The Use of Cloud Technologies in the Process of Professional Training of

Future Mathematics Teachers

Vladyslav Ye. Velychko

1 a

, Elena G. Fedorenko

1 b

, Nataliia V. Kaidan

1,2 c

and Vadym P. Kaidan

3 d

Donbas State Pedagogical University, 19 Batiuk Str., Sloviansk, 64122, Ukraine

Limited Liability Company Technical University “Metinvest Polytechnic”,

80 Pivdenne Hwy, Zaporizhzhia, 69008, Ukraine

University of Economics and Entrepreneurship, 13 Heroiv Maidanu Str., Khmelnytskyi, 29013, Ukraine

Keywords:

Cloud Technologies, Training of Mathematics Teachers, e-Learning,e-Learning Resources.

Abstract:

During the restrictions caused by the COVID-19 pandemic, the education system was forced to make an urgent

transition to e-learning. The transition revealed problems not only in the material part of the organization of

education (lack of access to digital technology in a signiﬁcant number of both students and teachers) but also in

the lack of accessible electronic educational resources needed for educational activities. There were problems

with the appropriate training of practical teachers. It cannot be said that this issue was not addressed by the

educational community and state authorities, however, the results speak of the formality of the solution of these

issues. The general level of teachers’ mastery of information and communication technologies remains the

main problem, and mathematics teachers are not an exception to this problem, the readiness to use electronic

educational resources is at a low level due to the speciﬁcity of the methodological design of the educational

process. The problem of joint activities of students and teachers during the synchronous and asynchronous

interaction remains unsolved. The solution of these problems lies not only in the widespread use of information

and communication technologies during the practical training of future mathematics teachers and during the

practical re-training of practicing teachers, but also in the construction of an open educational environment

on the basis of cloud technology, which solves the problem of interaction between the participants in the

educational process.

1 INTRODUCTION

Due to the introduction of quarantine measures as a

result of the spread of COVID-19 in 2020-2021 and

the military aggression of Russia in 2022, distance

learning has been introduced in Ukrainian schools

(Hamaniuk et al., 2022). The organization of distance

learning is possible through the use of distance learn-

ing systems. Despite the active use of distance learn-

ing systems by educators, there are some problems

in organizing teaching and improving teachers’ qual-

iﬁcations. One of the main problems is the lack of

methods of using copyright services, which are not lo-

calized, but free for use in educational activities. We

https://orcid.org/0000-0001-9752-0907

https://orcid.org/0000-0002-1897-874X

https://orcid.org/0000-0002-4184-8230

https://orcid.org/0000-0003-2008-3539

assume that the use of cloud systems will make the ed-

ucational process not only possible, but will also gain

the traditional form of education, as much as possible.

News reports, social media, reports on webinars

of educators and teachers speak not only about the

possibility of achieving the goal of learning through

e-learning, but also about the problems that arise in

this process. These problems affect each of the sub-

jects of educational activity: students, teachers, lec-

turers, parents, and the administrations of educational

institutions. General problems are related to the low

level of knowledge of information and communica-

tion technologies among teachers, especially older

teachers, lack of understanding of the speciﬁcs of

the educational process when using electronic edu-

cational resources, the speciﬁcs of the ongoing and

post-summary control through the use of information

and communication technologies, etc.

Let’s look at the problems that arise for teach-

378

Velychko, V., Fedorenko, E., Kaidan, N. and Kaidan, V.

The Use of Cloud Technologies in the Process of Professional Training of Future Mathematics Teachers.

DOI: 10.5220/0012064700003431

In Proceedings of the 2nd Myroslav I. Zhaldak Symposium on Advances in Educational Technology (AET 2021), pages 378-389

ISBN: 978-989-758-662-0

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

ers and educators during the transition to e-learning.

The transition to the online space and transfer of even

one course read at a higher education institution or a

secondary educational institution is a long and com-

plex work of a whole team of professors. Some ex-

amples are the Coursera or Prometheus courses. All

video materials are recorded by professional opera-

tors in the studio, with the lecturer’s voice overlaying

on the video presentation, with a very accurate and

competent distribution of information on the hour in-

tervals. The course consists of 10-15 minutes infor-

mative videos with tests, a large amount of materials

for self-study, peer to peer review assignments, and so

on (Velychko and Fedorenko, 2020).

To implement such a feature, the teacher has to be

competent in information and communication tech-

nology, have the necessary material base for shooting,

editing, data processing, etc. Such work takes much

more time than preparing for lectures or lessons and

making presentations to them. The results of mas-

tering the material in most subjects, disciplines or

courses cannot be assessed by taking tests, there is a

need to check the written work, listening to auditions,

etc. Thus, the amount of mechanical work added to

the duties of a teacher or educator is signiﬁcantly in-

creased. To practice this you can, for example, open

20 e-mail letters, download 20 completed tasks and

store them in the appropriate folder. We can see that

even such primitive mechanical operations are very

costly in terms of working hours.

In addition, there is a great psychological problem

if during the lessons online listener is just a passive

observer, who does not even appear on the screen.

Not every teacher or trainer can talk for 40 minutes

to a black screen without any feedback from the audi-

ence, even if from time they can hear the responses in

the form of voice messages from time to time. Also

no less important is the problem of keeping the audi-

ence’s attention during the lesson.

The results of scientiﬁc and methodological work

are still one of the unresolved issues. This is required

by the duties of educators. It is very important to get

good results in the conditions of urgent creation of

electronic educational resources. It is also important

that the electronic educational resources should have

no less functionality of teaching in comparison with

the planned, veriﬁed face-to-face education.

Research conducted by the scientists and statisti-

cal data of mass open online courses have shown that

distance learning is effective for 20-25% of listeners.

This is the exact number of those who successfully

complete the distance learning course as compared

to the number of registrants. There is also another

side which is the speciﬁcity of the selection of mate-

rial for distance learning, which is determined by the

professional competence of the curator of the content.

The search for educational material, its systematiza-

tion, processing in accordance with the level, goals

and content of the training makes it necessary to pro-

cess a large amount of existing digital content.

Students and learners are also exposed to a lot of

stress. This category of participants in the educational

process has several problems:

• the procedure for evaluating the submitted assign-

ments is incomprehensible, especially those of a

summary nature;

• learning at home is more likely to be relaxing than

mobilizing and motivating for learning activities,

and it deprives the teacher of visual supervision;

• lack of own room to work remotely (presence of

parents, younger or older siblings) and, therefore,

the simplest decision that students make is not to

participate in online classes;

• cases of lack of access to Internet resources both

due to the lack of devices and communication

channels.

Secondary and higher education institutions are

making great efforts to organize distance learning

by means of distance platforms. Distance learning

seminars, webinars, trainings on forms, methods and

tools of distance learning for teachers and educators

are held on an urgent basis. Teachers and educators

communicate with their students by any means possi-

ble, from e-mails, messengers, and available distance

learning platforms to phone calls. It should be noted

that due to the different activity of participants in the

educational process there is a great difference in the

results. Thus, the need to investigate the professional

preparation of future mathematics teachers for edu-

cational activities in a distance form of organization

of the educational process is extremely interesting for

us.

2 RESEARCH METHODS

The research work is carried out on the basis of the

systematic approach as a methodological basis for

studying pedagogical and social facts, phenomena,

processes; the position of psychological and pedagog-

ical science in the use of information and communi-

cation technologies in the educational process of edu-

cational institutions.

To solve the tasks set in the work such theoretical

and general scientiﬁc methods were applied: the anal-

ysis of psychological and pedagogical, philosophical

The Use of Cloud Technologies in the Process of Professional Training of Future Mathematics Teachers

379

databases on the problem of research to understand

the state of development of the formation and devel-

opment of the cloud-oriented systems of learning sup-

port, the identiﬁcation of the research directions, the

principles and approaches to the cloud-oriented sys-

tems of learning support; the analysis of existing stan-

dards and regulations on the use of digital services

in the learning process and informatization of educa-

tional institutions; the generalization of the national

and foreign experience in the use of digital services

and technologies in the institutions of higher educa-

tion to identify development trends, the clariﬁcation

of the basic conceptual and terminological apparatus,

establishing the conceptual foundations of the study;

theoretical analysis, system analysis.

The empirical research methods were used to

solve the set tasks: the experimental study of the use

of cloud services in the institutions of higher educa-

tion in Ukraine, the expert assessment of the results of

the study, the monitoring of the initial activities with

the use of cloud technologies in the educational activ-

ities.

3 LITERATURE REVIEW

The issue of using information and communication

technologies, and cloud technologies in particular, in

the process of professional training of future teachers

of mathematics is the subject of active research of the

scientists. We will consider in more detail some of

those that are relevant to our research.

The professional preparation for the wide use of

information and communication technologies in the

educational activities of future mathematics teach-

ers begins long before the entering the institution of

higher education. Even during the study at the sec-

ondary general education institution, future mathe-

matics teachers can extensively observe the forms

and methods of using information and communica-

tion technologies. The research of the results of us-

ing information and communication technologies dur-

ing the teaching in general secondary education insti-

tutions was presented by Vakaliuk et al. (Vakaliuk

et al., 2020). The answer to question 13 of their sur-

vey is critical for our study: “Did any information

and communication technology tools (curricula, mul-

timedia, simulators, games, virtual laboratories, etc.)

get used in the school/college by non-CS teachers?”.

Unfortunately, only 48.5% of respondents answered

this question positively. The result suggests that ev-

ery other future teacher has not acquired the neces-

sary experience of using information and communi-

cation technologies in educational activities during

his/her studies at a secondary general education in-

stitution. However, the variability at the time of an-

swering question 14: “If the answer to the previous

question is “Yes”, in what lessons did the teachers

use such tools?” (Vakaliuk et al., 2020)) indicate that

some subject teachers, including teachers of language

and literature, mathematics, physics, history, chem-

istry, biology and geography not only found the pos-

sibility of using information and communication tech-

nologies in teaching, but also that the use of these

technologies was successful, otherwise they would

not have been understood by the students. So, thanks

to the introduction of information and communica-

tion technologies into the secondary school education

system, future mathematics teachers are already occa-

sionally familiar with the use of information and com-

munication technologies in educational activities.

Let’s review the approaches used in electronic

education based on the results of the research by

Proskura and Lytvynova (Proskura and Lytvynova,

2020) for Bachelors in Computer Science. The con-

tent analysis of the scientiﬁc paths of their colleagues

and their own experience allowed the authors to de-

velop “The Model of Web-based learning of Com-

puter Science Bachelors” (see ﬁgure 1 in (Proskura

and Lytvynova, 2020)). Considering such structural

components as learning environment, web-oriented

environment, control and evaluation unit, levels of

students’ educational achievements, the authors de-

ﬁne the content of e-learning. E-learning components

include cloud computing, working together class-

room, web-automated knowledge validation systems

and others. These components are realized through

web technologies as a mean of data transport. Their

appearance is conditioned by the presence of network

technologies and they belong to the high quality ser-

vices. Therefore, the need for the use of cloud ser-

vices to support the training of future mathematics

teachers is one of the current educational trends.

No less important for our research is the prac-

tice of online training of master’s degree students in

“Mathematics” for the practical training in universi-

ties. The research by Vlasenko et al. (Vlasenko et al.,

2021) was focused on the organization of students’

educational activities using the online course “Meth-

ods of Teaching Mathematics to Students of Techni-

cal Universities”. The study examines the problems

faced by the students during the internship in techni-

cal universities, and discusses the possibility of taking

into account students’ needs in developing the content

of the course. This study provides a description of the

activities of students while working on the course ma-

terials, requirements and recommendations to facili-

tate the learning process with the help of this course.

AET 2021 - Myroslav I. Zhaldak Symposium on Advances in Educational Technology

380

The active participation of students and their contri-

bution to the discussion of the course, its content, and

their support of the learning process implies that the

implementation of the online course and its integra-

tion into the training program for master’s students

will increase their readiness for the internship.

Markova et al. (Markova et al., 2015) conducted

a general review of the use of cloud computing in

educational activities. The authors made a histor-

ical analysis of the stages of using computer tech-

nology in education, starting with the study of the

problems of providing computer services conducted

by Alan O. Mann and to the prospects for the use of

cloud technologies in education in accordance with

the forecast of their development. Researchers came

to the conclusion that the consolidation of established

and promising cloud technologies in education re-

quires scientiﬁc forecasting of the development of

cloud technologies in education based on taking into

account historical trends in the development of ICT.

The logical continuation of our study is to iden-

tify speciﬁc tools to support the professional training

of future teachers of mathematics. The speciﬁcity of

professional training is determined by the speciﬁcity

of the ﬁeld of science. The speciﬁcity of mathemat-

ics is in its abstractness. The research of Vlasenko

et al. (Vlasenko et al., 2020) discuss the requirements

to modern web-based online courses for training of

future mathematics teachers. The authors analyzed

the ways of mathematical text presentation through

the peculiarities of its formation and creation of math-

ematical content with a focus on measure use.

The study by Zhaldak et al. (Zhaldak et al.,

2021) discusses the use of mathematical technologies

in cloud calculations using Remote Desktop Ulteo

OVD. As an example, the researchers examined in de-

tail some examples of using pedagogical software for

the educational purposes called Gran1. In particular,

the calculation of approximate value of the subsumed

integral; the graphical solution of two-dimensional

tasks, the so-called linear programming tasks; two-

dimensional problems including the convex program-

ming – ﬁnding the highest value of a convex func-

tion (or the highest convex function) on a convex set

of inequalities (including linear ones). However, the

use in the educational process of any technology, in-

cluding modern information and communication, as

well as the content of the teaching must be pedagogi-

cally balanced, which will allow to avoid any negative

inﬂuences on the formation of the personality of the

future member of society, his mental and physical de-

velopment.

Popel et al. (Popel et al., 2017) have investi-

gated reasonable ways of using cloud systems to sup-

port student collaboration in mathematics courses.

In particular, the researchers implemented a compo-

nent based on SageMath that combines electronic re-

sources for teaching several mathematical disciplines.

They proposed a training methodology for using the

SageMath in the training of educational personnel. In

addition, the usefulness of implementing this method-

ology for the active development of innovative ap-

proaches, forms and methods of teaching mathemat-

ics using modern devices was proved. The exper-

imental results of the implementation of the Sage-

Math educational component and methods of its use

which were developed in the course of the research

are presented. The use of the proof-based method to

improve the educational environment of the univer-

sity, expand the possibilities of access to electronic

teaching resources during mathematics teaching and

involvement of the educational community into this

process, as well as increasing their ICT competence

is explained.

Shyshkina and Marienko (Shyshkina and

Marienko, 2020) determined the content of necessary

general skills and speciﬁc skills necessary for future

mathematics teachers (see table 2 in (Shyshkina and

Marienko, 2020)) based on professional functions

and typical tasks that teachers of mathematics must

be able to perform (see table 1 in (Shyshkina and

Marienko, 2020)). The authors used Web-SCM

CoCalc (SageMath) to conduct an experimental

study that revealed the advantages of using the

CoCalc cloud service in the professional training of

prospective mathematics teachers.

Fedorenko et al. (Fedorenko et al., 2020) in-

vestigated the problem of studying virtual software

through the use of cloud services. The results of this

research had a positive result especially for the profes-

sional training of future mathematics teachers. Inves-

tigators have revealed the didactic capabilities of the

cloud services, which allow you to run free software

for mathematical orientation.

The GeoGebra environment is one of the most

powerful systems of computer mathematics for dy-

namic visualization, calculations during problem

solving, data processing and scientiﬁc and research

work. Dubovyk and Rudnytskyi (Dubovyk and Rud-

nytskyi, 2022) have looked at the peculiarities of us-

ing the author’s applets and other products of GeoGe-

bra during the teaching of disciplines of differential

geometry and linear algebra in the training of future

mathematics teachers, in particular, their research de-

scribes the possibilities of using this tool to study the

properties of space curves and to form practical skills

and abilities to perform operations on matrices, ﬁnd-

ing the invertible matrices. The advantages and dis-

The Use of Cloud Technologies in the Process of Professional Training of Future Mathematics Teachers

381

advantages of using the GeoGebra environment in the

educational process of linear algebra and differential

geometry are outlined.

Kholifah et al. (Kholifah et al., 2020) aimed to

show how effective the use of blended learning model

inﬂuences the learning motivation of vocational ed-

ucation students. The results obtained in the course

of the research show that the use of the model of

blended learning has a signiﬁcant impact on the mo-

tivation of students of vocational education. Attard

and Holmes (Attard and Holmes, 2020) investigated

the willingness of teachers to use blended learning ap-

proaches. The study was conducted on teachers who

taught mathematics classes in secondary schools. As

a result of the study it was found that the use of tech-

nology expands the possibilities for students to study

mathematics by providing multiple methods and ac-

cess methods, and students can use mixed teaching

approaches.

Kadirbayeva et al. (Kadirbayeva et al., 2022) have

assessed the methods of using blended learning tech-

nology in school mathematics from the viewpoint of

teachers. According to the results of the research,

most mathematics teachers stated that the attitude of

students to the mathematics lesson was ambiguous

and negative. Most of the teachers who participated

in the study stated that the combination of the won-

derful characteristics of the classroom environment,

the diversity of teaching and learning methods, the

variety of teaching materials and increasing student

success are the strengths of mathematics education

in a mixed learning environment. Most teachers who

participated in the study stated that a disadvantage of

mathematics education in the mixed environment is

the lack of educational programs suitable for mixed

learning, the disadvantages of software and hardware

that will be used in the course, and inability to use

technology effectively and efﬁciently, and the learn-

ing environment.

4 RESEARCH RESULTS

The normative and regulatory documents that deter-

mine the content and organization of the educational

process in institutions of higher education of Ukraine

provide the main requirements for lectures. The im-

plementation of these principles allows us to fully use

the signiﬁcant teaching and educational possibilities

of this form of learning, to increase the impact of each

lecture on the knowledge and feelings of the students.

However, the lecture in some way trains the student

to passive absorption of others’ ideas, does not stim-

ulate the desire for self-study, does not provide for

an individualized, differentiated approach to learning.

The leading role of the lecture in teaching disciplines

associated with their contents aspect, organizing prin-

ciples and methodological features. The main content

consists of the central methodological, theoretical and

practical problems. Not all questions of the topic are

revealed, but the most important, the most essential,

which require scientiﬁc discussion.

The pace of development of modern technologies

signiﬁcantly affects teaching methods and teaching

models in general. This allows you to expand ways

to implement the paradigm of competence in order

to improve the quality of education. The model of

blended learning has the greatest potential for opti-

mizing the educational process. This model allows for

the implementation of new technologies without dis-

regarding the generally accepted teaching methods.

Blended learning is a model for organizing the educa-

tional process because “it allows to increase the mo-

tivation of future teachers to learn, makes it transpar-

ent, interactive and guided, and ensures the constant

involvement of students in the educational process”

(Kukharenko, 2016) and continuous advancement of

qualiﬁcation. The aim of this form of learning is to

combine the advantages of face-to-face learning and

electronic educational resources through the combi-

nation of distance and traditional communication in

an integrated educational activity. The integration of

traditional and computer-based learning in the educa-

tional environment leads to a purposeful process of

developing knowledge and skills and abilities in the

classroom and post-attendance educational activities

of the subjects of the educational process on the basis

of the use of information and communication tech-

nologies. The existence of this form of learning is

possible due to the effective combination of differ-

ent ways of presenting educational content, teaching

models and styles. It is based on the interaction be-

tween all participants in the educational process.

One of the forms of blended learning is imple-

mented through Flipped Classroom. There are dif-

ferent ways of implementing the Flipped Classroom

model, all of which are based on one basic principle:

traditional learning takes place outside the classroom,

while practical work and application of the acquired

knowledge takes place in the classroom. In general,

the essence of inverted learning lies in restructuring

the key components of the educational process. With

the help of this teaching model “the content of new

educational material is mastered by oneself in the

electronic environment, and then the acquired knowl-

edge is used during practical lessons or discussions”

(Hlazova et al., 2018).

The shortening of classroom hours (lectures,

AET 2021 - Myroslav I. Zhaldak Symposium on Advances in Educational Technology

382

workshops, seminars, laboratory classes) leads to a

failure of the traditional logics of the educational pro-

cess. This fact leads to loss of quality of teaching.

One of the ways to “renew the balance of the edu-

cational process is to use the mixed teaching model

with Flipped Classroom technology” (University of

Queensland, 2017). In this case, the key components

of the learning process change places: the basic com-

ponents of the new material are studied independently

at home, and in the classroom the studied material is

consolidated and the more complex issues and practi-

cal use of the teaching information are discussed to-

gether with the tutor. The Flipped Classroom tech-

nology is characterized by the fact that the neces-

sary “theoretical knowledge is acquired in the class-

room, and in the classroom the individual tasks are

performed or a group project is developed” (Pieri and

Laici, 2017).

Flipped Classroom technology was used in the

study of mathematical disciplines “Mathematical

Logic and Theory of Algorithms” and “Elementary

Mathematics” at the 3rd year of the Physics and Math-

ematics Faculty of the Donbas State Pedagogical Uni-

versity. Taking into account the fact that the students

of the specialty “Secondary education (mathematics)”

specialization “Informatics” at the 3rd year of study

have signiﬁcant experience of educational activities,

and the level of self-awareness is already sufﬁcient

for the personal motivation to learn, it is not only log-

ical, but also quite proﬁtable to use the elements of

Flipped Classroom technology. Moreover, this model

of learning does not interfere with the educational

program, in which the main part of the years is al-

located for the students’ self-directed work.

Based on the practical experience of use, the fol-

lowing structure of the approach to each individual

topic was formed:

1. Formulation of the topic and the goal.

2. Identifying the place of the topic in the work pro-

gram of the discipline.

3. Suggesting sources of information.

4. Deﬁnition of types and content of control.

5. Control and evaluation.

Implementing the Flipped Classroom technology

while teaching the course “Mathematical Logic and

Theory of Algorithms” it is advisable to consider the

following topics:

• Boolean n-ary functions;

• Zhegalkin Polynomial;

• Complete systems of Boolean functions;

• Mathematical theories of the ﬁrst order.

For each of the proposed topics identiﬁed a place

in the work program and sources of information, se-

lected the content of tasks for the control of the type.

For example, to study the topic “Boolean function

systems” students receive the following information

(table 1).

The main reason for the implementation of the

Flipped Classroom model of teaching in the educa-

tional process of the university is the active cooper-

ation between the students and the teacher and, as a

result, increasing the success and motivation of the

students. A special feature of this model is the pos-

sibility of using group classroom sessions where stu-

dents can discuss key aspects of lecture materials, test

their knowledge and interact with each other. The task

of the teacher is to explain the problems, comment on

the work of students. Thus, we came to the need to

have in their arsenal of specialized mathematical plat-

forms for collaborative work. Through the use of col-

laborative platforms, or even mathematical platforms

that allow the exchange of output data, with the results

of calculations or developed electronic educational re-

sources for mathematics courses, we get a mechanism

for organizing blended learning with Flipped Class-

room technology for future mathematics teachers.

It is necessary to review the training tools for fu-

ture teachers of mathematics, which implement the

above-mentioned ideas. The CoCalc Cloud Platform

(https://cocalc.com/) is part of the SageMath Project

(https://www.sagemath.org/) developed by William

A. Stein. The main idea is that currently there is

a large number of open and complete mathematical

software already developed but they are implemented

in various programming languages. The SageMath

project integrates all existing developments and adds

its own, and integrates them with the Python lan-

guage. The cloud service CoCalc provides the oppor-

tunity not only to work online with SageMath work-

sheets via Jupiter notepad, but also to use the La-

TeX language for the design of documents. Merz-

lykin et al. (Merzlykin et al., 2022) investigated the

capabilities of the CoCalc system for scientiﬁc and

educational purposes and came to the following con-

clusions:

• the use of cloud services leads to the formation

and development of forms of learning focused on

collaborative learning activities on the Internet;

• cloud services can be used in the training of math-

ematics teachers as tools: communication; collab-

oration; storage and processing of data.

An example of supporting the process of profes-

sional training of future teachers of mathematics is

the use of computer-based mathematics systems in

The Use of Cloud Technologies in the Process of Professional Training of Future Mathematics Teachers

383

Table 1: Information resources of the topic.

Topic Complete Systems of Boolean Functions

Purpose To learn the concepts of classes of functions that store zero, functions that store one unit, self-

similar functions, monotonic and linear functions. To deﬁne the criteria of totality.

The sources

of information

(self-selection

of sources of

information is

encouraged)

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• Discrete Mathematics. Lecture: Completeness criterion of a Boolean function system.

Access mode: https://cutt.ly/jJaHyNj

• “Collect the Crystals” game simulator. Access mode: https://cutt.ly/SJaHK3c

• Simulator “Logical Elements”. Access mode: https://cutt.ly/0JaHMwY

• A Simulator for Minimizing Logic Functions by the Quine-McCluskey Method. Access

mode: https://cutt.ly/hJaH49v

• Creating the Truth Table. PDNF. PCNF. Zhegalkin polynomial. Access mode: https:

//cutt.ly/IJaJqzR

• Truth table. Access mode: https://cutt.ly/3JaJufo

Types of control Test for the comprehension of the material (carried out remotely during class time), individual

assignment to determine the completeness of the system

Date of control Conducting a self-study test

the form of cloud services. One of these free ser-

vices is Math Partner, available at http://mathpar.com.

This service allows you to create your own cloud

math “Workbook”, in which the user performs the

necessary mathematical calculations. To ensure qual-

ity and comfortable work this service provides access

to a large amount of guidance material with exam-

ples. The language of this service is Mathpar, based

on TeX, a widely used by mathematicians special-

purpose data layout language developed by Donald

Knuth. There is a possibility to save both the state-

ment of the problem and its solutions. You can save

both textual and pictorial views. Although for study-

ing mathematical logic Math Partner system does not

contain expanded functions, the basic logical opera-

tions the system is able to process (ﬁgure 1).

In particular, Math Partner allows you to efﬁ-

ciently perform such common tasks as ﬁnding the

shortest distances between all vertices of the graph

(\searchLeastDistances(A)) and ﬁnding the shortest

path between vertices (\ﬁndTheShortestPath(A,i,j)).

It should be noted that this service is convenient to use

for checking your own connections, because it gives

the answer itself, without access to the intermediate

results of the calculations.

The Graph Online service https://graphonline.ru/

en/), which is available under the MIT License, is

designed for graph visualization and execution of al-

gorithms on the created graph. The graph is created

using the complexity matrix or the incident matrix.

Besides searching for the shortest path, you can also

search for the connectivity component. The service

AET 2021 - Myroslav I. Zhaldak Symposium on Advances in Educational Technology

384

Figure 1: Calculation of logical function values in the Math

Partner system.

supports work with directed graphs (digraphs) and

non-directed graphs. You can save the result of the

work, i.e. the generated graph, and continue working

on it later. In addition, the cloud service Graph On-

line gives the user a number of additional functions

to make the work easier. Graph saving and loading is

available with maintenance of visual representation,

quick conversion between all supported types, vertex,

arc, background, constructor mode, etc.

Let us demonstrate the application of the pre-

sented services when solving the problem of ﬁnding

the shortest distances between graph vertices, which

are often encountered in practice. It is clear that for

the possibility to ﬁnd the shortest distances at least

one path from vertex 0 to each other vertex must ex-

ist, i.e. the graph must be connected. For this problem

the most well-known connectivity algorithm is the Di-

jkstra’s algorithm. The idea of this algorithm is that at

ﬁrst each vertex other than vertex 0 is given a distance

equal to +∞, and then we decrease these distances

step by step until we ﬁnd the minimum distance d(v)

and the shortest path p(v) for each vertex v.

Problem statement: In an arbitrary graph G =

(V, E), the set of vertices V = {0, 1, 2, 3, 4, 5}, and the

set of edges E is given by a matrix of weights:

E =







− 8 7 − 10 12

8 − 5 1 4 −

7 5 − 3 − 4

− 1 3 − 2 1

10 4 − 2 − 3

12 − 4 1 3 −







Use Dijkstra’s algorithm to build a spanning tree

Table 2: The progress of the Dijkstra algorithm.

1 2 3 4 5

8 7 10 10 12

9 11

0; 1 0; 2 0; 2; 3 0; 4 0; 5

0; 1; 3 0; 2; 5

0; 1; 3; 5

of the shortest paths from vertex 0 to all other vertices

of graph G and ﬁnd the shortest distances.

Solution: Let’s show the progress of the Dijkstra’s

algorithm in the table 2.

During the execution of the Dijkstra’s algorithm,

the vertices were in this order: 2, 1, 3, 4, 5. Thus, the

shortest distance to vertex 1 is 8, d(2) = 7, d(3) = 9,

d(4) = 10, d(5) = 10. The shortest path to vertex 1

is 0.1; p(2) = 0.2; p(3) = 0.1.3; p(4) = 0.4; p(5) =

0.1.3.5. Figure 2 shows these shortest paths in the

form of a tree.

Figure 2: The spanning tree of the shortest paths from ver-

tex 0 to all other vertices of graph G.

With the Math Partner service, we have the oppor-

tunity to check the correctness of the results in ﬁg-

ure 3. But the visualization of the graph is better per-

formed by Graph Online in ﬁgure 4.

The Robert Coleman SA403 FY17 project (https:

//cutt.ly/PKp7bJ4) on the CoCalc platform includes

the implementation of popular graph algorithms.

In particular, Dijkstra’s algorithm was implemented

(https://cutt.ly/FKp7eLy). Thus, we can conclude that

during the professional training of mathematics teach-

ers, the use of various cloud services is not only pos-

sible but also appropriate.

The students of the 2nd-4th years of Secondary

Education (Mathematics) at the Faculty of Physics

and Mathematics of the Donbas State Pedagogical

University were repeatedly interviewed (the ﬁrst one

was described in the study (Velychko et al., 2021)) in

order to ﬁnd out what means of computer teaching are

The Use of Cloud Technologies in the Process of Professional Training of Future Mathematics Teachers

385

Figure 3: The shortest distances from the vertex 0 to all

other vertices of graph G in the Math Partner service.

Figure 4: The visualization of the graph G and the shortest

distance from the vertex 0 to the vertex 5 of the graph G in

the Graph Online service.

used by future mathematics teachers in the process of

their professional training, including during the self-

study activity. A total of 80 students were involved

in the survey (in the ﬁrst one – 120 respondents). We

asked the respondents the following questions:

1. Have you used electronic educational resources

during your studies?

2. Did you use the cloud services during the study

period?

3. Did you use the cloud services during self-study

activities?

4. If the answer to the 2nd or 3rd question is positive,

then what cloud services did you use?

5. If the answer to the second and third questions is

negative, what kinds of software did you use dur-

ing the training?

We received the following results:

• 100% positive result on the ﬁrst question. This

result indicates that future mathematics teachers

understand the concept of “electronic educational

resources” without regard to their diversity.

• The second question was answered positively by

92% of respondents (in the ﬁrst survey – 82%).

The result of the answer to the second question

shows that not all respondents identify the concept

of a cloud service. We reached this conclusion

during the discussion of the results of the survey

with the respondents since during the practical ex-

ercises that we described in the study all the stu-

dents were involved in the use of cloud services.

• 76% of respondents (46% in the ﬁrst survey) re-

sponded positively to our third question. This

number is a good result of self-awareness activ-

ities.

The answer to the fourth question is shown in the ﬁg-

ure 5.

It should be noted that there was a signiﬁcant out-

pacing of cloud communication services due to quar-

antine measures caused by COVID-19 pandemic and

cloud ﬁle-saving and collaborative work servers. In

comparison with the ﬁrst study, tools for creating in-

fographics and interactive presentations appeared. A

positive result of the study is an increased use of

mathematical cloud services. The answer to our ques-

tion 5 is shown in ﬁgure 6.

By comparing the results of the study with those

described in the publication “The support of the pro-

cess of training pre-service mathematics teachers by

means of cloud services” (Velychko et al., 2021) we

get both quantitative and qualitative changes. First of

all, the respondents not only expanded their knowl-

edge of cloud technologies, but also expanded their

use in their own teaching activities. Software on the

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386

Figure 5: Cloud services which were used by the future mathematics teachers.

Figure 6: Software tools used by future mathematics teachers.

desktop system as well as on mobile systems has a

greater focus on learning, the advantages are given to

the available software. Respondents’ responses are

more demanding compared to the previous one. As

before, special software for mathematical purposes is

not used as often as desired, however, the rate of its

use is increasing.

5 CONCLUSIONS

The learning process at a high school is implemented

within a diverse and integral system of organizational

forms and teaching methods. Each form carries out

its own speciﬁc task, but the totality of forms and

methods of learning creates a single didactic complex,

the functioning of which is consistent with the spe-

ciﬁc psychological and pedagogical laws of the ed-

ucational process. The advantages of the cloud ser-

vices lead to their widespread use in educational ac-

tivities. Experience gained in implementing cloud

technologies in educational activities make it possi-

ble to begin to analyze the work carried out – what

results were obtained, which are the best practices of

their use. Quarantine restrictions and the rapid transi-

tion to e-learning has led to the creation of electronic

educational resources, development of methods of e-

learning, in which the cloud services play a signiﬁcant

role.

The results of the research allow us to conclude

that cloud technology is used in the process of teach-

ing mathematics in secondary schools. Future math-

The Use of Cloud Technologies in the Process of Professional Training of Future Mathematics Teachers

387

ematics teachers have the opportunity to use cloud

technology during their professional training. Mod-

ern cloud technologies have a wide range of use in

mathematics education. The training of future mathe-

matics teachers has begun to be based on new teach-

ing tools and technologies.

The next step of the research is the accumulation

of empirical data on the use of cloud computing in

the process of professional training of future teachers

of informatics and software for applied orientation.

We have analyzed the trends in the use of cloud com-

puting and software of mathematical orientation both

during the professional training of future mathemat-

ics teachers and during the professional activities of

practicing mathematics teachers. The problem of har-

monious combination of digital technology and clas-

sical methods of mathematics learning, even in mod-

ern pedagogical teaching technology, is not solved. In

our opinion, the feasibility of reduction of computing

operations through the use of information technolo-

gies in opposition to the loss of skills of operational

competence is interesting.

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