DOI: https://doi.org/10.62204/2336-498X-2023-2-16

ORGANIZATION OF DISTANCE LEARNING OF

MATHEMATICS IN INSTITUTIONS OF SECONDARY AND HIGHER EDUCATION

Natalia Shapovalova,

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

Mykhailo Dragomanov State University of Ukraine,

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

Larisa Panchenko,

Ph.D. in Pedagogy, Associate Professor,

Mykhailo Dragomanov State University of Ukraine,

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

Elena Volyanska,

Ph.D. in Pedagogy, Associate Professor,

Mykhailo Dragomanov State University of Ukraine,

elenavolyanska@ukr.net;  ORCID: 0009-0005-3136-5082

Annotation. The article discusses two ways of organizing distance learning – synchronous and asynchronous modes. The differences of the model curricula in the school education system “New Ukrainian School” (NUS) from the existing curricula are described. Some distance learning technologies are described, namely: cloud technologies, STEAM education, Case method, Crossense, Game technologies, or gamification, Health-saving technologies, Project method, storytelling and tools used by teachers during distance learning. The following services are provided for the organization of online learning: Gios, My class, EDERA, “Mathematics is simple” course. Changes in the organization of the process of teaching mathematics are described, which take place in accordance with the concept of the “New Ukrainian School” (NUS).

The methods, organizational forms and teaching tools used in higher education institutions when combining distance learning with face-to-face learning are described. The specific features of the educational process during distance learning in institutions of higher education are described. Various forms and measures are proposed for the organization of the educational process in order to maintain the appropriate level of knowledge acquisition and the formation of relevant skills in geometry during distance learning of future mathematics teachers.

Keywords: distance learning, methods of distance learning organization, distance learning technologies, learning methods, organizational forms of learning, learning tools, education, mathematics, geometry.

Problem statement. Laws of Ukraine «On education» and «On full general secondary education» [5] provide the opportunity for schoolchildren and students of higher educational institutions to receive educational services in the distance form of education. This follows from the right to education provided for by Art. 53 of the Constitution of Ukraine.

Depending on the conditions that have developed in Ukraine since 2020, education

in schools and universities will be conducted either in a mixed form of education or completely remotely. First, the country was forced to introduce a quarantine, and then from February 2022, martial law was introduced in the country.

Analysis of recent research and publications. The concept of distance learning in scientific and methodical literature is interpreted in different ways.

1. Bykov means distance learning as «a form, a system, according to which the interaction between the teacher and the student, students is carried out at a distance, and reflects all the components inherent in the educational process (goals, content, methods, organizational forms, means of learning), implemented by specific means of Internet technologies, or other means that involve interactivity» [2, p. 14].

Another scientist describes distance learning as «a new level of correspondence education, which provides the use of information technologies based on the use of personal computers, video and audio technology, satellite and optical fiber technology» [3].

Presenting main material. In general, if we analyze a large number of definitions of the concept of distance learning, we can conclude that it really differs from traditional learning, to which teachers and students are accustomed.

There are two ways of organizing distance learning – synchronous and asynchronous modes.

Synchronous mode is such an interaction between a teacher (teacher) and a student (student), when all of them are simultaneously in the web environment of distance learning. It can be a video conference, audio conference, chat, social networks, etc. i.e., classes are purely online.

Asynchronous mode involves participation in communication between students and teachers with a certain delay in time (e-mail, interactive educational platforms, social networks, forum). But distance learning involves both online learning and offline learning, that is, to be mixed.

From the 2021-2022 academic year, the reform of the school education system «New Ukrainian School» began to be implemented in secondary education (gymnasium).

In Ukraine, since 2021, various author teams have been working on the development of new model curricula specifically for basic secondary schools.

Model educational programs of the New Ukrainian School (NUS) differ from existing educational programs in that they do not determine the number of hours and the order in which the sequence of studying the topic occurs, and do not determine the toolkit for the teacher that he needs. The model program needs to be approved by the state, but it must determine the content of the subject and various types of student activities.

On the basis of the Model Educational Program, seven model programs for 5-6 grades of NUS were created and approved by Order of the Ministry of Education, Culture, Sports, Science and Technology of February 19, 2021 No. 235 «On Approval of the Model Educational Program for Grades 5-9 of General Secondary Education Institutions». From 2021, the 5th and 6th grades of Ukrainian schools began to work according to the new model programs of the New Ukrainian school (NUS). In accordance with the current model curricula, new textbooks in mathematics for grades 5-6 of different authors’ collectives were created, first in electronic form, and then paper samples of the textbooks were printed. Textbooks have been supplemented with additional electronic resources, such as workbooks, online materials and video lessons, which make the learning process more interesting and productive.

Mathematics textbooks by different authors for grades 5-6 differ from each other in terms of content, there are more problems with modern content, problems about Ukraine and the world, problems for financial calculations, about health preservation, environmental problems for traffic and its safety, applied and practical problems. Modern textbooks of NUS in mathematics, contains QR codes, digital tools for some topics can be downloaded from the link. Most textbooks are very convenient for students to use during distance learning.

The new Ukrainian school involves the use of new learning technologies, i.e.

organized learning methods and processes that ensure quality learning results. Distance learning technologies are one of these technologies:

1. Cloud technologies are the possibility of using computing resources and memory of remote servers (Gmail, Meta). The first to use the term «cloud computing» in the science of commercial products was the Amazon company, so in 2006 the Elastic Compute Cloud service was launched. At school, these are online services for educational communication, a library, file storage, electronic journals, diaries, creating presentations, tables, drawings, etc. In mathematics lessons, it is successfully used during the so-called «inverted learning». For example, the study of the topic «Pyramid» in grade 11 takes place online, and solving problems and checking homework is done in a real classroom. It can be video conferences, preparation for the thematic control work.
2. STEAM education includes science, technology, engineering, art and mathematics. During STEAM lessons, the focus is not on the teacher, but on the practical task to be solved. For example, one of the interesting tasks is the students’ construction of an automated greenhouse. Various disciplines are involved: mathematics, physics, computer science, biology, labor training.
3. The case method is a learning technique that uses a description of real economic, social, and business situations. This method has been known since ancient times. The analysis of «cases» was one of the favorite methods of Socrates’ time. Even in ancient Sparta, the discussion of situations that arose on the battlefield became widespread. As a method, it took shape 100 years ago at Harvard University. The concept of the case method is now used in today’s leading schools and universities. In class, students not only take part in the learning process, but also see the practical application of the learned new material.

Formulas of simple and complex percentages are studied by students in the algebra course.

In fiction, there are stories related to the calculation of simple and compound interest. In O. Balzac’s short story «Hobsec», one of the heroes, Mr. Derville, borrowed 150,000 francs from Hobsec for 10 years at 15% per annum. Pupils are given the task of calculating the amount that Dervil returned to Hobsek after 10 years.

Joint teamwork emotionally improves the level of knowledge and increases motivation. Students learn to apply mathematics to the realities of the world around them.

The task can be divided into two stages:

The first stage. Consider simple and compound interest formulas at home.

The second stage. Class work using the compound interest formula.

4. Crossense (from the English word «cross») is a visual associative chain consisting of 9 images, each of which is connected to the previous and next image (analogous to crosswords). For example, the «On the Lesson» and «Vseosvit» platforms contain various examples of cross-senses for the 5th grade on various topics (angles, percentages), which are interestingly used during distance learning at school.
5. Game technologies, or gamification. The Classtime platform contains a large number of team games that can be used both in class and given as homework. For example, in the 6th grade, students study the topic «Coordinate plane». At the stage of consolidating new material, it will be appropriate to offer the game «Sea Battle» to students.
6. Health-saving technologies – tasks for a healthy lifestyle, which are aimed at strengthening the physical, mental, emotional, moral and social health of students. Example:

Problem 1. The norm of salt consumption for a person is 6 grams per day. The salt content in boiled sausage is 2-4%. How many grams of cooked sausage should be eaten in order not to exceed the existing salt norm?

Problem 2. How many hours does a person shorten his life if he smokes 20 cigarettes every day? (One cigarette shortens a person’s life by 14 minutes).

The conditions of the tasks are accompanied by interesting presentations that help to better visually understand the conditions of the tasks.

7. The project method is an educational technology aimed at the acquisition of knowledge by students in close connection with real life practice, the formation of specific abilities and skills in them thanks to the systematic organization of problemoriented educational research.

Students in groups or individually receive a task – create a new product, solve a problem. In the implementation of the project, students are helped by existing pedagogical software tools and the Internet.

When studying the topic «Vectors and coordinates on the plane», prepare a presentation using the PS PowerPoint program on the topic «Vectors in science and technology», as well as «Rene Descartes». Pupils consider the use of vectors in physics, technology, illustrate interesting historical material about the founder of the Cartesian coordinate system, the scientist René Descartes.

Each slide of the presentation contains video information, graphic information, and animation that visually illustrate the research project. The demonstration method may contain a linear or branched structure.

The linear structure contains only message theses. It is, as a rule, used in sequential teaching of new material. In years of generalization and systematization of knowledge, a branched structure is used.

The most popular tools used by teachers during distance learning are Google Classroom – for the work of the teacher and the class, Google Sheets – tables for journaling, Google Forms – for communication with students, Gmail – for communication with students, electronic educational resources and services: Zoom – for conducting online classes, creating polls on the Vseosvita and Na urok platforms.

When conducting lessons, software tools should be used. For example, in geometry lessons, such programs help to build geometric models of objects and change their parameters. These are the so-called dynamic geometry packages. Some of these programs are GRAN 1, GRAN 2D, GRAN 3D, DG. They enable students to create images of planimetric or stereometric figures, or combinations of these figures, to transform them on the coordinate plane or in space.

The GRAN software tool was created by the Mykhailo Dragomanov State University of Ukraine team under the leadership of Professor M. I. Zholdak. GRAN 1 enables students to construct graphs, graphically solve equations, inequalities and systems, find definite integrals, surface areas and volumes of bodies of revolution.

8. Using the GRAN 2D program, students can solve construction problems on a plane. GRAN 3D is used in space to construct sections of polyhedra and bodies of revolution, helps to build combinations of polyhedra with polyhedra, or with bodies of revolution. It is convenient that the computer helps to perform cumbersome calculations that take a lot of time in the lessons.
9. Storytelling is one of the technologies, or the art of interestingly telling stories.

In mathematics lessons, the teacher can activate the educational and cognitive activities of students by telling a story about the importance of mathematics in people’s lives. For example, after studying the signs of equality of triangles in geometry lessons in the 7th grade, you can interest students in the question «How to determine the distance to an inaccessible point?», for example, a house located on the other side of the river.

The following services can be used to organize online training:

1. The platform has online mathematics courses for grades 5-9. The Gios service has the seal of the Ministry of Education and Science of Ukraine and fully corresponds to the mathematics program.

This platform contains video lessons on theory, examples of solving problems and a practical unit (tasks in the form of a test).

2. My class. The service contains online courses from the 1st to the 11th grade.
3. EDERA, Course «Mathematics is simple».

This course helps to systematize knowledge of mathematics. It is especially useful for high school students who have to take the NMT (national multi-subject test).

During distance education, the teacher must adapt the calendar and thematic plans, namely:

• highlight topics that can be taken out for independent study;
• reduce topics that are easier for students to learn and thereby save time for topics that require detailed study;
• topics that are planned to be studied independently should be divided into logical thematic blocks. The teacher should provide students with instructions and advice for mastering the material and criteria for evaluating the learned material.

According to the «New Ukrainian School» concept, there is a change in the organization of the process of teaching mathematics.

Under modern conditions, distance learning is a requirement of the time, therefore teachers need to develop their range of competences, create interesting lessons for students, be fluent in modern distance learning technologies, use software pedagogical tools and existing services that activate the educational and cognitive activities of students of different ages.

If we talk about studying in institutions of higher education, distance learning is mostly combined with face-to-face. This happens sequentially (they study sometimes in classrooms, sometimes remotely) or in parallel (some students study in classrooms, and some remotely).

At the same time, the purpose, goals and tasks of each academic discipline do not change. And the methods, organizational forms and teaching tools are improved and adapted to new requirements and conditions. The main means of distance learning are:

• modern pedagogical technologies;
• available technical capabilities; – capabilities of the subject of study.

Among the technical capabilities, the technical parameters of the computer used by the subject of study and the ability to connect to the worldwide Internet are especially highlighted. Distance learning systems use special communication technologies (electronic and non-electronic) to support the interaction of the subjects of the distance learning process, namely directly those who teach and those who study, as well as those who organize the technical process. Information and communication technologies (ICT) provide communication between all participants of distance learning. Among them, the following are distinguished:

1. symbolic or text educational information objects. These include an electronic whiteboard, a multimedia whiteboard, newsletters, teleconferences, etc.;
2. symbolic and graphic statistical and dynamic educational information objects: audio and video communications in real time, interactive audio and video.

Since we are talking about the distance form of student education, which is combined with face-to-face, the part of the course offered for consideration remotely can be:

• automated (training takes place without a person accompanying the training);
• under control. In this case, there is a person who monitors the learning process and provides consultations;
• It assumes the existence of a person who supports the course of training, not only controlling it, but also participating in discussions, group discussions, providing instant advice on complications;
• In this case, a personal learning environment is created, which requires

the additional use of technologies, for example, web cameras, Viber applications, YouTube, etc. [O. Rudenko. «Mathematics remotely»].

The use of information and communication technologies is a new level of mediating mental, creative, communicative, and executive activities and leads to a radical restructuring of various aspects of activity. The effectiveness of pedagogical action during the application of distance education using computer telecommunication networks is possible provided that the peculiarities of communication between the teacher and the student are taken into account.

t should be taken into account that:

• information in the process of communication is not only transmitted, but also assimilated, understood and realized;
• verbal communication is implemented with the help of actual, debatable types of dialogues;
• the interactive side of communication is manifested in joint activities;
• during communication, there should be mutual understanding between its participants.

Communication during distance learning has a certain specificity.

1. The teacher is a group. The main purpose of this communication is to set goals and analyze the results of students’ activities.
2. The teacher is a student. The coordinator manages the actions of an individual student, gives advice, recommendations and analyzes the results of activities.
3. A student is a teacher. The main form of such communication is a request from the listener, who tells the teacher what information he needs. In this case, the student becomes aware of his ignorance, the ability to distinguish it as a separate category. By formulating his difficulties, the student learns to anticipate possible complications in the work of others during the study of this material, independently pose a problem, and present the results of his work.
4. Student 1 – student 2. Similar interactions occur spontaneously at first. As a rule, the most active participants of the distance course are the first to come into contact.
5. A group is a student, a student is a group. In educational activities, cooperation with classmates is an essential moment of knowledge acquisition, because the polarization and coordination of points of view in the group is a way of recognizing the limits of each student’s knowledge and ignorance.

All of the above should be taken into account when transitioning to distance education in the process of studying all academic disciplines by students.

Let’s consider distance learning of the discipline «Projective geometry and methods of images». The final module of this discipline is module 3 «Projective geometry», which is studied remotely. An important place among the topics of this module is occupied by topics related to configuration theorems: «Theorem of Desargues. Principles of duality», «Complete tetrahedron and its harmonic properties», «Theorems of Pascal and Brianchon».

It is during the study of these topics that the acquired skills to solve construction problems are improved, and new skills are formed to solve construction problems with one ruler based on configuration theorems.

Distance learning is carried out with the help of teleconferences. Teleconferences

are a type of event in which group communication is carried out between geographically distributed participants using technical means. Computer conferences can be held using various forms of organization, the best online resources are Zoom and Meet. We use them when teaching projective geometry.

Let’s consider the forms of training classes. The lecture is one of the most important forms and forms the basis of the theoretical training of the students. Their purpose is to provide a systematized basis of scientific knowledge of projective geometry, to reveal the problems, to focus attention on more complex and knotty issues. A study of approaches to geometry lectures revealed the expediency of using text lectures («electronic lectures»). «Electronic lecture» is a set of educational materials in electronic form, the content of which includes, in addition to the text, additional materials (excerpts from scientific articles, historical references with portraits of outstanding mathematicians, whose names bear the theorems under consideration).

At the lectures, students learn and realize the knowledge that underlies the ability to perform specific constructions. In particular, while studying configuration theorems, students build Desargues and Pascal configurations according to the teachers step by step, but before that, drawings with a ready configuration are offered to their attention. The teacher first shows the picture on the computer monitor, and then uses the «Electronic board» resource to build the image on the computer screen, students build after the teacher in a notebook (ordinary) with a pencil and a ruler.

The ability to perform geometric constructions and solve construction problems is formed in practical classes. Practical classes are an active form of educational classes and are widely used during the teaching of geometry. They are conducted using computer video conferences. Students solve problems using the frontal form of work, or individually, and not only for construction.

When solving a construction problem, a drawing of the desired shape is shown, the condition is shown, and a solution plan is drawn up. The construction on the electronic board is performed step by step by a student (appointed by the teacher or who has shown activity himself), all this happens in the mode of a video presentation, other students build with ordinary means (a ruler and a pencil) in an ordinary notebook, as an option, the construction can also be carried out with the help of a computer on the screen of one’s own computer, but this (very often) delays the solution of the problem.

With such an organization of training, the teacher has difficulties with the organization of individual approaches to students. If the student does not succeed in something, then you can contact the teacher for consultation through a video call in the Viber application, show the constructed image and get a hint on how to complete it. For example, such situations arise when the lines of the specified configurations intersect at non-own or inaccessible points.

Very often, constructions are suggested to be performed according to a guideline rule, which is displayed as a sequence of steps on the screen. This is very convenient, because you can return to it every time you solve the corresponding basic task.

The degree of complexity of educational activities (assimilation of knowledge and formation of skills) during solving educational tasks in practical classes on projective geometry and image methods determines:

• activity according to the model in different situations and conditions, practice of operations;
• reproduction of actions in different conditions and situations;
• reproduction of actions with a separate addition of necessary conditions, modeling of activity;
• partially search activity, examination and resolution of problem situations.

One of the forms of guiding students’ work and providing them with assistance during independent study of educational material is consultation. Consultations can be individual or group. Consultations with the use of information and telecommunication technologies are carried out by telephone, e-mail, video and teleconferences. With the help of consultations, a person-oriented approach to student education is implemented.

Conclusions. Therefore, distance education at the current stage of the development of Ukrainian education is important, and in modern social conditions, it is the main form of education at various levels of its acquisition. Thanks to the successful organization, the appropriate level of knowledge acquisition and the formation of relevant skills in geometry is maintained during distance learning of future mathematics teachers at the pedagogical university.

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