=Paper=
{{Paper
|id=Vol-2879/paper26
|storemode=property
|title=Methodology of formation of modeling skills based on a constructive approach (on the example of GeoGebra)
|pdfUrl=https://ceur-ws.org/Vol-2879/paper26.pdf
|volume=Vol-2879
|authors=Marina G. Drushlyak,Olena V. Semenikhina,Volodymyr V. Proshkin,Serhii Ya. Kharchenko,Tetyana D. Lukashova
}}
==Methodology of formation of modeling skills based on a constructive approach (on the example of GeoGebra)==
https://ceur-ws.org/Vol-2879/paper26.pdf
Methodology of formation of modeling skills based on
a constructive approach (on the example of GeoGebra)
Marina G. Drushlyak1 , Olena V. Semenikhina1 , Volodymyr V. Proshkin2 ,
Serhii Ya. Kharchenko3 and Tetyana D. Lukashova1
1
Makarenko Sumy State Pedagogical University, 87 Romenska Str., Sumy, 40002, Ukraine
2
Borys Grinchenko Kyiv University, 18/2 Bulvarno-Kudriavska Str., Kyiv, 04053, Ukraine
3
Luhansk Taras Shevchenko National University, 1 Gogol Sq., Starobilsk, 92700, Ukraine
Abstract
Author’s methodology of forming modeling skills involves 4 steps: Step 1 – the teacher step by step
constructs the curve by means of cloud based service GeoGebra; Step 2 – the teacher offers a description-
definition of the curve and provides a ready-made algorithm by which students model the curve inde-
pendently in GeoGebra; Step 3 – the teacher offers an algorithm for constructing a curve model, and
students need to characterize the properties of the curve or give its definition based on the results, Step
4 – students are offered definitions of curves that they have to model in GeoGebra). An example of
realization of the author’s methodology is given, the pedagogical experiment on proof of its effectiveness
is described.
Keywords
modeling skills, skills development methodology, cloud service, GeoGebra cloud service, constructive
approach, modeling, interesting curves, professional training
1. Introduction
Modern science operates with various methods, among which modeling is one of the most
popular. Method allows you to move away from the object’s ideal representation and use its
analogue, which retains the most important characteristics that allow you to talk about the
object properties after certain changes or influences on it.
The development of computer technologies has contributed not only to the revival of the
modeling method, but also led to the emergence of specialized environments, where it became
possible to model various objects (processes) based on a constructive approach. At the same
time, the widespread use of smartphones and tablets that have access to the Internet has led
to the emergence of cloud services, which also allow you to model objects of different nature
[1, 2]. However, methods of using cloud services to develop modeling skills are just beginning
CTE 2020: 8th Workshop on Cloud Technologies in Education, December 18, 2020, Kryvyi Rih, Ukraine
" marydru@fizmatsspu.sumy.ua (M. G. Drushlyak); e.semenikhina@fizmatsspu.sumy.ua (O. V. Semenikhina);
v.proshkin2017@gmail.com (V. V. Proshkin); hk.sergey2014@ukr.net (S. Ya. Kharchenko);
tanya.lukashova2015@gmail.com (T. D. Lukashova)
0000-0002-9648-2248 (M. G. Drushlyak); 0000-0002-3896-8151 (O. V. Semenikhina); 000-0002-9785-0612
(V. V. Proshkin); 0000-0002-0310-6287 (S. Ya. Kharchenko); 0000-0002-1465-9530 (T. D. Lukashova)
© 2020 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073
CEUR Workshop Proceedings (CEUR-WS.org)
458
to be developed and implemented, and therefore are not well established and need experimental
confirmation.
The importance of developing constructive skills of youth is emphasized in the findings of
Laksha [3], Kononenko [4], Ivanina [5], and the formation of modeling skills by individual
scientists is associated with the formation of research skills [6].
Analysis of the problem shows that with the advent of specialized mathematics software, its
solution has focused not so much on understanding the algorithms of elementary constructions,
but on the ability to use computer tools and the ability to visualize the result. The latter, in
particular, is emphasized in the finding of Bilousova and Zhytyenyova [7].
Regarding the first (the ability to use tools) it should be noted the emergence of cloud services,
including specialized services. Analysis of the latter showed the following.
Different aspects of educational using of cloud technologies and services are examined in the
studies of national and foreign researchers. For example, Alkhansa et al. [8] explore advantages
of using cloud technologies in research, such as availability of various tools and applications
for analyses and collecting data, for managing and organizing references, for communication
with peers and experts, the absence of constraints, the access to cloud resources from anywhere
any time that has an active Internet connection, etc. They offer the approach to applying
of cloud services for enhancing the productivity of university research activities, increasing
competitiveness and flexibility of educational institutions. Smith et al. [9] core focus addresses
Education as an area of cloud application with a cloud based e-learning system developed to
demonstrate the capabilities and effectiveness of cloud technology.
Shyshkina and Popel [10] explore the problems of implementation of cloud technology tools
and services in the educational process, describe the current state of development and use of
cloud technology services in educational institutions and analyze the content of educational
and scientific components of cloud-based educational environment.
Scientific research on the involvement of cloud services for the formation of various skills in
specialists in various fields of knowledge is spreading [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23].
Currently, cloud versions of known environments with mathematical modeling capabilities
are available, including Maple, MATLAB, WebMathematica, Calculation Laboratory, etc. [24].
We will also add CoCalc [25, 26, 27, 28] and GeoGebra [29, 30, 31, 32, 33, 34, 35] cloud services
to this list.
Experience in using GeoGebra cloud service (visualization of mathematical objects; organi-
zation of not only analytical but also empirical search for answers in determining individual
characteristics of mathematical objects; organization of home computer experiment) allowed to
consider GeoGebra as means for formation of skills to model based on constructive approach.
However, the analysis of scientific findings confirmed the lack of established methods for its
use, and therefore the method of formation of skills to model interesting curves by GeoGebra
cloud service based on a constructive approach required testing and experimental confirmation
of its effectiveness.
The purpose of the paper is to describe the method of forming the skills to model interesting
curves through GeoGebra cloud service based on a constructive approach and to confirm its
effectiveness.
459
2. Material and methods
To test the effectiveness of the methodology of forming modeling skills, a pedagogical experi-
ment, which lasted 3 years (2018–2020) was conducted among pre-service mathematics and
computer science teachers.
Makarenko Sumy State Pedagogical University, Borys Grinchenko Kyiv University were the
experimental base. The total number of respondents was 86 people (pre-service mathematics
and computer science teachers).
The author’s methodology was implemented using the freely distributed GeoGebra cloud
service (https://www.geogebra.org) in the special courses. The experimental module “Modeling
of interesting curves” with a volume of 10 classroom hours (2 hours of lectures and 8 hours of
laboratory classes) was introduced.
Two tests of educational achievements (at the beginning and after studying the module) were
organized to confirm the effectiveness of the methodology.
Students were offered a typical test problem for modeling a conic by its geometric definition
(problems differed from each other in the input conditions, for example, the distance between
the foci was different or one of the foci was in the specific point in a given coordinate system,
etc.).
The solution of the problem was evaluated by the following indicators (table 1).
Table 1
Indicators of the formation of modeling skills
No Indicators Marks
1 Ability to take into account the analytical relationship between the elements 1
2 Ability to take into account the geometric relationship between the elements 1
3 Ability to use the Locus tool 1
4 Ability to use the Trace tool 1
5 Ability to demonstrate the change of the curve shape when changing the input 1
data
6 Ability to construct a model visually correct (location, color, size and style of 1
geometric objects)
7 Ability to add dynamic text to study numerical characteristics 1
8 Ability to write an algorithm for constructing a model 1
9 Ability to reproduce the steps of the algorithm to construct the model 1
10 Ability to interpret the result of a computer experiment 1
Since the results of the tests were dependent and each time provided for the accumulation of
marks, the sign test was used. The number of respondents, whose total score decreased (“–”),
did not change (“0”) and increased (“+”), was fixed.
In accordance with the purpose of the experiment and the sign test, the null hypothesis
was formulated: the author’s methodology does not contribute to the formation of modeling
skills by means of GeoGebra cloud service. Then the alternative hypothesis was “the author’s
methodology contributes to the formation of such skills”.
The constructed hypotheses define the one-sided sign test for checking the dependent samples.
460
According to the decision-making rule [36] the null hypothesis of inefficiency / effectiveness of
the author’s methodology was accepted or rejected.
3. Constructive approach
The constructive approach is characterized by the fact that acquaintance with the properties
of concepts begins with constructive activities for their “discovery” and assimilation with a
gradual transition to definitions and logical proofs. This, in particular, simplifies the perception
of the Geometry course, makes it more accessible, while raising the scientific level through the
intensification of educatees’ research activities.
The use of the constructive approach contributes to the fact that the activity is manifested
in the gradual transition of actions for the construction of objects from the executive level (is
characterized by external regulation), then to reproductive (is marked by internal regulation of
actions in the construction of known structures), then to the applied level (using the method of
construction), and, finally, the creative level (involves the construction of new objects) (figure 1)
[37].
Figure 1: Constructive approach activity levels.
We consider constructive tasks to be one of the means of implementing a constructive
approach. Many authors use the term “constructive tasks” in a sense identical to construction
problems. However, the peculiarities of the constructive approach require the use for its
implementation and other tasks, the solution of which requires the implementation of a particular
constructive activity. Therefore, for example, Lisimova [38] considers constructive tasks as tasks
for construction, imaging, measurement, geometric design and structural-geometric modeling.
Dalinger [39] considers constructive tasks as “problems in the process of solving which reveal
the material conditions of geometric figures. Their purpose is to identify the essential features
of ideas that are formed through the material conditions of their origin”.
Constructive tasks can be a base for establishing new properties of figures (concepts forma-
tion), for the consolidation of knowledge, for repetition, control, intensification of research
activities. “The peculiarity of constructive tasks is that they can be solved both logically and
figuratively or visually effective” [38].
When solving constructive tasks, there is a productive activity that motivates to think inde-
pendently (methods of constructing have to be developed independently), and not reproductive
activity, which is often not an independent mental process, but is a repetition of known steps
[40].
Mastering the system of knowledge and skills can take place on two levels: constructive and
461
analytical. The main feature of the constructive level of mastery of the material is its visual and
constructive awareness. This is manifested:
• in the ability to recognize objects that belong and do not belong to the content of this
concept, give examples, demonstrate the existence of the studied figures by construction;
• in the knowledge of their most essential properties and the ability to apply known
properties in solving problems.
The constructive approach provides “points of support” for Geometry study at a higher ab-
stract level, which is characterized by the ability to formulate definitions of concepts, statements
and prove already in formal language, rather than the language of geometric images.
4. Methodology of forming skills to model interesting curves
based on a constructive approach in GeoGebra cloud service
Any methodology involves the presence of five elements (figure 2): purpose, content, forms,
methods and means. Let us explain them in more detail.
Figure 2: Components of the author’s methodology.
The purpose of the author’s methodology is to form the skill to model interesting curves
using cloud services.
Interesting curves of the course of Analytical Geometry are the content on which the author’s
methodology is realized. Nowadays, their study in the classical course of Analytical Geometry
is possible in three ways:
462
• analytical description and further study of curves – at first analytical (parametric, implicit,
explicit) equations, usually of conics, are given and then they are investigated;
• study of curves as locus with a given property – at first the geometric definitions of
curves are given, on the basis of which their analytical equations are written (as a rule,
these are conics, conchoid of Nicomedes, limaçon of Pascal, strophoid, cissoid of Diocles,
lemniscate of Bernoulli, Cassini oval);
• study of curves generated in the kinematic way (as the trajectory of a point), usually
cycloidal curves, folium of Descartes, witch of Agnesi, logarithmic spiral.
Due to lack of time, there is only the first way of the study of interesting curves in the course
of Analytical Geometry at universities. This limits the possibility to develop the skills to model
mathematical objects.
The form of realization of the author’s methodology is virtual lecture-practical classes or a
combination of visualized lectures and virtual laboratory classes.
Among the teaching methods problem method and netnography method are used.
GeoGebra cloud service, constructive tasks and instructional materials are learning means.
Elementary knowledge in Analytical Geometry is required to use the author’s methodology.
In general, the methodology of forming the skills to model interesting curves is as follows: the
teacher on a common online platform informs about constructive approaches to the construction
of various curves, which were studied in ancient times (the netnography method). He briefly
tells about how such constructions were done (compass and ruler, one compass, two compasses,
one or two rulers, etc.), or gives an example of practically oriented problems that are solved using
interesting curves. After that, the teacher demonstrates one of the described constructions in
GeoGebra cloud service, and then asks students to write an algorithm for the above construction
(the problem method). Then the teacher asks students according to the algorithm (provided
to each student) to reproduce the construction and demonstrate the result, to analyze errors,
and if not, to analyze possible limit cases. After that, students are offered the definition of
curves generated in mechanical way together with the algorithms of their construction in
the cloud service; students must model this type of curve according to the existing algorithm.
Then together with students, the task of the following type is carried out: algorithms of curves
construction are offered and after their construction, students need to characterize properties
of the modeled curves, to give them definitions or kinematic characteristics. After completing
this type of task, students are offered only the definition of the curves or their kinematic
characteristics, and they must model the curve themselves.
The methodology should be briefly described as follows.
Step 1 – the teacher offers an example of a curve model through the definition (the teacher
step by step models the curve), and students must independently compile an algorithm for
constructing the model.
Step 2 – the teacher offers a description-definition of the curve and provides a ready-made
algorithm according to which students model the curve independently.
Step 3 – the teacher offers an algorithm for constructing a curve model, and students need to
characterize the properties of the curve or give its definition based on the results.
Step 4 – students are offered definitions of curves that they have to model them.
463
We will briefly demonstrate the application of the methodology of forming the skiils to model
interesting curves based on a constructive approach in GeoGebra cloud service on the example
of the hypocycloids study.
The teacher formulates the problem. The calculation of the action of the kneading machine
consists of a description of the trajectory of the point belonging to the working body. The
trajectory of the working body does not always fill the container with the substance, as a result,
the mixing time increases, which leads to unnecessary energy consumption. Determine the
optimal trajectory of the point of the mixer working body [41].
History of the curve discovery (netnography method). Hypocycloid (from Greek words 𝜐𝜋𝑜 –
“under, below” and 𝜅𝜐𝜅𝜆𝑜𝜁 – “circle”) – is a curve “under the circle”. The most famous hypocy-
cloids are the astroid (from Greek “star-shaped”) and the deltoid (Steiner curve). The name of
the astroid (from “cusp” – exacerbation) was proposed by the Austrian astronomer Carl-Ludwig
von Littrow (1811–1877). The shape of the deltoid resembles the capital Greek letter delta, its
properties were studied by Leonard Euler in the XVIII century, and then by Jacob Steiner in
1856, after whom it was renamed the Steiner curve.
Definition of the curve. A hypocycloid is a curve generated by the trace of a fixed point on a
small circle that rolls within a larger circle.
Next, the teacher demonstrates an example of a ready-made hypocycloid model (figures 3-4),
and then offers to describe the algorithm for its construction to students.
Figure 3: Construction of a hypocycloid using the Trace tool.
Let the circle 𝜔2 roll inside the circle 𝜔1 . Let’s construct a curve generated by a point on the
circle 𝜔2 .
464
Figure 4: Construction of a hypocycloid using the Locus tool.
Algorithm for hypocycloid model constructing
1. Construct a circle 𝜔1 with center 𝐴 passing through point 𝐵.
2. Construct the segment 𝐴𝐵.
3. Construct an arbitrary point 𝐶 on the radius 𝐴𝐵.
4. Construct the circle 𝜔3 with center 𝐴 passing through point 𝐶 (dotted line).
5. Construct an arbitrary point 𝐷 on the circle 𝜔3 .
6. Construct the circle 𝜔2 with center 𝐷, the radius of which is equal to the length of the
segment 𝐶𝐵.
7. Construct the ray 𝐴𝐷.
8. Constract the point 𝐸 – the intersection of the circle 𝜔2 with the ray 𝐴𝐷.
9. Hide the ray 𝐴𝐷.
10. Let the center 𝐷 of the circle 𝜔2 coincides with the point 𝐶 in the initial position. If
we move the point 𝐷 around the circle, it will rotate around the point 𝐴, and remains
tangent to the circle 𝜔1 . Let the circle 𝜔2 rotate around it when moving the center to
create the illusion that the circle 𝜔2 rolls on the circle 𝜔1 . Only one point of the circle 𝜔2
should be investigated. Let this point coincides with the point 𝐵 at the initial moment,
and is the point of tangency of the circles 𝜔1 and 𝜔2 . If the circle 𝜔2 rolls on the arc 𝐵𝐸,
then the center 𝐷 of the circle will return around the point 𝐴 at the angle 𝜓 = 𝐵𝐴𝐸.
Simple calculations show that all points of the circle 𝜔2 will rotate around its center at
the angle 𝜓 * 𝐶𝐵𝐴𝐵
.
11. Construct the point 𝐺, which is obtained from the point 𝐸 by rotating around the point
𝐷 by the angle 𝜓 * 𝐶𝐵 𝐴𝐵
.
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12. Hide the point 𝐸.
13. a) Make the point 𝐺 leave a trace by rolling the point 𝐷 on the circle 𝜔2 (Fig. 3).
b) Use the Locus tool: the point 𝐺 is the “pencil point”, the point 𝐷 is the “driver point"
(figure 4).
Next, the lecturer at the visualized lecture offers students to explore the analytical equation
of the curve, which is described through the parameter.
Let 𝑅 be the radius of a fixed circle, 𝑟 be the radius of a rolling circle, 𝑘 = 𝑅𝑟 , then the
parametric equation of a hypocycloid is
𝑐𝑜𝑠(𝑘 − 1) * 𝜓) 𝑠𝑖𝑛(𝑘 − 1) * 𝜓)
𝑥 = 𝑟 * (𝑘 − 1) * (𝑐𝑜𝑠𝜓 − ), 𝑦 = 𝑟 * (𝑘 − 1) * (𝑠𝑖𝑛𝜓 − ).
𝑘−1 𝑘−1
The modulus of 𝑘 determines the shape of the hypocycloid: 𝑘 = 2 the hypocycloid is described
2 2 2
by Tusi couple (diameter of a fixed circle), 𝑘 = 4 by the astroid 𝑥 3 + 𝑦 3 = 𝑅 3 .
Study of the curve and establishment of properties (problem method). When the hypocycloid is
constructed, we pass to establishment of its properties at the level of practical research.
Note that the curve constructed in this way is open. Let us investigate the conditions under
which the constructed hypocycloid will be a closed curve. By changing the radius of the circle
𝜔3 , thereby changing the ratio 𝑘 = 𝑅𝑟 of the radii of fixed and rolling circles, we conclude that
the curve will be closed when the ratio 𝑘 = 𝑅𝑟 ∈ 𝑍 (figure 5). In addition, in this case, the
hypocycloid consists of 𝑘 identical arcs, the ends of which lie on a fixed circle (figure 6).
Figure 5: Closed hypocycloid, 𝑘 = 𝑅𝑟 ∈ 𝑍.
466
Figure 6: The hypocycloid consists of 𝑅𝑟 = 6 of identical arcs.
If the radius 𝑟 of the rolling circle is twice smaller than the radius 𝑅 of the fixed circle,
then the hypocycloid degenerates into a segment (figure 7). This statement is well known as
Copernicus theorem.
After the lecture, students are first asked to construct curves models according to the given
definition and algorithm, for example, to costruct a cycloid, which is a curve generated by a
point of a circle rolling on a straight line, reproducing all the steps of the algorithm. Not only
instructions but also possible consultations are provided.
The next lesson tasks such as: “Establish the relationship between the number of arcs of the
epicycloid, generated by a point of a circle rolling on the outside of another circle, and the ratio
of the radii of these circles”, are offered.
Then, at the next two lessons, tasks such as: “A point and a line, the distance between which
is equal to 𝑎, are given. A line is drawn through an arbitrary point 𝑋 of the line and the point.
The points at a distance 𝑏 from point 𝑋 are marked. Find the locus of such points”.
The described methodology differs from the traditional one, as the latter usually involves only
step 2 and step 4, which does not allow students to understand the algorithms involved in the
constructing the model, to develop the ability to compare step-by-step construction ideas and
the algorithm steps. There is also no practice of qualitative analysis of the algorithm (which is
provided by step 3 and which is supported by the “step-by-step playback service” in GeoGebra).
This is what we consider fundamentally important for effective formation of skills to model and
what GeoGebra service makes possible.
Let us note also that in terms of distance learning, it is GeoGebra service that allows the
demonstration of constructions and the ability to work with models at anytime from anywhere.
467
Figure 7: Degenerate hypocycloid, 𝑅𝑟 = 2.
5. Statistical analysis of experiment results
Verification of the effectiveness of the author’s methodology was carried out based on two tests.
Students had to solve the problem 1 (for example, to construct a curve, for each point of which
the sum of distances to two given points is constant) before studying the topic and the problem
2 (for example, to construct a curve, each point of which is equidistant from the point lying in
the third quarter, and a line that forms an acute angle with the abscissa) after studying the topic.
The evaluation of solutions was carried out through the assessment of skills: to take into
account the analytical and geometric relationship between the elements; to use the Locus
and Trace tools; to visualize the model successfully; to add dynamic text to study numerical
characteristics; to write an algorithm for constructing a model; to reproduce the steps of the
algorithm to construct the model; to interpret the result of a computer experiment (in more
detail in table 1).
The results of the controls are presented in table 2.
These marks were used to determine the number of respondents whose total score decreased
(“–”), did not change (“0”) and increased (“+”) (table 3).
According to the decision-making rule [36] we have: the value of 𝑇𝑒𝑥𝑝 = 46 (this is the
number of “+” signs in the sample), 𝑛 = 67 (this is the number of respondents who have
changed the results), the acceptance interval of the hypothesis 𝐻0 is [26, 41] at the significant
level of 0.05.
Since 𝑇𝑒𝑥𝑝 is not included in the acceptance interval of the hypothesis 𝐻0 , we reject the
hypothesis 𝐻0 and accept the alternative one with the conclusion that the author’s methodology
contributes to the formation of modeling skills. Since the value of 𝑇𝑒𝑥𝑝 is beyond to the segment
on the right, it is necessary to conclude there is the positive dynamics of the number of such
468
Table 2
The results of students’ tests
No Test 1 Test 2 No Test 1 Test 2 No Test 1 Test 2
1 3 4 30 3 5 59 4 5
2 4 5 31 4 4 60 3 4
3 5 4 32 4 6 61 3 4
4 4 3 33 4 6 62 5 4
5 5 3 34 4 7 63 5 5
6 5 5 35 5 4 64 5 4
7 5 6 36 5 3 65 6 4
8 6 6 37 5 3 66 5 6
9 4 5 38 5 5 67 4 7
10 3 4 39 6 6 68 3 7
11 3 3 40 5 7 69 2 6
12 3 3 41 4 5 70 7 5
13 4 2 42 4 4 71 5 8
14 4 6 43 6 4 72 4 8
15 5 6 44 7 5 73 3 9
16 4 7 45 6 6 74 3 8
17 5 6 46 5 7 75 4 7
18 4 5 47 7 7 76 5 6
19 7 4 48 8 7 77 5 6
20 6 5 49 8 6 78 5 6
21 6 3 50 3 6 79 6 7
22 5 5 51 4 7 80 5 6
23 7 6 52 4 7 81 5 8
24 7 6 53 4 7 82 6 7
25 7 7 54 1 7 83 5 6
26 4 4 55 2 7 84 6 6
27 5 3 56 4 5 85 5 5
28 5 5 57 4 4 86 3 6
29 5 5 58 3 4 - - -
Table 3
Dynamics of marks according to the results of students’ tests
Dynamics Negative “–” Without changes “0” Positive “+” Changes 𝑛=“–”+“+”
Members 21 19 46 25
students who have developed the skills to model interesting curves using GeoGebra cloud
service.
469
6. Conclusions
1. Author’s methodology of forming modeling skills involves 4 steps: Step 1 – the teacher
offers an example of a curve model through the definition (the teacher step by step
constructs the curve), and students must write the constructing algorithm independently;
Step 2 – the teacher offers a description-definition of the curve and provides a ready-made
algorithm by which students model the curve independently; Step 3 – the teacher offers an
algorithm for constructing a curve model, and students need to characterize the properties
of the curve or give its definition based on the results, Step 4 – students are offered to
define curves that they have to model). The methodology effectiveness was confirmed by
the sign test at the significant level of 0.05.
2. GeoGebra cloud service is effective for being perceived as a means for the formation of
students’ (pre-service mathematics and computer science teachers) modeling skills. At
the same time, we consider it is also possible to use this methodology in the conditions of
traditional training with GeoGebra installed on PC.
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