DOI: https://doi.org/10.62204/2336-498X-2023-1-13

TOPOLOGY AS A DIDACTIC TOOL FOR THE DEVELOPMENT

OF STUDENTS’ MATHEMATICAL THINKING

Natalia Shapovalova,

Ph.D. in Physico-Mathematical Sciences, Associate Professor,

Mykhailo Dragomanov National Pedagogical University of Ukraine

n.v.shapovalova@udu.edu.ua; ORCID: 0009-0000-7084-1460

Larisa Panchenko,

Ph.D. in Pedagogy, Associate Professor,

Mykhailo Dragomanov National Pedagogical University of Ukraine

larpan97@gmail.com; ORCID ID: 0009-0001-8156-286X

Annotation. The article is aimed at revealing main methodical aspects of teaching topology for students of mathematical specialties of pedagogical universities.

The article emphasizes the special importance of topology as a didactic tool for the development of the mathematical worldview of students of mathematical specialties at universities.

Topology is a section of mathematics studying the properties of figures (or spaces) which they preserve in continuous deformations such as stretching, compression or bending. Continuous deformation takes place when there is no ruptures (that is integrity of a figure is not broken) or patching. Topological problems are solved through methods that are radically different from methods of geometry.

The studying of topology plays an important role due to broad theoretic and practical application of topological regularities in such branches of science as physics, astronomy, anthology, cybernetics, geology, architecture, geography, cosmology, genetics etc. Topology is particularly useful in improving students’ intellectual and cognitive abilities and developing their mathematical skills.

Keywords: topology, differential geometry, competence, interdisciplinary ties, studying process, teaching, scientific approach, physics.

Problem statement. In the conditions of the development of new technologies, the demand for people with non-standard thinking, who know how to set and solve new tasks has increased sharply. This determines the importance of conducting methodological work to acquaint students with new methods of non-standard thinking, aimed at the development of logical mathematical thinking of a new generation.

Therefore, the concept of a person-oriented approach to learning is gaining more and more popularity at the moment, which involves the maximum possible individualization of the process, ensuring the possibility of realizing the requests and gifts of the student’s personality. Another no less important milestone in the process of learning and development of new methodological approaches to its organization is encouraging students to constantly develop and improve their knowledge, skills and expand the field of interests. After all, self-development in the future often leads to the emergence of new, creative ways of solving problems.

Such an apparatus of cognition and expansion of the space of thinking is topology, which, like no other science, contributes to the development of students’ intellectual abilities. The course of topology has not yet gained clear recognition, but observing the development of the science, it is safe to say that it will flourish in the near future. The secret of topology’s success is hidden in its versatility and ability to manifest itself in almost all milestones of human activity and science.

This article is devoted to the study of the role of topology as a didactic tool for the development of students’ mathematical thinking in the process of learning the educational discipline “Differential Geometry and Topology” as a mandatory part of the cycle of professional training of students of mathematical specialties of universities and the disclosure of the main methodological aspects of this process.

Analysis of recent research and publications. Questions of the theory and methodology of teaching topology were developed in the works of A. N. Kolmogorov, A. D. Alexandrov, L. S. Pontryagin, V. G. Boltyanskyi, V. A. Yefremovich, N. Burbaki, J. Shvarts, A. L. Werner, V. I. Ryzhik, B. E. Kantor, S. A. Frangulova, S. P. Novikova, A. S. Mishchenko, Yu. P. Solovyov, A. T. Fomenko, M. I. Kovantsova, O. A. Borysenko, S. G. Kononova, A. V. Prasolova, V. L. Tykhmokhovych, V. I. Glizburg, M. E. Sangalova, N. V. Timofeeva, and others.

The purpose of the article. The purpose of the article is to reveal the main methodological aspects of teaching topology as a didactic tool for the development of mathematical thinking of students of mathematical specialties at universities. For this purpose, the purpose, content and main provisions of the topology are first considered. Then the peculiarities of topology are analyzed and modern approaches and methods of its learning are proposed.

Topology is studied in the course of the educational discipline “Differential geometry and topology” which is a mandatory part of the cycle of professional training of students of mathematical specialties of universities.

The subject of study of the educational discipline “Differential Geometry and Topology” is geometric images, primarily curves and surfaces, as well as families of curves and surfaces in Euclidean space by methods of mathematical analysis, metric and topological spaces, mapping of topological spaces, topological manifolds, polyhedra.

The organization of the educational course “Differential Geometry and Topology” involves the active use of interdisciplinary connections with such disciplines as “Mathematical Analysis”, “Analytic Geometry”, “Linear Algebra”, “Elementary Mathematics”, “Differential Equations”, “Theory of Invariants” , “Physics”, as well as when studying separate sections of general and special courses in physics, mathematics, computer science and astronomy, etc.

The purpose of teaching the educational discipline “Differential geometry and topology” is to teach students techniques and methods of solving differential geometry and topology problems, to develop the ability to use methods of mathematical analysis, to study the basic facts of differential geometry and topology, and to be able to apply these geometric and topological facts as in solving solving geometric and topological problems, as well as problems of an applied nature, researching their connection with problems and methods of differential and integral calculus, with a school geometry course.

According to the requirements of the educational and professional program, students must know topology:

Basic concepts. Metric, metric space; open and closed sets in metric space; interiority, closure and limit of a subset; convergent sequences in metric space; topology; topological space; topological environment; comparison of topologies; discrete topology; antidiscrete topology; metric-induced topology; subspaces of the topological space; closed sets in topological space; convergent sequences in topological space; axioms of separability; Hausdorff topological spaces; axioms of countability; topology base; separable topological spaces; metric outer topological spaces; normal topological spaces. Display of topological spaces; continuity of mapping of topological spaces at a point and “as a whole”; homeomorphism; topological properties; connectivity; compactness; components of topological space, linear connectivity; topological dimensionality, Hausdorffness; inherited topological properties. Locally Euclidean topological space; dimensionality of the topological space; n-dimensional topological manifold; edge of multispecies; Möbius sheet; torus; Klein’s bottle; model surfaces; pens, tubes, films; multispecies orientation; spheres with handles; spheres with holes; spheres with films; triangulation; cellular divisions of surfaces; Euler characteristic of the manifold; regular polyhedra; surface sweeps.

Basic formulas and theorems. Properties of open and closed sets in metric space. Properties of interiority, closure and limit of a set. Theorem on the structure of a topological subspace. Theorems on the structure of the boundary and interior of a set of topological space. Properties of Hausdorff spaces. Criterion of metrization of topological space. Criterion of continuity of mappings of topological spaces “as a whole”. Properties and signs of continuous mappings. Properties of homeomorphisms. Connectivity criterion. Connectivity component properties. Properties of compact topological spaces. Criterion of compactness in Euclidean spaces. Criterion of homeomorphism. Theorem on topological classification of one-dimensional manifolds. Theorem on topological classification of two-dimensional manifolds. Euler’s formula. Theorem on the classification of topologically regular polyhedra.

Students should be able to:

Check the fulfillment of the axioms of topological and metric spaces. Metrize the basic set in different ways, check the equivalence of metrics. Topologize sets in different ways, compare topologies. Investigate the convergence of sequences in topological and metric spaces. Classify points by their position relative to a fixed set. Check the continuity of the display at the point. Check the continuity of the display “as a whole”. To prove homeomorphism (non-homeomorphism) of topological spaces. Calculate the topological dimension of subsets of the topological space. Investigate topological spaces and their subspaces for compactness, connectivity, linear connectivity, Hausdorffness. Determine the topological dimension of a manifold. Calculate the Euler characteristic of a manifold. Establish topological equivalence (non-equivalence) of one-dimensional and two-dimensional manifolds.

These knowledge and skills ensure the formation of the following competencies:

− basic ideas about the variety of geometric objects, understanding the meaning of the unity of geometry as a science, its place in the modern world and the system of sciences;

− mastery of methods of description, identification, classification and definition of geometric objects;

− the ability to apply basic analytical, geometric methods and methods of mathematical analysis, in particular differential and integral calculus, to the creation, analysis and research of mathematical models of real objects, processes and phenomena;

− the ability to apply modern information technologies to solve theoretical, practical and applied problems;

− basic ideas about the history of the development of differential geometry;

− ability to analyze educational and methodical literature on the discipline;

− ability to apply acquired theoretical knowledge in solving practical problems;

− the ability to use previously acquired knowledge when studying new theoretical material and solving practical problems;

− the ability to conduct deductive justifications of the correctness of solving problems and to look for logical errors in incorrect deductive reasoning;

− the ability to find invariant quantities, invariant elements and invariant properties and apply them in further research;

− the ability to use mathematical and logical symbols in practice.

Topology has great opportunities for the development of the cognitive activity of the future teacher of mathematics through the development of such methods of mental activity as analysis, synthesis, abstraction, comparison, generalization, analogy, intuition, etc. Taking into account the specialization and individual development of students in accordance with their abilities and capabilities, the content of the “Differential Geometry and Topology” course, in addition to theoretical material with mandatory and additional parts, task material that will ensure solid assimilation of basic knowledge, should also contain motivational material (system problematic and heuristic problems and questions, creative and research questions, problems of interdisciplinary content, historical materials for studying relevant course topics, etc.).

At the first lectures, it is necessary to explain the general purpose of topology as a separate module of the course, to clarify the structure of this module as a whole system. Attention should be paid to the dialectical nature of the module as a whole. It is necessary to draw the attention of students to a wide range of applied and practical problems that are solved by the methods and means of topology.

Topology studies those properties of geometric objects that are preserved under continuous transformations. Topology is a branch of mathematics that deals with the study of properties of shapes (or spaces) that are preserved under continuous deformations, such as, for example, stretching, compression, or bending. Continuous deformation is a deformation of a shape in which there are no breaks (that is, the integrity of the shape is not violated) or gluing (that is, identification of its points).

The subject of studying topology as an educational discipline is metric and topological spaces, mapping of topological spaces, topological manifolds, polyhedra. Topology as an educational discipline consists of three content modules: “Topological and metric spaces”, “Mapping of topological spaces”, “Topological manifolds”.

The main approach in the process of studying topology is a scientific approach.

In the process of studying topology, students learn to check the fulfillment of the axioms of topological and metric spaces, to metricize the basic set in different ways, to check the equivalence of metrics, to topologize sets in different ways, to compare topologies, to investigate the convergence of sequences in topological and metric spaces, to classify points according to their position relative to a fixed set , check the continuity of the mapping at a point, check the continuity of the mapping “as a whole”, prove the homeomorphism (non-homeomorphism) of topological spaces, calculate the topological dimension of subsets of the topological space, examine topological spaces and their subspaces for compactness, connectivity, linear connectivity, Hausdorffness, determine the topological dimension of a manifold, calculate the Euler characteristic of a manifold, establish the topological equivalence (inequivalence) of one-dimensional and two-dimensional manifolds.

The study of topology plays an important role in connection with a wide range of theoretical and applied applications of topological properties in such sciences as anthology, raceology, cybernetics, computer science, cartography, geology, pottery, architecture, geography, cosmology, biology, natural science, genetics, bacteriology, mathematics, physics, mining, astronomy, etc.

In the process of teaching topology students, you can successfully use historical examples, introduce them to the main stages of development of topology and the main historical problems and tasks. It is important to introduce students to the concept of topological dimension, which plays a central role in the development of many theories.

For the first time, the famous mathematician Leonard Euler encountered a purely topological problem, solving the so-called problem “about the seven Königsberg bridges”. Analogous to the problem about the seven Königsberg bridges are the problems about the ability to draw some geometric figure in a continuous movement (without taking the pencil off the paper) so that the tip of the pencil does not pass a second time along any of the already drawn lines. It turns out that the question of the possibility of solving such problems is determined not by the complexity, not by the dimensions of the figure itself, but by the mathematical dependencies that topology investigates.

Topological problems are solved by methods fundamentally different from those of geometry. Topology studies the conditions under which some shapes can be transformed into others in a smooth, continuous motion.

Studying topology contributes to the development of not only topological, but also intellectual, physical, mathematical skills and abilities of students, is a necessary element for self-improvement, self-education of students, development of erudition, abstract thinking, enrichment of knowledge in different areas of science at the same time, intellectual growth.

Taking into account the experience of teaching geometry and topology at a pedagogical university, we believe that topology should be studied as a separate academic discipline for at least one semester with one lecture and one practical lesson per week. Because solving topology problems causes significant difficulties for students, as they have their own specifics and require relevant knowledge and skills not only in topology, but also in geometry, mathematical analysis.

As an option, you can consider the possibility of studying the topic of topology as a special course not only for mathematicians, but also for physicists (due to the strong connections of topology with certain problems of physics, such as the question of clarifying the nature of the gravitational field, space, time).

Students and pupils should especially pay attention to the theoretical and applied applications of topological properties, which are manifested and applied in the various above-mentioned sciences. To do this, you need to correctly create presentations, choose tasks and examples of their application.

Conclusions. A system of purposefully designed problems, questions and tasks is an important condition for the development of cognitive motivation in the educational process and an effective means of developing productive heuristic thinking. Solving topological problems, students not only actively master the content of the module, but also acquire the ability to use analogy, generalization, and think independently and creatively. Along with tasks of a reproductive nature, associated with cognitive difficulties, to overcome which new knowledge or intellectual efforts are required. Such tasks form the basis of problem-based learning, the pedagogical conditions for success of which are: creation of cognitive difficulties corresponding to the intellectual abilities of students; provision of a set of knowledge on the subject content of the problem situation; formation of operational skills for solving problematic problems. The development of non-standard thinking will be facilitated by tasks that require creative mastery of educational material.

Diversification of methodical possibilities is provided thanks to the use of multimedia teaching tools, namely, the display of presentations and dynamic drawings. In particular, dynamic topological models provide a higher and clearer level of educational and cognitive activity of students. Drawings of this type serve as a substitute for textbook fragments and are especially useful for self-training. However, this requires the development of appropriate educational and methodological support, which necessitates further research in this direction.

Enrichment of the student’s topological culture takes place in the closest connection with the use of the apparatus of mathematical analysis, provides specific knowledge sufficient for teaching topology and qualified group classes.

Topology is of special importance as a didactic tool for the development of the mathematical worldview of university students of mathematical specialties.

The study of topological properties of figures in the course “Differential Geometry and Topology” provides wide opportunities for their practical application, increases the competence of future teachers of mathematics and physics and stimulates their own search for new topological, mathematical, geometric and physical ideas and theories.

References:

1. Aleksandrov A.D., Netsvetaev N.Yu. Geometry: textbook. allowance M.: Nauka, 1990. 672 p.
2. Atanasyan L.S., Bazylev V.T. Geometry. Part 2. M.: Prosveschenie, 1987. 352 p.
3. Borysenko O.A. Differential geometry and topology. Kh.: Osnova, 1995. 304 p.
4. Werner A.L., Kantor B.E., Frangulov S.A. Geometry: a textbook for students of physics and mathematics faculties of pedagogical institutes. Part 2. / St. Petersburg: “Special literature”, 1997. 320 p.
5. Werner A.L., Kantor B.E. Elements of topology and differential. M.: Prosveschenie, 1985. 112 p.
6. Kovantsov N.I. Differential geometry, topology, tensor analysis: collection of problems / N.Y. Kovantsov [and others]. K.: Higher School, 1989. 398 p.
7. Myshchenko A.S., Fomenko A.T. Course of differential geometry. M.: Factorial Press Publishing House, 2000. 448 p.
8. Myshchenko A.S., Fomenko A.T. A short course in differential geometry and topology. M.: FIZMATLYT, 2004. 304 p.
9. Novikov P.S., Fomenko A.T. Elements of differential geometry and topology. M.: Nauka, 1987. 432 p.
10. Shapovalova N.V., Panchenko L.L. The role of studying topology to increase the competence of future teachers of mathematics and physics // Problems of education: Nauk. coll. / Institute of Innovative Technologies and Educational Content of the Ministry of Education and Culture of Ukraine, K., 2015. Issue No. 80. Part 2. P. 285-290.
11. Shapovalova N.V., Panchenko L.L. A systematic approach to teaching topology to future teachers of mathematics and physics // Scientific developments, advanced technologies, innovations [a collection of scientific works and theses of scientific reports based on the materials of the II International Scientific and Practical Conference]. Budapest-Prague-Kyiv, K.: NDIER. 2015. P. 168-173. URL: http://sried.in.ua/05.2015.pdf