О ПРИВЛЕЧЕНИИ СТУДЕНТОВ К ИЗУЧЕНИЮ АБСТРАКТНОЙ АЛГЕБРЫ (НА ПРИМЕРЕ ОДНОЙ ЗАДАЧИ ТЕОРИИ ГРУПП И ЕЕ ПРИЛОЖЕНИЯХ)

Abstract

The authors analize the alterations in the content of the algebra course, which is a conventional part of the university curriculum aimed at training maths teachers. The changes in the algebra skills development resulting from the overhaul of the education system are highlighted. The importance of general algebra is underlined. The problem is considered to be the impossibility to create conditions for mastering abstract algebra under existing approaches to the algebra study. Given general educational guidelines and the significance of the general algebra component, the authors propose to apply a methodology-based solution to the stated problem. The authors notice some options for motivating actions in these aspects of developing the general algebra content in the course of mathematics teacher training. One of such options is the selection of a series of problems (ranging from educational to research related), which makes it possible to discover the methods of general algebra, formulate some concepts of abstract algebra and consider their applications. The article describes a proposed solution to a well-known problem for schoolchildren from the Kvant magazine. The description of the (main) problem is given. A set of exercises is proposed enabling one to achieve a solution to the main problem. The solution to the proposed tasks is of greatest interest. A discussion of these solutions with students of mathematics at a pedagogical university makes it possible to introduce new group-theoretic concepts, which is illustrated in the text of the article. In a certain sense, the algebraic operation ‘*’ artificially introduced on a set of natural numbers allows us to define a group, being it Abelian and periodic, which is isomorphic to the direct sum of two-element groups. These properties allow us to consider a number of applications, in particular, the game of nim. The material presented in the article was tested at the Pitirim Sorokin Syktyvkar State University in the course of mathematics teacher training.