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https://ceur-ws.org/Vol-2387/20190507.pdf
Cloud-oriented Training Technologies as a Means of
Forming the XXI Century Skills of Future Mathematics
Teachers
Mariia Astafieva1[0000-0002-2198-4614], Dmytro Bodnenko1[0000-0001-9303-6587] and
Vladimir Proshkin1[0000-0002-9785-0612]
1 Borys Grinchenko Kyiv University, Kyiv, Ukraine
m.astafieva@kubg.edu.ua, d.bodnenko@kubg.edu.ua,
v.proshkin@kubg.edu.ua
Abstract. The possibilities of cloud-oriented learning technologies based on
computer-based mathematical systems (SCM) to form the skills of the XXI cen-
tury of future mathematics teachers, in particular, critical thinking, have been
substantiated. It has been established that computer support for this process is
effectively provided by SCM GeoGebra and cloud-based learning technologies.
It has been established forms and methods of education in which SCM GeoGe-
bra and cloud-oriented learning technologies are used. It has been experimental-
ly proved the expediency of using cloud-oriented technologies on the basis of
SCM for forming the critical thinking for future mathematics teachers. The es-
timation of efficiency is carried out by methods of the theory of fuzzy sets.
Keywords: cloud-oriented learning technologies, systems of computer mathemat-
ics, cloud services, skills of the XXI century, ICT, future teachers of mathematics.
1 Introduction
One of the tasks of educational reforms is the formation of the skills of the XXI century of
a modern person. Researchers consider the skills of the XXI century (educational and
innovative, informational, media and computer, life and professional) in the context of
educational life skills, and their formation is considered to be the most important goal of
modern high school and secondary school. One of the key skills of the XXI century is
critical thinking. It is an important and universal instrument in the professional activity of
the teacher of mathematics, because mathematics is the basis of STEM-education that is
actively being introduced into the practice of the modern Ukrainian school for the purpose
of developing pupils with critical thinking and ability to research.
Today we are dealing with the so-called "digital generations" that were born and
growing in a high-tech information society, and has many significant differences from
their peers. Therefore, there is a need to seek new pedagogical methods to increase
the efficiency of the educational process, adequate to modern challenges and needs.
The purposeful use of SCM and cloud-oriented technologies is one of these ways.
2 Literature Review
Well-known foreign and Ukrainian scholars studied the problem of forming the skills
of the XXI century (J. Dewey, R. Paul, D. Halpern, B. Trilling, Ch. Feidl, D. Kun,
O. Pometun, I. Zymnia, etc.). However, presented scientific researches used mainly
the material of social and humanitarian disciplines. At the same time, in the previous
scientific works of M. Astafieva and V. Proshkin [1-3] it is proved that mathematics
can serve as an effective tool in shaping the critical thinking of future teachers; some
examples of the use of information and communication technologies for the increase
of the efficiency of this process are given.
Theoretical and methodical principles of the use of computer-oriented methodical
systems of mathematics training are covered in the works of scientists such as
M. Zhaldak, S. Semerikov, Yu. Tryus, O. Spivakovskyi, Yu. Ramskyi, and others.
Thus, the methodical features of the introduction of critical thinking technologies on
the basis of the use of computer algebra and dynamic geometry packages were high-
lighted by T. Oliinyk [4]. S. Lytvynova carried out the testing of cloud services for
the formation of skills of the XXI century students [5]. N. Morze disclosed the means
of ICT for developing the skills of the XXI century [6], and etc. Despite the diverse
scientific researches this problem remains relevant and insufficiently investigated in
the context of the use of cloud-oriented technologies (on the basis of SCM) in the
study of mathematical disciplines in higher education establishments.
The purpose of the article is to substantiate the possibilities of cloud-oriented tech-
nologies (on the basis of SCM) for forming the skills of the XXI century of future
mathematics teachers.
3 Research Methods
The achievement of the research goal was facilitated by the use of such methods: scientific
literature analysis; synthesis, generalization, systematization for the theoretical substantia-
tion of the pedagogical technology of the formation of critical thinking; diagnostic (con-
versation, content analysis, testing) for tracking the dynamics of the level of the formation
of critical thinking of students; a pedagogical experiment in order to prove the effective-
ness of the application of cloud-oriented training technologies for the skills of the XXI
century; mathematical methods (on the basis of the theory of fuzzy sets) for evaluation of
the results of experimental work. Experimental base: Borys Grinchenko Kyiv University
and A.S. Makarenko Sumy State Pedagogical University.
4 Research Results
4.1 Formation of Critical Thinking of Future Mathematics Teachers and Its
Support by Cloud-Oriented Technologies of Training
The results of external independent evaluation of Ukrainian school graduates in recent
years and the questionnaire for junior courses students of the specialties "mathematics",
"informatics" of some universities, as well as mathematics teachers indicate that the skills
of critical thinking of pupils, students and teachers are not formed or formed weakly [1-3].
Thus, today there is an urgent need, a public request for pedagogical technology aimed at
developing the critical thinking of students – future mathematics teachers.
From the point of view of tools and instruments for this purpose a mathematical
disciplines are advantageous. They teach to think logically (with the adherence to the
laws of mathematical logic) and responsible (understanding the limitations of own
knowledge and the possibility of mistake; the willingness to impartially consider a
point of view different from one's own, the internal need to be probative and truthful
in their own judgments, readiness to overcome obstacles for the sake of cognition
truths). And through visualization using ICT mathematics aspires to sensory percep-
tion, the formation of persistent associations in the consciousness of the individual
through visual images that promotes the development of spatial imagination and geo-
metric intuition. Solving tasks stimulates creative search, develops research skills,
teaches to organize and mobilize the knowledge, make you operate in conditions of
uncertainty, and promote communication. We have developed the pedagogical tech-
nology of forming the skills of the XXI century, in particular, critical thinking and
ICT literacy, of students when solving geometric tasks for the construction and exper-
imentally proved its effectiveness [1; 2].
Relying on the positive experience of introducing this technology into school prac-
tice [1] and taking into account the effective use of cloud-oriented technologies in
organizing student research [7], we have developed a pedagogical technology for the
formation of critical thinking of future mathematics teachers by means of mathemat-
ics, which consists of the target, the content-procedural and control, and evaluation
components. The essential element of technology at all stages of its implementation is
the purposeful use of cloud-based learning technologies. Without going into the de-
tailed description of all the components of the proposed pedagogical technology, we
must mention only the key elements of content-activity (content, forms, methods, and
means of learning) and evaluative-reflexive stages of technology implementation, role
and place of SCM and cloud services.
The content-activity stage involves the design of a holistic pedagogical process which
ensures the formation of critical thinking of future mathematics teachers. In fact, we are
talking about what exactly needs to be changed in the content of the training, which forms
and methods of training to choose in order to increase the level of formation of the investi-
gated competence. Contents of training. As noted above and as practice confirms, mathe-
matical disciplines have wide possibilities for forming critical thinking through the organic
combination of formal logic (abstraction) and constructivism (visibility, imagery). Unfor-
tunately, in recent years, the prevalence of formal-analytical approaches which does not
contribute to the formation of students’ imagination and figurative thinking, complicates
the perception and understanding of concepts and facts, exhausting it itself essence. Our
approach assumes resistance to natural sensory perception as the first and necessary link of
knowledge, its absolute basis, which helps to form students with a correct idea of the es-
sential characteristics of the subject of the study. Only on this basis the thinking and
speech of the individual develops; there is a successful transition from partial to general,
from concrete to abstract.
Forms of training. When selecting any of them, there are conditions for the subject,
active communication and interaction of the teacher-student, student-student (during
the classroom lecture or practical classes and, especially, outside of them, both on-
the-spot and virtual through cloud services storage and sharing of data, drawings,
joint documents. Learning methods based on research are focused on problem learn-
ing, heuristics. The task of research-based learning is to provide conditions in which
the student communicating directly with the object or comparing certain facts with an
appropriate degree of independence states a new, unknown to him previously fact or
comes to a new conclusion for him. Among the means of training is expected to use
various cloud services, such as those aimed at working together with text, tabular,
multimedia data, and the use of computer mathematics and WEB-SCM systems. Note
that students can choose arbitrary SCM in the implementation of the project task (new
applications are constantly appearing on the network, which eventually become cross-
platform), but with a compulsory wish to interpret the advantages and disadvantages
of the chosen SCM.
The cloud-oriented learning technologies are used to: organization of communication in
the process of solving mathematical tasks, discussion of ideas, opposition, discussion,
collective search for solutions, joint projects, etc., examples of cloud services – forums,
chats, groups, bitrix24; implementation of research training, in particular, conducting
experiments for the promotion and testing of hypotheses, working on common documents
in the clouds, examples of services – documents, cloud storage, presentations, SCM Geo-
Gebra; creation of presentation and multimedia materials, examples of cloud services –
presentations, Moovly, Pixel Expres, Picassa, Wewideo, etc.; monitoring of the education-
al process using questionnaires, diagnostics, surveys, in other words, the interactive rela-
tionship with the students, examples of cloud services – forms, Learning Apps, Kahoot,
etc.; evaluation of tasks solutions online (using cloud services, access to an online resource
is provided which contains criteria for evaluating the problem and individual sheet of each
group), an example of cloud services – tables.
Thus, learning becomes a kind of intellectual partnership between a computer and
a student, which enhances students' knowledge, develops their research competencies,
develops critical thinking. The determination of the effectiveness of the technology of
forming critical thinking takes place at the evaluative-reflexive stage.
4.2 Implementation of the Developed Pedagogical Technology in the Practice
of Bachelor's Degree in Mathematics Training
Our technology of forming the skills of the XXI century and critical thinking in par-
ticular with the use of cloud-oriented learning technologies and SCM GeoGebra was
tested experimentally. The experiment was conducted during September-December
2018. The experimental group included 44 people: students of the first - third year of
the specialty "Mathematics". The control group consisted of 62 persons: students of
the second and third year. Participants of both scientific groups, in particular, imple-
mented a collective project: developed 29 dynamic models (with elements of anima-
tion) to various topics of geometry and mathematical analysis in SCM GeoGebra. For
example, dynamic model for the Pascal theorem (Fig. 1a, b).
Fig. 1. a, b Two positions of the dynamic model for the Pascal theorem
After four months of work, we conducted diagnostic testing in the control and exper-
imental groups. The test [8] contained 20 tasks. Testing did not check the level of
academic achievement, but aimed at assessing how much a person would be able to
use elementary knowledge and skills for possible life difficulties and challenges, that
is, what is actually the most important sign of critical thinking. A comparative analy-
sis of the results is presented by a histogram (Fig. 2) in which we see that all the ques-
tions of the test students of the experimental group responded better.
Fig. 2. A comparative analysis of the results is presented by a histogram
The multicriterial assessment of the critical thinking level was carried out by the
methods of the theory of fuzzy sets [9] and is represented by the terms of the linguis-
tic variable "Critical Thinking" (Fig. 3).
Fig. 3. Terms of the linguistic variable "Critical Thinking"
We have identified five levels (so-called terms) of critical thinking: E1 – initial, E2 –
low, E3 – average, E4 – sufficient, E5 – high. The terms are given in the form of trape-
zoidal membership functions of the quadruple of numbers. Given the verbal-
numerical Harrington scale we will take for our task. The calculation of the coeffi-
cients of the formation of critical thinking has shown that for the control group it is
0.46 and is average (belongs to term E3), for the experimental group it is 0.77, which
is 67% higher and corresponds to more than sufficient level (belongs to terms E4 and
E5), Consequently, the level of critical thinking of students of the experimental group
can be characterized, for example, as "more than sufficient".
5 Conclusions
The expediency and the possibilities of cloud-oriented learning technologies (based on
SCM) are grounded for forming the skills of the XXI century for future mathematics
teachers. It has been established that computer support of the developed pedagogical tech-
nology is effectively provided by cloud-oriented learning technologies and SCM GeoGe-
bra, which has all the above-mentioned capabilities. The analysis of the results of the in-
troduction of the developed technology on the basis of the theory of fuzzy sets confirmed
the feasibility of using cloud-oriented technologies (on the basis of SCM) for developing
the skills of the XXI century for future mathematics teachers. Studying the possibilities
and effectiveness of computer support for the key skills of the XXI century building
through the development of open online courses at LMS MOODLE using other cloud-
oriented tools and technologies will be the subject of further research.
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