=Paper=
{{Paper
|id=Vol-2740/20200416
|storemode=property
|title=The Use of Digital Visualization Tools to Form Mathematical Competence of Students
|pdfUrl=https://ceur-ws.org/Vol-2740/20200416.pdf
|volume=Vol-2740
|authors=Mariia Astafieva,Dmytro Bodnenko,Oksana Lytvyn,Volodymyr Proshkin
|dblpUrl=https://dblp.org/rec/conf/icteri/AstafievaBLP20
}}
==The Use of Digital Visualization Tools to Form Mathematical Competence of Students==
https://ceur-ws.org/Vol-2740/20200416.pdf
The Use of Digital Visualization Tools to Form
Mathematical Competence of Students
Mariia Astafieva[0000-0002-2198-4614], Dmytro Bodnenko[0000-0001-9303-6587],
Oksana Lytvyn[0000-0002-5118-1003] and Volodymyr Proshkin[0000-0002-9785-0612]
Borys Grinchenko Kyiv University, Kyiv, Ukraine
m.astafieva@kubg.edu.ua, d.bodnenko@kubg.edu.ua,
o.lytvyn@kubg.edu.ua, v.proshkin@kubg.edu.ua
Abstract. The advantages of digital visualization to form mathematical compe-
tence of students are exploded in the article. Using a test experiment the prob-
lematic field of using the digital visualization tools in process of teaching high-
er mathematics was identified. Possibilities of involvement of digital visualiza-
tion tools for conduct mathematical experiment using dynamic geometry envi-
ronments (DGEs) GeoGebra and advanced graphing calculator Desmos in the
study of the fields of “Projective geometry and methods of image”, “Mathematics
analysis”, “Analytical geometry”.
Keywords: Visual Thinking, Digital Visualization Tools, Dynamic Geometry
Environments, Mathematics Competences, Mathematics Experiment.
1 Introduction
Modern educational technologies aimed to form knowledge and skills, developing cogni-
tive techniques and accelerating student learning are impossible without the use of digital
tools. The technology of visualization of educational material, which provides, in addition
to the transfer of educational information (images, animations, presentations etc.), psy-
chology-pedagogical techniques of engaging students in the process of learning so called
“visual thinking” (Rudolf Arnheim) provides considerable opportunities here. Unlike the
use of ordinary spotting, visual thinking is thinking through visual operations, the activity
of the mind in special environment, which makes it possible to translate information from
one language to another, to understand the connections and relation between objects [1].
This is especially important for conceptual understanding of mathematics, the subject of
which is abstract mathematical structures. Modern digital tools allow us to design an
interactive environment for the use of visual thinking and thereby influence the think-
ing “in general”; create favorable conditions for the formation of mathematical com-
petence of students.
However, despite the wide range of digital tools with various functionalities for visu-
alization of mathematical content teaching materials, there is not enough scientific and
methodological support for the use of these tools. Therefore, the theoretical substantiation
and methodological support of the use of digital visualization tools in teaching mathemat-
ics at the University is an urgent task.
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
2 Literature Review
The use of visualization tools in the educational process of higher school is receiving
considerable attention of scientists. Thus, some methodical aspects of the use of visu-
alization in mathematics are considered in works of [2-5]. The possibility of using
cloud based learning technologies, in particular, for visualization of mathematics
material was substantiated by the authors of this article [6,7]. Despite the large num-
ber of studies mentioned and not named here, there is still no scientific justification
for the feasibility, description of practices and methodological support for the use of
digital visualization tools to shape students’ mathematical competence. The purpose
of our article is to attempt to partially fill the gap, in particular, to justify the use of
digital visualization tools for the formation of mathematical competence of students.
3 Research Methods
In a course of the research, the following methods were used: analysis of scientific
literature, determination of categorical-conceptual apparatus; synthesis, generaliza-
tion, systematization; diagnostics (questionnaire and statistical processing). The re-
search was carried out within the framework of the complex scientific topic of the
Computer Science and Mathematics Department of Borys Grinchenko Kyiv University
"Theoritical and practical aspects of using mathematical methods and information
technologies in education and science", SR № 0116U004625.
4 Research results
Mathematical competence as the ability to formulate, apply and interpret mathematics
in a variety of contexts is recognized today as one of the key competences of a person.
Eight components of mathematical competence can be divided into two groups [8]:
1) the ability to ask and answer questions in and with mathematics (thinking mathe-
matically; reasoning mathematically; posing and solving mathematical problems;
modelling mathematically), 2) the ability to deal with mathematical language and
tools (representing mathematical entities; handling mathematical symbols and formal-
ism; communicating in, with and about mathematics; making use of aids and tools).
An analysis of the functionality of basis computer math packages has shown that
using them greatly helps to form all those components in students. Nevertheless, are
teachers in higher education institutions ready to make full use of these capabilities
within fundamental mathematical disciplines? In January 2020 we offered a question-
naire to university teachers on the use of digital visualization tools in a process of teach-
ing mathematics. 41 professors of higher education institutions in Kyiv, Vinnytsa,
Ivano-Frankivsk, Nizhyn, Sloviansk, Sumy, Uman, Kharkiv, Kherson and Cherkasy
provided answers to the questionnaire. Their analysis showed that almost half of the
respondents rarely use or do not use these digital tools at all (31.7% and 14.6%, re-
spectively). This clearly demonstrates the scientific problem of our study, which is to
justify the feasibility of using digital visualization tools with the subsequent develop-
ment of appropriate methodological support.
Among the mentioned reasons for non-systematic use of digital visualization tools
are most often indicated: lack of methodological support (73.7%); high time spent on
preparation for such classes (47.4%). 26.3% of those, who answered this question,
admit that they do not have enough digital technology, and 15.8% believe that it is
unnecessary to use digital tools when teaching mathematics. It was also found that
the overwhelming majority of respondents (63.4%) use the relevant software products
solely to identify study material or increase student interest. Thus, the survey con-
firms the lack of consensus on the usefulness or benefits of using digital visualization
tools to successfully teach mathematics.
Do teachers need methodical support for the use of digital visualization tools in the
teaching of mathematical disciplines? Questionnaire replies indicate that only 13.5%
of respondents do not think that they need some methodical help (and this is taking
into account those who basically do not use digital technology). Instead, 73% see this
problem as relevant for them.
After analyzing the results of the survey, we focused on one of the most relevant
but little-used practices – the use of digital visualization tools for mathematical exper-
imentation.
The undeniable advantage of software rendering environments is that they allow to
create dynamic images, as well as automate bulky calculations and constructions
within mathematical experiment. Digital visualization at the stage of observing the
behavior of some mathematical object or their totality allows us to intuitively come to
a hypothetical conclusion. The next step us to find a way to prove or disprove a hy-
pothesis (visual representation often helps to “see” the idea of proof), rigorous math-
ematical proof, exploration of the realm (limits) of truth, special cases, etc.
As an example, let’s consider the mathematical experiment in DGEs GeoGebra
when studying the topic “Projective Coordinates of a point” (discipline “Projective
Geometry and Image Methods”).
The notions of projective coordinate system and projective coordinates of a point on
an extended Euclidean line are not intuitively felt for students, as they differ significant-
ly from the usual Cartesian coordinates. To prevent these concepts and their proper-
ties from being a misunderstood mathematical abstractions for students, it is advisable
to use a computer-aided geometry experiment. Not only will it help to create a visual
image of projective line (p) as a projection of affine plane to the line, but it will also
allow students to express hypothesis about the characteristic features and properties of
the projective coordinate system and coordinates of a point on the projective line.
Thus, by changing the position of the points of the projective rapper on the line,
the points O, the length of the basis vectors and the vector m, students see (Fig.1):
1) that a projective coordinate system can be an arbitrary three points, since it is al-
ways possible to construct its generating basis and the vector e in it;
2) if the projective coordinate system on the projective line is given, then the coordi-
nates of the point M in it do not depend on the choice of the point O – the center
of the basis that generates this projective coordinate system;
3) that the coordinates of a point on a projective line are determined to a constant
factor, since all vectors collinear to the vector m, produce the same point M on the
projective line p.
Fig. 1. Projective coordinates of a point
Here is another example of using a dynamic geometric model when studying a topic:
“Continuous Function” (discipline “Mathematical analysis 1”). Let’s solve the problem: is
there a versatile triangle in which the smallest median is equal to the highest height?
Suppose in the triangle АВС (Fig. 2): АВ < ВС < АС; BM – is the smallest median;
CH – is the highest height. Fix the lengths of sides AB and BC. Then, moving the point C
in a circle with center B, “come across” such a position at which BM = CH (≈0,55). This
suggests a hypothetical affirmative answer to the question of the problem.
Fig. 2. A dynamic model of the dependence of the median on height
To prove this hypothesis let’s continue the experiment. Note, that with small oscillators
of point C (a slight change in the magnitude of the ABC angle) we have slight changes in
both the height of CH and the median BM and therefore in their differences. That is there
is a reason to believe that у = CH – BM is a continuous function of the angle AВС (find an
analytic expression of this function and make sure that the assumption is correct).
By changing position of point С, we can see that if an angle АВС is expanded, then
CH = 0 and у < 0. If however the angle АВС is straight, then CH coincides with ВС,
BM < CH and у > 0. Due to the continuity of the function, there exists a position of point
С (and hence the triangle is АВС), that у = CH – BM = 0, or the same as BM = CH.
A computer mathematical experiment can be effectively used to verify the solution
obtained. Here is an example of studying the topic “Trigonometric Fourier series”
(discipline “Mathematical analysis 2”).
As is known when finding Fourier coefficients, we have to calculate cumbersome
integrals, so there is a high probability of technical errors. Graphic visualization of the
found feature development in the Fourier series allows to control the correctness of
the analytical result. In Fig.3 we have visualization in the graphing calculator Desmos
of the development of the function ( ) in Fourier series in sinus (Fig. 3a),
cosines (Fig. 3b), sines and cosines (Fig. 3c) on segment [ ]. The images allow us
to conclude that the row of cosines coincides with a given function much faster than
the row of sinuses itself, and an error is made in the schedule for sines and cosines.
Fig.
3. Fourier series functions ( )
Finally, give the last example (discipline “Analytical geometry”, topic: “Curves of
the second order”). The dynamic model of conic sections (section of a cone by plane)
allows static, apparently completely different objects with significant qualitative dif-
ferences (curves of the second order: circle, ellipse, hyperbola, parabola, pair of
straight lines, point) to be interpreted only as separate manifestations (states) of one
dynamic phenomenon.
Note also that there is no clear scientific evidence that mathematics students’ educa-
tion outcomes depend on the use (or non-use) of digital visualization tools or the ben-
efits of visual-technical ways of presenting information over others. However, the
practice has shown that visualization of mathematical abstractions using digital tools
help “weak” students to better understand and learn these abstractions. Students with
a high level of mathematical competence are easier to do without “physical” visuali-
zation, but are able to use it when they needed.
5 Conclusions
Based on the elucidation of the essence and structure of mathematical competence of
students and the analysis of the functionality of the basic systems of computer math-
ematics, it is established that their use contributes to the formation of all components
of mathematical competence. At the same time, the survey of university mathematics
teachers outlined the current problem of substantiating the feasibility of using digital
visualization tools and the development of appropriate methodological support.
The implementation of the first part of the task is carried out based on examples
that reveal the possibilities of using digital visualization tools GeoGebra for a mathe-
matical experiment.
The experiment was carried out using computer mathematics systems (Desmos,
GeoGebra) within the disciplines “Projective Geometry and Image Methods”, “Ana-
lytical Geometry” and “Mathematical Analysis”. Specific examples show the feasibil-
ity of using digital visualization tools in teaching the following topics: “Projective
coordinates of a point”, “Continuity of function”, “Trigonometric Fourier series”,
“Second-order curves”.
Thus, judicious use of digital tools in the study of mathematical disciplines im-
proves conceptual understanding of mathematics, supports intuition about predicting a
possible result, hypothesis, facilitates the search for a method (idea) of formal proof
(of refutation), replaces (if necessary) technically sophisticated calculations, trans-
formations, computer-based constructions, allows to check (confirm) results obtained
analytically. All this contributes to the formation of mathematical competence of
students.
The development of methodological support of use of digital visualization tools to
form mathematical competence of students will be the subject of our further scientific
researches.
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