping concepts, proving theorems, and solving problems. The synthetic methodology of elementary geometry must not be squashed in school-level instruction by any coordinate-based methodology, vector-based methodology, or any other methodology. The direct geometric methodology is simpler, more important, and more natural for the purposes of a general secondary education and corresponds to the very essence of geometry. It is needed by anyone who deals with three-dimensional objects.
• The school course in geometry must be connected with contem- porary science, must include, as far as this is possible, elements of contemporary mathematics.In addition, the course in geometry, as a logical system in which everything is proven, is impor- tant for developing the rudiments of a scientific worldview, which demands proofs rather than references to authoritative sources.
• But sincethere is simply no such thing as absolute rigor, a certain level of rigor must be selected and established, and this level of rigor must be maintained through the entire course. The course must not have logical gaps, at least in its basic sequence of topics. Otherwise, it will lose its systematic aspect, the logic of the exposition will become blurred, and students will be exposed not to a unified science — geometry — but only to its fragments.
In this way, the three foundations of the textbook, according to Alexandrov’s way of thinking, were supposed to be visual explanations, logic, and connections with the real world and practical applications.
Such a conception of the author’s task led Alexandrov to present
many sections in a new way. For example, the fundamental object that he chose was not the straight line, as was the norm in other school textbooks, but the segment, since it was precisely this that people dealt with in practice. For the same reason, the traditional axiom of parallel lines — which states that through a point outside a given straight line, only one straight line may be drawn that is parallel to that line — was replaced by the axiom of the rectangle. This axiom postulates that it is possible to construct a rectangle whose sides are equal to given segments (the possibility of such a construction is confirmed by everyday practice).
Alexandrov used the congruence of segments — visually apparent and “testable” — to define other concepts, including the congruence of figures. Untraditional definitions (although equivalent to traditional ones) were given in the textbook for other concepts as well — for example, the similarity of triangles.
All of this made the presentation shorter and more visual. At the same time, although Alexandrov’s approach was based precisely on a deep understanding of the classical tradition, the novelty of many of the ideas scared off some teachers when they suddenly discovered that from now on they would have to teach the congruence (equality) of triangles in a way that differed from what they had been accustomed to for years.