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MATHEMATICS AND CREATIVITY: HOW DO GEOMETRIC PROBLEMS DEVELOP CREATIVE
THINKING?
Pulatova Kamola Nuralievna
National Institute of Pedagogical Education named after Kori Niyazi 2nd-year doctoral student,
Uzbekistan
AB O U T ART I CL E
Key words:
Geometry, creative thinking,
geometric problems, creative abilities, logical
reasoning, teaching methods, analytical thinking.
Received:
15.12.2024
Accepted
: 20.12.2024
Published
: 25.12.2024
Abstract:
This article analyzes the role of
geometry in developing creative thinking. It
explores methods for fostering students' analytical
and creative abilities through geometric problems.
In particular, the importance of using various
approaches to solve geometric problems and
seeking new and original solutions in cultivating
creativity is highlighted. The article elaborates on
the use of geometric problems as a tool to
stimulate creative thinking, broaden students'
cognitive abilities, enhance logical reasoning, and
apply these skills in solving real-life problems.
INTRODUCTION
Geometry, as one of the important branches of mathematics, is not limited to
calculations and studying formulas, but also plays a significant role in developing creative thinking.
Geometric problems, particularly challenging and complex tasks, test students' analytical and creative
abilities. These problems encourage students to find new, unconventional solutions, as well as teach
them to approach issues from a different perspective. Through geometric problems, we can see the
possibility of developing not only mathematical skills but also creativity. In this article, we will discuss
how geometric problems can enhance creative thinking and how they stimulate creative thinking in
students. Geometry, due to its complexity and inclusion of various shapes, fosters creative thinking in
students. For instance, exploring multiple methods in geometric problems and analyzing them from
new perspectives encourages students to discover novel approaches. This, in turn, leads to the
development of creativity and innovative thinking. Geometric problems also allow students to
independently draw their own conclusions and compare solutions. Consequently, this develops a
student's mathematical creativity, as they learn to solve each problem using different methods rather
than from a single perspective. In mathematics, creativity is the ability to find and apply new, unique
approaches in the process of solving problems, tackling tasks, and creating new knowledge. Creativity
requires solving mathematical problems in unconventional ways, which in turn broadens students'
VOLUME04 ISSUE12
DOI:
https://doi.org/10.55640/eijp-04-12-34
Pages:145-152
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thinking horizons. This, in addition to following basic rules, demands a deeper understanding of
mathematics and the ability to express one's thoughts. Creativity helps make mathematics more
interesting and effective, as new ways of thinking are necessary to discover novel methods and simplify
challenging problems. Creative thinking is crucial in mathematics because it aids in finding distinctive
and efficient solutions. This skill allows for examining problems from various perspectives, encourages
experimenting with new methods, and helps to think beyond traditional solutions. In mathematics,
many problems can have multiple solutions. Through creative thinking, students can experiment with
various methods and discover the most effective and straightforward solutions. Creative thinking is
beneficial not only in solving mathematical problems but also in grasping complex issues, developing
logical reasoning, and making decisions in practical life. The development of creative thinking enhances
a person's overall cognition, imagination, and ability to analyze and solve problems. Creative thinking
fosters students' intellectual capabilities, enabling them to tackle more challenging and engaging
problems. This, in turn, increases their interest in learning and motivates them to achieve success. This
increases their interest in learning and motivates them to achieve success.
Creative thinking typically allows for discovering new ideas and solutions, circumventing limitations,
and viewing complex situations from different perspectives. This approach is especially crucial in
geometric problems, as these often require going beyond simple formulas or classical solutions.
Geometric problems are those related to geometric shapes, their dimensions, areas, volumes, and
interrelationships. These problems help students delve deeper into mathematical and geometric
concepts, develop logical thinking, and apply creative approaches. Various methods, formulas, and
concepts are employed in solving geometric problems. For example, creative thinking helps in solving
problems related to polygons, circles, and other geometric shapes by applying new methods and
identifying connections between geometric objects to find effective solutions. We can categorize
geometric problems into several types, such as surface area measurements, perimeter calculations,
angles, volume, and symmetry. Calculating the area of geometric shapes is often the simplest and most
fundamental task, and finding the perimeter by measuring the edges is also crucial. Furthermore,
geometric problems frequently involve angles, and it is essential for students to calculate angles and
understand their relationships with one another. In geometric problems, calculating volume,
understanding measurements, and applying formulas help students develop mathematical imagination
and logical thinking. In geometric problems, symmetry represents a distinct characteristic of a shape or
object. Symmetry allows for the repetition of a shape or image through certain fixed points or by
reflection. Geometric problems not only aid in learning mathematical knowledge but also contribute to
the development of logical thinking and creative approaches. They enable students to understand
shapes and their interrelationships, as well as to solve problems using various methods. Studying
geometric problems is also beneficial in addressing issues that arise in everyday life. Geometric
problems for elementary school students begin with simple shapes and concepts, but as they become
more complex, new concepts and solutions are introduced. Consequently, geometric problems
progressively increase in difficulty according to age, allowing for the step-by-step development of
students. Geometric problems are tasks related to shapes and their properties. Each geometric problem
has its own unique characteristics, and various methods and approaches are employed to solve them.
In geometric problems, we deal with determining the dimensions of shapes. For example, we find the
sides of a triangle, the area of a rectangle, and similar measurements. Each shape has its own unique
dimensions and properties. Geometric problems often require logical thinking. When solving a problem,
in addition to basic rules and formulas, analytical thinking and searching for structured solutions are
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necessary. This encourages students not only to memorize formulas but also to learn how to apply them.
Furthermore, geometric problems demand not just traditional approaches but also creative thinking.
Often, problems cannot be solved by simple methods, so students strive to find new and innovative
approaches. This teaches them to expand their knowledge and discover solutions through various
methods. When solving geometric problems, visualizing shapes is crucial. Students frequently attempt
to understand the problem by drawing figures and using diagrams. Visual imagination and sketching
help to solve the problem more easily and accurately. Geometric problems help not only to acquire
mathematical knowledge but also to develop logical and creative thinking. Through them, students gain
skills such as studying shapes, understanding their relationships, and finding solutions. Geometry is an
important tool that helps students solve problems not only in mathematics but also in everyday life.
Working on geometric problems develops children's creative thinking, allowing them to view problems
from different perspectives and find their own new approaches. Here, we may ask the question: how do
mathematical problems develop creative thinking? Creative thinking is the process of generating new,
original, and useful ideas, thoughts, or solutions. It is primarily a skill necessary for problem-solving,
developing new ideas, and adapting to changes. The process of creative thinking encourages a person
to adopt new, unconventional approaches and decisions, unlike logical thinking. Below are several
stages of creative thinking:
•
Problem identification:
Identify a problem or situation and define questions related to it.
•
Information gathering:
Collect necessary data to better understand the problem or situation.
•
Idea generation:
Create new ideas and consider various thoughts.
•
Idea evaluation:
Analyze the generated ideas and select the best solution.
•
Solution implementation:
Apply the chosen idea or solution into practice.
Now let's examine these steps using an example of geometric problems.
Problem
: A triangle has two sides measuring 5 cm and 7 cm. If its perimeter is 18 cm, find the third
side of the triangle.
Identifying the problem
: The problem provides two sides of a triangle (5 cm and 7 cm) and we know
its perimeter (18 cm). Our task is to find the length of the third side.
Gathering information
: We know that the perimeter of a triangle is the sum of all three sides: P = a +
b + c. The perimeter is given as 18 cm, two sides are 5 cm and 7 cm, and we know this. We need to find
the third side.
Developing the idea
: Using the perimeter calculation formula, we construct the following equation to
find the third side:
5 + 7 + c = 18
12 + c = 18
c = 18 - 12 = 6
Evaluation of the idea
: Let's check the result. If the sides of a triangle are 5 cm, 7 cm, and 6 cm, their
sum is: 5 + 7 + 6 = 18
Implementing the solution
: We found the length of the third side to be 6 cm.
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Through this problem, students can be encouraged to formulate equations, gather information, and
think creatively. Finding an unknown side using the perimeter of a triangle is a problem that develops
creative thinking for elementary school students.
Problem
: A rectangle is 8 cm long and 4 cm wide. If we divide this rectangle into two, what shapes are
formed? Are the resulting shapes symmetrical to each other? Are the shapes symmetrical?
Identifying the problem
: We are given a rectangle that is 8 cm long and 4 cm wide. The rectangle needs
to be divided along its length, width, and diagonal. Will the resulting shapes be symmetrical?
Information gathering
: There are three ways to divide a rectangle:
1. Dividing by length: In this case, two rectangles are formed.
2. Dividing by width: In this case, two rectangles are formed.
3. Dividing by diagonal: In this case, two triangles are formed.
Developing the idea
: 1. Dividing by length: If we divide the rectangle lengthwise, we get two rectangles.
Both rectangles are 4 cm long and 4 cm wide. These shapes are symmetrical because they have the same
dimensions and shapes.
2. Dividing by width: If we divide the rectangle widthwise, we get two squares. Each square has sides of
4 cm. These shapes are also symmetrical because they are equal and of the same shape.
3. Dividing by diagonal: If we divide the rectangle along its diagonal, we get two right triangles. Each
triangle has a length of 8 cm, a width of 4 cm, and a hypotenuse. These shapes are symmetrical because
the two triangles divided by the diagonal are congruent and identical in shape.
Evaluating the idea
: The shapes formed in all three methods are symmetrical. Dividing by length
results in two equal rectangles. Dividing by width results in two equal squares. Dividing by diagonal
results in two equal right triangles.
Implementation of the solution
: When dividing a rectangle along its length, two symmetrical
rectangles are formed. When dividing a rectangle along its width, two symmetrical squares are formed.
When dividing a rectangle along its diagonal, two symmetrical right triangles are formed. This problem
explains three different ways of dividing a rectangle and the symmetry of the resulting shapes. It is
effective in teaching the concept of symmetry to elementary school students.
The connection between mathematics and creativity is remarkably strong, as mathematics serves as an
excellent tool for developing creative thinking, especially in the process of problem-solving and
exploring various approaches. Mathematics is not merely about numbers and formulas; it teaches
individuals to think analytically, logically, and creatively. For instance, solving complex problems
requires trying multiple methods and discovering new approaches. This process nurtures students'
creative thinking by encouraging them to seek alternatives beyond traditional solutions. Mathematics
demands logical and systematic problem-solving. As students break down problems and solve them in
small steps, they are compelled to make creative decisions at each stage. This process, in turn, enhances
their analytical and creative thinking abilities. Many mathematical problems, especially in geometry,
encourage students to visualize concepts through shapes, pictures, and diagrams. This process develops
creative thinking, as students learn to enhance their imagination and engage in abstract thinking. For
example, identifying and differentiating shapes in geometric problems and comparing them with other
shapes fosters students' creative thinking abilities. In mathematics, particularly when solving problems,
there arises a need to explore various options and methods, as well as to introduce innovative
approaches. This necessity allows learners to experiment and discover new ways of thinking, which in
turn supports and nurtures creativity. The development of creative abilities and the formation of
creative activity are particularly significant in teaching geometry. The possibilities for developing
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creative abilities and various aspects of fostering creative activity in geometry instruction are reflected
in the scientific and methodological works of mathematician-educators such as A.Ya. Tsukar, S.
Alikhonov, M.V. Egupov, G.D. Gleyzer, V.A. Gusev, S.V. Maslova, I.S. Bekesheva, D.M. Mahmudova, A.K.
Nasybulina, and others [2.184]. In the process of teaching mathematics, solving problems serves a dual
purpose: it not only achieves one of the main goals of mathematical education - forming mathematical
knowledge, skills, and abilities as outlined in the mathematics curriculum and reflected in textbooks -
but also naturally fosters creative activity in school students [4.444]. The following methods can be
employed to develop students' creative thinking abilities through geometric problems:
Creating new ideas through problem-solving
: Present students with simple geometric problems and
ask them to find solutions using various methods.
Example: Find the area of a rectangle.
Simple method: The area of a rectangle is S = length × width. If the length is 8 cm and the width is 4 cm,
then the area S = 8 × 4 = 32 cm2.
Creative approach: Suggest that students divide this rectangle into other shapes (for example, two
triangles or a square). Ask them to calculate the area of each shape and then use these to recalculate the
total area. This method allows solving the problem in different ways, encouraging creative thinking.
Generalizing and expanding problems
: Present geometric problems to students at a basic level using
simple shapes and ask them to transform these shapes into more complex ones. For example, ask
students to divide a shape into two parts and discuss their ideas. Then, you can ask them to divide it
into three parts or combine shapes. This develops students' creative thinking.
Example: Find the area of a square.
Simple method: The area of a square is
S = a
2
If the side is 6 cm, the area
S = 6
2
= 36sm
2
.
Creative approach: Encourage students to analyze the square in various ways. For instance, dividing the
square into two triangles or splitting it into smaller squares. This method helps students understand
how area can be represented in different forms.
Visualizing problems:
Encourage students to connect mathematical shapes with real-life situations.
For example, comparing shapes such as circles and rectangles with everyday objects (balls, windows,
curtains, etc.) expands their imagination.
Место для уравнения.
You can also ask them to modify
problems as they wish. For instance, asking, "What changes would occur if you transformed this shape
into another?" helps develop the student's creative thinking.
Example: Divide a rectangle into two congruent triangles. The rectangle is 8 cm long and 6 cm wide.
Calculate the area of the rectangle and determine the area of each triangle after dividing it into two
equal triangles.
Simple method: The area of a rectangle is
S
rectangle
=
length × width.
S = 8 × 6 = 48m
2
. If we divide the rectangle into two equal triangles, the area of each triangle will be
S
triangle
=
S
rectangle
÷ 2 = 48 ÷ 2 = 24cm
2
Creative approach: Ask students to divide a rectangle into various shapes. For example, you can suggest
dividing the rectangle into shapes such as triangles, squares, or parallelograms. Students can determine
the dimensions of the rectangle and decide which shapes to divide it into themselves. Through this
method, students learn to apply creative approaches in dividing rectangles into different shapes and
combining them. This, in turn, helps develop geometric visualization skills.
Analyzing problems and finding new approaches:
Geometric problems can have multiple solutions,
so teach students to solve problems using various methods. For example, calculating the area of a
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rectangle using different formulas or determining a circle using different techniques. Encourage
students to answer the question "How can you approach this differently?" This approach compels them
to think creatively.
Example: How can you calculate the area of a triangle with a base length of 6 cm and a height of 2 cm?
Simple method: The problem provides the base and height of the triangle, so we apply the formula.
S = (base × height) ÷ 2 = (6 × 2) ÷ 2 = 6 sm
2
Creative approach: Challenge students to transform the triangle into a rectangle. To do this, the triangle
can be divided into two equal parts: Draw a line along the middle of the triangle. By converting these
two halves into a rectangle, we can calculate its area.
Visualization and modeling:
Encourage students to think more deeply by modeling geometric
problems in real-life situations. For example, demonstrate how these shapes are used in life, such as
geometric shapes found in construction, art, or nature. Creating, constructing, or depicting shapes with
their own hands stimulates creative thinking.
Example: Calculating the area of a triangle.
Simple method: The area of a triangle is
S = (base × height) ÷ 2.
If the base is 8 cm and the height
is 5 cm, then the area
S = (8 × 5) ÷ 2 = 20 sm
2
.
Creative approach: Invite students to make or construct a triangle with their own hands. This helps
them understand how shapes can be used in real life. For example, students can learn how the triangle
they are studying is applied in construction or art.
Collaboration and group work
: Encourage students to solve problems in groups. Exchanging ideas on
a specific issue and discussing various approaches develops creative thinking. Allowing each student to
propose their own method and contribute to reaching a common group solution reinforces students'
creative approaches.
Example: Creating a new shape by combining three different shapes.
Simple method: Provide students with rectangle, circle, and triangle shapes and ask them to create a
new shape. For instance, combining these shapes to create new geometric figures in various forms.
Creative approach: Divide students into groups and give each group different shapes, asking them to
create their own solutions. Then, have each group present their approach and discuss all solutions. This
gives students the opportunity to develop creative thinking.
Learning from trial and error
: Give students the opportunity to find and correct mistakes when
solving problems. Finding the answer to the question "Why didn't this method work?" develops their
critical thinking skills. Identifying and analyzing various errors in solving geometric problems further
enhances creative thinking.
Example: Correcting errors when calculating the area of a rectangle.
Simple method: There can be several errors when calculating the area, such as incorrectly inputting
measurements or using the wrong formula.
Creative approach: Ask students to calculate the area of a rectangle, but have each of them use different
methods. Then analyze the errors and demonstrate how to arrive at the correct solution. This method
develops students' thinking and teaches them how to learn from mistakes during the learning process.
Integrating topics
: Combine geometry with other subjects, such as art, history, and natural sciences,
to offer students new and engaging problems. For example, by integrating geometry and art, you can
suggest that students study the works of famous artists and create artworks using geometric shapes.
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Example: Integrating geometry and art.
Simple method: Teaching students how to create geometric shapes.
Creative approach: Challenge students to transform geometric shapes into works of art. For instance,
they can create mosaics or pictograms using various geometric shapes. This provides them with an
opportunity to connect mathematics with art.
Developing students' creative thinking through geometric problems enables them to solve complex
problems in interesting and diverse ways. This not only enhances mathematical skills but also expands
students' general cognitive abilities. To date, numerous scientific studies have been conducted on the
development of creative capabilities. For instance, Dilnora Ziyadullaevna Ergasheva, in her article
"Utilizing the PISA Method in Solving Geometric Problems," identifies the development of creative
thinking as the primary goal that meets modern educational requirements. This article emphasizes the
significance of the PISA methodology in fostering students' creative and critical thinking skills. Solving
geometric problems teaches students not only to rely on theoretical knowledge but also to approach
problem-solving through various methods. Through the PISA program, creative abilities are developed,
and students become prepared to solve real-life problems. For example, students learn how to apply
their mathematical knowledge in everyday life. The approaches discussed in the article demonstrate
that education is focused on developing creative and logical thinking. Particularly through teaching
geometric problems: creative thinking is enhanced, and students are able to apply their knowledge in
various situations. Logical thinking is formed, which increases students' mathematical competencies.
The effectiveness of the primary education process improves because the PISA methodology focuses
not only on theoretical but also on practical development [1.195]. Additionally, in Nargiza Toshboeva's
article "Developing Students' Creative Abilities Based on Geometric Problem-Solving," several key
conclusions and analyses are presented regarding the development of students' creative abilities. The
article highlights the important role of geometric problems in developing students' creative abilities.
Geometric tasks help students develop non-standard thinking, which shapes their creative competence.
The article also emphasizes the necessity of integrating innovative and traditional teaching methods. It
demonstrates that geometric problems in elementary school mathematics lessons can enhance
students' intellectual activity, develop spatial imagination, and strengthen logical thinking. The
integration of information technology can further boost students' creative abilities in solving geometric
problems [2.185]. The role of geometric problems in developing creative thinking has been highlighted
in numerous studies. An independent researcher from Termez State University, Aytuvganov Urol, in his
article "Fundamentals of Developing Students' Creative Thinking in Teaching Geometry," provides
extensive scientific analyses on the development of students' creative abilities, particularly focusing on
fostering these skills in geometry. The development of creative and intellectual abilities is considered
one of the crucial directions in education. It is noted that geometry lessons serve as an effective tool for
developing students' abilities such as intuition, imagination, and visual thinking. The selection and
solving of creative tasks foster students' independent thinking, problem analysis, and approaches
aimed at finding new solutions. Particularly in elementary school mathematics lessons, developing
creative and analytical thinking through geometric problems enables the practical application of future
knowledge. Creative abilities are developed through "creative tasks." In these problems, the boundaries
of students' thinking expand, they experiment with new methods, and strive to create diverse solutions.
Especially when solving a problem using multiple methods is required, new levels of creativity emerge.
The development of mental abilities is enhanced through the synthesis of creative tasks. Students gain
the opportunity to apply their knowledge to new situations by learning to solve geometric problems
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using both algebraic and visual methods simultaneously[3.90.]. These studies are aimed at developing
creative thinking and innovative abilities in mathematics and geometry education.
CONCLUSION
In conclusion, it can be said that geometric problems are important in shaping students' analytical and
creative abilities. Geometry is important not only for mathematical calculations, but also for the
development of creative and logical thinking. Solving geometric problems in various ways allows for
the development of creative abilities. The application of new approaches to solving geometric problems
and the application of non-traditional thinking methods is required. This expands students' thinking
and encourages them to make independent decisions. The application of geometric problems in
mathematics lessons not only increases theoretical knowledge, but also serves to develop practical
skills. The article emphasizes the importance of geometry in shaping creative thinking in the education
system and proposes the widespread use of methods that develop creative abilities in pedagogical
practice. In developing creative abilities in elementary school, students should be given simple but
deeply thought-provoking tasks. For example, tasks aimed at finding new geometric representations by
combining or partially analyzing shapes make this process effective. The use of geometric problems to
develop creative and intellectual abilities is an effective method. By solving problems in various ways,
students develop critical thinking, logical analysis, and creative approach skills. Developing students'
interest in solving problems it is a very important means of fostering interest in mathematics and its
study, as well as an effective means of engaging students in creative learning activities in mathematics.
Such work is possible and necessary not only in extracurricular activities, but also during students'
direct study of the program material. To do this, in the process of teaching mathematics, it is necessary
to consistently include specific problems that are easily designed from the problems placed in
mathematics textbooks. The use of tasks that develop students' creative abilities is an important
condition for improving the quality of mathematical training for schoolchildren.
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