Volume 02 Issue 12-2022
6
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
03
I
SSUE
01
Pages:
06-14
SJIF
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FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
ABSTRACT
The article reveals the trajectory of organizing and teaching a facultative course in order to familiarize students with
non-Euclidean geometries in schools. It shows why students should be introduced to non-Euclidean geometries in
schools, and what goals can be achieved by teaching non-Euclidean geometries. The most important aspect is the
trajectory of the organization of the optional course. When organizing a facultative course, it is indicated what topics
to choose, how many hours to allocate, and the program of the facultative course is developed.
KEYWORDS
Distance, metric, coordinate system, Euclidean geometry, non-Euclidean geometry, Lobachevsky geometry,
Minkovsky geometry, Galilean geometry.
INTRODUCTION
Geometry is one of the fastest growing subjects. The
school geometry course is also being enriched with
modern innovations of geometry science. But even so,
it is difficult to say that the geometry course of the
school, which is enriched by these changes, is able to
provide students with knowledge in accordance with
the requirements of the present time.
At the time when new fields of modern science,
athematics, cosmonautics relativity theory are being
Research Article
THE TRAJECTORY OF ORGANIZING AND TEACHING A FACULTATIVE
COURSE ON NON-EUCLIDEAN GEOMETRIES AT SCHOOL
Submission Date:
January 14, 2023,
Accepted Date:
January 19, 2023,
Published Date:
January 24, 2023
Crossref doi:
https://doi.org/10.37547/ajast/Volume03Issue01-02
Ibrohim M. Khatamov
Researcher, Jizzakh State Pedagogical University, Uzbekistan
Journal
Website:
https://theusajournals.
com/index.php/ajast
Copyright:
Original
content from this work
may be used under the
terms of the creative
commons
attributes
4.0 licence.
Volume 02 Issue 12-2022
7
American Journal Of Applied Science And Technology
(ISSN
–
2771-2745)
VOLUME
03
I
SSUE
01
Pages:
06-14
SJIF
I
MPACT
FACTOR
(2021:
5.
705
)
(2022:
5.
705
)
OCLC
–
1121105677
Publisher:
Oscar Publishing Services
Servi
opened and improved, we believe that it is not enough
to teach students only Euclidean geometry at school.
In order to live in harmony with such developing
scientific achievements and to acquire knowledge in
accordance with these scientific achievements, to
provide students with information about non-
Euclidean geometries and to teach them the
axiomatics of modern geometry and the existence of
geometries other than Euclidean geometry in the
plane, all geometries in the plane there is a need to give
an understanding of.
THE MAIN FINDINGS AND RESULTS
If we pay attention to the teaching of geometry in
secondary schools, we can see that knowledge about
non-Euclidean geometries is given in textbooks based
on brief information only in upper grades. So, in the
school geometry course, students get detailed
information only about Euclidean geometry, and about
non-Euclidean geometries they can only get
information about the history of its emergence.
In the usual teaching of geometry taught in schools (a
point is the trace of a pencil on a sheet of paper, or the
trace of a chalk on a blackboard, a straight line is a trace
drawn by a pencil or chalk using a ruler, etc.) They are
familiar with only one interpretation (imagination) of
Euclidean geometry. In this case, their spatial
imagination will be limited. Modern science and
technology development requires comprehensive
development of students’ spatial imagination. It is very
difficult to fulfill this requirement without getting
acquainted with the elements of non-Euclidean
geometries in general education schools. In particular,
the introduction of the theory of relativity, which is the
next
achievement
of
mathematics,
physics,
astronomy, and cosmonautics, into general education
schools shows the need to familiarize students with
non-Euclidean geometries. For this purpose, it is
appropriate to teach the elements of non-Euclidean
geometries as a facultative course in general education
schools [1], [2].
As a result of teaching the elements of non-Euclidean
geometries, students’ geometric imagination develops
and the following qualities are brought up in them:
1. Students’ general understanding of the axiomatic
method, Euclidean geometry and non-Euclidean
geometries will grow;
2. Pupils’ ideas about basic geometric figures will
expand;
3. Pupils’ understanding of quantities, measure
ment
and distance develops;
4. Pupils’ understanding of methods of solving
geometric problems will expand;
5. The relationship of geometry with other sciences
and the understanding of its reality will be developed;
6. Pupils’ outlook on real life and geome
try develops.
Now, we will reveal the meaning of each of these
qualities through examples.
1. The axiomatic method, through which getting
acquainted with Euclidean geometry and non-
Euclidean geometries, creates great opportunities to
develop the students’
geometric imagination in
accordance with the requirements of the present time:
a) After getting acquainted with the axiomatic
method, students’ usual geometric ideas develop,
because different interpretations of Euclidean
geometry are shown using the system of axioms. For
example, it is possible to take the interpretation of
Euclidean geometry using a set of parabolic spheres.
By “point” in this case, we mean all points of the usual
Volume 02 Issue 12-2022
8
American Journal Of Applied Science And Technology
(ISSN
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2771-2745)
VOLUME
03
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Pages:
06-14
SJIF
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(2021:
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OCLC
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Publisher:
Oscar Publishing Services
Servi
Euclidean space, except for one point S. As straight
lines we consider all circles passing through this point
S, as well as ordinary straight lines (circles with center
at infinity) passing through point S. As a plane, we
consider all spheres passing through point S as well as
normal planes (spheres with center at infinity) passing
through point S. In this case, the new “space” created
will consist of a parabolic connection of spheres with a
radical center (this center is removed from space).
Here, the fulfillment of all axioms of Euclidean
geometry is checked (in the 11th grade geometry
course) [3].
If we replace some geometric shapes with another,
students will get acquainted with non-Euclidean
geometries at this time. Students should understand
that if we accept all axioms of Euclidean geometry
except postulate V, and instead of postulate V we
accept
Lobachevsky’s
axiom
of
parallelism,
Lobachevsky’s geometry is formed. Or if we replace
the concept of parallelism in Euclidean geometry with
the sentence that any two straight lines intersect (in
this case some axioms of groups I, II, III also change),
students should understand well that Riemannian
geometry is formed. Acquainting students with these
geometries reveals to them the content and essence of
science and technology innovations
b) An understanding of absolute geometry is given,
and special features geometry is studied. It is proved to
the students that the theorems of absolute geometry
(perpendicularity of straight lines, equality of vertical
angles, theorems about the signs of congruence of
triangles, etc.) are appropriat
e in Lobachevsky’s
geometry and Galileo’s geometry, and their fulfillment
is checked on a pseudosphere and a sphere. When
teaching Lobachevsky, Galilean and Minkovsky
geometries, one should pay attention to their special
features. Examples showing schematic connections of
theorems, axioms and definitions clearly show
students the essence of proving theorems.
c) By comparing the fulfillment or non-fulfillment of
certain concepts, a precise definition of these concepts
in Euclidean geometry is achieved. For example, if we
consider the theorem that a circle can be drawn
outside any triangle in Euclidean geometry, its proof
ignores the intersection of the altitudes transferred to
the middle of the sides of the triangle. However, when
it was seen that in Lobachev
sky’s geometry there are
triangles that cannot be circled outside, importance is
attached to the intersection of these heights in
Euclidean geometry. After that, it is shown that there
are no squares, rectangles, rhombuses and trapezoids
in Lobachevsky’s g
eometry.
d) Students’ ideas about geometric interpretations will
expand. Various interpretations of Euclidean geometry
and non-Euclidean geometry were shown.
2. Familiarity with Euclidean and non-Euclidean
geometries expands students’ imaginations about
geometric figures. For example:
a) Students’ ideas about the shortest distance
expanded. Students connect two points on a
pseudosphere and a sphere with the help of a regular
(elastic) line, and come to the opinion that the shortest
distance between two points, that is, the geodesic
character of a straight line segment, depends on the
plane. Then depicting figures of Lobachevsky’s and
Galileo’s geometries on a regular blackboard does not
cause misunderstanding and internal contradiction in
students.
b) Students’ ideas
about asymptotic convergence lines
will expand. In Lobachevsky’s geometry, the distance
between parallel straight lines decreases in the
direction of parallelism. But these straight lines never
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VOLUME
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OCLC
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Publisher:
Oscar Publishing Services
Servi
intersect, so students’ ideas about asymptotically
converging straight lines are broadened.
c) Pupils’imagination about different spaces expands.
Doing everything in the usual Euclidean plane limits
students’ spatial imagination. Due to familiarity with
non-Euclidean geometries, their spatial imagination
will expand and an opportunity will be opened for
them to familiarize themselves with multidimensional
spaces.
3. After learning about Euclidean geometry and non-
Euclidean geometry, students’ concepts of measuring
quantities and distance will expand, that is, comparing
sections in different geometries, adding and
subtracting them, will develop students’ concepts of
measuring quantities.
4. As a result of getting acquainted with Euclidean and
non-
Euclidean geometries, pupils’ ideas about solving
geometric problems and its methods develop:
a) along with solving problems related to geometry,
they also solve problems related only to Lobachevsky
and Galilean geometries on the pseudosphere and
sphere;
b) their ideas about solving calculation and proof
problems are expanded. Along with the fact that the
students will check the implementation of the
calculation and proof problems of absolute geometry
in Lobachevsky and Galilean geometries, they will be
able to solve the problems related to Lobachevsky and
Galilean geometries themselves (for example,
measuring the sum of the interior angles of a triangle,
a rectangle, etc.) also solve.
5. By getting acquainted with non-Euclidean
geometries, students will clearly see the connection of
geometry with other sciences. Currently, it is known to
science that Euclidean geometry is not appropriate in
real physical space. The hypothesis that Lobachevsky's
geometry is valid in the macro world is being proven
day by day. The meridian lines of the globe studied in
astronomy and geography are Riemannian lines, or the
trajectories of electrons are also good examples of
Riemannian lines. Knowledge of the theory of relativity
is further expanded by studying Galilean geometry.
6. Familiarity with Euclidean geometries develops
students’ worldview ab
out real life and geometry. The
scientists who lived and worked in the late 19th century
and the first half of the 20th century believed that
Euclidean geometry is the only possible geometry, and
were against other geometries. The great Russian
mathematician N.I. Lobachevsky attacked such views
and created his geometry, which fully reflects the vital
physical space. In addition, there is Galilean geometry,
one of the non-Euclidean geometries, which can be
considered as a mathematical model of the theory of
relativity [4], [5].
Taking into account the above, we have set the topic
“Elements of non
-
Euclidean geometries” as an
optional course for students at the school.
Now, if we pay attention to the purpose of teaching
facultative courses in schools, the following two
purposes can be indicated:
1. Expanding and deepening the existing knowledge;
2. Teaching topics that are not included in the
textbook, but which help to acquire more perfect
knowledge of science through teaching, and then
include these topics in the curriculum.
Providing knowledge about non-Euclidean geometries
through optional courses enables the implementation
of the above two objectives required of optional
courses.
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Servi
Therefore, the selection of topics for teaching the
elements of non-Euclidean geometries as an optional
course and the development of teaching methods are
one of the most important issues that bring school
geometry closer to modern geometry.
When discussing the issue of teaching the elements of
non-Euclidean geometries at school, many of our
scientists emphasized that teaching should start from
the upper grades. We do not agree with this opinion.
We believe that it is enough for students to have
knowledge of the 7th grade level to teach the elements
of non-Euclidean geometries. So, the elements of non-
Euclidean geometries can be taught starting from the
7th grade. This is due to the fact that in the 10th or 11th
grade, when students think of geometry, only
Euclidean geometry is embodied in their minds, and
this may cause a little difficulty in accepting the new
geometry. To teach non-Euclidean geometries, it is
enough for students to know concepts such as point,
straight line, coordinate system, parallelism and
perpendicularity, distance, angle.
We follow the following guidelines for the selection of
topics for the organized optional course and its
organization:
1. If students are introduced to non-Euclidean
geometries starting from the 7th grade, students will
study both Euclidean and non-Euclidean geometries in
parallel. By comparing the differences and common
aspects of each geometry, knowledge is further
expanded. As a result of skepticism about the concepts
of non-Euclidean geometries, the desire to engage in
scientific activities is formed in students. Even so, we
consider it appropriate to teach this optional course in
upper grade.
In this optional course, students will be introduced to
Lobachevsky
and
Galilean
geometries.
These
geometries are studied sequentially.
2. In order to organize the teaching of Lobachevsky
and Galilean geometries, introducing various new
concepts, students should not have insecurity or
internal contradictions. Here, it is appropriate to use
real-life examples and problems for students.
3. As each concept is learned (angle size, distance,
parallelism, triangle, area, etc.), these concepts are
taught in comparison with the concepts of Euclidean
geometry, which motivates a deeper study of the
essence of these concepts.
During the teaching of this course, students must be
presented with various issues, especially real-life
examples, and relevant examples should be
developed.
We plan the optional course “Elements of non
-
Euclidean geometries” in schools for 34 hours. We will
focus on the teaching trajectory of the optional course
on non-Euclidean geometry at school in terms of
content and methodology. Experience shows that the
optional course “Fundamentals of non
-Euclidean
Geometry” should be implemented on the basis of the
following program.
Elective subject (optional course) “Fundamentals of
non-Euclidean geome
try” (34 hours)
10th grade
1. Letter of explanation.
This course is aimed at deepening and expanding
knowledge of geometry. For many students, the
transition from plane geometry to space geometry is
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difficult. Developing spatial imagination requires a lot
of time and a variety of exercises and tasks, from the
simplest to the most difficult.
“Fundamentals
of
non
-
Euclidean
geometry”
competition subject is one of the most amazing and
huge discoveries in the history of mathematics - the
great Russian scientist N.I. Lobachevsky is called non-
Euclidean geometry. The creation of non-Euclidean
geometry not only helped open up new horizons in
mathematics, but also made a huge contribution to the
development of modern space theory. The formation
of geometric images is an important part of intellectual
education, polytechnic education, and is of great
importance in all cognitive activities of a person.
The purpose of the course is to generalize the
properties of geometric shapes and to study them in
depth, to develop logical thinking. The course is aimed
at developing ideas about the leading mathematical
method of knowing reality - mathematical modeling,
forming a holistic natural-mathematical component of
the image of the world.
Examining the questions of the course is aimed at
students’ understanding of the multidimensio
nal
nature of mathematics, the organic combination of
theoretical and practical aspects, which helps to
establish fundamental internal connections, the
opportunity to choose an independent field of activity;
and preparing for research work at the intersection of
different departments of mathematics.
The course is characterized by a judicious combination
of logical rigor and geometric clarity. The theoretical
importance of the studied material increases, the
internal logical connections of the lesson expand, the
role of deduction increases, and the level of
abstraction of the facts under consideration increases.
Students learn analytical and synthetic methods of
activity.
Objectives:
- to educate the understanding of the importance of
mathematics for scientific and technical development;
-
expanding students’ ability to adapt to the modern
world;
-
formation of students’ understanding of the role of
mathematical knowledge as a means of seeing the
diversity and uncertainty of the surrounding reality;
- mastering the basics of mathematical culture by
students, forming a personality.
Main objectives:
1. To help students determine their own destiny by
being in a situation of independent choice of an
individual educational trajectory.
2. Activation of cognitive activity of schoolchildren.
3. Increasing students’ information and communicative
competence.
4. Providing pedagogical conditions for the flourishing
of the student personality and his creative potential.
5. Increasing the amount of mathematical knowledge.
Solving the set tasks is an additional factor in the
formation of positive motivation in learning
mathematics, as well as students' understanding of the
philosophical postulate about the unity of the world
and the position about the universality of
mathematical knowledge.
2. Content of topics.
Euclidean geometry - 10 hours.
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- System of axioms in Euclidean geometry;
-
Properties of the system of axioms. Euclid’s
“Fundamentals”;
- The fifth postulate and parallelism axiom;
-
Theorems equivalent to Euclid’s fifth
postulate;
- Different ideas in solving the problem of the fifth
postulate;
Some geometries in the affine plane - 24 hours.
Methods of measuring distance and angle. Modern
definition of geometry. Lobachevsky geometry. Basic
concepts of Minkowski geometry. Concept of circle
and angle in Minkowski geometry. Planimetry of the
Galilean plane. Circle and Angle in Galilean Geometry.
Properties of triangular elements. Polygons. Basic
concepts
and
their
applications.
Projective
conjunction. About nine geometries in the plane.
Examples and problems.
3. Requirements for the level of preparation of
students.
As a result of studying the course, students will:
- concepts of axioms, postulates. According to Euclid,
they can distinguish these concepts;
- model concept;
- properties of the system of axioms;
-
Euclid’s system of axioms;
-
Gilbert’s system of axioms;
- the role of axiomatics in geometry;
- Distance and angle measurement methods;
- Basic concepts of Lobachevsky geometry;
- Concept of Minkowski geometry;
- circle and angle in Minkowski geometry;
- planimetry of the Galilean plane;
- Circle and angle relations in Galilean geometry;
- being able to independently solve examples and
problems;
- being able to model life issues;
- Differences and common aspects between Euclidean
and Galilean geometries;
- they should understand the common aspects and
differences in the system of axioms under
consideration.
They are required to have the following skills:
- mastering methods of distance and angle
measurement;
- to distinguish existing geometries in the affine plane;
- to be able to draw each geometric shape based on
clear ideas about it;
- construction of polynomial sections based on
accepted axiomatics and proven course theorems;
- proving theorems in solving problems, reasoning
based on evidence;
- to be able to clearly imagine these concepts by
learning concepts such as surface area and angle size
based on each geometry;
- being able to model life issues based on the
knowledge learned in geometry;
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- to be able to distinguish different geometries using
distance and angle measurement methods;
- to be able to understand 9 geometries in the plane;
- in practical activities and modeling life situations in
geometric language;
- checking the presented situations, predicting the
result;
- it is necessary for them to have the opportunity to
develop ideas about space in the universe.
4. Calendar - thematic planning.
Euclidean geometry - 10 hours
1. When and how did geometry appear? Euclid’s
attempts to create the first model of geometry - 1 hour;
2. Gilbert geometry model
–
1 hour;
3. Consistency of the system of axioms - 1 hour;
4. Independence and completeness of the system of
axioms - 1 hour;
5. Equivalence of axioms - 1 hour;
6. Euclid’s work “Fundamentals” and its content
- 1
hour;
7. The fifth postulate and axiom of parallelism - 1 hour;
8. The theorem about the sum of the angles of a
triangle is equivalent to the fifth postulate - 1 hour;
9. The Pythagorean theorem and the fifth postulate - 1
hour;
10. Different opinions on solving the problem of the
fifth postulate - 1 hour;
Some geometries in the affine plane - 24 hours
1. The concept of distance and its measurement
methods - 1 hour;
2. The concept of an angle and its measurement
methods - 1 hour;
3. Modern definition of geometry - 1 hour;
4. Basic concepts of geometries in the affine plane - 1
hour;
5 Basic concepts of Lobachevsky geometry - 1 hour;
6. Basic concepts of Minkowski geometry - 1 hour;
7. The concept of circle angle in Minkowski geometry -
1 hour;
8. Basic concepts of Galilean geometry - 1 hour;
9. Angle concept and its measurement in Galilean
geometry - 1 hour;
10. Circle and its properties in Galilean geometry - 1
hour;
11. Equivalence properties of triangles in Galilean
geometry - 1 hour;
12. Triangle height, median, bisector - 1 hour;
13. Signs of triangle equality - 1 hour;
14. Polygons - 1 hour;
15. Properties of interior angles of polygons - 1 hour;
16. Concept of surface - 1 hour;
17. Triangle surface - 1 hour;
18. Concept of cycle
–
1 hour;
19. Features of the cycle - 1 hour;
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20. Problems on the mutual location of straight lines - 1
hour;
21. Problems related to polygons - 1 hour;
22. Circle, equidistant, oricycle issues - 1 hour;
23. Projective conjunction. Nine geometries in the
plane - 1 hour;
24. Interrelationship of geometry departments - 1 hour;
CONCLUSION
As a result of the completion of the optional course
based on this program, students’ knowledge of
modern geometry will be expanded, as a result of
comparing different geometries with each other,
students will have a better understanding of geometry
and will be motivated to understand each concept
more deeply.
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Artikbayev A., Khatamov I. Nine geometries in the
plane. -
Tashkent.: “Smart Reader”. 2021.
3.
Adajonov N.D., Yunusmetov R., Abdullayev T.
Geometry part 1-2. -
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4.
Artikbayev A. Modern view of ancient science. -
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magazine, issue 3, 2004.
5.
Artikbayev A., Berdiyeva O. Geometry in
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Tashkent.: “Physics, mathematics
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Khatamov I. Development of pupils' ideas about
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Khachaturyan A.V. Geometry of Galileo. - Moscow.:
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Yaglom I.M. Galileo’s principle of relativity and non
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Atanasyan L.S. Geometry of Lobachevsky. -
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