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Extracurricular Work in Mathematics
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by taking classes at a mathematics correspondence school. Mathematics
correspondence schools were originally created in the 1960s under
the supervision of one of the greatest Russian mathematicians, Israel
Gelfand. Together with his collaborators, Gelfand personally developed
the programs for these classes and wrote textbooks for students. The
idea was not to allow students who lived in regions that were far from
the academic centers of the Soviet Union to slip through the cracks. The
work of such schools is based on a simple principle. Students who enroll
in them receive pamphlets in the mail with expositions of various areas
of mathematics, examples of problems with solutions, and problems
to solve on their own. The students solve these problems and send
them back to schools, where they are usually checked and graded by
students from the universities under whose aegis the correspondence
schools operate; after which, the graded homework assignments are
sent back to the students. Gradually, a framework developed in which
not just individual students could enroll as students in correspondence
schools, but entire classes or groups of students could do so as
well (as a “collective student”). Within such a framework, teachers
at ordinary schools could inform and organize their students, and
at the same time learn together with them and continue their own
education.
Since it is impossible for us to cover all details here, we can do no
more than simply mention correspondence mathematics Olympiads
(Vasiliev et al., 1986), which became an important form of Olympiad
activity — and quite distinctive in character, since problems that were
assigned for solving over an extended period of time at home needed
to be somewhat different from problems used in ordinary Olympiads,
which had to be solved on the spot. We will, however, say a few words
about the “ordinary” assignments given in correspondence schools.
As an example, we will use one of the assignments of the Petersburg
Correspondence School (centers of correspondence work also sprung
up outside of Moscow).
The pamphlet Problems in Algebra and Calculus (Ivanov, 1995)
is mainly devoted to solving problems, whose formulations resemble
ordinary school-style problems, by using ideas from calculus and
combining these ideas with standard ideas from the school curriculum.
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Russian Mathematics Education: Programs and Practices
The exposition begins with an analysis of several problems and a
discussion on the intermediate value theorem, which is employed
in their solutions. Among the variety of problems analyzed is the
following: “For what values of the parameter a does the equation
√
2
− x +
√
2
+ x = x
2
+ a
have a solution?” (p. 2). The solution becomes obvious if one uses the
derivative to sketch the graph of the function y
=
√
2
− x+
√
2
+ x and
determines its maximum and minimum. Another section of the pam-
phlet is devoted to function composition and the concept of the inverse
function. Here, a certain theory is presented (again in the form of
solutions to several problems), and then different ways of utilizing it are
demonstrated.
Based on the analyzed material, several problems are posed. Among
them are the following:
• Prove that the equation sin x = 2x + 1 has a single solution.
• How many solutions, depending on the value of a, does the
following equation have
√
x
2
− 4 = a − x
2
?
• Is it true that function f is invertible if the function g(x) = f(x
3
)
is invertible?
The pamphlet contains 24 analyzed examples and 40 unsolved
problems. Its material forms the content for two gradable assignments
(15 problems each). To receive the highest grade (5), students must
solve no fewer than 11 problems in each assignment, and to receive a
satisfactory grade (3), they must solve no fewer than 7 problems.
The content of the pamphlet described here has a pretty close
resemblance to the curriculum of so-called schools with an advanced
course in mathematics. The topics in the pamphlets for correspondence
schools, however, have varied: some pamphlets have dealt with tradi-
tional topics studied in ordinary schools, such as linear and piecewise
linear functions, while others have addressed topics traditionally found
in mathematics Olympiads (for example, the same invariants) or still
other, untraditional subjects [the very title of one of the sections in
Vasiliev et al. (1986) is noteworthy in this respect: “Unusual Examples
and Constructions”].
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Russian Mathematics Education: Programs and Practices
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Extracurricular Work in Mathematics
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