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Visual approach to higher mathematics
Sunnatullo DO‘STOV
1
,
Axtamqul A’ZAMQULOV
2
, Anvar YUSUPOV
3
Denau Institute of Entrepreneurship and Pedagogy
ARTICLE INFO
ABSTRACT
Article history:
Received January 2021
Received in revised form
30 January 2022
Accepted 20 February 2022
Available online
15 March 2022
The article discusses the possibilities of visualizing abstract
mathematical concepts with the help of computer programs for
educational purposes. The use of the interactive geometric
environment GeoGebra in the study of certain topics of higher
mathematics at the university contributes to the formation of
visual representations of the studied mathematical objects.
2181-
1415/©
2022 in Science LLC.
https://doi.org/10.47689/2181-1415-vol3-iss2/S-pp
This is an open access article under the Attribution 4.0 International
(CC BY 4.0) license (https://creativecommons.org/licenses/by/4.0/deed.ru)
Keywords:
higher mathematics,
visualization,
complex numbers,
interactive geometric
environment,
GeoGebra,
cognitive-visual approach,
computer programs.
Oliy matematikaga vizual yondashuv
ANNOTATSIYA
Kalit so
‘
zlar:
oliy matematika,
vizualizatsiya,
kompleks sonlar,
interaktiv geometrik muhit,
GeoGebra,
kognitiv-vizual yondashuv,
kompyuter dasturlari.
Maqolada mavhum matematik tushunchalarni o'quv
maqsadlarida kompyuter dasturlari yordamida vizualizatsiya
qilish imkoniyatlari muhokama qilinadi. Universitetda oliy
matematikaning ayrim mavzularini o‘rganishda GeoGebra
interaktiv geometrik muhitidan foydalanish o‘rganilayotgan
matematik obyektlarning vizual tasvirlarini shakllantirishga
xizmat qiladi.
1
lecturer, Department of Higher Mathematics, Denau Institute of Entrepreneurship and Pedagogy of the Republic of
Uzbekistan, Surkhandarya, Uzbekistan
2
lecturer, Department of Higher Mathematics, Denau Institute of Entrepreneurship and Pedagogy of the Republic of
Uzbekistan, Surkhandarya, Uzbekistan
3
lecturer, Department of Higher Mathematics, Denau Institute of Entrepreneurship and Pedagogy of the Republic of
Uzbekistan, Surkhandarya, Uzbekistan
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212
Визуальный подход к высшей математике
АННОТАЦИЯ
Ключевые слова:
высшая математика,
визуализация,
комплексные числа,
интерактивная
геометрическая среда,
GeoGebra,
когнитивно
-
визуальный
подход,
компьютерные
программы
В статье рассматриваются возможности визуализации
абстрактных математических понятий с помощью
компьютерных программ в образовательных целях.
Использование интерактивной геометрической среды
GeoGebra при изучении отдельных тем высшей математики
в
вузе
способствует
формированию
наглядных
представлений об изучаемых математических объектах.
The study of higher mathematics at a university for students of engineering
specialties and areas of training is a rather complicated process. On the one hand, the
difficulties of studying mathematics among first-year students are caused by their
insufficient school mathematical preparation. On the other hand, the number of
classroom hours allotted for the study of mathematical discipline at the university is
decreasing every year. It should also be noted that the studied sections of higher
mathematics are, as a rule, abstract in nature, which causes great difficulties with the
perception of the material being studied by students.
Currently, there is an active development of information technology. They have
found their application in almost all spheres of human life, including in the educational
process of school and university.
In this regard, the requirements for graduates of engineering specialties and areas
of training in the field of IT competence, as well as for knowledge in the field of higher
mathematics, have increased. A modern graduate must have a wealth of mathematical
knowledge that will allow him to create his own software tools, as well as apply software
in the performance of engineering and technical developments.
The studied topics of higher mathematics cause certain difficulties in the
perception and understanding of educational material among first-year students. On the
one hand, the difficulties in studying higher mathematics are caused by the insufficient
school mathematical preparation of applicants entering the university for engineering
specialties and areas of study.
Studies show that the traditional approach to teaching mathematics at a university
is based, first of all, on the possibilities of abstract-logical thinking of students. Which also
does not contribute to the achievement of the proper level of knowledge of higher
mathematics among future engineers.
The solution to these problems and the deep understanding and assimilation of
mathematical knowledge can be the use of a cognitive-visual approach to teaching higher
mathematics. The implementation of the approach to the theory and methodology of
teaching certain sections of higher mathematics at the university is reflected in the works
of V.A. Dalinger, A.I. Ryzhkova, N.V. Schukina and others.
In his publications, V.A. Dalinger talks about the need to build the process of
teaching mathematics on the basis of a cognitive-visual (visual-cognitive) approach. He
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notes that: “One of the central provisions of this approach is the wide and purposeful use
of the cognitive function of visualization. The implementation of the cognitive-visual
approach in the process of teaching students mathematics allows us to construct a visual
learning environment - a set of learning conditions in which the emphasis is on using the
reserves of visual thinking of students
”
[1. P. 297].
Currently, there are a large number of different educational computer programs
that contribute to the creation of the necessary visualization of abstract mathematical
concepts. Computer programs, as a rule, allow you to perform all possible algebraic
calculations, geometric constructions of various mathematical objects, and have
animation capabilities. Among the programs for educational purposes, the following are
especially widely used: GeoGebra,
“
Live Geometry
”
, GONExT, Cabri Geometry and others.
The interactive geometric environment GeoGebra, in addition to performing
various algebraic calculations, allows you to perform geometric drawings of
mathematical objects and, by changing the required parameter, modify the finished
drawing.
Using the capabilities of the GeoGebra interactive geometric environment in the
educational process allows students to demonstrate a mathematical object, depending on
the chosen learning approach. So, for example, considering the concept of "ellipse" most
often use its classical definition. Using the GeoGebra program, you can draw a curve
drawing based on its definition.
n=11.000
Fig. 1. The classical definition of an ellipse
As you know, the largest amount of information that a person receives in his life is
visual in nature. It is worth noting that “the advantage of a visual image in comparison
with motor or auditory ones is that it allows you to simultaneously highlight many
aspects in the model-image, instantly penetrate the essence of the problem in all its
complexity. In a visual image, it is possible to fix various theoretical connections and
dependencies (spatial, structural, functional, temporal)” [1
. P. 63].
The visual, representation of abstract concepts of the university course of
mathematics can be facilitated by various computer programs for educational purposes.
Among them, we should highlight: GeoGebra, C.a.R., Cabri Geometry, GEONExT,
“
Live
Geometry
”
and others. The listed interactive geometric environments allow you to
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perform algebraic calculations, geometric drawings of mathematical objects, manipulate
the finished drawing by changing parameter values, have animation capabilities, etc.
The concept, of an ellipse can also be introduced through the use of a conical
surface. Let the plane intersect the conical surface. If the cutting plane is not parallel to
any of the generators of the conical surface and does not pass through the top of the cone,
then the line of intersection is called an ellipse (Fig. 2).
Fig. 2. Definition of an ellipse through a conical surface
An ellipse can also be represented as a line of a plane. Let
’
s add on the finished
drawing (Fig. 2) the image of two spheres inscribed in a conical surface that touch the
cutting plane: the upper one at the point F1, and the lower one at the point F2 (Fig. 3).
Fig. 3. Definition of an ellipse as a line of a plane
Of the above computer programs, GeoGebra occupies a special place. The
peculiarity of this geometric environment lies in the interactive combination of
geometric, algebraic and numerical representation. GeoGebra has the ability to create
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structures with points, vectors, lines, conic sections, mathematical functions, and also
allows you to dynamically change them.
In the process of studying the topic
“
Complex Numbers
”
there are a number of
abstract concepts related to the performance of actions: multiplication and division of
complex numbers, raising to a power and extracting a root from a complex number. All of
the above actions, as a rule, are communicated to students in the form of appropriate
formulas. Therefore, performing actions with complex numbers for students of
engineering specialties and areas of training is very abstract. Using the interactive
geometric environment GeoGebra, you can clearly demonstrate to s
tudents’
what kind of
object is obtained as a result of the multiplication of complex numbers.
To this end, two complex numbers should be given
a
=
r
⋅
(cos
α
+
i
sin
α
) and
z
=
m
(cos
β
+
i
sin
β
), complete the necessary constructions and inform students that
point
K
is the result of multiplying complex numbers
a
and
z
(Fig. 4)
Fig.4. Product of complex numbers
Similarly, using the GeoGebra interactive geometric environment, you can visualize
the process of dividing complex numbers:
a
=
r
⋅
(cos
α
+
i
sin
α
) and
z
=
m
⋅
(cos
β
+
i
sin
β
).
Having completed all the necessary constructions in the computer program, we get
that the point
M
is the result of dividing the complex number
a
by
z
(Fig. 5).
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Fig.5. Division of complex numbers
The use of the interactive geometric environment GeoGebra and other computer
programs for educational purposes allows the visualization of abstract mathematical
concepts, which contributes to the speedy perception of the material being studied, its
deeper understanding and increases interest in the discipline being studied.
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