# Mechanics and Mathematics Education

## Mechanics and Mathematics Education

a system of training advanced specialists for scientific research and teaching careers in mathematics, mechanics, and allied branches of science, engineering, economics, industry, and agriculture. In the USSR a distinction is commonly made among general mathematics education, which is provided by the secondary general-education school, where the fundamentals of mathematics are studied beginning with the first grade; special mechanics and mathematics education; and supplementary mechanics and mathematics education.

Special mechanics and mathematics education is provided by the mathematics and mechanics department and the mathematics and physics department of a university or pedagogical institute. In Russia special mechanics and mathematics education was first provided at the Academic University in St. Petersburg (founded in 1726) and later at Moscow University (1755) and the Teachers’ Gymnasium in St. Petersburg (1803). As early as the 18th century, prominent figures in Russian mathematics and education emerged from the universities, including S. E. Gur’ev, S. la. Rumovskii, and T. F. Osipovskii. They were greatly influenced by the pedagogical views of L. Euler. In the 19th century special mechanics and mathematics education was developed at various universities, including Novorossiia University (in Odessa) and the universities of Kazan, Kharkov, Kiev, St. Petersburg, and Dorpat (now Tartu). Among the students at these universities were N. I. Lobachevskii, M. V. Ostrogradskii, P. L. Chebyshev, N. E. Zhukovskii, and A. M. Liapunov, who became the founders of new fields and branches of mechanics and mathematics and helped improve general and special mechanics and mathematics education in Russia. In the early 20th century, the Russian mathematical school was represented by such scientists as A. M. Liapunov, A. A. Markov, and A. N. Krylov (St. Petersburg); N. E. Zhukovskii, D. F. Egorov, N. N. Luzin, and S. A. Chaplygin (Moscow); and S. N. Bernshtein (Kharkov). The physics and mathematics faculties of universities primarily trained mathematics instructors for *Oymnasien, Realschulen*, and higher and secondary specialized educational institutions. University courses quite fully reflected the content and level of development of mathematics and mechanics of the times. During this period, mechanics constituted a natural part of special mechanics and mathematics education.

In the first years of Soviet power, universities became the leading educational and scientific mathematical centers. The industrialization of the country required that the mathematical training of specialists better meet the needs of developing industry. In the early 1930’s university mechanics and mathematics education underwent considerable reorganization; specialities in mechanics—chiefly aerodynamics, hydrodynamics, elasticity theory, and general mechanics—were distinguished. The curricula reflected contemporary scientific ideas (in particular, functional analysis and tensor geometry). The university physics and mathematics department was separated into two departments: a physics department and a mechanics and mathematics department. At several universities scientific research institutes of mechanics and mathematics were established. In the 1950’s and 1960’s departments of computer mathematics, cybernetics, and automatic control systems were organized at universities, and departments of applied mathematics at many higher technical educational institutions.

Universities train mathematicians and specialists in theoretical mechanics for various sectors of the national economy, teachers for secondary and higher schools, and researchers for scientific research institutions. In addition to general scientific subjects, including such mathematical disciplines as mathematical analysis, higher algebra, and analytic geometry, mathematics students also study theoretical mechanics, the theory of functions of a complex variable, the theory of functions of a real variable, functional analysis, mathematical logic, probability theory, mathematical statistics, differential equations, and mathematical physics.

In the 1950’s courses in computer programming were added to the curricula, and training in computer mathematics was intensified. Computer centers were established at most universities. The training of specialists in mechanics was expanded greatly, especially in connection with the study of outer space, the development of automation and automatic control systems, and the necessity of studying the mechanical properties of both old and new synthetic materials.

Students in mechanics receive substantial mathematical training (similar to that given to mathematics students) and study such fields as elasticity theory, plasticity theory, hydrodynamics, aerodynamics, and the strength of materials.

Mathematics teachers for secondary schools are trained chiefly at pedagogical institutes. General mathematical, general pedagogical, and methodological subjects are an important part of the curricula. Students study the foundations of arithmetic and geometry, as well as probability theory, mathematical logic, mathematical machines, computer programming, general physics, and astronomy. Much attention is focused on elementary mathematics, the methodology of teaching mathematics, and teaching practice in schools. In some pedagogical institutes teachers are trained by specialization: mathematics and physics, mathematics and programming, and mathematics and drafting. The periods of study in mechanics and mathematics specialties are five to six years at universities and four to five years at pedagogical institutes.

In 1974 specialists in mechanics and mathematics were trained in the following areas: mathematics (58 universities with 38,200 students, 8,800 admissions, and 5,600 graduates and about 200 pedagogical institutes with 129,900 students, 27,100 admissions, and 23,300 graduates), mechanics (more than 20 universities with 4,300 students, about 1,000 admissions, and 700 graduates), and applied mathematics (more than 60 various specialized higher educational institutions and universities, with 23,900 students, 7,400 admissions, and 1,900 graduates). Advanced graduate training (*aspirantura*) in mathematics and mechanics is conducted at higher educational institutions, the Scientific Research Institute of Mathematics and Mechanics of the Academy of Sciences of the USSR, the academies of Union republics, and the Academy of Pedagogical Sciences of the USSR.

The goal of supplementary mechanics and mathematics education is to provide students with the mathematical knowledge needed to study special subjects and to use mathematical techniques in various types of research and in the daily routine. Supplementary mechanics and mathematics education refers to mathematics and mechanics courses that are offered (1) at higher technical institutions; (2) at specialized institutes and in university departments, for example, the economics, chemistry, biology, and geology departments; and (3) at secondary specialized educational institutions. The Moscow Physical Engineering Institute and the Moscow Physicotechnical Institute have been established to provide mathematics training for engineers, economists, and physicists. For such persons, mathematics provides deep insight into the principles that underlie processes in various fields, such as production, engineering, and economics. A number of engineering mathematics departments at higher technical educational institutions and departments of mathematical economics and mathematical linguistics at universities in Moscow and Leningrad have also been established.

In the 1950’s and 1960’s the number of hours devoted to mathematics instruction was significantly increased in the curricula of higher technical educational institutions, special courses in mathematics were introduced, and probability theory, mathematical statistics, and elements of computer programming, of linear programming, and of optimal process control were included in the general curriculum. The use of computer technology is mandatory in course and diploma design projects at many higher technical educational institutions. In the 1960’s departments for the advanced training of specialists in mechanics and mathematics were established at the largest higher educational institutions.

Abroad, research mathematicians, statisticians, computer operators and programmers, mathematics teachers, and other personnel are trained primarily at universities. In many European countries and in the United States, national committees on mechanics and mathematics education have been organized to improve this field. The International Commission on Mechanics and Mathematics Instruction operates in cooperation with UNESCO, and Soviet mathematicians work on this commission. International congresses on mathematics education are held every four years. An international journal devoted to mechanics and mathematics education has been published in Great Britain since 1970; special collections dealing with problems in mathematics instruction at higher educational institutions are published in the USSR.

### REFERENCES

Gnedenko, B. V.*Ocherki po istorii matematiki v Rossii*. Moscow-Leningrad, 1946.

Lankov, A. V.

*K istorii razvitiia peredovykh idei v russkoi metodike matematiki*. Moscow, 1951.

Prudnikov, V. E.

*Russkie pedagogi-matematiki XVIII-XIX vekov*. Moscow, 1956.

Kolmogorov, A. N.

*O professii matematika*, 3rd ed. Moscow, 1960.

*Voprosy istorii fiziko-matematicheskikh nauk*, sec. 1. Moscow, 1963.

B. V. GNEDENKO