Digital Technologies in Specialized Mathematics Education: Application

of GeoGebra in Stereometry Teaching

Tetiana H. Kramarenko

1 a

, Olha S. Pylypenko

1,2 b

and Olga Yu. Serdiuk

3 c

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

State University of Economics and Technology, 37A Vatutin Str., Kryvyi Rih, 50096, Ukraine

Kryvyi Rih National University, 11 Vitaly Matusevich Str., Kryvyi Rih, 50027, Ukraine

Keywords:

GeoGebra 3D Graphing Calculator, Geometry, STEM Competencies, Stereometry Teaching, Methodology of

Teaching Mathematics, Cloud Technologies in Education.

Abstract:

The purpose of the paper is to improve methodology of teaching Mathematics via the use of digital technolo-

gies. The task of the paper is to identify the issues that require a theoretical and experimental solution. The

objective of the paper is the educational process in the higher education institution, the subject of the paper is

modern ICT. The result of the study is the learning tools of pedagogically considered and adequate bending

of conventional and modern learning environment implemented into the educational process. The possibilities

of using cloud technologies and Dynamic Mathematics system GeoGebra in the educational process through

Stereometry specialized training have been revealed. The use of GeoGebra Dynamic Mathematics in Stereom-

etry teaching will favourably inﬂuence the formation of students’ STEM competencies. In order to encourage

Mathematics and Computer Science teachers to implement effectively the elements of STEM education, it is

suggested that cloud-based learning tools such as GeoGebra be used in the teaching process.

1 INTRODUCTION

Velikova and Petkova (Velikova and Petkova, 2019)

point out that every society needs STEM profession-

als, talented people who design new technologies,

materials, constructions and processes. One of STEM

training related ﬁelds is an integrated problem-based

Mathematics training.

The results of the research study regarding the

development of teachers’ STEM competencies have

shown that most of them are not aware of the pe-

culiarities of students’ STEM competencies forma-

tion. And some contradictions between the necessity

to form the students’ STEM competencies and insuf-

ﬁcient attention to the training of future teachers of

Mathematics occur.

One of the most effective tools for STEM-based

Mathematics teaching is the system of dynamic math-

ematics, e.g. GRAN 2D tool and cloud-based Ge-

oGebra. Lavicza et al. (Lavicza et al., 2020) claim

that GeoGebra was designed to integrate arithmetic,

https://orcid.org/0000-0003-2125-2242

https://orcid.org/0000-0003-0493-8429

https://orcid.org/0000-0003-0505-0800

algebra, geometry, calculus, statistics. In recent times

it is supposed to support STEM subjects as a single,

integrated system available on most technology plat-

forms. Moreover, teachers and students are offered

free access to it all over the world. GeoGebra was

originally created to integrate Algebra and Geometry

into a single environment. GeoGebra enables to use

such new technologies as Augmented (Striuk et al.,

2018) and Virtual Reality (Lavrentieva et al., 2020),

3D Printing (Hevko et al., 2020) and MID (Modlo

et al., 2019) in the learning Mathematics.

The use of GeoGebra Dynamic Mathematics in

Stereometry teaching will enhance the formation of

students’ STEM competencies. It can also be help-

ful in the formation of key mathematical and digi-

tal competences (Astaﬁeva et al., 2020; Moiseienko

et al., 2020), self-study competences, as well as in the

development of spatial thinking.

In order to encourage Mathematics and Computer

Science teachers to implement the elements of STEM

education, it is suggested the cloud-based learning

tools, such as GeoGebra, be used in the learning pro-

cess.

The 3D Graphics application (3D feature graphs,

surface, and 3D geometry) can be used while develop-

576

Kramarenko, T., Pylypenko, O. and Serdiuk, O.

Digital Technologies in Specialized Mathematics Education: Application of GeoGebra in Stereometry Teaching.

DOI: 10.5220/0010926300003364

In Proceedings of the 1st Symposium on Advances in Educational Technology (AET 2020) - Volume 1, pages 576-589

ISBN: 978-989-758-558-6

ing the visuals with AR. The application is now avail-

able not only for users of gadgets in the iOS operating

system, but also for Android version 4.0 and higher.

This program includes some examples of 3D mathe-

matical objects that can be placed, such as on a desk-

top, ﬂoor, or any other ﬂat surface. The ﬁxed models

will be managed. They can be resized. Such visual-

ization will allow you to see more mathematics in the

surrounding world.

More detailed information about the application

3D Graphics can be found in our article (Kramarenko

et al., 2019b), teaching aids visualization during lec-

tures and practical classes, in particular in Mathemat-

ics, allows students to understand the learning mate-

rial better, to increase the applied orientation of learn-

ing and the communication competence both learners

and teachers.

Gartner attributes Artiﬁcial Intelligence Edu-

cation Applications, Conversational User Inter-

faces, Blockchain in Education, Immersive Tech-

nology Applications in Education, Design Thinking,

Competency-Based Education Platforms and Adap-

tive Learning Platforms to the main tendencies of us-

ing ICT in education (Panetta, 2019).

Since augmented reality technology already has

an important place in innovative development, it can

also have signiﬁcant potential for implementation in

Mathematics learning. That is why this technology

needs more detailed study. Because augmented real-

ity is intrinsically linked to 3D construction, its usage

in conjunction with Dynamic Mathematics systems

like GeoGebra, can signiﬁcantly increase the level

of visualization in Mathematics and enhance students

learning. In addition, Augmented Reality can be-

come a tool for enhancing STEM-based learning for

students majoring in Mathematics and Computer Sci-

ence.

2 LITERATURE REVIEW

According to the recommendation of the European

Commission (EC, 2018), competence is deﬁned as a

combination of knowledge, skills and attitudes. In the

updated list of key competences of a personality eight

major ones are emphasized (EC, 2018). We are keen

to foster the development of students’ STEM compe-

tencies: mathematical competence and competence in

science, technology and engineering.

Rakov et al. (Rakov et al., 2009) explores the

possibilities of mastering mathematical competencies

in the learning by future teachers through the re-

search approach using the system of dynamic math-

ematics DG. The scientist distinguishes procedural,

logical, technological, research and methodological

competencies of Mathematics teacher. In particular,

methodological competence implies the ability to as-

sess the appropriateness of using mathematical meth-

ods to solve individual and social issues.

Consider in detail the opportunities which GeoGe-

bra gives to a teacher. Hohenwarter et al. (Hohen-

warter et al., 2008) emphasize a signiﬁcantly impor-

tant role of free open-source software for teaching

mathematics world-wide. Open-source software do

not only offer opportunities for teachers and students

to use them both at home and in the classroom with-

out any restrictions, but they also provide the tools

for developing support and user communities reach-

ing across borders. Such collaboration as well con-

tributes to the equal access to technological resources

and democratization of learning and teaching mathe-

matics.

GeoGebra is available in multiple languages. Us-

ing GeoGebra a teacher can create interactive mate-

rials to solve the main tasks of teaching Geometry –

the development of spatial imagination, practical un-

derstanding and logical thinking.

GeoGebra can be used both in standalone and on-

line modes on mobile Internet devices (MID) which

offer new opportunities to improve learning and

teaching either inside or outside of classrooms. Fur-

ther advantage of GeoGebra is the Resources tab con-

taining a number of teaching materials: textbooks, ar-

ticles, instructions, and illustrations. Also, a regis-

tered users can share their own educational materials

on the ofﬁcial website via GeoGebraTube (this is a

kind of ﬁle-sharing site). Its essential feature is that

the created dynamic constructions can be accessed

online via the use of View Sheet tool. Any user is

allowed to comment on GeoGebraTube content.

A user is provided with a possibility to create their

own GeoGebraBook, a folder where selected materi-

als can be added and subfolders can be created. A

user can publish a game, a collection of visuals, lesson

workbook that may include text messages, GeoGe-

bra models, web pages, questions, videos, images and

manuals public to anyone. GeoGebraTube content

can be shared via social networks Facebook, Twitter,

emailed and uploaded to Google Drive. Cloud-based

tools affect favourably teacher-student interactions in

an informal learning environment.

Dikovi

c (Dikovi

c, 2009) statistically conﬁrms the

fact that the use of GeoGebra applications in learning

calculus had a positive effect on the development of

students’ reasoning and visualization skills.

The issues of using GeoGebra are highlighted

in (Drushlyak et al., 2020; Semenikhina, 2017; Se-

menikhina and Drushliak, 2014). Application of Ge-

Digital Technologies in Specialized Mathematics Education: Application of GeoGebra in Stereometry Teaching

577

oGebra 5.0 to the solution of solid geometry problems

has been analyzed in (Semenikhina and Drushliak,

2014). Examples of solid geometry problems with

detailed solution and learning clues have been pre-

sented. Among them there are problems on auxiliary

section, polyhedron net, locus problems, space trans-

formations etc. Semenikhina (Semenikhina, 2017)

discusses the relevance of the study of the dynamic

Mathematics software for future teachers of Mathe-

matics in Ukraine.

The purpose of study by Zetriuslita et al. (Zetrius-

lita et al., 2021) was to identify improvements in Self

Efﬁcacy and selfregulated through GeoGebra Based

Teaching seen from the level of students’ initial math-

ematical abilities. The research method used was a

mixed-method with a sequential explanatory strategy,

while the research design was an untreated control

group design with pretest and posttest. the experi-

mental group obtained direct learning using GeoGe-

bra software while the control group obtained conven-

tional learning. The population in this study were uni-

versity students of mathematics education department

at the Islamic University of Riau who took the ﬁeld

and space analytic geometry course. The research in-

struments used were self-efﬁcacy questionnaire and

self-regulated questionnaire for getting quantitative

data and interview guidelines for getting qualitative

data. It was concluded that GeoGebra based teach-

ing was effective in improving students’ self-efﬁcacy

and self-regulated. Judging from the level of students’

initial mathematical abilities, there was an increase

in self-efﬁcacy and self-regulated through GeoGebra

based teaching from both high, medium and low lev-

els.

Modelling implies simplifying thus providing the

model with its own characteristics, independent from

the original reality. Consequently, geometry, in its

interpretation and processing, acts like a new reality

both abstract in its logic and concrete in its represen-

tation modes.

Richard and Blossier (Richard and Blossier, 2012)

begin with classical theories of modelling representa-

tion processes in order to apply them to 3D dynamic

geometry software, GeoGebra3D, computing devel-

opment seeking to bring geometric models closer to

the reality of a school setting. The authors intro-

duce three key notions, developed from Mathematics

teaching, in order to support the exploration of two

interaction situations between mathematics and their

teaching. Finally, they ﬁnish on a few general consid-

erations for the teaching of mathematics.

A number of works of scientists and software de-

velopers are devoted to the research of integration is-

sues of the augmented reality technology into the ed-

ucational process. In particular, (Brzezinski, 2019;

Striuk et al., 2018), focus on the general trends and

special issues of the augmented reality application in

education.

In teaching Stereometry, the assignment work-

sheets on the construction of sections through the

trace method, internal projection and combination

method by Sidoruk (Sidoruk, 2018) should be taken

into consideration. Each of the assignments is pro-

vided with the possibility of step-by-step procedure

of the construction building. Thus, students are given

an opportunity to develop both mathematical compe-

tence and the ability to learn independently. Some as-

signments include constructions created on a 3D can-

vas.

Fariha and Lestari (Fariha and Lestari, 2019) de-

scribe in detail how to realize the Dandelin Sphere

more realistically, using GeoGebra Classic 5, which

supports the creation of 3D images.

At a profound methodological level, a number

of assignments on combinations of geometrical solid

shapes was created by Rykovskyi (Rykovskyi, 2018).

These models are designed as constructions of plane

geometric shapes through the use of parallel projec-

tion properties. They are intended to be used to vi-

sualize the properties of geometrical shapes. How-

ever, students might face a challenge with reproduc-

ing models as the algorithm of the construction and

details of the construction are not available for them.

The models are designed as constructions of plane

geometric shapes with the use of the properties of par-

allel projection. In particular, a book / collection of

visuals covering the topic “Prism” is recommended.

There is a number of visuals that represents combi-

nations of stereometric shapes: sphere and pyramid,

sphere and prism, sphere and cylinder etc. Consider

the visual “There is a cone inscribed in the sphere.

And there is the right triangular pyramid inscribed

into the cone. A pyramid is inscribed in the sphere”.

Even the name itself suggests that integrity is inher-

ent to visuals. It can be used in the process of teach-

ing Stereometry in school or teaching Methodology

of Mathematics on various topics. While using vi-

sual aids, users are advised to study the properties of

shapes inscribed into the sphere, to look into the re-

lationship between the sphere radius, radius of cone

base, the height of the cone and element of the cone.

These visuals are equipped with the Checkbox

tool which allows a user to do the revision and answer

to the questions: Which cone is called inscribed into

the sphere? What element of the cone is the center

of the sphere? Which pyramid is called inscribed in

the cone? Which pyramid is called inscribed into the

sphere? How to ﬁnd the axial section of the cone?

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578

Which circle of the sphere is called a great circle?

Which properties of the diameter of the sphere per-

pendicular to its great circle?

Mobile applications such as Geometry, Graphing

Calculator, 3D Graphing Calculator have been rela-

tively recently launched and now they are available

on the ofﬁcial GeoGebra website. These applica-

tions are still undergoing improvements. However,

the methodology of their use, and especially in teach-

ing Stereometry, is not yet well developed. Therefore,

this paper aims at elucidating the features of creating

visuals with GeoGebra and their use in teaching how

to solve Stereometric problems.

3 RESULTS

3.1 GeoGebraBook “Models for

Stereometric Problems” as Training

Manual on the Methodology of

Teaching Mathematics

The methodical materials of the paper authors related

to the use of the GRAN software are presented in the

manual “Innovative information and communication

technologies of teaching Mathematics”. The issue of

GeoGebra use has been considered in an updated ver-

sion of the manual (Kramarenko et al., 2019a). On

the GeoGebra site, users can ﬁnd the materials which

contain relevant visuals at GeoGebraBook “Models

for stereometric tasks” (Kramarenko, 2019). One can

ﬁnd here problem situations to be solved by students

independently, self-assessment tests.

The study of the effectiveness of the use of Ge-

oGebra mobile applications in the process of teaching

Stereometry was conducted by us within the prepara-

tion of future teachers of Mathematics during Mathe-

matics teaching methodology classes, advanced train-

ing courses at Kryvyi Rih State Pedagogical Univer-

sity, in teaching students of Kryvyi Rih State Col-

lege of Economics and Technologies, and students

of classes with mathematical specialization of Kryvyi

Rih schools.

Particular interest in using GeoGebra was shown

during the advanced training courses by Kryvyi Rih

teachers of Mathematics who have been working at

school for a considerable time. The study was also

conducted on the effectiveness of the use of GeoGe-

bra applications by students in order to enhance stu-

dents’ motivation during extra-curricular vocational

activities at the university. And it was studied as well

during the course of Sereometry lessons conducted by

masters majoring in Mathematics education complet-

ing professional practice in secondary education insti-

tutions.

When using applications, it is useful for a student

to be able to reproduce the construction in the browser

window, open other temporarily hidden drawing ele-

ments in case of necessity, and receive text notiﬁca-

tions. However, a considerable number of construc-

tions should be created later or simultaneously with

the viewing via the mobile application.

Consider the peculiarities of the implementation

of speciﬁc constructions and provide recommenda-

tions concerning their use.

One of the main issues that school students and

students of higher education establishments face is

the recognition of a 2D drawing representing the im-

age of a 3D shapes. In order to develop the spatial

imagination of students at schools and universities it

is necessary to provide them with the assignments on

constructing sections of polyhedron by planes.

3.2 Construction of Sections of

Polyhedron With Plane via the

Applications GeoGebra Geometry

and 3D Graphics

Consider examples of problems for constructing a

section of a polyhedron with a plane passing through

three given points that are not on the same line;

through a straight line and a point that does not belong

to it etc. It is necessary to construct a linear angle of a

dihedral angle and measure them; the angle between

the straight line and the plane.

Look into what four approaches to constructing a

section of a polyhedron in a plane which should be

distinguished by a teacher in order to be able to pro-

vide their students with the most appropriate one for

their educational trajectory (Kramarenko et al., 2020).

If a student has not fully mastered GeoGebra

tools, it is advisable to provide them with a sheet of

paper guiding the algorithm to follow in order to build

a construction. Moreover, a student should be sup-

plied with information regarding the icons that cor-

respond to the tools in question. It is appropriate to

place QR codes on printable worksheets for demon-

strations.

1. The construction on a 2D canvas is rather cum-

bersome and requires logically justiﬁed steps for

the construction. At ﬁrst, it is necessary to fol-

low step-by-step construction procedures to build

several drawings, created via “the trace method”

on a 2D canvas. To intensify the material dissem-

ination, students may be at once provided with a

constructed polyhedron. However, in the course

Digital Technologies in Specialized Mathematics Education: Application of GeoGebra in Stereometry Teaching

579

of the study, there were cases when the students

constructed the polyhedron wrongly. That is, after

its construction on the plane, they were not able

to change it dynamically. For example, to build a

prism image on a 2D canvas made of an n-sided

polygonal base (n > 3), it is necessary to follow

the following steps. Firstly one should build an n-

sided polygon via the use of the appropriate tool.

Then it is preferable to build a vector along one of

the edges of the prism. Further, one should apply

a parallel translation of the n-sided polygon to the

constructed vector. The given vertexes are con-

nected. Such a construction is represented taking

into account the properties of the parallel projec-

tion.

2. Next it is necessary to compare the previous draw-

ing with the one presented according to the same

problem situation but on the 3D canvas. In this

case, all straight lines for the construction of the

section can be presented step by step. The advan-

tages of such a construction include the possibil-

ity to change the constructed polyhedron dynam-

ically and the location of points which the secant

plane passes through. Via the GeoGebra 3D Ge-

ometry (GeoGebra Team German, 2018) app, one

can simulate external actions aimed at geometri-

cal solid shapes which are necessary for a stu-

dent to be able to apply internal thinking regarding

them and as a result develop spatial thinking. In

our opinion, this approach is the most appropriate

for the development of students’ spatial imagina-

tion, logical thinking. In ﬁgure 1 the construction

of the section of the prisms with a secant plane

which passes through the point on the lateral edge

and a straight line drawn in the plane of the base

is presented.

3. The third option for constructing a section on the

3D canvas. Firstly, a polyhedron, and a secant

plane should be built, and then the tool “Inter-

section of two surfaces” should be applied. A

student, by “turning” the polyhedron, will have

an opportunity for the better perception and un-

derstanding of the construction. The approach

in question is appropriate when augmented real-

ity tools are used enabling to align the drawings

with the image of a real object (Kramarenko et al.,

2019b). It is advisable to encourage students to

ﬁnd examples of such sections among the sur-

rounding objects, in architecture etc.

4. The use of both 2D and 3D canvases simultane-

ously. Such a combination is advisable if there is

some kind of difﬁculty in constructing a polygon

that is for a base of the pyramid or specifying the

position of the vertex of the pyramid, etc. For in-

stance, if a trapezoid, in particular, a rectangular

or equilateral one, is for a base of the pyramid or

prism (ﬁgure 2).

It is worth pointing out that such constructions can

only be built in the classic version of GeoGebra, and

the construction preview may also be available via

MID after uploading the ﬁle to the GeoGebra cloud

repository.

3.3 The Peculiarities of the Deﬁnition

and Angle Construction between

Planes and Linear Angle of the

Dihedral Angle

Via the assistant of GeoGebra (3D canvas) it is pos-

sible to build the base of the pyramid (OXY) on the

canvas, and in the process the constructions can be

synchronously displayed on the 3D canvas (ﬁgure 3).

Attention should be drawn to the convenience and

ease of the construction with the help of the linear an-

gle of the dihedral angle at the base. According to

the deﬁnition, to determine the linear angle of the di-

hedral angle at the base a plane perpendicular to the

edge of the dihedral angle should be drawn. Then

the angle between the rays formed as a result of the

intersection of this plane with the edges of the dihe-

dral angle will determine its linear angle (Bevz et al.,

2011).

It should be stated that the measure of the angle

between the planes varies from 0

◦

to 90

◦

. While the

measure of the dihedral angle can vary from 0

◦

360

◦

(Rykovskyi, 2018). Here is an example of the

problem from the textbook with major in Mathemat-

ics (No. 836, (Bevz et al., 2011)).

At the base of the pyramid there is an isosceles

triangle with angle β at the vertex and radius R of the

circle described. The plane of each face of the pyra-

mid forms an angle α with the plane of the base. Find

the area of the side surface.

It is recommended to use 2D and 3D canvases si-

multaneously to represent the drawing corresponding

to the set problem (Kramarenko, 2019).

The reason why mistakes are often made is that

only one case is under consideration. Whereas the

concept of “angle between the plane of the lateral face

and the plane of the base” is substituted by “dihedral

angle at the base”. If the problem situation said that

“dihedral angles at the base were congruent”, then we

would consider the pyramid the vertex of which is

projected into the center of the circle inscribed into

the triangle of the base (Pyramid of Type 1).

Since it concerns only congruent angles between

planes, the problem will have two solutions. It is also

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580

Figure 1: Construction of a section of the prisms with a plane.

Figure 2: Construction of a prism, using both 2D and 3D canvas simultaneously.

necessary to take under consideration the case when

one of the dihedral angles at the edge of the base is

obtuse (Pyramid of Type 2). In this case, the vertex

of the pyramid is projected into the center of the in-

scribed circle lying outside (ﬁgure 3).

Using the problem mentioned above and similar

ones we have conducted a research on the basis of

the two 11th grade classes specialized in Mathemat-

ics, and in particular with the teachers of Mathemat-

ics Svitlana Shakhmatova and Ruslan Kaluhin (Kra-

marenko et al., 2019a). 16 students of the experimen-

tal group (EG students) and 18 students of the con-

trol group (CG) have participated in the research. The

similar research has also been conducted at the ad-

vanced training courses among the teachers of Math-

ematics in Kryvyi Rih State Pedagogical University

Digital Technologies in Specialized Mathematics Education: Application of GeoGebra in Stereometry Teaching

581

Figure 3: Image of the pyramid and linear angles.

(group 1 – 22 teachers, group 2 – 26 teachers). Some

teachers among the participants of the courses did not

teach students in high school that is why they do not

deal with solving these very problems in their teach-

ing.

The suggested questionnaire consisted of several

questions and corresponded mainly to the Pyramid of

Type 2. The participants were supposed to deal with

the questions during the problem solving. Alongside

the questionnaire we present the results of the survey.

1. On reading the problem situation, have you distin-

guished two types of the Pyramid?

A positive answer: EG – 1 student (6%), CG – no

students (0%), group 1 – 5 teachers (23%), group 2 –

6 teachers (23%).

2. The participants received a clue about the way it

is possible to construct the Pyramid with the ob-

tuse dihedral angle at the base (Pyramid of type

2). The question was the following: Is it possible

to construct the vertex of the pyramid projection

on the plane base and in fact the Pyramid itself?

A positive answer: EG – 6 students (28%), CG –

5 students (28%), group 1 – 12 teachers (55%), group

2 – 13 teachers (50%).

3. First, a dynamic model, constructed with GeoGe-

bra, was demonstrated to the participants. The al-

gorithm of the Pyramid construction via the use

of the tool GeoGebra Construction Steps was dis-

played. The task was to construct the Pyramid

of Type 2 and create a problem solving algorithm

individually. Next, a paper Pyramid model was

demonstrated to the participants. The algorithm

of the construction was discussed. The task was

to construct the Pyramid of Type 2 and create a

problem solving algorithm.

A positive answer: EG – 12 students (75%), CG –

8 students (44%), group 1 – 19 teachers (86%), group

2 – 17 teachers (65%). Fisher’s angular transforma-

tion (Stevens, 1953) was used for the research results

processing. For this purpose, the task performance

proportions in the groups during the ﬁrst and second

stages were compared.

ϕ (EG, CG) = 0,61 < 1,64; ϕ (group1, group2) =

0,31 < 1,64: the received results do not differ consid-

erably both in the groups of students and teachers.

After stage 3, the proportions of successful task

performance in groups were compared one more time.

ϕ (EG, CG) = 1,85 > 1,64; ϕ (group1, group2) = 1,73

> 1,64: the results differ signiﬁcantly this time.

It has been deﬁned that the use of dynamic models

created with GeoGebra was more effective in compar-

ison with the static paper ones.

On discussing the use of dynamic models created

with GeoGebra, participants pointed out the follow-

ing advantages: variability; dynamic visuals which

enhance the development of spatial imagination; mul-

tiple reproducibility of the construction, which will

contribute to the development of algorithmic think-

ing.

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582

3.4 Tasks on Combinations of

Polyhedron and Solids of Revolution

Consider the way it is possible to inscribe a sphere

into the right rectangular pyramid via the use of 3D

Geometry. In order to construct the base of the pyra-

mid, it is necessary to use the Right Polygon tool, by

pointing two points on the 3D canvas – adjacent ver-

texes of the base, and indicating that the right polygon

has 4 vertexes. Then one should construct the diago-

nals of the square (the Segment tool) and deﬁne the

center (Intersection point). Then through the center

of the square, which is also the center of the circle

inscribed in the square, one draws a straight line per-

pendicular to the plane of the square. On this straight

line, one chooses an arbitrary point (Point on the ob-

ject) and constructs a polyhedron (Pyramid). The per-

pendicular to the plane of the square straight line is

the geometric location of points, equidistant from the

sides of the base of the right pyramid.

To determine the position of the center inscribed

sphere in the pyramid, one constructs a geometric lo-

cation of points that are equidistant from the edges of

the dihedral angle at the base of the pyramid. Since

there is no construction of the bisector plane in the

GeoGebra tools, it is necessary to construct a linear

angle of the dihedral angle at the base and then bi-

sector of the very angle. The plane passing through

the vertex of the pyramid perpendicular to the edge of

the base is built (Plane through the point perpendicu-

lar to the straight line; Intersection point). Instead of

a plane, it is possible to draw a straight line from the

vertex of the pyramid perpendicular to the edge of the

base (straight, perpendicular to straight). Next, one

should ﬁnd the intersection point of the constructed

plane / perpendicular with the edge of the base (Inter-

section point of the straight line and the plane / Inter-

section point of two straight lines). Then one builds

the bisector of the obtained linear angle.

The point of its intersection with the perpendicu-

lar to the base of the pyramid, drawn from the top of

the pyramid, will determine the center of the inscribed

sphere (Point of intersection). Finally, one constructs

the inscribed sphere (Sphere outside the center and ra-

dius), specifying in sequence the center of the sphere

and the point of intersection of the diagonals of the

square (Kramarenko et al., 2020).

For better understanding and mastering of the

algorithm the construction of the sphere inscribed

around the pyramid the students setting of the can-

vas are adjusted to be able to show the step-by-step

procedure of the construction.

With AR, the students can understand the basic

concepts of 3D geometrical shapes, their relationships

and ways to construct the 3D shapes and the objects

in 3D space. Importantly, AR can provide a dynamic

visualization of 3D structures of geometrical shapes.

This feature helps the students to understand a com-

prehensive background of 3D geometrical shapes and

improve the abilities of geometrical structures. More-

over, the hand gesture based interactions furnish an

intuitive and convenient way for the students to di-

rectly control and interact with geometrical shapes in

3D space.

GeoGebra Augmented Reality application allows

you to transfer the constructed ﬁgure into the space of

the room (ﬁgure 4). Having built a ﬁgure, we press

the “AR” button. Next, you need to use the camera

to select the environment in which we plan to move

the object. For example, on the table. By tapping on

the screen, the ﬁgure will be transferred to the real

world (Kramarenko and Pylypenko, 2021b) where it

can be explored. The phone camera will serve our

eyes. Immersing the phone in a virtual ﬁgure we will

see it from the inside, we can bypass it, also the ap-

plication allows you to resize, color (Kramarenko and

Pylypenko, 2021a).

With the experiences of interacting with the 3D

shapes using their own hand gestures, the students

can improve their own awareness of the relationships

of the 3D shapes and easily remember or retain the

knowledge about the 3D shapes.

3.5 Stereometric Problems of Applied

Content

Geometry is an abstract science, often taught without

proper implementation of its applied orientation. This

leads to the fact that a signiﬁcant part of students do

not feel the need to study this subject, because they

do not see the possibility of using the acquired geo-

metric knowledge, in particular in stereometry, in the

future. And so there is a need to connect stereometric

problems with life. We propose to consider two prob-

lems of applied direction, for the solution of which

we consider it expedient to involve the GeoGebra 3D

application. We offered these tasks to students of the

State University of Economics and Technologies.

Problem 1. What percentage of wood goes to

waste when made of wooden logs, 5 m long and 20

cm and 15 cm in diameter, beams with a rectangular

cross-section of the maximum cross-sectional area?

Problem 2. Calculate the volume of the largest

beam with a base in the shape of a rectangle, which

can be carved from a log of cylindrical shape. The

length of the log is 5 m and the thickness is 20 cm.

What percentage of wood will go to waste?

Using these tasks, we conducted research on the

Digital Technologies in Specialized Mathematics Education: Application of GeoGebra in Stereometry Teaching

583

Figure 4: GeoGebra AR demos.

basis of two parallel groups majoring in “Finance and

Credit”. 18 students of the experimental group (EG)

and 17 students of the control group (CG) took part in

the study. In the experimental group, the task was to

solve problems based on a dynamic ﬁgure, the control

group solved the same problems, but with the help of

static.

The proposed questionnaire consisted of several

questions that students answered while solving prob-

lems.

1. What ﬁgures will we work with? Positive answer:

CG – 6 students (35%), EG – 7 students (39%).

2. How are the ﬁgures relative to each other? Posi-

tive answer: CG – 7 students (41%), EG – 7 stu-

dents (39%).

3. What shape should be the cross section of the

beam to maximize its size? The volume of the

beam will be the largest if the cross section of the

beam is square. It is not necessary to compose a

function and study it to the extreme, it is enough

to use the formula to calculate the area of a quadri-

lateral inscribed in a circle. Positive answer: CG –

4 students (24%), EG – 3 students (17%).

In the second stage, the CG group was shown a

ﬁgure for the problem on paper, the EG group was

shown a ﬁgure in GeoGebra (ﬁgure 5) and consid-

ered in dynamics.

4. After that, the groups were asked the last question

about the cross-section again, the statistics of pos-

itive answers improved: CG – 6 students (35%),

EG – 9 students (50%).

5. What is meant by waste from the manufacture of

logs? The positive answer that this is the differ-

ence between the volume of the truncated cone

and the volume of the parallelepiped was given

by: CG – 9 students (53%), EG – 12 students

67%.

The dynamic image in GeoGebra helped the EG

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584

Figure 5: A parallelepiped inscribed in a truncated cone.

group to improve the statistics of responses, after the

demonstration of the ﬁgure on paper this effect could

not be achieved. The results of the survey showed that

the highest efﬁciency is achieved when demonstrating

dynamic models.

Optimization tasks using Geogebra were proposed

by us in the textbook (Kramarenko and Pylypenko,

2021b). We supplemented the sets of tasks using Ge-

ogebra with visual aids for specialized teaching of

mathematics, realization of interdisciplinary connec-

tions of the beginnings of mathematical analysis and

stereometry. In this case, you can use the expressions

to calculate the volume of the body to track the change

in this value and ﬁnd the optimal size of the beam. It

is also advisable to use the “Function Inspector” tool

in GeoGebra to ﬁnd the extreme values of the function

and visualize the abstractions.

It is convenient to write the formulas on the canvas

at once, and then open them step by step during the

discussion. To make such a blank in the application

GeoGebra 3D, you must ﬁrst build a truncated cone

(by crossing the cone plane), then through the center

of a smaller circle and a point on it build a line. Draw

a perpendicular line to the obtained line, choosing the

center of a smaller circle as a point. Mark the points

of intersection of the lines with the circle and through

the obtained 4 points build a square (using the Poly-

gon tool), connecting the points in series. From the

vertices of the square we lower the perpendiculars to

the lower base of the cone (larger circle) and mark the

points of intersection of the perpendiculars with the

plane of the base of the cone, through the obtained 4

points we build a square, connecting the points in se-

ries. Using the Prism tool, build a prism by selecting

a polygon of the base (square) and the vertex at one

of the points of the smaller circle.

During the in-depth study of mathematics at the

Kryvyi Rih Pokrovsky Lyceum, we offered students

the problem of stereometry for optimization accord-

ing to the textbook (Skanavi, 1990). After calculating

the optimal dimensions of the prism / pyramid, the

polygon scan was drawn and glued. Models in dy-

namics created by means of system of dynamic math-

ematics were offered for demonstrations. Here are ex-

amples of mathematical problems that students had to

reformulate as problems of applied content.

1. (15.194) What are the dimensions of the base ra-

dius and the height of the open cylindrical tank, so

that at a given volume V for its manufacture was

spent the least amount of sheet metal?

2. (15.195) The side face of a regular quadrangu-

lar pyramid has a constant given area and is in-

clined to the plane of the base at an angle α. At

what value of α is the volume of the pyramid the

largest?

Digital Technologies in Specialized Mathematics Education: Application of GeoGebra in Stereometry Teaching

585

3. (15.196) In a regular quadrangular pyramid with

the edge of the base a and the height H, a regular

quadrangular prism is inscribed so that its lower

base is located at the base of the pyramid, and the

vertices of the upper base are placed on the side

edges. Find the edge of the base and the height of

the prism that has the largest side surface.

4. (15.197) The side edge of a right triangular pyra-

mid has a constant given length and forms an an-

gle α with the plane of the base. At what value of

α will the volume of the pyramid be the largest?

5. (15.198) In a regular triangular pyramid, the side

face has a constant given constant area and forms

an angle α with the plane of the base. At what

value of α is the distance from the center of the

base of the pyramid to its side face the largest?

6. (15.199) A pyramid is inscribed in a cone with

a given constant volume, which is based on an

isosceles triangle with an angle at the vertex equal

to α. At what value of α is the volume of the pyra-

mid the largest?

7. (15.200) The generating cone has a constant

length and forms an angle α with the height of the

cone. A regular hexagonal prism with equal edges

is inscribed in the cone (the base of the prism is

located in the plane of the base of the cone). At

what value of α is the side surface of the prism the

largest?

Solving problems of applied content will provide

an opportunity to motivate, intensify the educational

and cognitive activities of students and promote the

practical application of acquired knowledge.

3.6 Project Work in GeoGebra 3D

One of the effective means of developing students’

cognitive activity is the project method. After all,

the project method includes a set of research, search,

problem, creative approaches, promotes the creative

development of students, prepares them to solve prob-

lem situations in everyday life. Therefore, it is advis-

able to offer students to perform mini-projects while

studying the section of stereometry.

The task of the project will be to build a

playground in the GeoGebra 3D application, using

the maximum number of studied geometric shapes:

prisms, pyramids, spheres, cones, cylinders, etc. (ﬁg-

ure 6). Performance appraisal is a mandatory element

of the organization of project work. The effectiveness

of the project lies in the ratio of planned expectations

with the ﬁnal results. Created designs can be designed

in the yard with an augmented reality application.

There are three stages of self-regulated, namely

the Planning Phase, at this stage students set steps

for learning, namely (1) Analyzing learning tasks,

(2) Determining learning objectives, and (3) Planning

learning strategies. In the analyzing stage, students

implement a plan that is constantly monitored to en-

sure it leads to learning goals. In the determining

stage, students determine how well the learning strat-

egy is chosen and how to achieve these learning goals

(Tomaschko et al., 2018).

Students were also asked to develop a project

“Artist’s Room”, in which students will model a room

from improvised means, and before that it is advis-

able to offer to make a layout in GeoGebra. In this

way, students will already know where to start, what

sizes of objects to take, what colors will impress, what

shapes are needed to create a room, they will learn

to break an object into simple geometric bodies and

shapes.

Project work interests students in the subject, in-

creases mental activity and creative thinking, helps

to mobilize knowledge in practice and quickly adapt

to unusual situations. During the construction of a

playground or an artist’s room, students use inno-

vative abilities, invention, STEM competencies are

formed, such as critical thinking, creativity, organiza-

tional skills, teamwork, emotional intelligence, ability

to interact effectively, cognitive ﬂexibility.

4 CONCLUSIONS AND

PROSPECTS FOR FURTHER

RESEARCH

1. Investigating the possibilities of using GeoGebra

in the learning calculus and geometry, found out

that engaging students to research using GeoGe-

bra helps to expand the range of educational tasks,

including STEM problems. This allow to achieve

the high level of learning motivation and individ-

ualize the learning process.

2. Implementation of applied aspect in teaching

mathematics using GeoGebra 3D Calculator with

AR will help to solve one of the main problems of

STEM education –individualization. We can ex-

plore AR objects because this application brings

3D math to the real world. Systematic using of

GeoGebra 3D Calculator with AR can help to de-

velop students’ research skills, enhance their so-

cialization opportunities through the acquisition

of ICT, which should lead to the systematic de-

velopment of universal STEM competencies.

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586

Figure 6: Sample implementation of the project “Playground”.

3. The use of dynamic geometry software GeoGe-

bra as a modeling tool makes it possible for scien-

tiﬁc experimental logic to engage students in dis-

covering new mathematical facts. Teachers who

are concerned by the issues of instrumented learn-

ing help to create activities which imply that the

main students’ motivation is in the understanding

of mathematical concepts.

4. In the process of Stereometry teaching, it neces-

sary to divide the students into subgroups accord-

ing to the type of spatial thinking. This will con-

tribute to the implementation of the individual ap-

proach to the development of the spatial imagina-

tion, taking into account individual peculiarities,

gradually complicating the task, supplementing

the teaching material with visual aids, focusing on

the practical application of knowledge. Actions

with the models created with GeoGebra serve as

an intermediate link between external actions with

geometrical solid shapes and internal mental pro-

cesses. Thinking must precede the external ones

to engage and develop a person’s imagination.

5. In order to increase the efﬁciency of the percep-

tion and assimilation of stereometric material, to

overcome the difﬁculties in transcoding the sym-

bols of a spatial body and to create an adequate

spatial image, it is necessary to supplement the

theoretical material with multimedia demonstra-

tion models created by the means of ICT, to en-

courage students and to engage them in the pro-

cess of creating such models for classes indepen-

dently. After all, the acquisition of mathematical

competence depends not on memory, but on the

activity in which the person is involved, on the

system of mental operations that they perform in

the process of knowledge acquisition.

6. In general, the result of the research study is the

improvement of teaching methods of Mathemat-

ics via the use of cloud technologies, the imple-

mentation in the educational process of method-

ological materials of pedagogically balanced and

appropriate blending of traditional and innovative

learning tools, which contribute to the intensiﬁca-

tion of teachers’ readiness for the use of STEM

training in teaching.

In the future, it is necessary to explore the condi-

tions for the effective use of other GeoGebra mobile

applications in STEM learning.

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