Integration of natural science and humanities knowledge in the teaching of applied mathematics to students in the conditions of informatization of education

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Problem statement. Achievements of applied mathematics are used in many areas of human activity - space exploration, research of the air space, the earth's environment, the bowels of the world ocean; nuclear power, thermonuclear syn- thesis, etc.; industrial production, economics, agriculture; biology, medicine, chemistry, physics, computer science, engineering, etc.; sociological and humanitarian research, and other areas of human activity. A great contribution to the formation and development of applied mathematics was made by the research of A.A. Andronov, S.N. Bernstein, O.M. Belotserkovsky, E.P. Velikhov, V. Velkovich, N.M. Gunter, J.L. Dalembert, N.E. Zhukovsky, M.V. Keldysh, A.N. Kolmogorov, S.P. Korolev, N.E. Kochin, N.N. Krasovsky, A.N. Krylov, M.A. Lavrentiev, A.M. Lyapunov, O.E.H. Love, D.K. Maxwell, G.I. Marchuk, I. Newton, M.V. Ostrogradsky, Yu.N. Pavlovsky, L. Prandtl, S.D. Poisson, A.A. Samarsky, L.I. Sedov, S.L. Sobolev, V.A. Steklov, D.G. Stokes, A.N. Tikhonov, J.B.J. Fourier, S.A. Chaplygin, V.N. Chelomey, E. Schrodinger, L. Euler and other authors. Modern applied mathematics has been enriched with new features, including the scientific and cognitive potential of mathematical models, humanitarization, distribution of optimality ideas, algorithmization, increasing the role of general mathematical structures, and other features (see, for example, [1-8]). New methods and directions of applied mathematics are being developed that allow to study objects, processes and phenomena that may be located at large distances from the researcher and are inaccessible for direct study. This can be water space, land space, air space, or outer space. At the same time, new methods of applied mathematics allow to determine not only their location, but also, for example, their shape, structure of inclusions, and cause-and-effect relationships (see, for example, [5; 6; 9; 10]). It is also obvious that information is a mobile means of studying both analytical and discrete mathematical models and allows to visualize their solutions. Method of research. Currently, Russian universities provide training for bachelors and undergraduates in such areas as “Applied mathematics and computer science” (01.03.02 - bachelor's level, 01.04.02 - master's level), “Mechanics and mathematical modeling” (01.03.03 - bachelor's level, 01.04.03 - master's level), “Applied mathematics” (01.03.04 - bachelor's level, 01.04.04 - master's level), “Applied mathematics and physics” (03.03.01 - bachelor's level, 03.04.01 - master's level) [11]. The relevant state educational standards specify the characteristics of such areas of training and professional activity, requirements for the results of educational programs, the structure of educational programs, and other requirements. In modern conditions, the system of higher mathematics education focuses on the training of creative highly qualified specialists who have not only solid and deep scientific knowledge, analytical and informational thinking, but also are able to show interest in modern problems of world science (see, for example, [12-21]). Integration of natural, humanities, and information technologies in applied mathematics classes helps motivate students to take an interest in learning and acquire deep and solid knowledge of applied mathematics. Students develop the skills to apply effective mathematical methods and approaches to solve sometimes complex mathematical problems. In addition, students gain invaluable experience in conducting humanitarian analysis of mathematical models that can be used to explore a variety of processes. Students develop a scientific outlook, creative abilities, and acquire deep knowledge not only in the disciplines that are included in the training program in applied mathematics, but also in some disciplines that are not included in the training program. For example, philosophy. The fact that using modern methods of applied mathematics, namely with the theory and practice of inverse and ill-posed problems it is possible to identify the causal relationships of the studied processes. So by studying cause-and-effect relationships and new scientific information, students master important methods of understanding the world around them, such as theory, experiment, and philosophy. Of course, students pay attention to the fact that new scientific information obtained as a result of research of an applied problem is organically intertwined with fundamental philosophical questions of natural science. Integration of natural, humanitarian knowledge, and information technologies in teaching applied mathematics makes it possible to implement environmental education of students. The fact is that today more than ever, specialists in the field of ecology are required. Therefore, higher education institutions train specialists in such areas of training as “Ecology and natural use”, “Geoecology”, “Environmental management”, etc. In these areas of training, students are introduced to disciplines whose content is developed on the basis of such scientific fields as ecology, geoecology, applied ecology, biosphere, hydrosphere and other scientific fields. Of course, students are also taught to apply modern environmental technologies in their professional activities. Currently, a great interest in environmental problems is shown by specialists of various professions-ecologists, physicists, biologists, mathematicians and other specialists. Note such authors as N.V. Bolotelov, Yu.I. Brodsky, A.V. Gagarin, M.M. Elanova, A.V. Ivashchenko, I.S. Ilyasova, G.I. Kushnikova, L.V. Mantatova, 1. Muraveva, Yu.N. Pavlovsky, A.P. Petrov, E.V. Rakhmatullina, S.A. Stepanov, S.M. Fayrushina and other scientists (see, for example, [22]). For example, it can also be noted that the integration of natural, humanitarian knowledge, and information technologies in the teaching of applied mathematics allows to identify fundamental concepts of computer science, such as information, modeling, formalization, algorithmization, computational experiment, syntax, semantics, computer graphics, information technology, and other basic concepts of computer science. Results and discussions. The integration of natural science, humanities, and information technologies in lectures, seminars, and laboratory classes dedicated to teaching applied mathematics allows students to master new subject knowledge not only in applied mathematics, but also in computational mathematics, information technology, and other areas of scientific knowledge. During the seminars, students are taught the skills to study mathematical models, followed by humanitarian analysis, which involves a deep analysis of environmental aspects, such as the environmental state of the air space, the earth's environment or the water environment. It is obvious that the knowledge and skills gained by students as a result of such training in applied mathematics will help them to be successful and qualified specialists. Integration of information technologies in the process of teaching applied mathematics is successfully implemented in laboratory classes, where it is possible to use a wide range of information technologies to solve applied mathematical problems. Computer algebra systems, or as they are also called - computer mathematical packages, are among such information technologies. Widely known and widely used in teaching are Mathematica, Maple, Matlab, MathCad and other computer mathematical packages (see, for example, [23-25]). Such computer technologies with a friendly interface have a great potential that allows to effectively and quickly find solutions to many mathematical problems from the fields of applied mathematics. Among the advantages of such computer technologies is undoubtedly the ability to demonstrate the obtained solutions in analytical, graphical, and numerical form. Of course, these opportunities are attractive when conducting lectures, since it is possible to implement a visual demonstration method of training. Such computer technologies allow students to quickly find analytical or approximate solutions to various mathematical problems by consistently applying commands. Conclusion. Integration of natural science and humanitarian knowledge, as well as information technologies in the teaching of applied mathematics in the process of teaching students of higher educational institutions allows students to form a system of fundamental knowledge in applied mathematics; to acquire the skills to apply various information technologies to solve applied mathematical problems and conduct their humanitarian analysis; to realize the role of information technologies in mobile research of applied mathematical problems; to develop their mathematical creativity.

About the authors

Viktor S. Kornilov

Author for correspondence.
Email: vs_kornilov@mail.ru

candidate of physical and mathematical sciences, doctor of pedagogical sciences, full professor, professor of the department of informatization of education

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