This and subsequent translations from Russian are by Alexander Karp.
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Russian Mathematics Education: Programs and Practices
systematic study of the properties of geometric shapes in the plane and
in space and through the use of these properties in solving problems
of a computational and constructional nature. A substantial role is also
assigned to the development of geometric intuition. The combination
of visual demonstrability and rigor constitutes an integral part of
geometric knowledge. The sections on “Coordinates” and “Vectors”
contain material that is largely interdisciplinary in nature and finds
application in various branches of mathematics as well as related
subjects. (Standards, 2009, p. 7)
Thus, the teaching of geometry is seen to be of great benefit
precisely because of the role that it plays in students’ development.
Geometry is undoubtedly useful as an applied discipline as well, as is
indicated by the conclusion of the quoted passage: natural scientists
speak a geometric language, and by failing to teach students this
language we compromise their comprehension of the natural sciences
and thereby also condemn them to a sort of second-class status in
the modern world (whatever the rhetoric employed to legitimize
this fact). Russian pedagogy, however, has traditionally harbored the
conviction that education is valuable not only and not principally
because it conveys various kinds of skills and knowledge that may
be subsequently applied directly in practical life, but also because it
facilitates the development of students’ reasoning skills [this tradition
found expression in the works of Vygotsky (1986), which in turn
became very influential].
So what is behind this general proposition concerning the devel-
opment of logical reasoning skills and why is geometry particularly
important in this respect? The tradition of major scientists being
involved in the writing of courses in geometry, which goes back
to Euclid and Legendre, was continued in Russia (USSR), where
many outstanding research mathematicians thought about school-level
education, wrote school-level textbooks, and, by doing so, have left us
their notions about the role and significance of geometry.
In his programmatic article “On Geometry,” A. D. Alexandrov
(1980) wrote:
The logic of geometry consists not only in separate formulations and
proofs, but in the entire system of formulations and proofs considered
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as a whole. The meaning of every definition, every theorem, every
proof, is defined in the final analysis only by this system, which
is what makes geometry a unified theory and not a collection of
isolated definitions and propositions. This idea of an exact science
with a rigorously unfolding system of deductive conclusions, which
geometry conveys, is as important as the precision of each conclusion
considered on its own. (p. 59)
In other words, geometry teaches students how to analyze and
comprehend a system of propositions — how to correlate separate facts,
how to look for connections and mutual influences between them.
Genuine understanding is possible only through an understanding of
the system as a whole. Conversely, although thinking in a fragmentary
fashion and ignoring various facts do not entirely preclude all kinds
of reasoning, such an approach inevitably makes reasoning more
primitive. It would be misleading, of course, to claim that only the
study of geometry can teach students a system-oriented approach, but
the historic role of geometry as the model for a systematic program
(see, for example, Spinoza, 1997) suggests that it would be wise to
consider, before rejecting geometry altogether, the possible substitutes
that might be found for it in this particular respect within the school
program (if any such substitutes exist). We should point out that
a comparably systematic course in algebra or the natural sciences is
likely impossible at the school level (at least we know of no large-scale
experiment with any course of this nature).
Another outstanding Russian geometer, A. V. Pogorelov (1974),
wrote in the introduction to one of his courses in Euclidean geometry:
In offering the present course, our basic assumption has been that
the main purpose of teaching geometry in school is to teach students
to reason logically, to support their assertions with arguments, to
prove. Very few of those who graduate from school will become
mathematicians, let alone geometers. There will be those who, in
their professional lives, will never once make use of the Pythagorean
theorem. However, it is unlikely that we would find anyone who will
not have to reason, analyze, prove. (p. 7)
At the same time, the logical aspect of geometry stands in a
complicated relationship to its visual aspect (as is indicated in the
