Representation of geometric space and its visual-figurative model in educational activities

Statement of the problem. In educational geometric activities, the problem of forming model representations of geometric space revealing the content of its visual-figurative model is investigated. The purpose of the article is to establish in historical and geometric analysis the content and methodological patterns of the formation of a visual-figurative model of geometric space and its subsequent integration with vector, arithmetic models in the representation of abstract geometric space. Methodology (materials and methods). The methodology of the research is based on a spatial-theoretical approach implemented at successive levels of mathematical abstraction to form a model representation of geometric space and, based on it, to define an abstract geometric space. Research results. The subject of research in educational geometric activities is the patterns of formation of model, abstract representations of geometric space. Since the time of Euclid’s Elements, the basis of the model representation of geometric space has been its visual-figurative model, characterized by constructive images of geometric shapes, spatial images of their classes. In addition to creating constructive images of geometric shapes, transformations of motion and similarity, the content of the visual-figurative model consists of a generic systematization of geometric shapes based on the operation of spatial images, the activity of mathematical proof, the study of constructive, spatial, and metric properties of geometric shapes. Along with constructive images of geometric shapes, their iconic images are also used in educational geometric activities: vector models of geometric shapes, analytical models of geometric shapes. Vector, arithmetic models of geometric space are created in the system of vector, analytical images of geometric shapes in the subject’s consciousness. In the integration of visual-figurative, vector and arithmetic models, a visual-figurative level of spatial geometric thinking is formed.