Methodological aspects of professional training of future mathematics teachers in studying integral calculus of a single variable function

Abstract

The article theoretically substantiates the current methodological aspects of the fundamental training of future mathematics teachers in higher education institutions in the context of the development of specialized upper secondary school. The conceptual framework of the study is built around solving a fundamental problem of pedagogical education – overcoming Felix Klein’s «double discontinuity» between academic university mathematics and the school curriculum. The authors thoroughly reveal the essence and didactic potential of the backward transfer mechanism, which ensures the integration of students' fundamental knowledge into their future professional and pedagogical activities. The paper presents the results of a comprehensive comparative analysis of current educational programs and modern mathematics textbooks for specialized and advanced levels. Based on this analysis, the key cross-cutting content lines of the topic «Integral and its Applications» are identified, and the levels of requirements for students' mathematical preparation are differentiated. The expediency of introducing a system of special tasks with «double context» into the educational process of pedagogical universities is substantiated. Using specific mathematical examples, the advantages of shifting emphasis to the study of the form invariance of the indefinite integral and the technique of integration by substitution in the form of bringing a function under the differential sign are demonstrated, which allows rationalizing calculations and overcoming the school algorithmic pattern of linear substitution. Special attention is paid to the methodology of using the inverse geometric technique (calculating an integral through the area of known geometric shapes) and the analysis of improper integrals of the second kind. Through the philosophical categories of «form» and «content», the nature of typical errors associated with the unlawful application of the Newton-Leibniz formula to discontinuous functions is revealed. It is proved that the formation of such mathematical and didactic vigilance is a reliable tool for preventing formalism in students' knowledge and protects the future teacher from scientific and methodological errors when independently designing didactic materials for high school students.