Russian Mathematics Education: Programs and Practices
passage from the Standards quoted above). As A. D. Alexandrov
wrote:
The distinctive feature of geometry, which distinguishes it from
other branches of mathematics and from all sciences in general,
consists precisely in the indissoluble organic conjunction of lively
imagination and rigorous logic. Geometry in its essence is spatial
imagination, permeated and organized by rigorous logic. In any
genuinely geometric sentence, be it an axiom, a theorem, or a
definition, these two elements of geometry are inseparably present:
the visual picture and the rigorous formulation, the rigorous logical
deduction. Where either of these sides is absent, there is no genuine
geometry. (Alexandrov, Werner, Ryzhik, 1981, p. 6)
The student is in a sense invited to retrace the footsteps of the
ancients, who were able to pass from observation to interpretation
and abstraction. This experience of systematic mathematical modeling
also renders geometry particularly important in the eyes of Russian
mathematics educators.
Visual ideas, even visual ideas that are not subsequently proven,
are naturally very valuable. A. N. Kolmogorov, perhaps the greatest
Russian mathematician of the 20th century, criticized the then-
standard textbook by N. N. Nikitin (1961) as follows:
[The textbook] does not sufficiently distinguish between the two
levels at which the material is presented: the logical-deductive level
and the visual-descriptive level. The combination of these two levels
in textbooks for grades 6–8 seems to me unavoidable. In my opinion,
the body of geometric facts with which students become acquainted
purely through description might be somewhat expanded.
And he went on:
But this must not obscure the notion of geometry as a deductive
science in the minds of the students. This notion must already become
quite clear to them as a result of their study of geometry in grades 6–8.
This duality of the school course in geometry must be understandable
to the students themselves. They must always know what they are
proving and on the basis of which assumptions, what they are simply
told on faith, and which conclusions they themselves reach on the
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basis of visual arguments without a clear proof. (Kolmogorov, 1966,
p. 26)
Alexandrov saw the opportunity frankly to indicate about virtually
all propositions examined in school geometry, whether they were
accepted as unproven or rigorously grounded, as well as the oppor-
tunity for all students to establish the truth for themselves, without
trusting to the authority of a teacher or a textbook — as the enormous
potential benefit that geometry had to offer for developing students’
minds and worldviews. (Indeed, it is impossible to deny that in other
school subjects students must constantly or at least very often trust cited
facts, while in geometry classes they become convinced of everything
or almost everything on their own.) As Alexandrov (1980) wrote:
The deep objective of the course in geometry consists of the assimi-
lation of the scientific worldview, of the formation of its foundations.
It is shaped by an unequivocal respect for established truth, the need
to prove that which is put forward as truth, the refusal to substitute
faith or references to authoritative sources for proof. The striving for
truth, the search for a proof (or a refutation) — this is the active,
and therefore the dominant, aspect of the foundation of the scientific
worldview ….
The respect for truth and the demand for proofs convey an
extremely important ethical message. In its simplest but very impor-
tant form, it consists of the imperative not to judge without proving,
not to succumb to impressions, moods, and slander where it is neces-
sary to get to the bottom of the facts. Scientific commitment to truth
consists precisely of the striving to justify one’s convictions about
any issue with observations and conclusions that are as objective,
as unsusceptible to subjective influences and passions, as is humanly
possible. (p. 60)
Below, we will focus on differences between conceptions of the role
of geometry and approaches to its teaching; here, we have addressed
that side of geometry about which there may be said to be a consensus.
Naturally, such complex issues as “the scientific worldview” are almost
never mentioned in geometry classes. What an ordinary lesson looks
like to working teachers may be imagined, for example, by looking
at the methodological recommendations put forward by Glazkov,
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