A general description of the Kolmogorov reforms is given in another chapter of this two-volume set (Abramov, 2010). Kolmogorov himself and the subject committee of which he was the chair devoted great attention to the teaching of geometry. Criticizing existing programs for being outdated, Kolmogorov emphasized that this was especially true of geometry (Kolmogorov, 1967). He envisioned the restructuring of the course in geometry as follows:
The basic objectives of restructuring the school course in geometry, which have now won the widest acceptance, may be formulated in terms of three propositions:
1. The formation of elementary geometric concepts should take place in the first years of school.
2. The logical structure of the systematic course in geometry in the middle grades should be substantially simplified by comparison with the Euclidean tradition. At this stage, students should become habituated to rigorous logical proofs while the right to accept a redundant system of assumptions without proof should also be openly recognized.
3. The course in geometry in the higher grades should be founded on vectorial concepts. In this respect, it would also be natural to rely on the coordinate method (but only in an auxiliary fashion, so that the presentation does not become less “geometric” as a result of the reliance on this approach). (Kolmogorov, 1967, p. 11)
Some of these assertions may give rise to objections (for example, it is by no means an established fact that the vector-based approach to geometry instruction is simpler or in any way superior to the traditional approach). What is important, however, is that Kolmogorov envisioned
the creation of the new textbook as an open process that would rely — just as the creation of Kiselev’s textbook had relied — on international findings. Kolmogorov wrote:
In order to make it possible to work calmly and confidently on new geometry textbooks, preliminary work must be carried out at once: one or several working groups of scholars and teachers, using foreign findings, must put together and publish the outline (or several outlines) of a “logical skeleton” of a school course in geometry (the basic assumptions and the basic sequence of theorems with proofs) in a form that will be open to criticism and experimental use by sufficiently experienced teachers. (Kolmogorov, 1967, p. 13) Unfortunately, this was not done.
An idea of some of the aims set by Kolmogorov during the writing of the textbook (which he himself oversaw) is conveyed by the following statement made by him:
We have decided to retain separate geometry textbooks for grades 6–10. By comparison with a system of unified textbooks in mathe- matics, which is the norm in many countries, the existence of a separate geometry textbook has some advantages, but only if the logic of the construction of the geometry course is rigorously coordinated with the courses in algebra and elementary analysis. (Kolmogorov, 1971, p. 17)
It was expected that suchrigorous coordinationcould be achieved, in part, by organizing the presentation of the material around geometric transformations.
The new course in geometry was structured on the basis of set theory. This led to the appearance in schools of the term “congruence,”
which became perhaps the most frequently mentioned example of the difficulty of Kolmogorov’s course — prior to it, as well as afterward, people spoke about the “equality” of figures. Since in Kolmogorov’s course figures were seen as sets of points, and a set was “equal”
only to itself, it was impossible, in the opinion of Kolmogorov and his coauthors, to talk about “equal triangles,” as had been done before (Kolmogorov et al., 1979). Triangles that could be superim- posed through a geometric transformation that preserved distances
(rigid motion) began to be characterized as “congruent.” It seems unlikely that the introduction of one new term by itself could have exceeded students’ capacities sufficiently to warrant discussions about their suffering (which were not unusual for the pedagogical periodicals of the time and indeed are not unusual today). On the other hand, the introduction of a new term always creates certain difficulties, and if it could have been avoided, for example, by specifying the precise mean- ing that was being ascribed to the old term, then fighting so hard for the new term, and turning it into a rallying cry, hardly seems worthwhile.
What probably happened to be more important was that many proofs turned out to be fundamentally new and unfamiliar. For example, Kiselev and his followers had proven the classic theorem that the diagonals of a parallelogramABCDbisect each other (Fig. 1) by examining the trianglesAODandBOC(Ois the point of intersection of the diagonals). It is not difficult to see that these triangles are congruent (or “equal,” to use the term of that time), from which everything immediately follows.
Kolmogorov’s approach was to examine the midpoint O of the diagonal BD and point reflection with respect to this point. Since it was stated at the outset that a point reflection maps a straight line to a parallel straight line, and since it is clear that pointB, under such reflection, is mapped to point D, while point D is mapped to point B, it was possible to conclude that, under the point reflection being examined, the straight lineAD←→ is mapped to the straight line←→BC (as the only straight line which passes through pointB and is parallel to
←→AD). In an analogous manner, it was proven that the straight line←→AB is mapped to the straight lineDC←→. Thus, it was concluded that, under
Fig. 1.
the given point reflection, pointAis mapped to pointC, which proves thatOis the midpoint of the diagonalAC.
Kolmogorov’s proofs, which were in their own way beautiful and vivid, were nonetheless often difficult to grasp. In addition, if a student using Kiselev’s textbook had the impression that all of the propositions to which reference was made were completely proven (whether this impression was correct or not is another matter), then Kolmogorov’s textbook did not foster such an impression, if only because it attempted to set a much higher level of rigor than Kiselev’s textbook did.
Discussing the axiomatic approach, Kolmogorov (1968) wrote:
But in schools it has become common practice merely to indicate
“examples of axioms.” The actual list of these examples of axioms is usually laughably short. Apparently, the students are never asked to analyze a proof by identifying all of the axioms on which it is based.
Meanwhile, such an exercise should be insistently recommended: the proof of theoremTrelies on theoremsT1andT2, the proof of theorem T1relies on axiomsA1andA2, while the proof of theoremT2relies on axiomA3and theoremT3, and so on, until only axioms remain.
(p. 22)
It may be objected, however, that such an exercise is quite difficult for ordinary public school students if they are dealing with a theorem that has any substance. Even more significantly, such an exercise might give rise to a misguided notion of geometry as a subject in which there is a strange ritual of explaining what is obvious at great length for unknown reasons (this is especially the case if, as unfortunately often happens in Western textbooks, the theorem being examined is a very simple one, consisting of one or two steps).
The first chapter of Kolmogorov’s textbook Basic Concepts of Geometryformulates and enumerates 15 propositions. Nine of them are axioms. Five are proven; one is illustrated. Of the five proofs of the propositions, four are one step away from the axioms on which they are based, and only one (the derivation of a formula for distance between points on a coordinate line) contains more than one logical step.
The textbook Geometry 6 (Kolmogorov, 1972) contains 38 separate propositions in all, over half of which are not proven.
We will not discuss the other methodological innovations that provoked criticism — such as the approach to defining vectors in Kolmogorov’s textbook and the accompanying textbook by Klopsky et al. 1977 — or, on the contrary, met with success (such as the replacement of a separate problem book with sections on “Questions and Problems” in the textbook itself). Making the course at once more rigorous and more simple, which was Kolmogorov’s goal, is not an easy task. Kolmogorov and his coauthors took many revolutionary steps.
Possibly, given many years of further work, many difficult spots might have been smoothed over. At least, Kolmogorov (1984) himself later wrote:
The question of when it is proper to begin talking to students about geometry’s logical structure should be discussed again. The experience of working with different versions of geometry textbooks over the past decade has shown that doing so at the beginning of sixth grade is premature. (pp. 52–53)
But no more time was allowed for correcting, rethinking, and revising. A major campaign (Abramov, 2010) effectively resulted in the setting of a new agenda: to create new textbooks with the aim of replacing Kolmogorov’s.