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6
Combinatorics, Probability, and
Statistics in the Russian School
Curriculum
Evgeny Bunimovich
Editor in Chief, “Matematika v shkole,”
Moscow, Russia
1
Finite Mathematics in the School Curriculum
Prior to the Revolution of 1917
The debate over the role of statistics and probability theory in the
school curriculum goes back as far as the first half of the 19th century.
Interest in these subjects was informed in large part by the significant
contributions made to the field by Russian mathematicians: at the time,
foreign scholars jokingly referred to probability theory as the “Russian
science.”
By the mid–19th century, N. T. Scheglov, an instructor of algebra at
the Tzarskoselsky lycée, published a textbook covering several topics in
probability theory: “Simple or absolute probability. Conditional prob-
ability. Complex probability. Probability of interchangeable events.
Probability of events in repeated experiments.” (Scheglov, 1853). The
text covered the basic principles of these topics and offered sample
problems and solution strategies.
The popular textbook for elementary algebra by K. D. Kraevich
(1866) — as well as his exercise book (1867) — included chapters “on
probabilities.” Kraevich set out the material in an informal manner,
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Russian Mathematics Education: Programs and Practices
typically emphasizing practical application and avoiding rigorous
proofs and formulas (for example, “Mathematical advantage. On the
lottery. On the probability of human life. On insurance.”). A detailed
study of the probability curriculum, as envisioned in the textbooks of
the period, may be found in A. Kolmogorov’s (1947) work on the role
of Russian science in the development of probability theory.
The plans and methods of teaching probability theory in the
secondary school were actively debated in the early 20th century, as
part of the larger discussion of reforms in mathematical education. It
should be noted that at that time, curriculum reform — particularly
integration of probability theory into the general course of study —
was felt to be a necessity not only among mathematicians but also in
the natural sciences. A model syllabus in probability theory for the
secondary school, developed by P. S. Frolov, was published in 1902
in the Proceedings of the XI Conference of Russian Naturalists and
Physicians. By the XIII conference, delegates were considering two
distinct curricula, incorporating an introduction to probability and
statistics: basic-level and advanced-level.
A detailed account of the proceedings, along with a chronicle of
events and related documents, may be found in a publication released
by the Ministry of Education (Ministry, 1915). The plan of integrating
probability theory and the closely associated combinatorial analysis into
the secondary school algebra curriculum was raised once more in a
report delivered to the Ministry of Education by a special commission
charged with curriculum reform in mathematics, and debated at teach-
ers’ conferences. The Ministry gave the matter serious consideration,
soliciting the opinions of teachers and professors as well as expert
analysis. The ensuing publication remarks on “general educational
advantages of learning to calculate probability with combinatorial
analysis, as well as its practical applications in the areas of trade,
financing, management, and accounting)” (Ministry, 1915, p. 48).
Beyond educational benefits, the outlined curriculum was also seen as
an important formative tool. Consequently, the Ministry was interested
in the opinions of experts from a variety of disciplines.
Another innovation discussed at pedagogical conferences was the
establishment of a two-tier secondary school curriculum with a general
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233
track and an advanced track “adapted to a variety of individual learning
abilities and sensitive to the needs of educated people.” Accordingly,
the commission considered two proposals for a two-hour course and
a four-hour course in probability, the first submitted by one of its
members, Professor P. A. Nekrasov, and the other by the director of
the Uriupinsk real school, P. S. Frolov (Ministry, 1915):
Frolov Plan
Nekrasov Plan
I. A two-hour basic course:
1. Combinations
1. Combinations
2. Introduction to
probability
2. Introduction to probability
3. Newton’s binomial
theorem
3. Newton’s binomial theorem
4. Bernoulli’s theorem
4. Bernoulli’s theorem
5. Statistical correlation
5. Transformations of Bernoulli’s
theorem
II. Additional topics included in the four-hour course:
6. Multiplication of
probabilities
6. Multiplication of probabilities
7. Addition of probabilities
7. Addition of probabilities
8. Huygens’ problem
8. Huygens’ problem
9. Bayes’ theorem
9. Comparison of statistical
arithmetic means and
mathematical expectations;
Chebyshev’s theorem of means;
Statistical correlation
10. Witness testimony
10. Bayes’ theorem
11. Buffon’s problem
11. Witness testimony
12. Gambler’s ruin problem
12. Buffon’s problem
13. Mathematical expectation
13. Gambler’s ruin problem
14. Life insurance
14. Additional topics in
mathematical expectation; price
15. Life insurance
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It is evident that Russian scholars were largely in agreement: the
study of probability begins with combinatorial analysis, followed by
additional topics in statistics, historical aspects, and practical applica-
tions. The two versions of the basic course comprise essentially the
same elements, differing primarily with respect to the sequence of
presentation. The two-hour course would be integrated “into the
secondary school general curriculum. The new course may be accom-
modated into the curriculum by eliminating certain less significant
theories and the more abstract and futile exercises” (Ministry, 1915,
p. 35). The four-hour course was intended for the newly organized
“real schools” (or modern-language schools, as they were also called),
offering courses useful for the future study of economics, biology, or
other subjects dependent on the descriptive or comparative inductive
method and grounded in mathematical statistics and probability
theory.
Regrettably, the planned integration of probability theory into the
secondary school curriculum — first experimentally and then on a
mass scale — was never realized because of historical circumstances:
the breakout of the First World War, followed by the October coup,
which ushered in a new state, soon to be known as the USSR.
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