Modelling in GeoGebra in the Context of Holistic Approach Realization

in Mathematical Training of Pre-service Specialists

Liudmyla I. Bilousova

1 a

, Liudmyla E. Gryzun

2 b

, Svitlana H. Lytvynova

3 c

and

Valentyna V. Pikalova

4,1 d

1

Kryvyi Rih State Pedagogical University, 54 Gagarin Ave., Kryvyi Rih, 50086, Ukraine

2

Simon Kuznets Kharkiv National University of Economics, 9A Nauky Ave., Kharkiv, 61166, Ukraine

3

Institute for Digitalisation of Education of the National Academy of Educational Sciences of Ukraine, 9 M. Berlynskoho

Str., Kyiv, 04060, Ukraine

4

National Technical University “Kharkiv Polytechnic Institute”, 2 Kyrpychova Str., Kharkiv, 61002, Ukraine

Keywords:

Modeling Activity on GeoGebra, Mathematical Training of Pre-service Specialists, Holistic Education,

Computer Dynamic Model.

Abstract:

In accordance with its aim, the article represents students’ modeling activity (held within inter-university

projects of Kharkiv GeoGebra Institute) which resulted in the complex of GeoGebra models focused on holistic

learning of Mathematics at higher school and university. Proper theoretical background for the complex design

is elaborated and the stages of the students’ modeling activity are covered. The models in the developed

complex are grouped in the three sections. The ﬁrst group consists of the models which enable to facilitate

mastering basic essential mathematical concepts (objects) by the potential trainees. The second group is

focused on the realization of transdisciplinary connections between Mathematics and other subject domains.

The third group embraces models which provide real-life problems solving based on the models investigation.

All the groups are represented in the article along with speciﬁc examples of the models. In order to facilitate

potential trainees’ personal cognitive activity that is expected by holistic education, it was elaborated procedure

of cognitive activity which includes some tips on changing the parameters of the dynamic model, monitoring

the results, investigating, making conclusions etc. Such a procedure is aimed to streamline understanding the

essence of the concept (phenomenon). The didactic support for each model was developed by the students to

involve potential trainees into the solving special problems and real-life tasks which encourage them to obtain

holistic understanding of the basic concepts via special cognitive activity based on work with dynamic models.

The said didactic support is characterized in the paper. The prospects of further research are outlined.

1 INTRODUCTION

The analysis of the evidence of university and pre-

university mathematics training as well as the results

given in recent studies (Bobyliev and Vihrova, 2021;

Elkin et al., 2017; Semenikhina and Drushliak, 2014;

Singh, 1996; Vlasenko et al., 2019), testify the num-

ber of drawbacks of contemporary school mathemat-

ics training which then lead to the difﬁculties faced

by the university students and raise the problems of

increasing the level of mathematical education both

a

https://orcid.org/0000-0002-2364-1885

b

https://orcid.org/0000-0002-5274-5624

c

https://orcid.org/0000-0002-5450-6635

d

https://orcid.org/0000-0002-0773-2947

at school and university.

According to the (Bevz, 2003; Bilousova et al.,

2019; diSessa et al., 2004), the most common learn-

ing difﬁculties which brake the process of successful

mastering mathematics are the following. Students

ﬁnd mathematical concept to be difﬁcult to take in and

to apply them properly to practical tasks. It leads to

inability to achieve basic educational goals by the stu-

dents which results in their loosing interest to Math-

ematics. Finally, it becomes impossible for the stu-

dents to see the beauty of the science and to appreci-

ate its importance for mastering other knowledge do-

mains.

One of the essential problems of mathematical

training that can cause the learning difﬁculties is the

absence of holistic understanding of mathematics as a

Bilousova, L., Gryzun, L., Lytvynova, S. and Pikalova, V.

Modelling in GeoGebra in the Context of Holistic Approach Realization in Mathematical Training of Pre-service Specialists.

DOI: 10.5220/0010925700003364

In Proceedings of the 1st Symposium on Advances in Educational Technology (AET 2020) - Volume 1, pages 499-510

ISBN: 978-989-758-558-6

Copyright

c

2022 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

499

basis and tool for solving interdisciplinary tasks and

real-life problems. Unfortunately, for the most of

students, Mathematics remains too abstract curricu-

lum subject which is really complicated and distracted

from real world. Hence, the students are demotivated

to master it and feel true necessity to its deep under-

standing.

The said problems of mathematical training which

are faced nowadays by the pre-service specialists of

different majors cause the necessity of ﬁnding and ap-

plying new approaches to mathematics learning. One

of such approaches seems to be holistic educational

paradigm which aims to provide dynamic, harmo-

nized, and interconnected ways of learning.

According to the research on the holistic ap-

proach, the core concept of holistic education is the

cohesive development of the students’ personality

both at the intellectual and emotional levels (Singh,

1996). In addition, it is underlined that the said cohe-

sive progress should base on the links between real-

life problems and personal experience of a trainee.

Among basic principles of holistic education the

studies (in particular, (Mahmoudi et al., 2012; Miller,

2004, 1991; Miller et al., 2005)) point out some pil-

lars which seem to be essential in the context of the

problems of Mathematics training, mentioned above.

The ﬁrst principles expects students’ freedom and au-

tonomy. So, within the holistic paradigm any trainee

is considered to be really active participant of the

learning process who is ready to interact with reality

through his own cognitive activity with his own ups

and downs.

Next pillar of the holistic approach is necessity to

establish connections and relationships between the

object of learning and existing knowledge. The more

links trainees have, the stronger memories are formed

in their minds and better understanding of the whole

they obtain.

Similar to the establishing links is the principle of

transdiciplinarity which focuses teaching and learning

on ruining boundaries between subject ﬁelds them-

selves as well as between subject areas and reality.

Researchers also point out that holism helps both

the connection facet and transdiciplinarity, because it

seems to be fruitful to learn separate things which in

fact are not separate. However, at the same time it is

necessary to understand how they work together.

The analysis of the holistic education basis reveals

a need to apply efﬁcient learning tools enabled to

provide holistic approach to nowadays teaching and

learning.

One of such tool seems to be computer dynamic

models (CDM). The learning of recent studies on

their didactic facilities testiﬁes that CDM have quite

powerful potential as for revealing transdisciplinary

connections and facilitating their understanding by

trainees. In particular, researchers point out that CDM

are typically based on the mathematical model of a

concept (process, phenomenon, etc.), and enable to

visualize its essence at real time operation, learn dy-

namic changes, and investigate the concept or process

via active cognition. In such a way CDM help to

form and develop students’ techniques of mental ac-

tivity including transdisciplinary ones (Semenikhina

and Drushliak, 2014; Alessi, 2000).

Characterizing advantages and facilities of CDM

using in the context of holistic education, it is im-

portant to emphasize that they encourage students to

learn objects independently and actively. In addition,

they reveal and demonstrate in action the wholeness

of the learnt concepts (phenomenon).

In this context, it is essential to focus on the valu-

able potential of contemporary mathematical com-

puter environments which enable to create the mod-

els of different complexity, visualize changes of the

model behavior and do proper research. Despite the

great variety of the said software on the modern IT

market, we would like to focus on modeling facilities

of free GeoGebra software which provides a trainee

with convenient tools to develop a CDM, and do ef-

fective simulations and investigations with it. In par-

ticular, GeoGebra allows to create geometrical objects

and obtain easily their algebraic interpretation; get in-

teractive and dynamic visualization of the objects of

various essence; manipulate with the model parame-

ters to monitor the changes etc (Pikalova, 2018; Kra-

marenko et al., 2020). In addition, online service Ge-

oGebraTube grants the access to the variety of ex-

isting elaborations provided by the global GeoGebra

community which unites the educators and students

all over the world.

Nowadays, GeoGebra Institutes work in many

countries and make together International GeoGe-

bra Institute (IGI) as a global organization that nour-

ishes and stimulates collaboration between practition-

ers and researchers, seeking to expand the community

of independent GeoGebra users.

GeoGebra Institute, Kharkiv, Ukraine, which has

been realizing its mission since 2010 within IGI, fo-

cuses on:

1) promoting the dissemination and productive use

of software, scientiﬁc, educational, methodolog-

ical developments of the international GeoGebra

community in professional activities of mathe-

maticians and other specialists;

2) encouraging students and teachers to conduct re-

search in mathematics, physics, computer science

and information technology;

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500

3) implementation of the concept of STEM-

education in educational practice;

4) involvement of students and teachers in cooper-

ation with the international GeoGebra commu-

nity via participation in the conferences and other

events initiated by the IGI.

One of the interesting and signiﬁcant inter-

university projects of Kharkiv GeoGebra Institute was

involving students of various specialties into mod-

eling activity focused on various GeoGebra models

elaboration and learning the modeled objects (pro-

cesses, dependences etc.).

The purpose of the article is to represent stu-

dents’ modeling activity which resulted in the com-

plex of GeoGebra models focused on holistic learning

of Mathematics at higher school and university.

2 THEORETICAL FRAMEWORK

During the research, the set of theoretical, empirical,

and modelling methods were applied.

Characterizing the arrangement of the said stu-

dents’ modeling activity within Kharkiv GeoGebra

Institute, we can describe all necessary stages of the

work.

At the ﬁrst stage, the core task was formulated

for the students as following: to develop the complex

of GeoGebra models for the maintaining learning of

Mathematics at higher school and university based on

holistic approach.

In addition, there were formulated necessary re-

quirements for the whole complex which determined

its potential functions and enabled to realize exactly

holistic paradigm according to its aim and principles

(covered above).

In accordance with Requirement 1, the students

had to develop different groups of models. The ﬁrst

group consists of the models which enable to facilitate

mastering basic and complicated mathematical con-

cepts (objects) by the potential trainees. The second

group of the models must be focused on the realiza-

tion of transdisciplinary connections between Math-

ematics and other subject domains. The third group

has to provide real-life problems solving, based on

the models investigation. Such models and simula-

tions should emphasize the meta-role of Mathematics

as well as demonstrate its practical value (rather than

pure abstract science).

Requirement 2 determined all the models to be

dynamic, to visualize immediately the results of the

trainee’s manipulation and encourage them to learn

the modeled concept actively, via their own experi-

ence.

Requirement 3 expected the complex to be cloud-

based, that is, to be available at www.geogebra.org for

the global GeoGebra community. According to recent

studies, cloud-based learning environment for teach-

ing and learning STEM disciplines opens wide hori-

zons for holistic education due to the realized support

for various processes of learning and research activi-

ties; great level of learning resources ﬂexibility; inte-

gration of variety of educational components based on

innovative technologies. On the other hand, it seems

to be powerful motivational factor for the students

evolved into the complex elaboration as their work

makes them participants of the global community.

Thus, at the ﬁrst stage of the modeling activity,

the students had to understand main features of holis-

tic theory, to realize their core task and the common

requirements to the complex, plan the work and allo-

cate sub-tasks.

Next stage of the work was analytical one which

created necessary theoretical background for develop-

ment of all three groups of dynamic GeoGebra mod-

els.

At this stage of the complex elaboration the

students made deep and comprehensive analysis of

Mathematics to reveal its key concepts and their po-

tential links with the notions of other subject domains.

In order to meet the main pillars of holistic educa-

tional approach (covered earlier) it is necessary to re-

veal key objects of learning in the subject areas, es-

tablish connections between them, and build chains

of proper transdisciplinary links.

Researchers distinguish different types of trans-

disciplinary connections. However, scientists (in par-

ticular, Bevz (Bevz, 2003), diSessa et al. (diSessa

et al., 2004), McDonald and Czerniak (McDonald and

Czerniak, 1994)) recommend to base the connections

classiﬁcation upon the set of three main grounds: in-

formation content of the subject domain, structure of

learning activity, and organization of educational pro-

cess.

As a result, considering the transdisciplinary con-

nections from the standpoints of holistic education,

the students had to reveal key concepts of subjects,

detect their place in the current curriculum, consider

peculiarities of their mastering and proper cognitive

activity.

These procedures were done through the learn-

ing main content threads of the said curriculum sub-

jects (Ministerstvo osvity i nauky Ukrainy, 2017a,b)

and didactic analysis of each subject domain (covered

in (Gryzun, 2018, 2016)). Main content threads of

Mathematics, Science subjects (Physics, Chemistry,

Biology) and Computer Science enabled us to reveal

some transdisciplinary chains. We would like to point

Modelling in GeoGebra in the Context of Holistic Approach Realization in Mathematical Training of Pre-service Specialists

501

out a paramount role of penetrating content threads

in revealing transdisciplinary concepts and links be-

tween them. According to the Concept of the New

Ukrainian School, there are four penetrating content

threads – “Ecology security and sustainable develop-

ment”, “Civil responsibility”, “Health and security”,

“Financial literacy” – which are seen as a mean of

key competences integration of all curriculum sub-

jects. The penetrating threads are considered to be

socially important meta-topics that focus teaching and

learning on the trainees’ holistic understanding of the

world. They are recommended to be regarded during

the learning environment creation at all the levels of

education (Elkin et al., 2017).

Finally, at the analytical stage of the described

modeling activity, the students obtained the set of

connection chains between the Mathematics and other

subject domains. In particular, there were revealed the

links:

• Mathematics – Computer Science;

• Mathematics – Physics;

• Physics – Mathematics – Biology;

• Mathematics – Economics;

• Mathematics – Building (design) and others.

Subsequent detailed analysis of the qualiﬁcation

standards, textbooks, and subject areas resulted in

establishing of transdisciplinary links between the

learning elements (LE), representing concepts and

phenomena which are co-explored by several subject

domains. In particular, the effective semantic analysis

was held with the help of specialized software, such

as: TextAnalyst 2.0, Text Miner 12.1 (its Text Parsing

Node), Trope 8.4.

Such a “smart” analysis of the subject areas en-

abled to distinguish the weightiest LEs of the speciﬁc

subject area along with their conceptual links. Basing

on the depicted analysis, for the revealed weightiest

LEs of Mathematics it was built a graph, representing

their transdisciplinary links with exact learning ele-

ments (LE1...LEn) of other subject domains (SD), ac-

cording to the chains of connections mentioned ear-

lier.

The general scheme of the graph representing

their transdisciplinary links with exact learning ele-

ments (LE1...LEn) of subject domains (SD) as well as

the example of the graph for selected LEs, represent-

ing the transdisciplinary links for the chain: Physics-

Mathematics-Biology (Used below for the transdisci-

plinary Model “Lens”), are given on the ﬁgures 1 and

2.

3 RESULTS AND DISCUSSION

The results of theoretical framework were used by the

students at the design stage of their modelling activ-

ity on the development of the complex of GeoGebra

models, focused on holistic learning of Mathematics

at higher school and university.

The process of the models elaboration provided

by the students (with accordance to the requirements

formulated at the preparation stage) embraces some

phases. At the ﬁrst phase mathematical model of the

future computer model is built. At this point it is

done:

1) revealing and learning of the transdisciplinary

essence of the proper concept (See theoretical

framework);

2) deﬁning of the mathematical dependencies which

can illustrate and investigate the concept;

3) determination of the ﬁxed model parameters and

changeable ones along with the range and step of

their changes;

4) picking up proper graphic elements which are able

to illustrate dynamic changes;

5) revealing of transdisciplinary tasks and real-life

problems which might be solved by the model;

6) elaboration of didactic support as a scheme of

work upon the ransdisciplinary tasks and real-life

problems directed on the forming holistic image

of the said concept (phenomenon).

At the second phase the mathematical model is

built in the environment of GeoGebra. In particular,

the set of standard GeoGebra tools are used (Points,

Lines, Special Lines, Polygon, Circle and Arc, Mea-

surement, Transformations) as well as the CAS com-

ponents (Calculations and Analysis Tools). For real-

ization of dynamic transformations, the Action Object

Tools and Movement Tools are used (Semenikhina

and Drushliak, 2014; Pikalova, 2018).

In order to make the use of the complex more ﬂex-

ible and available to a wide community of students

and teachers, it was organized it in the form of Ge-

oGebra Book. GeoGebra Book is a cloud service

which enables to gather GeoGebra resources, to en-

hance them didactically, and to share them easily. Due

to this fact, the complex of models is oriented to be

a component of a cloud-based learning environment

available to the global GeoGebra community.

The third phase is devoted to the testing, debug-

ging and improving of the model.

The models in the complex are grouped in the

three sections according to the Requirement 1: the

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502

Figure 1: The common scheme of the graph, representing their transdisciplinary links with exact learning elements

(LE1...LEn) of subject domains (SD).

ﬁrst group consists of the models which enable to fa-

cilitate mastering basic and complicated mathemati-

cal concepts (objects) by the potential trainees; the

second group is focused on the realization of transdis-

ciplinary connections between Mathematics and other

subject domains; the third group embraces models

which provide real-life problems solving based on the

models investigation.

Each of the models is presented in the complex

according to the general scheme.

It includes (see examples below):

• model title;

• chain of the transdisciplinary links which are il-

lustrated by the model;

• model description which explains concept (phe-

nomenon) that is a prototype of the model;

• dynamic model itself with a proper functionality;

• procedure of cognitive activity on the realizing the

essence of the concept (phenomenon);

• didactic support as a set of transdisciplinary tasks

and real-life problems for forming holistic image

of the said concept (phenomenon), and a scheme

of work upon them;

• graph of the revealed transdisciplinary links for

the visualization and remembering this holistic

representation.

As it was mentioned above, holistic education ex-

pects trainees’ personal cognitive activity. In order

to facilitate it, it was elaborated procedure of cogni-

tive activity which includes some tips on changing the

parameters of the dynamic model, monitoring the re-

sults, investigating, making conclusions etc. Such a

procedure is aimed to streamline understanding the

essence of the concept (phenomenon).

The didactic support for each model is developed

to involve potential trainees into the solving special

problems and real-life tasks which encourage them to

obtain holistic understanding of the basic concepts via

special cognitive activity based on work with dynamic

models. All of the tasks focus the trainees on the re-

vealing and realizing transdisciplinary links.

Some of the models with their description and

functionality are included into more than one subject

section. However, didactic support as a set of trans-

disciplinary tasks for each model is speciﬁc in each

section and focuses on different transdisciplinary con-

nections.

Below we demonstrate fragmentary some of the

models from various groups of the complex created

by the students within the modeling activity (accord-

ing to general scheme of model presentation depicted

above) and offer recommendations as for their using

to provide holistic learning of Mathematics at high

school and university.

As it was said above, the dynamic models of the

Modelling in GeoGebra in the Context of Holistic Approach Realization in Mathematical Training of Pre-service Specialists

503

Figure 2: The example of the graph for selected LEs, representing the transdisciplinary links for the chain: Physics-

Mathematics-Biology (Used below for the transdisciplinary Model “Lens”).

ﬁrst group enable to facilitate mastering basic and

complicated mathematical concepts (objects) by the

potential trainees. The models are accompanied by

special didactic support focusing them on investiga-

tion and holistic learning of the modeled concept. A

lot of the models expect the model transformation by

a trainee with the aim of its extension on different

class of problems.

Among the models of this group it is worth

mentioning the models Elementary functions inves-

tigation (ﬁgure 3), Triangular properties learning,

Graphical inequalities solution, Calculation of the

area limited by the curve (ﬁgure 4), Remarkable

curves investigation, Investigation of the approxima-

tion curve and others.

Example 1. Model “Remarkable curves investiga-

tion. Epicycloids”

Chain of the transdisciplinary links: Geometry-

Algebra-Mechanics.

Model description: According to deﬁnition,

epicycloids is a plane curve made by tracing the path

of the ﬁxed point P on the circumference of a circle

(called epicycle) which rolls without slipping around

another ﬁxed circle. R is the radius of the ﬁxed circle,

r is the radius of the rolling circle. The model is built

based on the kinematic deﬁnition of epicycloids and

illustrates its different types. Unlike cycloid, epicy-

cloids are not transcendental.

Procedure of cognitive activity with the model (se-

lected tasks of the elaborated didactic support by the

students):

1. Manipulate the model parameters to ﬁgure out

how the number of the curve lobes depends on the

ratio n of R and r. Answer the questions:

• What types of epicycloids is obtained at n=1,

n=2, n=3?

• What happens, when n is integer and when n is

rational? Make conclusions.

2. Monitore the model work, and calculate position

of the point P via radiuses (R, r), the radian from

the tangential point to the moving point and the ra-

dian from the starting point to the tangential point.

3. Transform the original model of epicycloids into

the model of hypocycloid, answering the ques-

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504

Figure 3: Episodes of cognitive activity on the model Elementary functions investigation.

Figure 4: Episodes of manipulation with the model Calculation of the area limited by the curve.

tions and doing proper steps:

• How to calculate R now, when the small circle

“rolls” along inside part the bigger circle?

• How to calculate the speed of the point P which

remain the trace? Why?

• How must animated points P and the center of

smaller circle move now? How are their move-

ment directions related?

4. Use obtained model of hypocycloid, monitor its

work and convince that its number of cusps is also

controlled by the ratio n. Investigate the curve be-

havior, answering the questions:

• How many cusps does the curve have, if n is

integer n?

• How is hypocycloid transformed at n=2?

• What value of n stops the work of the model?

Why? How does the curve look like?

• Investigate the curve behavior at 1<n<2 and

n>2. Make conclusions.

Selected episodes of the students’ cognitive activ-

ity with the dynamic model “Remarkable curves in-

vestigation. Epicycloids” are given on the ﬁgure 5.

The models of the second group are concentrated

on the realization of transdisciplinary connections be-

tween Mathematics and other subject domains. In

particular, it contains the models Clock (connections:

algebra, geometry, trigonometry, physics, sociology,

history, philosophy); Mathematical pendulum (con-

nections: mathematics, physics) (ﬁgure 6); Number

systems (connections: algebra, computer science, dis-

crete mathematics, history); Binary tree (connections:

discrete mathematics, computer science) (ﬁgure 7)

and many others.

Work upon the transdisciplinary models of the

second group is selectively shown in the Example 2

below.

Example 2. Model “Lens”

Chain of the transdisciplinary links: Physics-

Mathematics-Biology.

Model description: The model illustrates princi-

ple of operation of a lens as a simplest optical device

that focuses or disperses a light beam. A lens consists

of a single piece of transparent material (e.g. glass

or plastic). A lens can focus light to form an image

which differs it from prism (See Model “Optical dis-

persion”). A lens has its optical axis, two focuses,

main optical center and plane (you can ﬁnd their def-

initions in your textbook). Lenses are classiﬁed by

the curvature of the two optical surfaces. The model

demonstrates the operation of exactly biconvex lens.

Procedure of cognitive activity with the model (se-

Modelling in GeoGebra in the Context of Holistic Approach Realization in Mathematical Training of Pre-service Specialists

505

Figure 5: Episodes of the students’cognitive activity with the dynamic model “Remarkable curves investigation. Epicycloids”.

Figure 6: Episodes of the students’cognitive activity with the transdisciplinary model Mathematical pendulum.

lected tasks):

1. Operate the model. Change curvature with the

slider. Monitor the focuses positions and image

positions. Find and formulate dependences.

2. Fix the lens curvature and change the object posi-

tion relative to the focus. What is happening with

the image of the object?

3. Fix the object at the distances: d=2F, d>2F, d<2F.

Analyze changes and make conclusions.

4. Analyze changes of the image’s size and position

when the object is between 2F and F, between F

and lens center.

Fragment of didactic support as a set of transdisci-

plinary tasks and real-life problems for forming holis-

tic understanding of the optical device from the stand-

point of Mathematics, Physics, and Biology):

1. Manipulate the model parameters. What is math-

ematical dependence between object distance to

the lens and focus distance? How is it called?

Write the formula of the dependence.

2. What geometrical ﬁgures describe the object, its

image, light beams and the phenomena of light

penetration through the lens?

3. What geometrical facts and properties are re-

vealed by the device operation?

4. Which angles are equal at any values of the model

parameters? Why? Which rays are parallel?

Why?

5. Working with the model, detect the parameters

of the model which provide the highest optical

power of the lens.

6. Manipulating the model and using the scheme of

the optical system of a human eye (ﬁgure 8), an-

swer the questions: (1) what are the components

of the eye optical system? (2) what is the differ-

ence between real and virtual image? (3) what are

the basics of a human eye functioning from the

standpoint of physics? (4) can you explain eye-

sight disorders (short sight, long sight, etc.) via

physical concepts and phenomena? (5) compare

the principles of human eye operation and work

of a digital camera.

Episodes of transdisciplinary tasks doing and the

model operating are shown on the ﬁgure 9.

Graph of the revealed transdisciplinary links for

the visualization and remembering this holistic repre-

sentation presented on ﬁgure 2 above.

The third group of the created models embraces

ones that provide real-life problems solving based on

the so called STEM investigation. As it is expected by

the requirements, the models of the third group en-

able simulations which help realize the meta-role of

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506

Figure 7: Episodes of the students’cognitive activity with the transdisciplinary model Binary tree.

Figure 8: Scheme of the optical system of a human eye.

Mathematics as well as demonstrate its applied value

for practical daily needs. It includes the models such

as Investigation of shooting path; Lift work; Geomet-

rical transformations in real-life measurements; Re-

markable triangular points; Bridge approximations

and many others.

Selected fragments of the real-life tasks solving

within the dynamic models Investigation of shoot-

ing path and Geometrical transformations in real-life

measurements are given on the ﬁgure 10.

Below we are giving the example of STEM

investigation which it is recommended to build,

maintaining the model Fermat-Torricelli points

investigations.

Example 3. Real-life problems solving on the

model Fermat-Torricelli points investigations

Investigation 1. Construct the second Fermat-

Torricelli point by constructing right triangles on the

sides inward. Investigate the properties of the Fermat-

Torricelli point: the sum of the distances from the

point to the vertices of the triangle is minimal, and

all the vertices are visible from it at an angle of 120°.

Investigation 2. Using the obtained dynamic

model, investigate the position of the Fermat-

Torricelli point (with to the triangle), when the tri-

angle has one angle greater than 120°. Determine

whether it will have its properties in this situation.

Determine how the point will behave when there is

an angle of 120°.

Investigation 3. Elaborate the model and try to

ﬁgure out how to use the properties of the point to

solve current problems in your city.

1. Imagine that you need to place the emergency

medical center so that it was at a minimum dis-

tance from the three points of the city A, B, C.

Using a digital map of the city as a working geo-

metric ﬁeld GeoGebra.

2. Match the points of the city to the vertices of

the triangle and determine if there is a Fermat-

Torricelli point for this triangle, ﬁnd this point for

it.

3. Determine which geographical point on the map

corresponds to the found Ferma-Torricelli point.

Investigate whether it is possible to locate an

emergency center at this point in terms of social,

economic and geographical conditions.

4. If this is not possible, manipulate the model pa-

rameters, change the position of the points A, B,

C and ﬁnd out their geometric location so that the

built triangle and its Ferma-Torricelli point meet

the current social needs of the city.

Investigation 4. Formulate a mathematical prob-

lem about the use of the properties of the Ferma-

Torricelli point, which can arise when building roads

507

Figure 9: Episodes of transdisciplinary tasks solving, operating the model “Lens”.

between settlements in your region in order to save

resources. Use the dynamic model to solve and inves-

tigate this problem.

Summarizing the presentation of the complex

of GeoGebra dynamic models created by the stu-

dents (pre-service specialists of different specialities)

within the realization of holistic approach to their

mathematical training, we would emphasize that be-

sides models, the students developed special didactic

support as a set of transdisciplinary tasks and real-life

problems for forming holistic image of the said con-

cept (phenomenon).

The prepared didactic support includes the trans-

disciplinary tasks of various types. In particular, there

are tasks on establishing connections between math-

ematical concepts and notions of other subject areas.

The aim of these tasks is to specify and generalize

mentioned connections; to form the system of the no-

tions of different level of generalization and subor-

dination; to illustrate casual relations of phenomena.

This type of the problems are directed on the forming

of the set of potential trainees’ skills of integrative

properties: to understand meta-role of Mathematics

for other domains of knowledge; to explain processes

and phenomena of one domain with the help of con-

cepts of other branch; to make outlook conclusions

based on common concepts, and others.

Besides, the developed didactic support can offer

transdisciplinary tasks for potential trainees on the de-

termination of community of the facts from differ-

ent subject domains. They allow to specify learn-

ing material, to form new mathematical concepts and

explain them from the standpoints of other branches

of science, to use some mathematical facts to illus-

trate other ones. Such tasks are aimed at the forming

students’ skill of facts’ analysis, generalization and

AET 2020 - Symposium on Advances in Educational Technology

508

Figure 10: Selected fragments of the real-life tasks solving within the dynamic models Investigation of shooting path and

Geometrical transformations in real-life measurements.

explanation from the standpoint of general scientiﬁc

ideas; skill to integrate generalized facts into the ex-

isting knowledge system; skill to apply generalized

knowledge into practice.

In addition, into the didactic support there are in-

cluded the tasks on the establishing connections be-

tween theoretical knowledge and methods, and their

practical use. Mostly they are real-life problems

which focus on the ruining boundaries between Math-

ematics, other subject domains and reality. They

might help to form the trainees’ ability to see scien-

tiﬁc subtext in pure practical tasks, to attract general-

ized knowledge from related areas, and to apply them

to resolving the problem.

Thus, the cloud-based complex of GeoGebra

models (created by the pre-service specialists of dif-

ferent specialities) as for their functionality provides

main principles of the holistic education, such as con-

nections establishing, personal cognitive activity, fo-

cus on the ruining boundaries between subject ﬁelds

and reality.

It seems to be relevant to predict positive inﬂuence

of the complex application on the forming of potential

trainees’ holistic system of mathematical knowledge.

In addition, we would like to point out that the

complex is a result of modeling activity of the stu-

dents within the realization of holistic approach to

their mathematical training. In this context, our ob-

servations and monitoring all the stages of the stu-

dents’simulation work in the process of the complex

development, allow to predict not only raising the

level of their mathematical knowledge. Our monitor-

ing programs and regular surveys also testify deﬁnite

impact on the level of the students’ investigative (en-

quiry) skills. In particular, there was detected pos-

itive dynamic of cognitive, motivational and behav-

ioral components of the said skills. Generalization

and statistical analysis of the obtained empirical re-

sults make prospects of our research.

4 CONCLUSIONS

In accordance with its purpose, the article repre-

sents students’ modeling activity (held within inter-

university projects of Kharkiv GeoGebra Institute)

which resulted in the complex of GeoGebra models

focused on holistic learning of Mathematics at higher

school and university.

Proper theoretical background for the complex de-

sign is elaborated and the stages of the students’ mod-

eling activity are covered. The models in the devel-

oped complex are grouped in the three sections. The

ﬁrst group consists of the models which enable to fa-

cilitate mastering basic essential mathematical con-

cepts (objects) by the potential trainees. The sec-

ond group is focused on the realization of transdisci-

plinary connections between Mathematics and other

subject domains. The third group embraces models

which provide real-life problems solving based on the

models investigation. All the groups are represented

in the article along with speciﬁc examples of the mod-

els.

In order to facilitate potential trainees’ personal

cognitive activity that is expected by holistic educa-

tion, it was elaborated procedure of cognitive activity

which includes some tips on changing the parameters

of the dynamic model, monitoring the results, inves-

tigating, making conclusions etc. Such a procedure is

aimed to streamline understanding the essence of the

concept (phenomenon). The didactic support for each

model was developed by the students to involve po-

tential trainees into the solving special problems and

real-life tasks which encourage them to obtain holistic

understanding of the basic concepts via special cogni-

tive activity based on work with dynamic models. The

said didactic support is characterized in the paper.

The prospects of further research are outlined.

509

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