=Paper=
{{Paper
|id=Vol-2732/20201082
|storemode=property
|title=Providing the Fundamentalisation of Operations Research Learning
Using MAXIMA System
|pdfUrl=https://ceur-ws.org/Vol-2732/20201082.pdf
|volume=Vol-2732
|authors=Uliana Kohut,Mariya Shyshkina
|dblpUrl=https://dblp.org/rec/conf/icteri/KohutS20
}}
==Providing the Fundamentalisation of Operations Research Learning
Using MAXIMA System==
https://ceur-ws.org/Vol-2732/20201082.pdf
Providing the Fundamentalisation of Operations
Research Learning Using MAXIMA System
Uliana Kohut1[0000-0002-2861-2274] and Mariya Shyshkina2[0000-0001-5569-2700]
1
Drohobych Ivan Franko State Pedagogical University, 24 I.Franko Str., Drogobych
2
Institute of Information Technologies and Learning Tools of NAES of Ukraine
9 M.Berlynskoho St., Kyiv, Ukraine
1ulyana3001@gmail.com
2shyshkina@iitlt.gov.ua
Abstract. In the article, the problems of using the systems of computer mathe-
matics (SCM) as a tool to provide the fundamental component of operations re-
search learning and students research activities support are considered. The role
of SCM in the process of bachelors of informatics training and special aspects of
pedagogical applications of these systems in the “Operations research” study is
defined. The analysis of the basic concepts of the fundamentalisation of educa-
tion and in particular the basic concepts of the fundamentalisation of informatics
disciplines learning is summarized. The attempt to distinguish explicitly and
specify the fundamental concepts in the content of “Operation research” learning
is made. The method of “Operation research” study using Maxima system as a
tool to support the basic concepts learning and an investigation is approved. The
results of the pedagogical experiment on MAXIMA application to support the
fundamental component of learning in the course of “Operation research” study
and the analysis of its results are reported.
Keywords: “Operations research”, MAXIMA, learning tools, fundamentalsa-
tion of learning, informatics disciplines, learning environment, educational uni-
versity.
1 Introduction
1.1 Research objectives
In course of information society formation as scientific and technological progress is
currently enhancing, there is a challenge for an educational system to provide training
of specialists for their immediate inclusion in the technological processes at the pro-
duction level. After all, it is impossible to predict accurately at the time of enrollment
at the university the state of the art of technological achievements that could be reached
at the moment of graduation [14].
The way out of this critical situation in the education system is in the fundamentali-
zation of education, which is due to the orientation of the education system to create
holistic, generalized knowledge that would be the core of all the knowledge acquired
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
by the student, which would unite the knowledge gained during the training process
into a single system [11].
According to V. G. Kinelev, the purpose of fundamental education is to create fa-
vourable conditions for the development of flexible and multifaceted scientific think-
ing, various ways of perceiving reality, the formation of an internal need for self-reali-
zation and self-education throughout life [4].
Over time, the rapidly growing amount of diverse information leads to "the need for
their adequate structuring and reflection in the disciplines. Mathematics and informatics
disciplines play an important role in the process of mastering some of the most basic
knowledge that is the basis for the formation of general and professional culture, rapid
adaptation to new professions, specialities and specializations " [3].
The aim of the article is the justification of Maxima system use of in the process of
"Operations research" learning in a pedagogical university as a toll for fundamentalisa-
tion of learning and providing investigative approach through the analysis of the basic
concepts in the course of study.
1.2 Problem statement
The role of fundamental knowledge in modern scientific studies is mentioned in the
works of many authors regarding the foundations of classical science. In particular,
B.G. Kuznetsov notes that the style of physical thinking radically changed in the twen-
tieth century. In particular, he mentions the loss of uniqueness, the erosion of the con-
tent of classical physical concepts in their relativistic interpretation [5]. “According to
many scientists, in our time it is impossible to say where physics ends and technology
begins, where mathematics ends and physics begins” [14], notes L. S. Khizhnyakova.
This has a significant impact on the development of teaching methods in many disci-
plines.
In many studies, the fundamentalisation of education is associated with equal access
to education. It is also considered as a "fundamentally-knowledgeable" frame of per-
sonality development, provides systematic knowledge, holistic perception of the world
and the person in it, the creation of a basis for professional culture and mastery [10],
[12], [12].
Nowadays it becomes necessary to form not only specific but also generalized
knowledge and skills. Such knowledge and skills, formed in the process of a certain
discipline study, are then available for the use in the course of the other disciplines
study or for other professional activities [1], [2], [9]. The fundamentalisation of educa-
tion is facilitated by the consideration and use of interdisciplinary relations, the research
work of teachers and students at the intersection of basic and applied sciences [12].
1.3 The Research Methods
In the course of the study, the scientific and methodological foundations of using the
Maxima system are substantiated and analyzed for the unchallenged fundamentalisa-
tion of the training of computer science disciplines for computer specialist. The study
is based on the methods of theoretical analysis, generalization and systematization of
scientific facts about the pedagogical processes and phenomena, methods of system
analysis and modelling, pedagogical observations and generalization of pedagogical
experience, as well as the results of the pedagogical experiment. The study was carried
out in the framework of the implementation of the planned research undertaken in the
Institute of Information Technologies and Learning Tools of NAES of Ukraine and the
Department of Informatics and Computing Mathematics of the Drohobych Ivan Franko
State Pedagogical University.
Such interdisciplinary methods and procedures are used in informatics as analysis
and synthesis, induction and deduction, visualization and formalization, algorithmiza-
tion and programming, informative-logical, mathematical and computer modelling,
program management, expert evaluation, identification and others. It is necessary to
acquire them in complex, otherwise, there is not a sufficient level of mastering the ma-
terial of informatics disciplines.
2 The Research Results
The combination of education and science is a condition for the modernization of the
education system, the main factor for further development should be provided by the
fundamental education, the intensification of scientific research in higher educational
institutions, research institutions [13].
S. O. Semerikov, determining the fundamentalisation of education by the totality of
interrelated functions (methodological, vocational, developmental, prognostic, integra-
tive), determines the appropriate ways to provide the educational process with funda-
mental components [12]:
─ saturation of the content of higher education with systemic theoretical knowledge,
fundamental theories, concepts, ideas;
─ the dominance of research methods of teaching, creative activity, integration of ideas
and methods of science, teaching and scientific creativity;
─ self-development of a student as a subject of mobile educational, professional and
research activities.
In the process of computer science specialists training in fundamental disciplines, it is
necessary to attribute primarily philosophical, informatics and natural-mathematical, as
well as professional and practical training disciplines. Along with the relevant
knowledge, the opportunity to study professionally oriented disciplines should be pro-
vided, so the fundamental knowledge that is the most basic and stable in time provide
the possibility of further professional growth of a specialist [12], [13].
The essential feature of the fundamental disciplines is that in the process of study
the mechanisms of cognition and the basics of understanding the processes and phe-
nomena of the world are formed. The pragmatic necessity of applying a certain mathe-
matical apparatus or understanding the essence of a certain physical effect when per-
forming a professional task requires additional study of mathematical and natural dis-
ciplines.
Fundamental learning provides the theoretical foundations of the speciality follow-
ing the requirements for the level of theoretical training of a teacher of the correspond-
ing profile and is based on the latest achievements of science [6].
According to M. I. Zhaldak, the use of modern ICTs plays an important role in the
fundamentalisation of knowledge, a comprehensive and thorough study of the domain,
the formation of knowledge necessary for a valid explanation of cause-effect relation-
ships of the processes and phenomena studied. Fundamental knowledge is important
for applied research, and the needs of everyday practical activity of people cause and
stimulate the corresponding cognitive activity, aimed at the disclosure of laws of a fun-
damental nature [16].
Higher education in informatics is largely based, as before, on the foundations of an
accumulative model of new knowledge, when skills are formed to solve standard pro-
fessional tasks, act in known situations [10].
We consider the organization of information and educational space at universities as
the basis for fundamental teaching of mathematics and computer science, while the task
is not the formation of pragmatic, highly specialized knowledge, but methodologically
important, invariant knowledge, based on a holistic perception of the world, intellectual
development of personality being able to be adapted to rapidly changing socio-eco-
nomic and other general processes. The fundamentalisation of informatics specialists
training is based on the emphasis in the content of instruction on philosophical and
mathematical foundations of educational disciplines. The practical implementation of
this process in the preparation of future specialists in computer science should be car-
ried out using computer mathematics systems that arise through supporting the teaching
of mathematical and computer sciences, by combining the theoretical and applied com-
ponents of student training, strengthening the professional orientation of their education
and the implementation of inter subject communications [12], [15], [16].
By the fundamentalisation of informatics disciplines learning, we understand the se-
lection of the basic concepts, fundamental theoretical principles, concepts, ideas under-
lying the system-forming knowledge and skills in the field of mathematical and infor-
matics disciplines, the implementation of interdisciplinary communications, providing
a competency-based approach to improve the level of training students, their full-
fledged activities in the information society [15].
An analysis of the basic concepts of the fundamentalisation of education is summa-
rized in Table 1.
In the process of teaching first-year students of speciality 014 Secondary Education
(Informatics), Ivan Franko Drogobych State Pedagogical University revealed their con-
scious orientation for work in the field of computer science. Most of them feel confident
when working with popular software environments and quickly perform typical opera-
tions. But the need to deviate from the usual technological schemes causes difficulties.
The experience of introduction of the systems of computer mathematics (in particular,
MAXIMA) into the learning process obtained during the educational experiment that
had been conducted in Ivan Franko Drogobych State Pedagogical University in 2016
was disseminated into the learning process of the several educational institutions of
Ukraine (Ternopil Volodymyr Hnatiuk National Pedagogical University, Kryvyi Rig
National University, Kherson State University and others) [15].
Table 1. Analysis to understand fundamental
Fundamental large, strong, stable, deep, basic, main (Ozhegov SI)
Fundamental stable and universal general theoretical knowledge,
knowledge the content of which is noted by generalization, struc-
turedness, in which the internal-no and external con-
nections of various subject areas are revealed, based
on which the person's ability is formed to learn new
knowledge, to navigate problems (a tool to achieve
scientific competencies) ( Laptiev V.V.) [6].
Fundamental strengthened interconnections of theoretical and
preparation practical training of a specialist for professional ac-
tivity, aimed at the formation of a holistic scientific
picture of the world, at the individual and profes-
sional development of a student, together provide a
high level of education (S. Semerikov) [1210].
Fundamentalization a qualitative change in higher education based on the
of education principle of fundamentality, the introduction of theo-
ries of a high degree of generalization, with increased
information capacity and universal applicability into
the educational process (A. Rostovtseva).
Fundamentalization improving the quality of the fundamental preparation
of informatics education of the student, his system-forming and invariant
knowledge and skills in the field of computer science,
makes it possible to form the qualities of thinking
necessary for full-fledged activity in the information
society, for the dynamic adaptation of a person to this
society, for the formation of the internal need for con-
tinuous self-development and self-education, ac-
counting for corresponding changes in the content of
academic disciplines and the methodology for the im-
plementation of the educational process (S. Seme-
rikov) [12].
Fundamentalization of in- the selection in the content of the discipline of basic
formatics training concepts, fundamental theoretical principles, con-
cepts, ideas underlying system-forming knowledge
and skills in the field of mathematical and informatics
disciplines, the implementation of intersubject com-
munications, providing a competency-based ap-
proach to improve the students’ level of training, their
full-fledged activity in the information society (Shy-
shkina M.P., Kohut U.P.) [14].
The learner begin to iterate through the available actions to obtain the desired result.
The reason for this is the lack of knowledge of those fundamental theoretical and tech-
nical principles on which these environments are built. And as a result, the system con-
struction of a new algorithm is practically impossible.
There is a shift in the content of knowledge to the technological side. This is the fact
that in real information processes it is objectively difficult to distinguish explicitly and
specific fundamental components. The training of mathematical and computer science
disciplines as fundamental can be carried out as follows:
─ for any level of education, a system of fundamental theoretical principles, concepts,
methods and means is being developed, which are studied in this discipline and suc-
cessfully assimilated. According to such a system, the boundaries of fundamental
knowledge for a chosen level are determined;
─ the system of each level is used as the basis for the system of the next level and is
supplemented by new components and theoretical justifications of previous compo-
nents;
─ the study of each theoretical component is necessarily accompanied by its practical
use in the most accessible form. In this case, the student will understand not only the
content of the component but the fact that knowledge of the theoretical foundations
of computer science is very important for solving practical problems will be obvious.
It is very important to show the existence and ways of using the fundamental com-
ponents of knowledge in modern computer programs and technologies. In this case,
the material will be better absorbed.
In mathematical and computer science disciplines three components must be presented
in unity: scientific, technical and technological. But they are implemented differently
depending on the level and goals of training. At each level, a place must be found for
fundamental knowledge.
The role of the fundamental component is often underestimated. In pedagogical
practice, training will be introduced primarily in the technological direction. The meth-
ods and techniques used are theoretically substantiated and not analyzed. Students have
a poor understanding of the fundamental component of computer science courses com-
pared to mathematics and physics. This is because in real information processes it is
objectively difficult to identify, clearly and characterize specific fundamental compo-
nents. At the same time, fundamental concepts play a key role in the process of funda-
mentalisation of training, which is also closely related to the basic concepts of related
disciplines.
In computer science, such interdisciplinary methods and procedures are used as anal-
ysis and synthesis, induction and deduction, visualization and formalization, algorith-
mization and programming, information-logical, mathematical and computer model-
ling, program management, expert evaluation, identification and others [12]. They must
be mastered comprehensively, otherwise, there will not be a sufficient level of mastery
of the material of information disciplines. All this testifies in favour of fundamentality
of the content of the training.
At the same time, fundamental concepts play a key role in the process of fundamen-
talisation of education, which is also closely related to the basic concepts of related
disciplines.
For example, the fundamental concepts of an algorithm and operation are closely
related to the concept of a function, which can be associated with an operation, the
implementation of which implements this function. Considering the concept of algo-
rithm and operation only from the procedural side, that is, as a prescription character-
izing the transformation that must be performed, the process side of this concept re-
mains aside. Then the algorithm acts as a process for solving a specific problem, the
result of which is a solution. This concept acquires applied to content in solving prob-
lems arising in practice. At the same time, excessive bias towards the applied applica-
tion of the algorithm does not contribute to understanding its relationship with the math-
ematical foundations of this theory.
Therefore, the selection of the fundamental concepts of informatics disciplines, their
awareness and consolidation of the experience of research activities is an integrable
component of the organization of training, the creation of intersubject communications,
the formation of a holistic system of knowledge and ideas among students about both
the theoretical basis and the ways of applying the acquired knowledge in practice.
It is the relationship with the mathematical foundations that are an essential factor in
the fundamentalisation of the teaching of information disciplines. In particular, a pos-
sible reason for misunderstanding in many cases is that it is not possible in the right
sense to consider the relationship between the various aspects of solving the problem -
constructing an analytical relationship, and based on the mathematical laws governing
the description of the phenomenon, the very phenomenon that a computer program im-
plements are obtained. Using a computer program, you can simulate the dynamics of
the system or the manifestation of a phenomenon.
In various disciplines consider modelling various phenomena. In this sense, it is ad-
visable to focus further attention on one of the disciplines where it would be possible
to demonstrate the advantages of a systematic approach. For this, the discipline "Oper-
ations Research" was chosen, which later became the subject of an experimental study.
Operations research is the theory of using scientific quantitative methods to make
the best decisions in various fields of human activity. This science gives objective,
quantitative recommendations for managing targeted human actions.
The following fundamental concepts arise in the study of operations: an operation, a
system, a model, modelling, a systematic approach, a task, an optimality criterion (qual-
ity, efficiency), as well as closely related concepts of a method, procedure, function, in
general form the fundamental core of learning mathematics - of general and informatics
disciplines. Besides, the so-called fundamental algorithms (methods) play an important
role in the content of training in the study of operations, which must be mastered when
solving a certain set of classical problems: resource allocation problems (transport
problem, assignment problem) network planning problem; the task of choosing a route
(the travelling salesman problem) problems of game theory.
By the example of teaching this discipline, one can demonstrate the relationship of
mathematical methods and the implementation of the corresponding operations and al-
gorithms with the visualization of results, which reflect the relationship of certain ob-
jects and their properties. Based on the curriculum of the course "Operations Research"
for the specialities 014 Secondary Education (Informatics) of the Drogobych State Ped-
agogical University, the fundamentalisation of operations research training can be gen-
eralized as follows (Fig. 1).
Operations research
Modelling
game optimization dynamic
theory problems on programming
graphs
fundamental theorems
concepts fundamental algorithms fundamental theorems
concepts concepts
model, task Neumann model,
building a Bellman's
model, system,
the operation, theorem modelling,
framework of systematic
theorem
optimal option,
the best option the performance minimum cost, approach,
criterion finding the optimal variant,
shortest paths the criterion of
in the network efficiency
Fig. 1. Fundamental concepts of operations research
To uncover the essence of the problems that arise in the search for the advantages of
using ICT in teaching operations research, we need to consider the concept of an edu-
cational task.
The training task is aimed at mastering a certain mode of action, while the practical
task is to obtain the result contained in the task condition. When solving any of these
problems, the subject acquires certain knowledge, but it is the educational tasks that
have an exceptional impact on the functioning and development of educational activity
[7].
By the example of training in the study of operations, one can demonstrate the rela-
tionship of mathematical methods and the implementation of their corresponding oper-
ations and algorithms with the visualization of results, which reflect the relationship of
certain objects and their properties. This relationship is reflected in the model of the
educational task "Operations Research".
The model of using the training task with “Operational Research” (Fig. 2.) reflects
the interconnection of fundamental concepts, mathematical method, fundamental algo-
rithm, algorithm and operations that are implemented in the process of solving. This
shows the role of fundamental concepts as an integrative component of the training
“Operational Research”.
Forming part Permitting part
mind method fundamental algo- algorithm of
decoupling
rithms
tasks
mathematical model implementation of the algo-
rithm by SCM
Integrable components
Mastering operations research performs an integrable and complex function, providing the
fundamental basis for the competence of future computer scientists .. Fundamental
concepts of operations research: operation, system, model, modelling, system approach,
task, optimality criterion (quality, efficiency)
Form of professional competencies in computer science
Fig. 2. The model of primary tasks with the previous operation
The use of ICT, in particular SCM as a means of training, is associated with problems
of increasing the level of fundamental training of future specialists in computer science:
─ combination of theoretical, applied and practical aspects;
─ presentations in a systematic form of theoretical information about methods and the
main provisions of decision theory, computer mathematics systems and the for-
mation of practical skills for their application to solving real practical problems;
─ deepening students' knowledge on issues related to the study of the effectiveness of
solving applied problems using computer mathematics systems, analysis and inter-
pretation of the results;
─ development of the algorithmic style of students' thinking through the development
of algorithms and their software implementation;
─ the formation of students' skills of independent work with theoretical material and
computer mathematics systems.
Correctly selected tasks of professional orientation for laboratory work enable the
teacher to use SCM as a means to ensure inter subject communications between com-
puter science and mathematics. The foundation is laid for the formation of ICT compe-
tencies [15].
For each laboratory lesson, individual task options have been developed that are di-
vided into three difficulty levels. The difficulty level of the assignment for completion
is selected by the student. The task of the first level of complexity corresponds to the
reproductive level of assimilation of knowledge and is evaluated by a small (up to 5
points) number of points. To solve problems of the second level of complexity, the
heuristic nature of the intellectual activity is required; tasks are estimated by the average
(up to 15 points) number of points. The third level of complexity includes tasks that
require a creative approach. The tasks are formulated in such a way that for their solu-
tion it is necessary to have elements of divergent thinking. Divergent thinking is usually
inherent in creative individuals, inclined to create new combinations of those elements,
others use only the usual way. With the successful completion of tasks of this level, the
student deserves the most (up to 20 points) number of points. Thus, differentiation of
training is implemented, the student sees the results of his work and can evaluate the
objectivity and accuracy of rating points [15].
The most important element of laboratory studies in the course "Operations Re-
search" appropriately selected tasks. Tasks are given to students taking into account the
fundamentals of the theory presented at the lecture. As a rule, in a laboratory lesson,
the main attention is paid to the formation of specific abilities and skills, based on which
the content of students 'activity is determined - solving problems, graphic works, clari-
fying categories and concepts of the studied discipline. When analyzing tasks with stu-
dents, the teacher should pay special attention to the formation of abilities to compre-
hend and understand the material on the topic.
Considering the system of individual tasks for laboratory work, as well as tasks for
the practical protection of modules, one should analyze the problems and advantages
of using such tasks. When preparing tasks, you should carefully approach the determi-
nation of the level of difficulty. This can only be helped by the teacher's experience, his
ability to identify key points in the training material, and understanding the relationship
of the tasks with other disciplines. Also important is the question of the relationship of
difficulty levels in one task. For laboratory work, it is more appropriate to set tasks
where the fulfilment of tasks of a higher level of complexity is possible provided that
the tasks of the previous level of complexity are completed. Otherwise, students often
overestimate their capabilities, take on complex tasks immediately, cannot complete
them, and they don't have enough time to complete simple tasks. Thus, they do not gain
those rating points that they could gain by correctly assessing their capabilities. It is
advisable to use tasks of different difficulty levels when conducting modular controls
and exam tickets, where you need to cover all the training material.
Each laboratory work is accompanied by a list of questions for self-examination and
several tasks to perform during students' independent work. The main task is to form
practical skills for future specialists in formalizing tasks and solving them using SCM
tools.
On the advantages of a system of multi-level individual tasks, the accuracy and ob-
jectivity of the assessment come to the fore here. The classical four-point student com-
petency assessment system, despite its usual simplicity, had some drawbacks regarding
the objectivity of assessment. A 100-point rating system gives greater accuracy in the
assessment, but here the problem arises of ensuring this accuracy - what maximum error
can a teacher make when setting rating points. Differentiation of the complexity of
tasks, and accordingly the number of points for their implementation, allows to some
extent to ensure acceptable accuracy and objectivity of the assessment.
Differentiation of the rating of students is also carried out according to a disciplinary
indicator. It would be wrong to give equal scores to students who complete the curric-
ulum on time and to those who, for no good reason, are significantly late. In this case,
students, although successfully, did the laboratory work on time, passed the module
control, etc., receive only 1 point for the control element, regardless of the level of
complexity of the tasks performed.
Table 2 shows an example of a variant for determining the semester rating of student
performance.
Table 2. Scores ratings per semester
Number of con-
Number of Total
Form of control trol measures per
points points
semester
Attending classes 1 10 10
Performing laboratory work
1st level of difficulty 0,5 10 5
2nd level of difficulty 1,5 10 15
3rd level of difficulty 2 10 20
Writing a report for each la-
1 10 10
boratory work
Practical protection of each labora-
1 10 10
tory work
test 30 1 30
Total for completing tasks:
1st level of difficulty 65
2nd level of difficulty 80
3rd level of difficulty 100
3 Implementation and Evaluation
The results of the pedagogical experiment on the use of Maxima system in the
process of “Operations research” teaching.
During 2016-2019 the experimental research has been conducted. During the exper-
iment, the SСМ MAXIMA was implemented in the process of "Operations research"
teaching concerning the students of the Institute of Physics, Mathematics, Economics
and Information Technology of the Drohobych Ivan Franco State Pedagogical Univer-
sity (education and qualification level "Bachelor", for the specialities 014 Secondary
Education (Informatics). In the experiment, the specially worked out methodology of
"Operations research" teaching using Maxima system was tested. In the experiment, on
his forming stage, 50 students participated. The experiment confirmed the research hy-
pothesis concerning the increase of the level of professional competences development
in the process of studies according to the worked out methodology. It also showed that
using cloud technology students can achieve greater access to the means of research
activities (it is possible to attain expansion of access to research activity facilities).
In the experiment, they involved both the local version of the system, installed on
the student computer desktop and the cloud-based version that was posted on the virtual
desktop.
Results formative stage of the pedagogical experiment in the control and experi-
mental groups and comparative histogram distribution educational achievements stu-
dents on the results of the final exam discipline "Operations Research" is shown in
Fig. 3.
55,36%
60,00%
50,00% 40,63% 42,19%
40,00%
23,21% CG
30,00% 16,07%
20,00% 9,38% 7,81% EG
5,36%
10,00%
0,00%
low middle sufficiency high
Fig. 3. A comparison of educational achievements of students on the results of final control the
course "Operations Research" after the forming stage of the experiment
Processing of the experiment results and evaluation of the efficiency of the developed
technique was carried out by methods of mathematical statistics [15]. The objective of
the experiment was to identify differences in the distribution of certain characteristic
(the level of formation of individual components of professional competence) compar-
ing two empirical distributions according to the χ 2 - Pearson criterion, λ – Kolmogo-
rov-Smirnov criterion [15].
χ2 - Pearson criterion. The samples in the study are random and independent. The
measurement scale is С = 7 categories (1-39, 40-59, 60-66, 67-74, 75-81, 82-89, 90-
100). The number of the degree of freedom v = C – 2 = 5.
The null hypothesis H0: the distribution of the estimates for the student residual
knowledge concerning the use of systems of computer mathematics in the control (n1
= 56) and experimental samples (n2 = 64) to the forming stage of the experiment do not
differ (і = 0, 1, …, 6).
Q1і – number of participants in the control group who scored i points;
Q2і – number of participants in the experimental group who scored i points.
Alternative hypothesis H1: the distribution of the estimates for the student re-
sidual knowledge concerning the use of systems of computer mathematics in the control
(n1 = 56) and experimental samples (n2 = 64) to the forming stage of the experiment
differ (і = 0, 1, …, 6).
The value of χ2 is calculated according to the formula
1 C 1 n1Q2i n2Q1i
2
Texp Q Q
n1n2 i0 1i 2i (1)
The results of calculating statistics of these samples are given in the Table. 3.
Table 3. The calculation of the χ2 for the control and experimental groups before the forming
experiment
I Q1i Q2i S12i
0 (F) 0 0 0
1 (FX) 28 36 3136,00
2 (E) 36 38 1674,38
3 (D) 18 24 3510,86
4 (C) 16 18 30,12
5 (B) 10 6 23104,00
6 (A) 4 6 2560,00
Т 2,372723
From the table of values, χ2 for the level of significance α=0,05 and number of degrees
of freedom of v = С – 2 = 5 determine the critical value of statistics of Тcritical = 11,07.
Since the obtained value Тexp< Тcritical (2,372723< 11,07) does not fall in the
critical region [χ2, +∞], this suggests that before the forming stage of the experiment
the level of students’ residual knowledge concerning SCM using in the control and
experimental groups do not differ significantly.
The level of students knowledge on the course "Operations research" as well as pro-
fessional disciplines was checked according to the results of complex state examination
to justify the influence of methodology of SCM using as “Operations research” teach-
ing tools on the increase of the level of some components of professional competence.
Null hypothesis H0: distribution of students estimations on “Operations research” in
the control (n1 = 56) and experimental samples (n2 = 64) after the formative forming
stage of the experiment do not differ (і = 0, 1, …, 6).
Q1і – number of participants in the control group who scored і points;
Q2і – number of participants in the experimental group who scored і points.
Alternative hypothesis H1: distribution of students estimations on “Operations re-
search” in the control (n1 = 56) and experimental samples (n2 = 64) after the formative
forming stage of the experiment differ (і = 0, 1, …, 6).
The calculation results of the statistics of these samples are given in Table 4.
The calculation of χ2 criterion for the experimental and control samples after con-
ducting the formative stage of the experiment showed that Тexp >Тcritical (30,20408>
11,07). This is the reason for rejecting the null hypothesis.
The acceptance of the alternative hypothesis suggests that these samples have statis-
tically significant differences, i.e., the experimental method is more effective than the
traditional one.
Table 4. Calculation of χ2 for the control and experimental groups after the formative experi-
ment on the course "Operations research"
I Q1i Q2i S12i
0 (F) 0 0 0
1 (FX) 18 12 30720,00
2 (E) 50 22 215168,00
3 (D) 12 30 79213,71
4 (C) 14 34 84672,00
5 (B) 12 20 15488,00
6 (A) 6 10 7744,00
Т 30,20408
Taking into account that in the experimental groups the training of students was per-
formed according to the developed methodology, it can be assumed that this contributed
to the achievement of better results. Therefore, it is possible to speak of experimental
confirmation of the hypotheses.
Summarizing, we conclude that the pedagogical experiment confirmed the hypothe-
sis of the study. Analysis of the results indicates the increase in the level of formation
of individual components of professional competence using the developed methodical
system and, consequently, its effectiveness.
4 Conclusions and Prospects for Further Research
SCM implementation in the process of teachers training and also the process of com-
puter science professionals training provides an opportunity to intensify the educa-
tional-cognitive activity of students, assists to development of their creative abilities,
mathematical intuition and skills of research activities realization. SCM systematic us-
ing contributes to students attitude toward a computer as to the means of solving pro-
fessional problems. Such students gain more knowledge not only in mathematical dis-
ciplines but also in computer science. As a rule, they have no psychological barrier to
using sophisticated software tools. On the contrary, they are attracted by the programs
created at a high professional level, and they notice the unique application possibilities
of such systems. SСМ is an environment for learning tools projecting and, conse-
quently, can be used for the creation of innovative pedagogical technologies.
References
1. Hrytsenchuk O.O,, Ivaniuk I.V., Kravchyna O.Y., Malytska I.D., Ovcharuk O.V.,
Soroko N.V.: European experience of the teachers' digital competence development in the
context of modern educational reforms. Information Technologies and Learning Tools,
65(3) 316 - 336 (2018)
2. Ivaniuk I.V.: The development of teachers’ digital compenebce: the Scandinavia countries
experience. Information Technologies and Learning Tools, 72(4), 81-90 (2019) (In Ukrain-
ian)
3. Kinelev V. G.: On the results of the work of the higher school in 1994 and the main direc-
tions of its activities in 1995. Vyisshee obrazovanie v Rossii, 1 7-27 (1995)(in Russian)
4. Kinelev V. G.: Fundamentalization of university education, Vyisshee obrazovanie v Rossii,
4, 6-13 (1994) (in Russian)
5. Kuznetsov B. G.: On the style of physical thinking of the twentieth century, Eynshteynovskiy
sbornik, Nauka,Moscow, 121-133 (1967) (in Russian)
6. Laptev V. V.,Ryizhova N. I., Shvetskiy M. V.: Methodical theory of teaching computer sci-
ence. Aspektyi fundamentalnoy podgotovki, Izd-vo S.-Peterb. un-ta, SPb (2003) (in Rus-
sian)
7. Lapchik M. P.: Methods of teaching computer science, Akademiya, Moscow (2001) (in Rus-
sian)
8. Mashbits E. I.: Psychological foundations of learning management. Vyischa shkola, Kyiv
(1987).(in Russian)
9. Ovcharuk O., IvaniukI., Soroko N., Hritsenchuk O., Kravchyna O.: The use of digital learn-
ing tools in the teachers' professional activities to ensure sustainable development and de-
mocratization of education in European countries. In: E3S Web of Conferences, 166, 10019
(2020)
10. Ovcharuk O.: Сurrent approaches to the development of digital competence of human and
digital citizenship in European countries. Information Technologies and Learning Tools
76(2), P.1-13 ( 2020)
11. Semerikov S.O., et al.: Sustainability in Software Engineering Education: a case of general
professional competencies. In: E3S Web of Conferences. The International Conference on
Sustainable Futures: Environmental, Technological, Social and Economic Matters (ICSF
2020). EDP Sciences, 1-13 (2020).
12. Semerikov S. O.: Fundamentalisation of teaching of information disciplines in higher edu-
cation. NPU im. M. P. Drahomanova, Kyiv, (2009) (in Ukrainian)
13. Striuk A., Semerikov S.: The Dawn of Software Engineering Education. In: Proceedings of
the 2nd Student Workshop on Computer Science & Software Engineering (CS&SE@SW
2019) Kryvyi Rih, Ukraine, November 29 (2019) http://ceur-ws.org/Vol-2546/
14. Hizhnyakova L. S.: Some laws of teaching physics methodology. Vestnik MGOU. Seria
"Pedagogika", 1, 110-116 (2011).(in Russian)
15. Shyshkina M. P., Kohut U. P., Popel M.V.: The Systems of Computer Mathematics in the
Cloud-Based Learning Environment of the Educational Institutions. In: Proceedings of the
13th International Conference on ICT in Education, Research and Industrial Applications.
Integration, Harmonization and Knowledge Transfer. CEUR-WS, 1844, 396-405 (2017).
16. Zhaldak M. I.: Problems of fundamentalization of the content of the teaching of informa-
tional disciplines,http://www.ikt-cn.org/images/zhaldak_2014.pdf (2014) (in Ukrainian)