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Mathematical training of an engineer, ways to improve it


Математична підготовка інженера: шляхи вдосконалення

A graduate of modern higher technical educational institution must have a high level of professional knowledge as well as knowledge and skills of science subjects cycle, and especially math. Improving the training of specialists is impossible without improving their mathematical training.

What disciplines future engineers must learn?

As you know, the basic course of higher mathematics studied in higher technical education, almost entirely (with a few exceptions in the form of elements of linear algebra and analytic geometry) is based on classical mathematical analysis. This is understandable because most of our knowledge about the world it is obtained by means of this mathematical tools known to mankind for more than three centuries. Without belittling the importance of the foundations of mathematical analysis in the system of basic training of future professionals it should be, however, recognized that at the present stage of social development it could not be limited by the scope of mathematical analysis, on which, unfortunately, most often confined course of higher mathematics that is taught in universities . This course reflects only the current understanding of the role and importance of various mathematical concepts and ideas in engineering education. A substantial amount of course important and necessary mathematical concepts in a number of areas of engineering are not reflected in the standard course of higher mathematics and only occasionally are the subject of special courses or electives.

However, the progress is possible only on the basis of new knowledge. In mathematical science (on the memory of one generation) new ideas, theories and trends were generated, new mathematical methods have been developed, it was enriched with outstanding results. The central object was the notion of a mathematical model, and which actually originated with the development of mathematics. Mathematics has become a powerful tool for analysis and forecasting of technical and technological processes of nature, public situations. In combination with the enormous possibilities of computer technology, it engendered a new direction of scientific knowledge - mathematical modeling and mathematical experiment. Therefore, it is not possible to train specialists of tomorrow without including in the curriculum of basic mathematical training, developed in recent decades new branches of mathematics.

Depending on the audience and their specialization, it could be group theory and the theory of wavelets, matrix analysis or methods for solving nonlinear equations, etc. New math courses may be introduced in the educational process: 1) partially due to some sealing applications from standard course of higher mathematics (this is possible because of the key concepts of mathematical analysis the derivative and the definite integral students are introduced, albeit poorly, still in high school); 2) partly through optional, unclaimed (specialists for) material and, therefore, appropriate reallocation of teaching hours between topics inside the course; 3) partly due to new courses.

If this is not done soon, then talk about qualitative fundamental training of future engineers is unlikely to be appropriate. It should be stopped immediately, but rather to reverse the process of reduction of teaching hours for fundamental discipline.

A possible implementation of the program of modern mathematics education is seen as part of a two-level training, which we gradually turn. At the undergraduate level, for example, though it is not quite right, to limit the prevailing standard math courses that provide the necessary minimum of mathematical preparation. At the level of Master - to give advanced mathematical methods and theories. This may to some extent provide a modern basic training of future specialists.

Particular attention should be given to the mandatory in universities course of mathematical physics. To this day it is given the same way as in the beginning (or middle) of the last century. Today it is not enough, this course must be supplemented with the latest methods for solving nonlinear equations, the theory of solitons, etc. Or comprise this section of mathematical physics, which is booming in recent decades, in a separate course. Paradoxically, the fact is that today's graduates physics and mathematics departments do not know the meaning of the word "soliton" or "strange attractor".

How to train a future specialist?

It is known that the learning process is characterized by a contradiction between reproductive character training of future specialists and the need for creative manifestation of his professional qualities. Obviously, the course of higher mathematics, which systematically sets out the main issues and content rich set of new concepts and ideas, the student is unable to master (with very rare exceptions) without the help of a teacher. However, using a combination of dogmatic and heuristic approaches, traditional teaching methods are characterized by their weak focus on the formation of student skills to solve specific practical problems.

Indeed, the dogmatic approach means that a teacher, the carrier of knowledge, conveys knowledge to students, usually in a concise, synoptic form. Students have an opportunity only understand and remember. Of course, not all of them are capable of this, often unjustified, for lack of time, "reception" of new knowledge. And the material is not fully absorbed, and most often superficial. Some of the students in this approach remain passive spectators performing a set amount of inert learning tasks. The educational process in this case is aimed primarily at students transferring some amount of knowledge, and to a lesser extent - on the formation of future professionals in independent, creative approach to teaching material.

Heuristic approach does not block the creativity of students, rather it involves putting to the fore the acquisition of new knowledge through the solution of the problems by the students. The teacher only unobtrusively manages this process, "pushing" him leading questions and thoughts in the right direction. In the foreground, motivation and creativity are the main things of the process. Teacher seeks to develop students' thinking, gradually leads them to a full understanding of educational material.

The first approach is faster, but not effective, other - more effective, but slow. Given the factor of time during which a course in higher mathematics in technical universities is taught, the qualified teacher forced in different periods of learning to use different approaches and their specific combination.

Given the extremely low level of mathematical preparation of current graduates of technical university, it can hardly be limited by the mentioned approaches in teaching students. There is a need for a set of actions that would allow the teacher, to the extent possible, to manage the educational process, activating it and demonstrating the importance and necessity of conscious study of the subject.

In this regard, the teaching must necessarily take into account future professional activity of the listener. If a student does not see the connection of mathematical concepts and methods of future profession, he does not see the point in an educational context, this information is not transformed in his mind into system-knowledge, it becomes a formal knowledge, superficial. Therefore, the possibility of improving the quality of mathematical education, based on traditional content of the above mentioned studies, is limited. It should be clearly delineated focus of study – who is a listener (eg, engineer or teacher), and pay attention to the intricacies of mathematical proofs or on methods of solving problems that are used in the practice or profession. Of course, for future teachers, mathematicians are significant different subtleties in evidence, for example, for them is crucial to the concept of integral and Lebesgue measure, as a generalization of the Riemann integral, respectively, and measures of Jordan, while practicing engineer as well as in applied physics are essential only the methods of integration.

One important way to improve the situation is a coordination the programs in mathematics with the programs of the related disciplines, establishing real, not declarative interdisciplinary connections. Unlikely to be understood by first year students solutions of the Schrödinger equation, if they have little understanding of the differentiation of functions and have no idea of differential equations and functions of several variables.

One must use every opportunity to illustrate the connection of the learning process with the future production activities of students. Even up to the integration of basic mathematical courses, specialized, organic inclusion in the basic discipline specific examples that are understandable and interesting to students as they relate to their future profession.

An analysis of instructive examples and tasks of practical content is crucial in the system of training future engineers. Of course, there is difference between examples. If at the beginning of the study some topics it may be examples to practice a particular method or algorithm admission decision, subsequently, the development of the theme, it should make the task of general character that require mathematical intuition and ingenuity. At the final stage it is very desirable: a) verification of the results obtained against the physical meaning and dimension, b) assumptions about possible changes result in some changes of the task or the initial conditions, c) detailed analysis and conclusions. It is most important that to the best of their knowledge and skills, the students were trained to do.

The tasks of practical content combine learning activities and scientific research (especially if the content relates to task for future specialty trainees or used as guiding considerations, knowledge of the sphere), form mathematical and engineering intuition (finding the optimal solution method using known algorithms for solving, principle analogies or other heuristic methods), creativity (ability to roughen task to get a reasonable engineering decision or finding acceptable solutions with incomplete or redundant data), form logical thinking.

The use in educational process carefully selected tasks where along with the subject studied in mathematical methods the techniques of knowledge from other fields are used, as here, geometric representations or physical considerations much more effective formal theorem proving. After all, "in the study of science the examples are more useful than the rules" (I. Newton).

In no case it should be considered as exhaustive the proposals and answers to the problems of basic training of future specialists - we only outline them. For example, it must be noted among them such important problems of our engineering education as a low demand of mathematical knowledge in the study of special subjects. Reasoned response on this occasion I would like to hear from professionals graduating department.

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