The Use of ICT Tools in Teaching Mathematical Modeling to Students

Olha I. Matiash

1 a

, Liubov F. Mykhailenko

1 b

, Natalia Y. Dmitrenko

1 c

Halyna D. Kateryniuk

2 d

and Ihor V. Kalashnikov

1 e

Vinnytsia Mykhailo Kotsiubynskyi State Pedagogical University, 32 Ostrozkoho Str., Vinnytsia, 21001, Ukraine

Regional Scientiﬁc Lyceum of the Municipal Higher Education Institution “Vinnytsia Humanitarian Pedagogical

College”, 13 Nahirna Str., Vinnytsia, 21001, Ukraine

Keywords:

Mathematical Modeling, Information and Communication Technologies, Distance Learning.

Abstract:

The study reveals that for the development of students’ mathematical competency it is important to provide

effective conditions for the formation of mathematical modeling skills. It is considered that methodologically

balanced use of information and communication technologies (ICT) in the process of teaching mathemati-

cal models is a key means in the formation of students’ mathematical modeling skills. In the article, the

authors described the possibilities of using the electronic manual “Methodological Tool for Developing Stu-

dents. Ability to Mathematical Modeling” and the mathematics teacher’s website in today’s conditions of

forced distance learning. According to the results of the expert survey, it was elucidated that there was a re-

quest from the mathematics teachers for methodologically adapted and substantiated electronic manuals. For

mathematics teachers, the most important aspects in such manuals were: methodologically successful selec-

tion of applied problems; modern, relevant content of tasks for students; methodological assistance to teachers

in order to increase the effectiveness of teaching. The research on the effective use of ICT in the process of

formation of students’ mathematical modeling skills was based on the same principles as the development of

computer-oriented methodical learning systems were based: the principle of interest in learning, the principle

of adaptability to the individual characteristics of students, the principle of the student’s search activity, the

principle of self-evaluation and self-actualization, the principle of individualization of learning, and the prin-

ciple of cooperation and mentoring. It is concluded that it is important to form skills of using ICT for creation

and study of mathematical models to prospective teachers of mathematics.

1 INTRODUCTION

All recent Ukrainian educational documents on math-

ematics education emphasize that mathematical com-

petency is the ability to see and apply mathematics in

real life, understand the content and method of math-

ematical modeling, build a mathematical model and

study it by using mathematical methods, and interpret

the obtained results. We consider the ability to mathe-

matical modeling as a direct feature of students’ prac-

tical competency, which is formed in parallel with the

formation and development of skills to use mathemat-

ical modeling in educational activities and practice.

https://orcid.org/0000-0002-7149-9545

https://orcid.org/0000-0001-5051-5561

https://orcid.org/0000-0002-3556-0003

https://orcid.org/0000-0002-5794-7743

https://orcid.org/0000-0001-7961-8134

The analysis of scientiﬁc and methodological publica-

tions shows that research on the problems of teaching

mathematical modeling to students in different coun-

tries around the world has evolved from simple qual-

ity research cases to large research projects. The anal-

ysis of the Ukrainian scholars’ pedagogical research

reveals that special attention should be paid to the

problem of providing the effective conditions for the

formation of students’ mathematical modeling skills.

We completely agree with opinion of Semerikov et al.

(Semerikov et al., 2010), formulated more than ten

years ago, that “today it is no longer necessary to

prove the obvious fact that increasing the effective-

ness of the study of all school subjects is based on the

systematic use of ICT”. The intensive development

of computer technologies is leading to the emergence

of new approaches to the educational process. The

quarantine requirements for COVID-19 have acceler-

ated this process. The program of education reform

Matiash, O., Mykhailenko, L., Dmitrenko, N., Kateryniuk, H. and Kalashnikov, I.

The Use of ICT Tools in Teaching Mathematical Modeling to Students.

DOI: 10.5220/0012066900003431

In Proceedings of the 2nd Myroslav I. Zhaldak Symposium on Advances in Educational Technology (AET 2021), pages 675-687

ISBN: 978-989-758-662-0

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

675

activities developed by the Cabinet of Ministers of

Ukraine provides the creation of modern electronic

educational resources, in particular, electronic text-

books and manuals. In the context of this article,

the particular interest is paid to the scientiﬁc heritage

of Myroslav I. Zhaldak as to the methodological ac-

tivities of teachers, in particular, “pedagogically bal-

anced, theoretically and experimentally justiﬁed use

of innovative ICT in the educational process harmo-

niously combining with the scientiﬁc-educational her-

itage of the past allow to form the knowledge that un-

derlies many contemporary professions related to new

information and production technologies” (Zhaldak,

2003).

The purpose of the article is an explanation of the

methodologically balanced use of ICT by the mathe-

matics teacher in the process of teaching mathemat-

ical modeling to students in the present-day forced

conditions of distance learning.

2 LITERATURE REVIEW

The analysis of scientiﬁc publications has shown that

the problem of formation and development of math-

ematical modeling skills is one of the most globally

recognized problems in the study of mathematical ed-

ucation. The most important arguments for this con-

clusion are:

• Since 1983, the International Conference on the

Teaching and Learning of Mathematical Model-

ing and Application (ICTMA) has been held ev-

ery two years. The conference discusses the state

and problems of teaching mathematical modeling

to students in the form of an international discus-

sion. The conference abstracts are regularly pub-

lished in the series Springer’s International Per-

spectives on the Teaching and Learning of Math-

ematical Modeling.

• At the CERME International Forum, organized by

the European Society for the Study of Mathemat-

ical Education, one of the traditional sections is

“Applications and Modeling”.

• Springer search system issued 58,613 search re-

sults for “Mathematical modeling in secondary

school”. Moreover, these results are publications

of recent decades, which address the problem of

formation and development of mathematical mod-

eling skills.

In order to obtain information about the areas of

current studies and their results related to the use of

ICT in teaching mathematical modeling, it is consid-

ered to observe some recent publications.

Tezer and Cumhur (Tezer and Cumhur, 2017) an-

alyzed the results of research connected with the im-

pact on mathematical achievement, problem-solving

skills and students’ views by means of the educational

model 5E and the method of mathematical modeling

in the process of studying the topic “Geometric Ob-

jects”. The results of statistical analysis proved that

learning with the 5E “Instructional Model” in experi-

mental group 1 and the method of mathematical mod-

eling in experimental group 2, improved students’

academic achievements. However, the method of

mathematical modeling was more effective for math-

ematical achievements and problem-solving skills.

Blum (Blum, 2011) and Kaiser (Kaiser, 2019),

the most authoritative German researchers on the for-

mation and development of mathematical modeling

skills, founded the ISTRON group. Since 2014, the

ISTRON group has published 20 volumes of pub-

lications to support teachers in solving real prob-

lems of teaching mathematical modeling to students

at school. In all these studies, various aspects of

the problem of modeling training were analyzed in

complex. In particular, one of the learning environ-

ments, designed speciﬁcally for beginners in model-

ing, examined the KOMMA (computer learning en-

vironment). The KOMMA learning environment in-

cluded four variants of heuristic activity samples. In

these samples, two ﬁction characters solved simu-

lation problems and explained their ideas, heuristic

strategies, and tools. All samples were structured by

means of a 3-step simulation cycle. The eight-grade

students’ competency of modeling (316 people) was

tested before the experimental training, immediately

after it, and four months after the experiment. The

results of the research have shown a signiﬁcant in-

crease in the competency of modeling immediately

after the implementation of the learning environment

and a slightly lower long-term effect (Greefrath and

Vorh

olter, 2016; Blum, 2015).

In recent years, in Germany, the methodological

activity of teachers in teaching mathematics (includ-

ing modeling) has changed mainly due to the active

development and implementation of digital technolo-

gies. Solving application problems, a computer or a

well-equipped graphing calculator can be especially

useful tools to support teachers and students. For ex-

ample, Henn (Henn, 2007) suggested the use of digi-

tal tools, such as notebooks with algebra software, as

it allows to incorporate complex programs and mod-

eling into everyday learning.

One of the possibilities of using digital technolo-

gies is research and experiments (Hilscher, 2002). For

instance, a real situation can be transferred to a ge-

ometric model, where one can experiment with dy-

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676

namic geometry software or spreadsheet analysis that

seems to be similar to experimental modeling. Mod-

eling in the process of experiments is designed to pro-

vide an idea of the real system presented in the model

(Greefrath and Weigand, 2012). Applied mathemati-

cal simulations performed by a computer can be per-

ceived as part of a simulation cycle in which a numer-

ical model, developed from a mathematical model, is

tested and validated by comparing it with measure-

ment results (Sonar, 2001). A common use of digi-

tal tools, especially computer algebraic systems, is to

calculate or estimate numerical or algebraic solutions

(Hilscher, 2002). Without the use of a computer, stu-

dents would not be able to make these assessments,

at least in a reasonable amount of time. In addition,

digital tools can perform visualization of the studied

object (Barzel and Hußmann, 2009; Hilscher, 2002;

Weigand and Weth, 2002). Digital tools also play a

useful role in controlling and testing the mathemati-

cal modeling skills (Barzel and Hußmann, 2009). If

computers with Internet connections are provided for

teaching mathematics, they can be used for research

(Barzel and Hußmann, 2009), for example, in the con-

text of programs.

German scientists state that there is currently little

empirical knowledge about the possibilities of learn-

ing modeling and recommendations for working with

digital tools in teaching mathematics. Only some case

studies have been conducted, there are no large-scale

experiments on the introduction of computer technol-

ogy in the process of teaching mathematical mod-

eling. Case studies (Greefrath et al., 2011; Geiger,

2011) note that digital tools can be useful for each

step of the modeling process, especially for interpre-

tation and validation. Open research questions can be

found in the studies of Niss et al. (Blum et al., 2007;

Niss and Højgaard Jensen, 2002): “How should dig-

ital tools be used in different classes to support mod-

eling processes? What is the impact of digital tools

on the range of modeling issues that need to be dis-

cussed? How does the use of digital technologies af-

fect the learning culture? When do digital tools im-

prove or hinder learning opportunities in the modeling

process?”

Based on the mentioned analysis of teaching

mathematical modeling in Germany, we indicate sev-

eral current issues and tasks in formation of students’

mathematical modeling skills for Ukrainian mathe-

matics education:

• modeling activities can be signiﬁcantly changed

due to the development of digital technologies;

• the activity and role of mathematics teachers in

the successful implementation of mathematical

modeling in mathematics lessons is important;

• the focus should be on research of particular mod-

eling lessons, as well as the entire modeling edu-

cation environment.

In the article (Matiash and Mykhailenko, 2021)

we analyzed the course of scientiﬁc discussions in

terms of the International Forum CERME-2021 (we

were lucky enough to be participants of it), which

related to the use of information technology in the

formation of mathematical competencies of students.

Nowadays, the attention of researchers of mathemat-

ics education in the world is focused on problems:

blended learning of mathematics students, blended

learning in the process of professional training of

mathematics teachers, and the developing tools, plat-

forms and learning equipment for online mathematics

education.

3 THEORETICAL BACKGROUND

Systematic application of the method of mathemat-

ical modeling at the process of teaching mathemat-

ics at school can be considered as a means of imple-

menting the applied orientation of the school course

of mathematics. In particular, it means: creating a

bank of mathematical models that describe real phe-

nomena and processes, have general cultural signiﬁ-

cance, and are studied in related courses; formation

of students’ knowledge and skills which are required

for the study of these mathematical models; teaching

students to build and study the simplest mathematical

models of real phenomena and processes.

3.1 Mathematical Modeling

The essence of the concept of “mathematical model-

ing” is explained in different ways in many modern

available sources:

1. Modeling is the study of knowledge objects on

their models; construction of models of real ob-

jects and phenomena (living organisms, engineer-

ing structures, social systems, various processes,

etc.). Mathematical modeling is the most mod-

ern comprehensive method of scientiﬁc research;

it is the process of creating mathematical mod-

els. A mathematical model is a system of mathe-

matical relations that describes the studied object,

process, or phenomenon (Mathematical model,

2023).

2. Mathematical modeling or mathematical simula-

tion is a research method of processes or phenom-

ena by creating mathematical models and study-

ing these models. Mathematical modeling allows

The Use of ICT Tools in Teaching Mathematical Modeling to Students

677

to replace a real object with its model and then

study it (Mathematical model, 2023).

3. Mathematical modeling is one of the main modern

methods of systems research. It usually involves

the creation of a conceptual model of the studied

object, its formalization and transformation into a

mathematical or computer model, veriﬁcation of

adequacy and further study of the model by means

of analytical or numerical methods and modern

computer technology (Semenova, 2014).

4. Modeling is a process of real system research,

which includes construction of a model, its re-

search and transfer of the obtained results to the

studied system. A model can be deﬁned as an ob-

ject that in some respects coincides with the pro-

totype and is a means of describing, explaining

and/or predicting its behavior. The mathematical

model of the real system (process) means a set

of relationships (formulas, equations, inequalities,

logical conditions, operators, etc.) that determine

the characteristics of the system depending on its

parameters, external conditions (input signals, in-

ﬂuences), initial conditions and time (Bakhrushin,

2004).

5. Modeling is the construction (or selection) and

study of an object of any nature (model) that can

replace the studied object (the original) and the

study of model provides new information about

the studied object. Mathematical modeling is the

highest form of modeling. It contributed to the de-

velopment of science and technology in industrial

society, and the advent of electronic computing fa-

cilities led to the rapid development of contempo-

rary – post-industrial society (Stanzhytskyi et al.,

2006).

Thus, mathematical modeling is considered

mostly as the study of the object properties in a math-

ematical model. Simultaneously, the mathematical

model is an approximate description of a phenomenon

or process of the external world, which is presented by

mathematical symbolism. Mathematical modeling is

one of the most up-to-date directions, which is closely

related to the introduction of modern computer equip-

ment and information technologies (Zahrai and Ko-

tovenko, 2007).

From the pedagogical point of view, it does not

matter where (mathematics, physics, computer sci-

ence) students learn to model. It is important to

understand that by means of modeling, the system-

combinatorial thinking and the ability to solve real

problems are formed (Teplytskyi, 2000). It is worth

mentioning that modeling forms the world view and

the scientiﬁc picture of the world not only for the stu-

dent but also for the teacher. Modeling is the method,

the most adequate to modern requirements for the ed-

ucation system, of including a computer in the learn-

ing process, which provides an active type of educa-

tional and cognitive activities (Morze et al., 2022). In

particular, a computer model is a software environ-

ment for a computational experiment. Based on a

mathematical model of a phenomenon or process, it

combines tools of the experiment object analysis and

information display. Using widely computer graph-

ics, 3D modeling programs help to turn individual

ideas into smart models and prototypes. 3D models

are used in various ﬁelds: cinema, computer games,

interior design, architecture, etc. Choosing software

for modeling is a rather difﬁcult process, as it is not

easy to ﬁnd a program that would have all the required

functionality.

The use of computer modeling in the learning

process (study of phenomena based on ready-made

models, constructing models by students themselves)

can increase the intensity of learning and the stu-

dents’ cognitive activity. The advantages of educa-

tional computer modeling are related to overcoming

the formality of knowledge acquisition, the develop-

ment of research and design skills, and the develop-

ment of students’ intellectual abilities (Morze et al.,

2022).

Teplytskyi (Teplytskyi, 2000) proposed a method-

ological system for studying computer modeling and

aimed to reveal the content of all school education

through the introduction of the concepts of “model”

and “modeling”, which have developed both in spe-

ciﬁc sciences and in the methodology of science in

general. The scholar considered the following deﬁni-

tions to be the most accessible to students: a model is

a mentally imagined or materially realized system of

reﬂecting or reproducing the studied object and the

study of the model provides new knowledge about

this object (Shtoff, 1963). Additionally, the model is

considered as a system that does not differ from the

studied object in some of its essential properties and

differs in all other insigniﬁcant properties (Biriukov

and Gutchin, 1982; Maliarchuk, 1997).

In our study, we reveal the content of the mathe-

matical model as a set of mathematical relationships,

equations, and inequalities that describes the basic

laws in the studied process, object, or system (Kviet-

nyi et al., 2012). Modeling in the teaching of mathe-

matics at school is understood as the process of build-

ing a model. Studying mathematics at school, stu-

dents should realize that the process of any applied

problem solving is divided into three stages: 1) for-

malization (transition from the situation described in

the problem to a formal mathematical model of this

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678

situation, and from it to a clearly formulated mathe-

matical task); 2) solving the problem within the con-

structed model; 3) interpretation of the obtained so-

lution of the problem and its application to the initial

situation.

3.2 Key Aspects of the ICT Use in

Teaching Mathematics to Students

In the publications of Zhaldak (Zhaldak, 2003; Zhal-

dak and Hrybiuk, 2014), ICT in education are con-

sidered as a set of computer-oriented educational and

teaching materials, software, and hardware for edu-

cational purposes, as well as a system of scientiﬁc

knowledge about the role and place of computer tech-

nology in the educational process, the methods and

forms of their pedagogically balanced, methodologi-

cally motivated, and appropriate use in order to im-

prove the educational process.

There is awareness among mathematics teachers

that the introduction of ICT tools can signiﬁcantly

help to create more effective conditions for students’

cognitive activity and contribute to the formation of

their competencies. “The basis of informatization of

the educational process is grounded on the creation

and widespread introduction of new computer-based

teaching methods into everyday pedagogical practice

based on the gradual and non-antagonistic principles,

omitting destructive reforms. Embedding informa-

tion and communication technologies in existing di-

dactic systems is considered as a harmonious com-

bination of traditional and computer-oriented learn-

ing technologies, without denying and rejecting the

achievements of pedagogical science of the past, but,

on the contrary, their improvement and strengthening

by means of including the use of advances in com-

puter technology and communications as well” (Zhal-

dak and Hrybiuk, 2014). According to Zhaldak (Zhal-

dak, 2003), the speciﬁc components of a teacher’s in-

formation culture are the ability to use ICT for train-

ing, support, analysis, adjustment, and management

of the educational process; the ability to choose the

most rational methods and tools of learning, and take

into account the individual characteristics of students,

their requests, inclinations, and capabilities; ability

to combine effectively traditional teaching methods

with new ICT. At the same time, the use of ICT in

the educational process (Zhaldak, 2003; Zhaldak and

Hrybiuk, 2014) should not promote only the study of

certain learning material, but, ﬁrst of all, the com-

prehensive and harmonious development of students’

personalities and their creative capabilities. Thus, in

the conditions of active use of ICT tools in teaching

mathematics, the requirements for the methodological

competency of the teacher increase signiﬁcantly.

Nowadays, in the era of mobility and globaliza-

tion, there is an urgent need to use the Internet, social

networks, and personal sites. Looking at modern stu-

dents’ level of informatization, it becomes clear that

the teacher needs “to keep up with the times”, as tra-

ditional teaching methods, in some way, lose their ef-

fectiveness. Currently, a mathematics teacher should

have to some extent universal, fundamental and mod-

ern knowledge in order to be able to use ICT in a me-

thodically competitive and effective way, and create

conditions to develop students’ inclinations and capa-

bilities, meet educational and cognitive needs.

The study and justiﬁcation of the required direc-

tions of ICT use in the educational process should

be considered as one of the most important ped-

agogical problems. Zhaldak (Zhaldak, 2003) note

that the problems of teaching mathematics in sec-

ondary schools by means of ICT are studied insufﬁ-

ciently. Despite a signiﬁcant amount of research on

this topic, there is a lack of computer-based scien-

tiﬁc and methodological support for teaching school

subjects in the context of systematic pedagogically

balanced and methodologically motivated use of ICT,

particularly, in mathematics. The methods of study-

ing the effectiveness of computer use in education

need to be improved (Zhaldak, 2003; Zhaldak and

Hrybiuk, 2014). Studying the effectiveness of teach-

ing mathematics to students by means of ICT, it is

necessary to identify its criteria and reasoning factors.

The teacher can achieve the set goals only if these

goals are accepted and achieved by the students. Eval-

uating the possibilities and expediency of using ICT

in teaching mathematics to students, it should be men-

tioned that a computer is only teachers and students’

tool for educational activities.

It was remarked principally by Razumovsky et al.

(Razumovsky et al., 2013) that with the introduction

of computers in the educational process, the possi-

bilities of many methods of scientiﬁc knowledge are

increased, especially the method of modeling, which

can dramatically inﬂuence the intensity of learning.

While modeling, the essence of phenomena is singled

out and their commonality is cleared up, i.e. scien-

tiﬁc and theoretical thinking is developed. However,

the fascination with the use of ready-made models

threatens to prematurely disconnect between the stud-

ied phenomenon and reality. It often happens when

students are asked to work with ready-made mod-

els without disclosing the process of model creation.

Since the objects of study must still be real phenom-

ena, their replacement, by abstract concepts and sym-

bols with an insufﬁcient base of observations and ex-

perience, often leads to pernicious formalism, when

The Use of ICT Tools in Teaching Mathematical Modeling to Students

679

the pretended knowledge lacks the essence.

The issue of computerization of school education

and the development of appropriate pedagogical soft-

ware is the subject of constant attention of Ukrainian

scholars (Zhaldak, 1989; Zhaldak et al., 2012, 2020;

Zhaldak and Franchuk, 2020; Leshchuk et al., 2022;

Kramarenko et al., 2019; Morze et al., 2022; Se-

merikov et al., 2021).

4 TEACHING MATHEMATICAL

MODELING TO STUDENTS IN

CONDITIONS OF ICT USE

4.1 Literature Review of Ukrainian

Scientists’ Research Results

The Ukrainian scientists study the problem of stu-

dents’ forming mathematical modeling skills in many

ways. In particular, the content of teaching stu-

dents the methods of mathematical modeling is de-

termined; the main stages of constructing a mathe-

matical model and their operational composition are

highlighted; the functions of modeling in the edu-

cational process are described; some methodological

recommendations for teaching mathematical model-

ing to students have been developed; the ways of us-

ing ICT in the process of teaching mathematical mod-

eling to students are proposed. The main method-

ological fundamentals of teaching mathematical mod-

eling to students are revealed in the studies of Vozniak

and Vozniak (Vozniak and Vozniak, 2003), Gnedenko

(Gnedenko, 2020), Slepkan (Slepkan, 1983), Shvets

(Shvets, 2009).

Filimonova and Shvets (Filimonova and Shvets,

2010; Shvets and Filimonova, 2010) developed the

main fundamentals of the methods of forming the

middle-class students’ knowledge, skills and abilities

in mathematical modeling. Among them are the fol-

lowing aspects: the formation of mathematical mod-

eling skills should be provided through successful use

of organizational and methodological tools. In partic-

ular: a rational combination of traditional and innova-

tive teaching methods. A special role should be given

to interactive and project methods.

Panchenko and Shapovalova (Panchenko and

Shapovalova, 2010), one of the goals of mathemat-

ical training of prospective mathematics teachers in

higher education is teaching the basics of mathemati-

cal modeling and training students for the introduc-

tion of ideas and methods of mathematical model-

ing in the course of mathematics in secondary school.

To achieve this goal, it is necessary to perform some

tasks, namely: to teach students and pupils to use ICT

in creating and studying mathematical models.

From the point of view of our research, the fol-

lowing position draws our attention: it is expedient

to include elements of mathematical modeling in sec-

ondary school education to a somewhat greater extent

and with greater penetration into the essence of mod-

els than it is done in many cases today. Obviously,

it should be done in classes of an advanced study

level of mathematics in secondary school. However,

in classes with a general academic level of mathemat-

ics, physics or computer science, the appeal to math-

ematical modeling should be moderately dosed and

balanced with the interests and abilities of students

(Krasnytskyi and Shvets, 1997).

Since the method of mathematical modeling is a

powerful tool for studying various processes and sys-

tems, according to many Ukrainian scholars, the con-

cept of mathematical model and some general funda-

mentals related to it should be illustrated in one form

or another at the learning process of the entire system-

atic course of mathematics in school. The samples of

this method application for solving speciﬁc problems

are presented in many well-known monographs and

textbooks.

Ukrainian scientists’ research emphasizes that for

older adolescents it would be appropriate to orga-

nize the educational process with lessons-lectures,

lessons-seminars, lessons-conferences, etc., involve-

ment in writing research papers, and projects. Units

of school curricula in various discipline courses re-

lated to solving problems for work, movement, inter-

est, progression, application of derivatives and inte-

grals can serve to develop the students’ mathemati-

cal modeling skills. At the same time, Ukrainian re-

searchers point out the current problems in the math-

ematical training of students, which often cause dif-

ﬁculties in the process of developing mathematical

modeling skills. As for the method of mathematical

modeling as a method of scientiﬁc research and learn-

ing cognition, in fact, it was not realized systemati-

cally and continuously in the school course of mathe-

matics in the last three decades.

4.2 Methods of the Research

The Ukrainian and foreign researchers emphasize the

need to use project technologies in the process of for-

mation of students’ mathematical modeling skills. We

consider the organization of project activities of stu-

dents at school as one of the priorities of modern ed-

ucation. Educational projects allow taking into ac-

count the individual characteristics of students that

contribute to the formation of their active and inde-

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680

pendent position in learning, and readiness for self-

development. The project method, as a component

of the education system, creates personal motivation

to solve an interesting problem. The project method

is associated with active practical activities and team-

work. We assign a signiﬁcant role to ICT in the orga-

nization of students’ project activities. In particular,

ICT (mathematics teacher’s website, social network

group, and the electronic manual) were actively used

in the process of teaching mathematics to students of

the 9th grade in order to improve the conditions for

the formation of mathematical modeling skills.

We tested the students’ implementation of a re-

search project on the topic “Geometry and Football”.

A group of nine-grade students from the Regional

Sports and Humanitarian Lyceum-Boarding School of

the Municipal Higher Education Institution “Vinnyt-

sia Humanitarian Pedagogical College” took part in

the project. The designed project was awarded the

third place at the Ukrainian Conference-Olympiad of

Geometric Creativity named after V. A. Yasinskyi. As

the participants of the project were students of the

sports and humanitarian proﬁle, the sports direction

was chosen for the project. Out of one hundred stu-

dents of the lyceum – 70 students were engaged in

various kinds of sport at that time. The most num-

ber of lyceum students played football – 22 students

(31% of all lyceum students). Therefore, the play of

football was stressed in the project. Students learned

with great enthusiasm that mathematical models can

be used to solve certain practical problems in foot-

ball, helping the team and coaches to achieve the best

results. Our experience has shown, that solving foot-

ball problems has increased interest, motivation, and

as a result, the effectiveness of the study of mathe-

matics, geometry in particular. Cooperation with stu-

dents was established by means of the teacher’s per-

sonal website and social networks. All students in the

class were asked to solve geometric tasks about foot-

ball. Each student sent the solution personally, hidden

from other project participants. If the student solved

the problem correctly, he received the next task, in

case he did not solve the task, he received instructions

about mistakes and tips on how to correct them. The

students were motivated by the fact that they did not

know at what stage (what task) the rest of the class-

mates were currently solving.

The project aim was not only to increase the moti-

vation of sports students to learn geometry with the

help of new technologies but also to show the im-

portance and practicality of geometric knowledge in

everyday life and future activities. It was important

to ensure that students learned to come up with their

ideas, were not afraid to express their opinions, and

were able to think logically and critically.

In order to do this, students were offered a re-

search task: to create problems in geometry, which

would be directly connected to football and things re-

lated to it (football ﬁeld, football goal, football ball,

etc.). Surprisingly, but not only the students with the

high level of academic achievements worked hard. As

samples, the students used the problem tasks that they

solved remotely in the ﬁrst phase of the project.

We managed to ensure that students learned to

come up with their ideas, were not afraid to express

their opinions, and tried to think logically, in partic-

ular, critically. In the process of implementing the

designed project, it became clear that we managed to

create conditions with good opportunities to convince

students that mathematics is the way to improve a lot

of circumstances of the surrounding reality. For fur-

ther diversifying the methods of motivation to master

the method of mathematical modeling, it is important

to select and accumulate effective teaching tools and

techniques by mathematics teachers. Among effective

teaching tools and techniques are applied tasks, meth-

ods of implementing interdisciplinary links, prepara-

tion and conduct of special practical work with an ap-

plied focus.

As we have already mentioned in this article, one

of the main problems in the process of forming stu-

dents’ skills of mathematical modeling in Method-

ology of mathematics is the lack of methodologi-

cal materials, especially for high school students.

Therefore, in order to help mathematics teachers,

we have developed and experimentally tested the

textbook “Methodological Tools for Developing Stu-

dents’ Ability to Mathematical Modeling” (Matiash

and Kateryniuk, 2019). In the textbook, it is pre-

sented and substantiated the theoretical aspects of

students’ ability to mathematical modeling, offered

a system of applied problems for the formation of

mathematical modeling skills (tasks to ﬁnd the small-

est or largest values; geometric problem-tasks; phys-

ical problems; stochastic problems; production prob-

lems; everyday and professionally oriented tasks), ex-

plained the methodological aspects of solving prob-

lems based on mathematical modeling. The method-

ological recommendations on the organization of in-

dependent cognitive activity of students in order to

form the skills of mathematical modeling, students’

project activities, and diagnostic tools for the forma-

tion of students’ mathematical modeling skills were

given. As a separate unit of the textbook, we have

provided teachers with a list of publications on the de-

velopment of students’ mathematical modeling skills.

Thus, we have systematized relevant material for

practical use in mathematics lessons at school, and

The Use of ICT Tools in Teaching Mathematical Modeling to Students

681

proposed authors’ tasks. In the textbook, we have

emphasized that at school, according to the learn-

ing outcomes identiﬁed in the mathematics curricu-

lum, mathematics teachers must provide conditions

for the development of students’ mathematical mod-

eling skills. The textbook is prepared for teachers of

mathematics and prospective teachers of mathematics

in order to provide methodological assistance in over-

coming the problem of developing students’ mathe-

matical modeling skills.

Based on the printed textbook “Methodological

Tools for Developing Students’ Ability to Mathe-

matical Modeling”, we have developed an electronic

manual. It is not just an electronic version of the

printed textbook; it is also an electronic supplement

to it. Simple and clear navigation allows turning

pages quickly and easily. The use of an electronic

appendix to the third section of the manual, which

is devoted to the systematization of publications for

further detailed study of materials on this topic, is of

great practical value. It is much more convenient to

use it as it contains all the publications we have men-

tioned, not just a list of them (as in the print version).

They are available for viewing and downloading in

pdf or djvu format. To download the e-manual, the

following link can be used: https://drive.google.com/

ﬁle/d/1K9B9VI6Yr12w0JHG4Y aSOSJVsm3cuGd/

view?usp=sharing.

According to the results of our research and ex-

perimental study aimed at ﬁnding ways to improve

the effectiveness of forming the students’ mathemat-

ical modeling skills, we can summarize that the per-

sonal website of the mathematics teacher is in great

use. The personal website of a mathematics teacher

should facilitate the exchange of experiences with col-

leagues, reduce the distance between teacher and par-

ents, as well as between students and parents, and

most importantly between students and teachers. The

teacher’s website allows implementing an individual

approach, and focusing on the development of the per-

sonality of a particular student. With the help of the

website, the teacher can distribute the required infor-

mation in a short time, share important news, post

educational and methodological hand-outs, creative

work of students, their achievements, and much more.

The personal website of the mathematics teacher has

the opportunity to convey information to students in

multimedia forms.

On the pages of the personal site, our teacher-

experimenter (co-author of this article) added, edited

and commented on materials, posted and analyzed

test tasks, discussed various issues, and communi-

cated with students. It is observed that modern stu-

dents spend increasingly much time online and it is

much easier for them to get the required materials on-

line than to use libraries, books, reference books, and

even writing home-assignment. According to most

teachers’ opinion, it is not very good. However, teach-

ers should understand that they work with students of

the new generation, and train them for life in a new

society, which requires a modern personality, who is

ready to live in a new information society. Out of 100

students (the survey was conducted at the Regional

Sports and Humanitarian Lyceum-Boarding School

of the Municipal Higher Education Institution “Vin-

nytsia Humanitarian Pedagogical College”), 95 have

mobile phones (95% of respondents). 68% of stu-

dents have mobile Internet access, and the rest stu-

dents use Wi-Fi. After classes, 100% of respondents

have access to the Internet. It conﬁrms the ability

of the mathematics teacher to communicate actively

with students via the personal website.

During the experiment studying, the mathemat-

ics teacher’s website provided an opportunity to draw

students’ attention by interesting learning materials,

in particular, to the teacher’s information competency.

It expanded the possibilities of conveying interest-

ing and important information to students and show-

ing the wide application of the acquired mathematical

knowledge in everyday life.

Such an environment allowed implementation of

educational functions. The teacher, for example,

placed the text of a problem task on the forum of the

website and set the instruction: to construct a math-

ematical model. In the process of completing the

task, students discussed, suggested their models, and

when everyone expressed the opinion, the teacher cor-

rected their steps, explained the incorrectness or in-

accuracy of their reasoning, if a mathematical model

did not correspond to the task in a result. On the con-

trary, the teacher could emphasize the right consider-

ations, encourage further work, and summarize which

model was still correct, or the most appropriate. The

next step was the process of collective research of

the mathematical model. Those students, who were

not able to participate in the online discussion, had

the opportunity to open the forum later and get ac-

quainted with the discussion and the whole process

of solving the tasks proposed by the teacher. Thus,

we consider the function of the mathematics teacher’s

website “teacher-student interaction” to be quite ef-

fective. While experimental training, the question

was put: “Why can’t the teacher’s personal website

be replaced by communication with students on social

networks?” According to the results of our research,

we came to the conclusion that it is better to leave so-

cial networks for personal communication and enter-

tainment, and everything that the mathematics teacher

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

682

wants to convey to his/her students is better to post on

the pages of the teacher’s personal website. In our

opinion, the modern mathematics teacher can create

a personal website by him/herself with the help of a

site designer, using a simple and free service Google

Sites, as well as due to information competency which

was formed in the process of professional training in

a higher educational institution.

We created the mathematics teacher’s per-

sonal website (URL: https://sites.google.com/view/

kateryniuk/). Its content and effectiveness in improv-

ing the conditions for the formation of students’ math-

ematical modeling skills were essential for us.

Thus, modern content management systems

(CMS), the availability of free hosting, and free web-

site designers allow teachers to create and develop

their personal websites. The personal pedagogical

website is not just a matter of time, but a necessity

for the fruitful work of any subject teacher who cares

about providing conditions for improving the effec-

tiveness of teaching. Distance learning and external

studies are developing extremely, so the exchange of

tasks between a student and a teacher is rapidly mov-

ing into the ICT medium. In our opinion, the main

tasks that are effectively solved with the help of the

teacher’s personal website are searching for new ways

to interact with students and the organization of dis-

tance learning and project activities on the Internet.

5 RESULTS AND DISCUSSION

Different research and experiment methods allowed

to construct an up-to-date method for the formation

of students’ mathematical modeling skills and to ex-

plain the place and role of ICT use in this method.

In order to determine the attitude of the mathematics

teachers to some results of our research (expediency

and quality of the electronic manual “Methodologi-

cal Tools for Developing Students’ Ability to Math-

ematical Modeling”), we used the method of expert

evaluation (Delphi method). The main stages of im-

plementation of the method of expert evaluations in

our study were: expert selection, identifying factors

for the survey, conducting the survey, analyzing the

results of the survey, and processing the results.

32 experts were chosen from a group of math-

ematics teachers of Vinnytsia and Vinnytsia region,

who took courses of continuing professional develop-

ment for teachers in-service based on the Department

of Algebra and Methods of Teaching Mathematics at

Vinnytsia Mykhailo Kotsiubynskyi State Pedagogi-

cal University. Before the presentation of the elec-

tronic manual “Methodological Tool for Developing

Students’ Ability to Mathematical Modeling” to ex-

pert teachers, we found out the experts’ preliminary

vision of the importance of certain characteristics of

the use of electronic manuals in teaching mathematics

to students.

The survey suggested 10 factors:

(1) quality of the manual design;

(2) the presence of clear scientiﬁcally grounded rec-

ommendations for the formation of students’

mathematical modeling skills;

(3) increasing the number of school mathematics

lessons;

(4) modern, relevant for students the content of tasks;

(5) the conviction of the mathematics teacher in the

need for special conditions for the formation of

students’ mathematical modeling skills;

(6) high teacher salaries;

(7) successful selection of applied tasks;

(8) the teacher’s methodological competency;

(9) students’ interest in learning material;

(10) teacher’s knowledge and understanding of the cri-

teria and indicators of the formation of students’

ability to mathematical modeling.

The used methods of mathematical statistics were

based on ranking. In our case, each factor was as-

signed a rank from 10 to 1 by each expert (in descend-

ing order to determine their relative importance). The

required condition for the reliability of the collective

assessment was sufﬁcient consistency of opinions of

the interviewed experts. To determine the consistency

of group assessments, we used the concordance coef-

ﬁcient – a common rank correlation coefﬁcient for a

group of experts. Statistical processing of the results

of the teacher survey allowed us to state that the most

signiﬁcant ranks were:

5. Conviction of the mathematics teacher in the

need for special conditions for the formation of stu-

dents’ mathematical modeling skills;

7. Successful selection of applied tasks;

4. Modern, relevant for students the content of

tasks;

9. Students’ interest in learning material;

8. Teacher’s methodological competency.

The consistency of experts’ opinions was assessed

by the concordance coefﬁcient:

W =

− n)

∑

j=1

∑

i=1

i j

−

m(n + 1)

where m – the number of experts, n – the number of

factors.

The Use of ICT Tools in Teaching Mathematical Modeling to Students

683

In our case, m = 32, n = 10. The obtained value is

W = 0.64, according to the scale for rank correlation

coefﬁcients, which falls into the interval (0.6; 0.8). It

demonstrates a good consistency of experts.

According to the results of the expert survey, it

was found that the designed electronic manual con-

tained the important characteristics of the use of elec-

tronic manuals in the teaching of mathematics to stu-

dents that were relevant for mathematics teachers,

and therefore would be useful for the formation of

students’ mathematical modeling skills. In order to

identify the needs of mathematics teachers in printed

or electronic manuals for the formation of students’

mathematical modeling skills, the survey was pro-

posed. The seventh question of the questionnaire was:

“Would you like to have a manual in printed or elec-

tronic version?”.

The response results of 500 surveyed teachers to

the proposed question are presented in the following

diagram (ﬁgure 1).

Analysis of respondents’ answers to other ques-

tions of the proposed survey allowed to assert that

78% of mathematics teachers (out of 500 participants)

need educational and methodological literature on the

formation of students’ mathematical modeling skills;

97.2% of teachers ﬁnd it useful to develop a system

of applied tasks; 66.4% of teachers need to master

the method of solving problem tasks based on the

mathematical modeling; 58.4% of teachers state that

it is appropriate to develop lesson outlines for teach-

ers oriented on the topic of mathematical modeling,

and 30.8% of teachers admit the usefulness of such

outlines. 90.4% of teachers are interested in diagnos-

tic tools for the formation of students’ mathematical

modeling skills. 95.6% of respondents express a de-

sire to get acquainted with the scientiﬁc and method-

ological publications on the formation of students’

mathematical modeling skills. 58.6% of teachers

want to have a textbook on teaching students math-

ematical modeling in the printed version and 68.3%

of teachers – in the electronic version.

In the process of pedagogical experiment, and ac-

tive communication with mathematics teachers, it was

conﬁrmed that “the use of ICT can signiﬁcantly im-

prove the efﬁciency of learning messages and data cir-

culating in the educational process, due to their timeli-

ness, usefulness, appropriate dosage, availability (in-

telligibility), noise minimization, and operational re-

lationship between the source of educational informa-

tion and the student, adapting the pace of presenta-

tion of educational material to the speed of its assim-

ilation, taking into account individual characteristics

of students, effective combination of individual and

collective activities, teaching methods and tools, or-

ganizational forms that to some extent contributes to

solving the problems of educational process human-

ization. Students and teachers remain the main par-

ticipants in the educational process. Computers, soft-

ware, and communication tools, are the only means of

their activities. The effectiveness and efﬁciency of ed-

ucational and cognitive activities of students depend

on the teacher’s awareness and skillfulness” (Zhaldak,

2003).

6 CONCLUSIONS

ICT and communication tools play an important role

in the forming technology of students’ mathematical

modeling skills. Nowadays, it is important to teach

prospective mathematics teachers to use ICT in the

creation and study of mathematical models, and in

the process of teaching mathematics to students. Due

to forced conditions, distance learning development

is accelerating, so communication between teachers

and students is rapidly moving into the ICT medium.

The results of teachers’ surveys and the pedagogi-

cal experiment lead to the conclusion that the task

of methodologically balanced use of ICT is compli-

cated by the necessity to change the personal atti-

tude of both teachers and students to self-education,

self-development, and cooperation. Simultaneously,

the Ukrainian scientists’ ideas are conﬁrmed that the

readiness of mathematics teachers to use ICT depends

signiﬁcantly on:

• understanding the effectiveness of the use of ICT

and communication tools in the educational pro-

cess;

• ability to use modern computer equipment and

tools of communication in professional activities;

• ability to competently assess the beneﬁts, oppor-

tunities, and limitations of the use of ICT in the

educational process;

• understanding the place and role of autonomous

work of students using ICT;

• understanding the changes in the functions of the

teacher in the organization of the educational pro-

cess with using ICT (Zhaldak and Hrybiuk, 2014).

Our research on the effective use of and other

methodological tools and techniques for forming stu-

dents’ mathematical modeling skills was based on the

same principles as the principles of the development

of computer-based teaching methodological educa-

tional systems were grounded by Zhaldak and Hry-

biuk (Zhaldak and Hrybiuk, 2014): the principle of

interest in learning; the principle of adaptability to

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

684

I would like to have this manual ...

100

150

200

250

300

in a printed version

in an electronic version

292

340

Figure 1: The results of the teacher survey.

the individual characteristics of students; the princi-

ple of the student’s search activity; the principle of

self-evaluation and self-actualization; the principle of

individualization of learning; the principle of cooper-

ation and mentoring. Thus, adherence to these prin-

ciples in the process of formation of students’ mathe-

matical modeling skills is an important component of

the methodological activities of mathematics teachers

at school.

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