The support of the process of training pre-service mathematics teachers by means of cloud services

The training of pre-service mathematics teachers is a complex process due to the specifics of the field. Informatization of education affects all the areas, and pre-service mathematics teachers can not be left out. The article is devoted to the problem of supporting the process of professional training of pre-service mathematics teachers by means of cloud services. Examples of the use of cloud technologies are given. The analysis of a survey of pre-service mathematics teachers on the use of information and communication technologies in the training process is done.


Introduction
During the constraints caused by the COVID-19 pandemic, education was forced to switch to e-learning as a matter of urgency [1,2,3,4,5].The transition revealed problems not only in the material part of the organization of education (lack of access to digital technologies for a significant number of both students and teachers) in the lack of electronic educational resources needed for educational activities.There were problems with the appropriate training of pre-service teachers.It cannot be said that this issue has not been raised by the educational community and government agencies, however, the results indicate a formality in resolving these issues.
Let's consider what problems teachers have when moving to the online space.Going online and transferring even one course taught in a higher education institution or a general secondary education institution is a huge long-term work of a whole team of professionals.The examples are Cousera or Prometheus resource courses.All videos are recorded by professional cameramen in the studios, with a professional overlay voice of the lecturer of the presentation on the video, with a very careful and competent distribution of information over time.The course consists of 20-30 minute informative videos, for which tests, a large amount of materials for self-study, tasks such as peer to peer review and so on are prepared.
In order to implement such functionality, it is not enough for a teacher to have information and communication technologies, to have the necessary material base for shooting, editing videos, informational material, etc.Such work takes much more time than preparing for lectures or conducting a lesson and creating presentations for them.The results of mastering the material from most subjects, disciplines or courses cannot be assessed by passing tests, there is a need to check written works, listen to auditions and so on.It significantly increases the amount of mechanical work that is added to the responsibilities of a teacher or lecturer.For clarity purposes, you can, for example, open 20 e-mails, download 20 completed works and save them in the appropriate folder.We see that even such primitive mechanical operations are very time consuming.
In addition, there is a big psychological problem if during online classes students are a passive cluster that is not even displayed on the screen.Not every teacher is able to tell 40 minutes on the black screen without any impact on the audience, even if from time to time there are single answers in the form of voice messages.It is also important to hold the attention of the audience during the lesson.
Another unresolved issue is the results of scientific and methodological work.It is required by the job responsibilities of educators.In the situation of excessive creation of electronic educational resources it is extremely difficult to get good results.It is also important that e-learning resources have no less functional teaching compared to planned, verified, proven full-time learning.
Researches and statistics from mass open online courses have found that distance learning is effective for 20-25% of students.This is the number of them who successfully complete the distance course compared to the number of registered ones.There is another side -the specifics of the selection of material for distance learning, which is determined by the professional competence of the content curator.The search for educational material, its systematization, processing in accordance with the level, goals and content of training forces to process a huge amount of existing digital content.
Students are no less stressed.There are several problems in this category of participants in the educational process: • the construction of the procedure for evaluating the completed tasks, especially those of a final nature, is not clear; • learning at home relaxes rather than mobilizes and motivates learning activities, and leaves visual control on the part of the teacher; • lack of own room for working remotely (presence of parents, younger or older siblings) and therefore, the simplest solution that students come to is not to participate in online classes; • cases of lack of access to Internet resources due to lack of devices and lack of communication channels.
General secondary and higher education institutions make great efforts to organize learning with the help of distance platforms.Seminars, webinars, trainings on forms, methods and means of distance learning for teachers are held in an accelerated manner.Teachers communicate with their students by any possible means, from e-mails, instant messaging, available distance learning platforms to telephone communication.It should be noted that due to the different activity of participants in the educational process there is a big discrepancy in the results.Thus, it is extremely interesting for us to explore the issues of professional training of pre-service teachers of mathematics for educational activities in the distance form of organization of the educational process.
Professional training for the widespread use of ICT in the educational activities of preservice mathematics teachers begins long before entering higher education.During their studies in a general secondary education institution, pre-service mathematics teachers can already clearly observe the forms and methods of application of ICT.The study of the results of the use of ICT during training in general secondary education was presented in a study by Vakaliuk et al. [35].Useful for our study is the answer to question #13 of their study: "Did any information and communication technology tools (curricula, multimedia, simulators, games, virtual laboratories, etc.) be used in the school / college by non-CS teachers?".Unfortunately, only 48.5% of respondents answered this question positively.The result suggests that every second pre-service teacher while studying in a general secondary education institution has not received the necessary experience of using ICT in educational activities.However, the variety in answering question #14: "If the answer to the previous question is "Yes", in what lessons did the teachers use such tools?"(see figure 14, [35]) indicates that some subject teachers, including teachers of language and literature, mathematics, physics, history, chemistry, biology and geography have found not only the possibility of using ICT in learning activities, and this application was successful, otherwise it would not be remembered by students.Thus, due to the introduction of ICT in the secondary school education system, pre-service mathematics teachers are occasionally familiar with the use of ICT in educational activities.
Let's consider the approaches used in e-learning based on the results of the study of Proskura and Lytvynova [36] for bachelors of computer science.The conducted semantic analysis of scientific achievements of colleagues and based on their own experience allowed the authors to develop the model of Web-based learning of Computer Science Bachelors (see figure 1 in [36]).Considering such structural components as learning environment, web-oriented environment, control and evaluation unit, levels of student's educational achievements, the authors determine the content component of e-learning.Among the components of e-learning there are such as cloud computing, working together classroom, web-automated knowledge validation systems and others.These components are implemented through network technologies as means of data transportation.Their appearance is caused by the availability of network technologies and they belong to cloud services.Thus, the need to use cloud services to support the training of pre-service mathematics teachers is one of the current educational trends.
A logical continuation of our study is to identify specific cloud tools to support the training of pre-service mathematics teachers.The specifics of professional training is determined by the specifics of the field of science.The specificity of mathematics lies in its abstractness.The study by Vlasenko et al. [37] examines the requirements for modern web-based online training courses for pre-service mathematics teachers.Researchers have analyzed the ways of presenting mathematical text through to the specifics of its formation, and the creation of mathematical content with a focus on network use.
Shyshkina and Marienko [38] determined the content of necessary general skills and specific skills needed for pre-service mathematics teachers (see table 2 in [38]) based on professional functions, typical tasks of mathematics teachers (see table 1 in [38]).Using the example of Web-SCM CoCalc (formerly called SageMathCloud), the authors conducted an experimental study that revealed the benefits of using the CoCalc cloud service in the training of pre-service mathematics teachers.
Fedorenko et al. [39] studied the problem of studying free software using cloud services.The results of this study had a positive result, for the training of pre-service mathematics teachers in particular.Researchers have discovered the didactic capabilities of cloud services that allow you to run a free mathematical software.

Research results
According to the Law of Ukraine "On Higher Education", the educational process at the university is carried out in the following forms: classes; individual work; practical training; control measures [40].The most common types of training are lectures.A lecture is a clear, systematic presentation of a particular scientific problem or topic.Its additional purpose is to help students with mastering the methods of independent work with textbooks, manuals, primary sources.This is one of the most important factors in the organization of educational activities of students and occupies a significant place in the training of pre-service professionals receiving higher education.
The normative-directive documents, which determine the content and organization of the educational process in higher education institutions of Ukraine, substantiate the main requirements for this type of classes.Their implementation allows to fully use the significant educational and upbringing opportunities of this form of education, to increase the impact of each lecture on the consciousness and feelings of students [41].However, the lecture to some extent accustoms the student to the passive appropriation of other people's thoughts, does not stimulate the desire for independent learning, does not provide an individual, differentiated approach to learning.
The leading role of the lecture in the teaching of academic disciplines is related to their content aspect, organizational principles and methodological features.The main content is the central methodological, theoretical and practical problems.Not all issues of the topic are revealed, but the most important, most significant ones that require scientific substantiation.
The pace of development of modern technologies has a significant impact on teaching methods and learning models in general [42].This allows to expand the ways of implementing the paradigm of competence in order to improve the quality of education.The model of blended learning has the greatest potential for optimizing the learning process [43].This model allows you to implement new technologies without abandoning conventional teaching methods.Blended learning is a model of organization of the educational process, as "it allows to increase the motivation of future teachers to learn, makes it transparent, interactive and manageable, provides constant involvement of students in the educational process" [44].
A blended form of learning is the individualization of learning, the creation of a favorable psychological climate, continuous training.The purpose of this form of learning is to combine the benefits of face-to-face learning and electronic educational resources through a combination of distance and traditional communication in integrated learning activities.The integration of traditional and computer-based learning in the educational environment will lead to a purposeful process of acquiring knowledge, skills and abilities in the classroom and extracurricular learning activities of the subjects of the educational process through the use of ICT.The existence of this form of learning is possible due to the effective combination of different ways of presenting educational content, teaching models and styles.It is based on the interaction between all participants of the educational process.
One of the forms of blended learning is realized through flipped learning (flipped classroom) [45,46].There are different ways to implement the model of flipped learning and they are all based on one basic principle: traditional learning is carried out outside the classroom, and practical work and application of knowledge is classroom.In general, the essence of flipped learning is to regroup the key components of the learning process.With the help of this model of learning "the content of new educational material is independently mastered in the electronic environment, and then the acquired knowledge is applied in practical classes or discussions" [47].
Reduction of classroom hours (lectures, practices, seminars, laboratory classes) leads to a violation of the traditional logic of the educational process.This fact leads to a loss of quality of education.One of the ways to "restore the balance of the learning process is to use a mixed learning model with Flipped Classroom technology" [48].At the same time, the key components of the learning process change places: the basic components of the new material are studied independently at home, and in the classroom the studied material is consolidated and together with the mentor more complex issues and practical application of the mastered information are considered.Flipped classroom technology is characterized by the fact that the necessary "theoretical knowledge is obtained outside the classroom, and in the classroom individual tasks are performed or a group project is developed" [49].
Flipped classroom technology was used by us in the study of mathematical disciplines "Mathematical Logic and Theory of Algorithms" and "Elementary Mathematics" in the 3rd year of the Faculty of Physics and Mathematics of Donbas State Pedagogical University.Taking into account the fact that students majoring in "Secondary Education (Mathematics)" specializing in "Computer Science" in the 3rd year have significant learning experience, and the level of self-awareness is sufficient to use elements of Flipped classroom technology in not only logical but also appropriate way.In addition, this model of study does not contradict the work program, in which the main number of hours is set aside for independent work of students.
Based on the practical experience of use, the following structure of the approach to each topic was formed: 1. Formulation of the theme and purpose.For each of the proposed topics, the place in the work program and sources of information are determined, the content of tasks for control and its types are selected.
For example, to study the topic "Complete systems of Boolean functions" students receive the following information in table 1.
The main reasons for the introduction of a blended learning model with flipped classroom technology in the educational process of the university are active cooperation between students and teachers and, as a consequence, increasing students' success and motivation.A feature of this model is the possibility of using group classes where students can discuss key aspects of lecture materials, test their knowledge and interact with each other.The task of the teacher is to explain the problematic issues, to comment on the work of students.
Another example of supporting the training of pre-service math teachers is the use of computer math systems in the form of cloud services.One such free service is the Math Partner computer math system, which is available at http://mathpar.com.This service allows you to create your own cloud mathematical "Notebook", in which the user performs the necessary mathematical calculations.To ensure quality and comfortable work, this service provides access to a large amount of reference material with examples.The language of this service is the Mathpar language, which is based on the widely used language of mathematicians and physicists T E X, which is usually used for typing mathematical texts.It is possible to save both the problem statement and its solution.You can save both text and images.Types of control test for mastering the material (conducted remotely outside the classroom), an individual task to determine the completeness of the system Date of control conducting a test for self-study of the material In particular, Math Partner allows you to effectively solve such common problems as finding the smallest distances between all vertices of the graph and finding the shortest path between the vertices.For the first task the command ∖ℎ() is used, and as a result the matrix of the shortest distances between vertices will be received.For the second task, use the ∖  ℎℎ ℎ(, , ) command, resulting in the shortest path between the vertices  and .It should be noted that this service is convenient to use to check your own solutions, because it provides the answer itself, without access to intermediate results of calculations.
Another relatively new service is Graph Online, which is available at http://graphonline.ru.This is a free cloud service designed to visualize the graph and find the shortest path on the graph, finding the Euler cycle.Creating a graph is performed on the adjacency matrix or incidence matrix.In addition to finding the shortest path, you can search for connectivity components.The service supports work with oriented (digraphs) and non-oriented graphs.The result of the work, the constructed graph can be saved thus you can continue working with it later.
In addition, the cloud service Graph Online provides the user with many auxiliary functions to facilitate work.Available to save and load a graph with support for saving the visual representation, fast conversion between all supported types, determining the appearance of vertices, arcs, background, constructor mode, etc.
Let's consider the example of the use of the presented services in solving the problem of finding the shortest distances between the vertices of the graph, which are often found in practice.It is clear that in order to be able to find the shortest distances, there must be at least one path from vertex 0 to every other vertex, i.e. the graph must be connected.For this problem, the most well-known solution algorithm is the Dijkstra algorithm.The idea of this algorithm is that first for each vertex other than vertex 0, we set the distance equal to +∞, and then step by step we reduce these distances until we find the minimum distance () and the shortest path () for each vertex .
Condition: Suppose that in a weighted graph  = (, ), the set of vertices  = {0, 1, 2, 3, 4, 5}, and the set of edges  is given by a matrix of weights: Using the Dijkstra algorithm, construct a skeleton tree of the shortest paths from vertex 0 to all other vertices of the graph G and find the shortest distances.
Using the Math Partner service we have the opportunity to check the correctness of the results in figure 1.But graph visualization is better performed by Graph Online (figure 2).The following results were obtained: • 100% positive result on the first question.This result suggests that pre-service mathematics teachers understand the concept of "electronic educational resources" despite their diversity.• 82% of respondents answered positively to the second question.The result of the answer to the second question shows that not all respondents identify the concept of cloud service.We came to this conclusion when discussing the results of the survey with the respondents.After all, during the practical classes that we described in the study, all students were involved in the use of cloud services.• 46% of respondents answered positively to our third question.This percentage is a good result of self-educational activities.
The answer to the fourth question is shown in figure 3. It is important to note the significant advancement of cloud communication services due to quarantine measures caused by the COVID-19 pandemic and cloud file storage services.A positive result of the survey is the availability of cloud services for mathematics.
The answers to our question 5 are presented in figure 4. At the first glance, pre-service math teachers use a regular calculator and calculator on mobile devices and make presentations and reports in a word processor.Note that the presented list includes purely mathematical programs, which are extremely necessary in the training of pre-service teachers of mathematics.

Conclusions
The learning process in higher education is implemented within a diverse integral system of organizational forms and methods of teaching.Each form solves its specific task, but the set of forms and methods of teaching creates a single didactic complex, the functioning of which is subject to the objective psychological and pedagogical laws of the educational process.
The results of the study provide an opportunity to claim that cloud technologies are used in the teaching of mathematics in high school.Pre-service mathematics teachers have the opportunity to use cloud technology during their training.Available cloud technologies have a wide range of applications in mathematics education.

2 .
Determining the place of this topic in the work program of the discipline.3. Offering sources of information.4. Definition of types and content of control. 5. Monitoring and evaluation.While introducing the flipped classroom technology during teaching the course of "Mathematical Logic and Theory of Algorithms" it is advisable to consider the following topics: 1. Boolean n-ary functions; 2. Zhegalkin polynom; 3. Complete systems of Boolean functions; 4. Mathematical theories of the first order.

Figure 1 :
Figure 1: The shortest distances from vertex 0 to all other vertices of graph  in the Math Partner service.

Figure 2 :
Figure 2: Graph  visualization and the shortest distance from vertex 0 to vertex 5 of graph  in the Graph Online service.

Figure 3 :
Figure 3: Cloud services used by pre-service math teachers.

Figure 4 :
Figure 4: Software used by pre-service math teachers.

Table 1
Tasks for studying the topic "Complete systems of Boolean functions"ThemeComplete systems of Boolean functions Meta Learn the concepts of classes of functions that store zero, store unity, self-dual functions, monotonic and linear.Master the criterion of completeness.

Table 2
The progress of the Dijkstra algorithm