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Russian Mathematics Education: Programs and Practices
Here, too, the solution cannot be reduced to an operation involving
radicals. The student must note that
10
√
35
<
1
√
26
+
1
√
27
+ · · · +
1
√
35
<
10
√
26
,
since each of the given ratios, beginning with the second one, is less
than that preceding it — after which it is not difficult to see that
5
3
<
10
√
35
and
10
√
26
< 2.
This section also covers the topic “Divisibility.” At this point,
we must explain in greater detail our understanding of what an
advanced course in mathematics in high school must achieve, and
the fundamental difference between such a course and a course that
results simply from the addition of certain topics to the basic course in
mathematics (unfortunately, there is a common but — in our view —
erroneous opinion that this latter type of course is just what constitutes
an advanced course in mathematics).
As an example, let us consider precisely the topic “Divisibility.”
Why was this topic included in the content of the curriculum? We can
point to many reasons for this, but the main, most “conceptual” one
apparently had to do with the fact that issues connected with divisibility
are far more important for mathematics than, say, solving irrational
equations; in other words, this topic brings the content of the school course
closer to real mathematics. In particular, knowledge of this material
makes it possible, in studying the topic “Polynomials with one variable”
(another topic that distinguishes the advanced course from the basic,
examined in greater detail below), to study questions connected with
the rational roots of polynomials with integer coefficients, i.e. to solve
a broader range of higher-degree equations, and in turn to make use of
this knowledge in studying rational and irrational numbers, and so on.
We might also mention that in solving problems pertaining to these
topics, students learn to use not so much the algorithms for solving
some narrow class of problems as the methods and techniques of
mathematical activity in general.
At present, problems connected with divisibility are generally
thought of as belonging to the category of so-called Olympiad prob-
lems; but this is so only because in the existing course in mathematics,
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this content is covered effectively only in grades 5–6 and essentially has
a narrowly directed aim — to develop certain well-defined arithmetic
abilities and skills.
The stylistic aspect of this topic’s presentation is determined first
and foremost by the objectives associated with studying mathematics
in basic school, where it is by no means assumed that most students
will take the advanced course in mathematics in high school. It is
also limited objectively by the age-dependent characteristics of the
students — the highly concrete nature of their thinking, which makes
it difficult for them to interact with abstract objects, and with letters in
particular, because of their insufficiently developed capacity for making
theoretical generalizations, and for understanding the essence of proofs
and their role in mathematics; because of their lack of any felt need
to prove propositions “in the general form” when confronted with
conclusive concrete examples; and so on.
However, these traits are no longer characteristic, by and large,
of 16–17-year-old teenagers, especially those who have gone through
three more years of schooling in a different style that is more in
harmony with the essence of mathematics and, above all, those who
have chosen an advanced course of study, designed essentially for the
formation of the country’s “technical–scientific elite.”
This position became more or less central in the general approach
of the textbooks and problem books of Dorofeev, Kuznetsova, Sedova,
and Troitskaya. “Divisibility” is the first topic presented in these
textbooks, mainly with a view of providing continuity with the content
of the basic school curriculum, but also in consideration of the objective
simplicity of its content and its proximity to experiences that students
already have. The difficulties with its assimilation (both on the level of
theory and, to an even greater degree, on the level of exercises) are
connected with a purely psychological barrier: the unfamiliarity of the
mathematical activity that corresponds to the content of the material.
In particular, in treating this topic, the authors of this textbook use
material that is quite simple to form and develop the students’ ability
to formulate proofs; this ability, as is well known, is one of the most
significant weak spots in the mathematical preparation of students. In
doing so, the authors have not deemed it necessary to fill in all the
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