ELEMENTS OF GRAPHIC INTERPRETATION DURING THE STUDY OF METRIC SPACES BY STUDENTS

Abstract

The article emphasizes that when studying the theory of metric spaces, students of higher education in physical, mathematical and engineering specialties have difficulties in understanding the main relationships between individual points and sets of points of a specific metric space. Attention is drawn to the fact that difficulties with students’ assimilation of the relevant material are, to a large extent, related to the actual lack of geometric interpretation of these concepts in different metric spaces. To facilitate learning of the basic concepts of this theory, the article proposes to use geometric interpretation and digital visualization of the properties of mutual placement of points in metric space. It is concluded that it is expedient to demonstrate the geometric properties of metric spaces to students using the examples of Euclidean spaces of the first, second and third orders. It is emphasized that the properties of the mutual location of the points of the space can change significantly with a change in its metric, which is explained by a change in the internal geometry of the space. The article provides examples of such a change.
Examples of geometric interpretation of certain concepts of the metric space depending on the change of the metric of this space are considered. Geometric interpretation, of course, comes down to comparing these concepts with their corresponding interpretations in Euclidean spaces of the first, second, and third orders. This makes it possible to visually see the change of the internal geometry of the metric space in the case when its metric is not the metric of the Euclidean space. Such an interpretation demonstrates the differences between Euclidean and non-Euclidean geometries, which will improve their understanding by students.
The material of the article is based on examples of simple metric spaces, for the understanding of which the content of the school mathematics course is sufficient. Therefore, this material can be used when teaching mathematics in institutions of general secondary education.