Russian Mathematics Education: Programs and Practices
• Find the value of 4 · d if d = 5, d = 6, d = 8, d = 3.
• Six envelopes cost 30 copecks. How much do three envelopes
cost?
Pictures and tables accompanied new problems, as in Fig. 1.
30
c
?
Price
Quantity
Cost
Same
6
3
30 c.
?
Fig. 1.
(Later, when solving similar problems, students would draw up
tables of their own.)
Plane geometrical figures were studied in all three years of the ele-
mentary school: segment, broken line, types of angles, and polygons —
triangles and quadrilaterals, including rectangles (and squares). Stu-
dents were asked to identify, construct, and transform figures (see e.g.
Moro et al., 1970, p. 213):
In the diagram below, find 2 pentagons, 2 quadrilaterals, and
2 triangles. In addition, find 6 right angles.
Cut out these figures and use them to construct new figures.
Fig. 2.
Textbooks gave special attention to exercises requiring comparison
and analysis, concretization and generalization, independent work, and
creativity. After a trial period in Russian schools, the textbooks were
translated into the languages of 11 Soviet republics.
Two other programs in elementary mathematical education
appeared at the same time, created by L. V. Zankov and V. V. Davydov.
They were used only in an experimental setting.
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Russian Mathematics Education: Programs and Practices
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The History and the Present State
59
Davydov’s program
12
placed primary emphasis on construing the
child as a subject of educational activity and developing his or her
theoretical thinking through a deductive approach to the structuring
of an elementary course in mathematics. Primacy of place was given
to the study of magnitudes, which, through comparison and practical
measurement, yielded number; the course made use of a generalized
notation to describe relations of magnitudes.
The system of Zankov
13
(1901–1977) aimed at maximizing stu-
dents’ general development and was based on the following principles:
intensive development of all children; systemic and comprehensive
content; primacy of theoretical knowledge; demanding, fast-paced
instruction; making the child aware of the educational process; tying
the educational process to the child’s emotional life; problematization
and variability of the educational process; and individualized approach.
These systems worked well in terms of general development, but —
according to their critics — were inefficient in furnishing students with
specific mathematical skills.
The widely adopted program and textbooks of Moro, Ban-
tova, and Beltiukova were new and unfamiliar to teachers. Despite
the tremendous effort through a variety of publications (Bantova
et al., 1976) to explain the methodological ideas that informed the
program, not all of them would be realized in general practice. Over
time, the program went through numerous changes, which eventually
undermined somewhat the original emphasis on general development,
reduced the role of theoretical knowledge, and underscored practical
application by increasing the number of practical exercises. The authors
of the textbook later wrote: “The changes made to the program in
mathematics over the past few years pursued a very important goal:
to give the course a more practical orientation” (Kolyagin and Moro,
1985, p. 3).
12
For a detailed account of Davydov’s system, see http://www.centr-ro.ru/school.
html
13
For a detailed account of Zankov’s system, see http://www.zankov.ru
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Russian Mathematics Education: Programs and Practices
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Russian Mathematics Education: Programs and Practices
After a series of experiments in the 1980s, textbooks were rewritten
to accommodate a four-year elementary education for children entering
school at the age of six (M. I. Moro, M. A. Bantova, and G. V.
Beltiukova; edited by Yu. M. Kolyagin).
It should be noted that the textbooks of Moro et al., which
had regrettably lost some of their developmental potential through
simplification, became the foundation for subsequent educational
programs and served to acquaint the average elementary school
teacher with their principles. The present author believes that the
educational program created by Moro et al. (especially prior to its
major simplifications) offered a thoroughly reasoned and structured
system of mathematical education (which may not hold true for other
programs).
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