Теория и методика профессионального образования | Мир педагогики и психологии №09 (62) Сентябрь 2021

Формирование первоначальных геометрических понятий в начальной школе

Биннатова Шалала Бахлул кызы
доктор философии по педагогике; старший преподаватель кафедры педагогики и методики начального образования педагогического факультета Гянджинского Государственного Университета, Гянджа, Азербайджан, azerin2@mail.ru

Аннотация: Статья посвящена проблеме изучения геометрического материала в начальных классах. В статье основное внимание уделяется развитию у учащихся графико – измерительных навыков. Излагается методика ознакомления учащихся со свойствами и единицами измерения геометрических величин, а также нахождение прообразов геометрических форм в окружающем мире. Особое внимание уделяется изучение геометрического материала в контексте существующей учебной программы начального курса математики. На этом обосновывается актуальность темы. В научной статье для закрепления и уточнения геометрических знаний представлены различные задания, которые положительно влияют на пространственные представления у учащихся. С этой целью в статье подробно объясняется изучение элементов и свойств отрезка, ломаной, угла и многоугольника. Анализируются конкретные примеры из учебников математики. Практическая значимость исследования заключается в том, что выявленные приемы преподавания геометрического материала могут быть полезны в работе учителей начальных классов. Исследование позволило автору сделать выводы, что формирование пропедевтических геометрических знаний и умений у школьников младших классов поможет в дальнейшем усвоить более сложный геометрический материал в старших классах.
Ключевые слова: начальная школа, урок математики, учебник, геометрическая фигура, измерения, линейка.

Shalala B.Binnatova
Phd in Pedagogy, senior lecturer, department of pedagogy and metholodogy of primary education, pedagogical faculty, Ganja State University, Ganja, Azerbaijan.

Abstract: The article is devoted to the problem of studying geometric material in primary school. The article focuses on the development of graphic and measurement skills in pupils. The methodology of familiarizing pupils with the properties and units of measurement of geometric quantities, as well as finding the prototypes of geometric shapes in the surrounding world is described. Particular attention is paid to the study of geometric material in the context of the existing curriculum for an initial mathematics course. This is the basis for the relevance of the topic. In a scientific article, in order to consolidate and clarify geometric knowledge, various tasks are presented that have a positive effect on the spatial representations of pupils. To this end, the article explains in detail the study of the elements and properties of a line, polyline, angle and polygon. Concrete examples from mathematics textbooks are analyzed. The practical significance of the study lies in the fact that the revealed methods of teaching geometric material can be useful in the work of primary school teachers. The study allowed the author to conclude that the formation of propaedeutic geometric knowledge and skills in primary schoolchildren will help in the future to assimilate more complex geometric material in senior classes.
Keywords: primary school, mathematics lesson, textbook, geometric figure, measurements, linear.

Currently, mathematics is taught in Azerbaijan on the basis of an approved educational program (curriculum). This curriculum provides key learning outcomes through the interaction of content and action lines to determine what pupils need to know and what they should be able to do. The curriculum in mathematics of I-IV classes consists of five lines of content. One of them is the “geometry” content line.

The study of the properties of spatial figures through a geometric line of content, the formation of spatial representations, the analysis and solution of mathematical problems using the properties of geometric figures and geometric methods. In the lower classes, recognition of basic geometric shapes (for example, triangles, circles, squares and cubes) through geometric lines of content. Further studies will be expanded and deepened to study the properties of geometric figures, including various geometric relationships and geometric transformations, as well as a more detailed study of spatial geometry [1, p. 9].

Geometric material is an integral part of the mathematics course. It is studied at the level of knowledge - acquaintance. Pupils practically distinguish figures, compare them, draw on paper. In the process of studying geometric material in mathematics lessons, such mental processes develop as: attention, memory, imagination, perception, logical thinking, spatial orientation, speech. During the study of geometric material, pupils develop the ability to design, transform shapes, develop cognitive interest [2].

In initial classes are given the opportunity to get acquainted with simple geometric shapes, study the properties of ordinary shapes, measure length, perimeter and area, establish geometric shapes and their similarities in the environment, solve geometric problems.

The technology of studying geometric material in primary school is aimed at

- development of logical thinking of pupils, instilling in them elementary skills in defining the simplest geometric concepts,

- development of spatial representations;

- familiarization with the simplest deductive reasoning based on observation, comparison, generalization. The concepts of "definition", "theorem", "proof" are not introduced;

- the formation of elementary skills and abilities of performing constructions with the help of basic tools: a compass, a ruler, a square;

- the formation of rational techniques for constructing geometric shapes (on lined paper in a cage);

- the formation of skills and abilities to measure geometric values [3; p.263].

When considering the role of a teacher in teaching geometric material in primary school, it is important to identify a methodology that facilitates the disclosure of geometric content. The main objectives of this study were identified:

1) the development of spatial thinking;

2) the development of reflective skills;

3) understanding the environment from a geometric perspective;

4) the formation of ideas about ordinary and spatial figures;

5) the willingness to study geometry in senior classes [4].

In the first class, pupils receive information about the polyline. In fact, although a specific name is not being prepared, the concept of a broken line is being prepared. The dashed line consists of several parts: the end of the first part, the beginning of the second, the end of the second, the beginning of the third, etc. is called a figure. These fabrics do not create new fabrics.

Pupils are taught the definition of the following new term: the fractures that make up the broken line are called its sides. If we mark one point into parts, it will divide this part into two parts. But these pieces do not form a broken line. It is important not to show pupils the model of the broken line, but to prepare it for the pupils themselves from soft wire. This model can be made of scratches and plastic rollers, or by breaking a piece (thin rod) into one or two points. Such illustrations are fully consistent with pupils' ideas about the broken line and are well supported by the term “cliff”. It is also important that pupils clearly display closed dashed lines. Closed fractures should be drawn on paper, scraps or wires modeled. This will greatly help pupils become familiar with the training ground in the future. This is because the boundary of the polygon is a closed line.

As a result, pupils of the 1st class should be able to determine the number of fragments that make up the geometric figure, show the number of lines, their sides and numbers in geometry and other figures depicted. 1st grade students learn to answer the question "show pieces in pictures." Pupils can show three or six pieces.

The practice of drawing broken lines on drum paper will not only help to form habits of formation, master the properties and the correct terminology of these figures, but also prepare students for future acquaintance with other geometric figures, including polygons.

Visual memory plays an important role in teaching geometry. Because when students are introduced to a concept for the first time, only students who are familiar with the standard situation cannot go beyond that stereotype. Therefore, in the inclusion of geometric concepts, it is necessary to create a unity of content between words and drawings. Because the definition of the concept expresses its main feature, and this is clearly reflected in the drawing [ 5; p.51].

Primary school children are faced with polygamy in school practice, both in life practice and in the process of teaching parallel disciplines. A study of the appearance and development of geometric images shows that most children are familiar with a form called a circle (capable of distinguishing it from other shapes). The teacher should use this introduction to provide the first information about the training grounds. You should look at the cardboard circle and the polygon, comparing them and displaying them together.

To explain the concept of a polygon, pupils should use geometric shapes known as dots, fabrics, and broken lines. The article provides instructions for drawing a polygon and modeling. In this case, it is recommended to use wetter paper. Let's give an example. Children (the teacher can also dictate) perform the following book in the book of veins: “Mark the point where two straight lines intersect. Pay attention to the second point after six checkers. Point the checkpoint four points below the second point and the third point to the right of the three checkers. Connect all three points in one piece. “As a result, all students should get the same or triangular shape. The teacher re-draws the drawing board and repeats the procedure for drawing a triangle with students. This is a closed line.

When constructing polygons, it is wrong to limit their actions only by drawing clear images. From drawings of various shapes, for example, a description of the State Emblem of the Republic of Azerbaijan, which is presented on the front pages of all textbooks (mathematics, life sciences, native language, etc.). Pupils need to be taught how to create polygons of various shapes, using As a result of proper work, pupils should be able to answer the following questions in the picture.

a) What are the numbers in the picture? 1 - curved line, 2 - straight line, 3 - point, 4 - fold curved line, 5 - circle, 6 - point dotted line, 7- rectangle, 8 - triangle, 9 - hexagon, 10 - straight line broken line.

(b) How many sides of the polygon have hills 7, (8, 9)? How many sides of the broken line are 6 (11)?

Gradually, pupils can complete tasks related to the formation of complex figures. Children answer (and point out) the question “What forms do you recognize?”, When it comes to environmental objects in different disciplines, this should be associated not only with the form (appearance), but also with the determination of the number (number of digits, number parts).

In the process of drawing fragments and polygons, children get acquainted with the relations “large”, “small”, “equal”, starting from the first class. Using a simple task system, children also gain experience comparing works. For example, pupils use a large number of objects using notebooks: top or middle; They try to determine which one is the smallest or the next: The lengths of the above and subsequent passages are called equal parts, because they are equal.

Pupils are then asked to identify and present equal parts to more complex versions, such as polygons. Such tasks are expected, and paper strips or ropes are used to verify the results. Pupils mark dots on a piece of paper and compare them by placing them on another sheet. The second piece is larger than the fourth (this is clearly visible). It's a little difficult to compare the first and third parts. On a strip of paper, mark the end points of the first piece and place it (these points) on the third so that the first is less than the third.

It is not recommended for children of the first class who do not have sufficient experience comparing fittings to provide information on “equal” and “unequal” sections. Children should first have a detailed experience comparing works. As you gradually move to high school, you will have to move on to comparing works presented in more complex forms in other textbooks. At this time, comparing fragments with linear, roller, curtain, etc. is practiced.

In the first class, pupils become familiar with the measurement of line segments, which allows you to establish a relationship between a line and a number. Acquaintance with the measurement of segments allows you to visually illustrate ideas about a natural number, a decimal number system (centimeter - one, decimeter - one hundred, kilometer - one thousand), about operations on numbers [6].

For younger pupils it is very important to get acquainted with the size of the works and their comparisons. Since this concept is used in the teaching of all disciplines, it is more appropriate to formulate and consolidate all disciplines. Therefore, the methodological direction that we are going to interpret can be attributed to all subjects of parallel education of primary classes. This approach is due to the fact that the geometric concept of fragment length is the first step in forming the general idea of quantitative measurements, as well as the importance of measuring fragmentation. At the first stage, a clear understanding of the size of the pieces should be formed.

Studies have shown that for this purpose it is more appropriate to use step-by-step measurements to visually represent the measurements. Measuring a board (or other part) step by step means counting steps from start to finish.

The teacher instructs one pupil (the highest) to measure the board. The student measures step by step, for example, 5 steps. Then the task is assigned to another pupil (the smallest) with a different result, completed in seven steps. This is repeated when measuring the width, width or length of the school gym. The teacher claims that people are units for measuring parts in different areas (length of the board, width and length of the gym, width and length of the yard, line length, tree height, street width, well depth, etc.). Pupils alternately (under the guidance and guidance of the teacher) measure the length of the board in meters and get the same number, for example, "4".

Pupils understand that everyone will get only the same number when measuring the board. In other disciplines, measurements of other parts (for example, classroom, gym, width and length of the landing area, various materials in the workshop, distance from the school, etc.) can also be made. At this stage of measuring pieces, you should also pay attention to the use of another centimeter.

One of the most important steps in forming a fragment measurement is the use of a 1 cm centimeter model in the teaching of other disciplines. Studies have shown that using a centimeter model to teach students how to solve the following types of issues is more appropriate: Task. Measure a long piece with a centimeter model.

In completing this task, the teacher should carefully monitor the following student actions: precisely place the centimeter model on one of the edges of the measured subject; mark the other end of the centimeter model with a pencil on the measured piece; Place one end of the centimeter model on the newly acquired point and get the other point in pieces, marking the other end. Then it should be explained that the second point is separated by 2 cm. It is necessary to act with the same rule, since the end point coincides with the last end of the piece. Thus, the student counts the centimeters on the piece, taking the length (in centimeters) of the piece. If the overlap is not taken, the student answers: "The length of this piece is more than 4 cm and less than 5 cm."

Testing can be performed after sufficient skills have been developed to use the centimeter model to measure figures.

The teacher dictates: “Pay attention to the point where two lines intersect on the left (right) side. From now on, mark 9 checkers on the left (left), and the second under 3 checkers. Combine these points with a piece. Measure the length of the piece obtained using the centimeter model. "; “Measure the width and length of our tricolor flag as shown. Draw it in your notebook and color it. Tell us what the colors mean.

When performing these tasks, the teacher must exercise special control over the following abilities:

1) draw a straight line through a line or a pencil or draw a line on a notebook;

2) mark the point (one of the ends of the piece) on one line and divide it into centimeters (each time with a pencil) in a certain direction;

3) mark the second end of the piece with a pencil.

Practice shows that at the first stage, such tasks can be difficult for students. This is due to the fact that the pupils do not yet have a small centimeter pattern and the ability to work with pencils (finger muscles are not trained enough). These tasks should be repeated for a long time and systematically when performing practical tasks while teaching other parallel disciplines.

At a later (more) stage in the formation of fragment measurement skills, questions similar to the two above are solved by using a non-numeric scale in other disciplines, especially in the process of learning technology. As instructed by the teacher, pupils mark the line with a centimeter model on a thick paper strip.

There are simple but very important tasks that help strengthen and form the initial skills of measuring pieces: "Measure the length of a sheet of paper." For this, the student must “read” each centimeter from one end to the other and read.

After developing the skills of measuring fragments on a drum and a smooth sheet, children should first be taught how to measure environmental objects using a centimeter model, and then a linear scale with their own design. It can be used as measuring objects with notes, notebooks, cyberspace, pencils and other small objects. It is recommended that particular attention be given to measuring the sides of the polygon. For example: “Find any triangular shape (rectangle, etc.) given in the textbook and measure the length of its sides.”

You do not need to rush to use the scale ruler with a numerical scale for measurement. Because, as can be seen from the study, pupils often make the worst mistakes when using such an incubator. One of the reasons students make mistakes is because they do not pay attention to the initial move (which is always on the edge of the line). The pupils overlap the point where the point is not the starting line of the scale and make a mistake. After the correct measurement of fragments has been mastered, the aforementioned approach to weighing fragments with its amplification will avoid such errors. If some students find such errors, it is necessary to return to centimeters and paper strips in order to measure the pieces again.

In the I class, it is better to use a unit of length (cm) as a numerical material when forming a measurement tool using a linear scale, but also use a linear scale as an illustration, and then as a computing tool for collecting and subtracting numbers. For example: “A part is divided into dots. Measure the length of each piece using a linear scale. Is it possible to find the full length of a piece without measuring? Check with measurement.” Using a linear scale, pupils combine numbers with this rule. You need to find: 2 + 4. Initially, “2” is written on the scale (two centimeters correspond to 2 units). At this point, the pupil counts 4 cm and writes “6”. According to this rule, the sum of numbers is replaced by the sum of the lengths of the pieces. Pupils are encouraged to move in the same direction (from start to right), collecting both pieces.

Then it is necessary to continue the relevant tasks in teaching the topics “collection and deduction of units” [7, p. 53-59]. At this stage, it is recommended that the student make a decision: “Solve 3 + 6 examples using the axis.” Solution: note the large assembly (6) on the readings of the modules. Then count the new one 3 steps forward until you collect the small one (3). Take 9 on the axis of the number. For example, 3 + 6 = 9. This method can also be used in preparing the shipment. With the collection, it’s more convenient to view the results at the same time. For example, you need to follow the withdrawal procedure: 8-5. “8” is recorded on a linear scale. This corresponds to a unit of 8 cm - 8 digits. Then the pupil counts up to 5 cm from this point to the left. This can be done sequentially, counting 1 cm or in groups. The student falls on the sign "3". One might ask: “How can you choose from 5 units, reducing 8 units to 2 units, 2 units and 1 unit?” As you can see, the lining of the scales (25 cm long) can be used for a long time as a “counting machine” (until the student has mastered the collection schedule completely). The study showed that it is desirable to begin the acquaintance of the student with a new unit of measurement, a descriptor, while studying the numbers of the second decade.

However, when studying existing textbooks in mathematics for elementary grades, inconsistencies were revealed. Thus, the first acquaintance of pupils with the concept of “decimeter” occurs in the textbook “Mathematics-2” on the topic “Unit of Measurement of Length” [8, p. 77]. The textbook shows that 1m = 100 cm; 1 dm = 10 cm; 1 cm = 10 mm. Unfortunately, when centimeters and millimeters are explained and regular work is done on these compounds, information about the decimeter is cut off, and by the end of the second grade, children are not exposed to the “decimeter” block. This interferes with the timely development of the "decimeter" and its practical application. The study showed the opposite: the information they extract from life experience and the content of other disciplines can be adapted for second-class pupils by remembering children. For this, the teacher can explain that the height of the building, the length of the fence and the width of the street are meters. There is another unit of measurement for smaller pieces than meters. It is more than a centimeter and less than a meter, they are called decimeters. Pupils get acquainted with a short 1-day recording of a decimeter: 3 dm, 5 dm, 15 dm and so on.

From the experience of weighing and constructing the pieces, it is clear that the length of the piece, for example, is about 12 cm in centimeters and less than 2 dm, as described in the decimeter. Pupils should know that “a pencil is 1 inch long and 2 centimeters long.” The teacher corrects: “The length of the pencil is 1 inch and 2 centimeters,” and he shows a short writing: 1 dm 2 cm. The manufacture is still in practice, for example, 1 cm 5 cm long, 1 cm 9 cm long. Customization of parts.

The concept of “angle” is used not only in life, but even in geometry in several different ways. Geometric figures in I class: exceptional, because they do not have enough information about the beam, they do not talk about the angle of inclination. Therefore, the concept of “corner” at the first stage should be used only in the sense of  “polygonal broken angles” based on information obtained from the environment illustrated in various textbooks. This approach is more informative and visual compared to the previous approach, but it also reflects a scientific view of concepts from other disciplines and provides the formation of the concept of a polygon based on knowledge of neighboring disciplines.

Information about the angles of pupils in I-IV classes should not be tiring. In the first lesson, during extracurricular activities or in technology lessons, pupils are encouraged to first show broken triangles and a rectangle without corners. In this case, the polygon must be divided so that each of these parts has a hill and two sides of the hill.

It should be clarified here that a hexagonal hill is also a (corresponding) hill. When doing this work in the learning process, it is advisable to first introduce students to the corner models made of paper. Children divide a paper polygon into pieces. Creating the right impression on pupils depends on their ability to represent them correctly.

To do this, place the pointer on the large end of the pole and rotate the pointer from one side to the other, rotating it with “special movements”. It is also possible that the content will be unlocked through movement: “corner” (small) and “growing” (large). For this purpose, a model developed by the pupils themselves (in technology lessons or at independent time) (two thin plastic plates fastened with plastic) can be used. Pupils are informed that the closer we are to the edges of the corner (models are shown), the smaller the angle, and the more we mix it, the larger the angle. The initial ideas related to the concept of angles are reinforced and formed in these practical exercises.

The interpretation of rectangular images should be carried out through the interaction of mathematical knowledge and skills that children still have with them, observation data collected from observations, and illustrated teaching materials on parallel subjects. One, two, three, and so on. It is better to start with a review of polygons with rectangles and illustrations in various illustrations from textbooks. It is also advisable to use linear vector lines to construct a rectangle with a rectangle in the

first class. It is important to draw students' attention to the fact that some straight lines form a rectangle. These corners are also used to build polygons. First, the classroom has inventory with right angles (board corner, window corners, table corners, etc.), and then rectangular objects in various illustrations from textbooks. right angle, tilt point and its sides are shown. Then a triangle with a right triangle is formed: first, the slope point of the rectangle is recorded, then the sides of the triangle forming a right angle, and, finally, the third.

A similar work continues with finding two right angles, showing their slope and side points. Attempts are being made to construct a triangle with two right angles. Pupils mark two points (two hot spots of a triangle) that connect the two rectangles, eventually realizing that the triangle is not “removed” because its lines (sides) do not intersect. From this it is concluded that the triangle has only one right angle.

Then comes the recognition of rectangular figures with one or two right angles, showing the angles of deviation and, finally, the construction. When constructing a rectangle with three right angles, pupils come to the conclusion that its fourth corner will be right. The accuracy of all corners of the cradle is also verified by students. Using different examples and drawings from different textbooks when constructing different polygons, students can come to the same conclusion that only all corners of the rectangle can be straight. Such a rectangle is called a rectangle. Pupils observe objects in the environment. They are rectangular names of objects in the form of boards, floors, window glass, etc. Comparing rectangles, children come to the conclusion that their opposite sides are in the same equation. It can also be checked without measurement. For example, children know that the booklet is rectangular. Using the drawn triangle, they make sure that all angles are straight. Then the right angle is obtained by folding the paper. By bending the sheets, the opposite sides overlap each other, that is, their equality. It indicates that all four corners are taken, and the opposite sides are the same.

Task. Draw a rectangle 4 cm long and 2 cm wide. The apprentice marks a hill with a dotted rectangle at the intersection of two lines. Children know that one side of this rectangular hill is its length and the other is its width.

Thus, it is necessary to draw two parts from this point:

1) take a right angle between them;

2) the length of one piece is 4 cm and the other 2 cm.

With the help of dams, children build a rectangle (this can be done without using scale, since the pupils already know that they are two centimeters). The three points obtained are the three points on the hill of the rectangle. By measuring drops without measurement, pupils find the fourth point of the hill. It connects the points of the hill in series (with a line) and sticks to a colored pencil. Pupils build a rectangle 4 cm long and 2 cm wide. This work continues in extracurricular activities in various disciplines and in the form of practical tasks of a different nature.

In the first year of study, geometric shapes and objects were used as graphs. Later elements such as objects (polygon sides, angles, height) are also used. In the first class, pupils become acquainted with the measurement of a work, which allows them to establish contact between the subject and the subjects. Familiarity with the measurement of the product provides a visual representation of them (children) about natural numbers, the decimal number system (cm-unit, dm-centimeter, km-thousand) [9, p.4].

Studies show that geometric skills acquired in mathematics are determined by other disciplines. Such related training can continue to work in the recognition of objects of various shapes, including character shapes, from objects of colorful geometric shapes, data in life and in other textbooks. Because these actions help develop appropriate skills and strengthen relevant mathematical knowledge. The formation of initial geometric knowledge and skills in primary school pupils will help in the future to master more complex geometric material in senior classes. At the same time, measurements, construction and cutting work in primary classes have a positive impact on the development of students' life skills.

1. Учебная программа (курикулум) по математике для общеобразовательных школ Азербайджанской Республики (I-XI классы). Министерство Образования Азербайджанской Республики, Институт Проблем Образования Азербайджанской Республики. Баку: 2013. - 138 с.

2. Янькова А. С. Изучение геометрического материала на уроках математики в начальной школе. 03.12.2019. Режим доступа: http://www.xn--80ajonoehe5g.xn--p1ai/2011-03-29-09-03-14/138-preschool-math/14798Izuchenie_geometricheskogo_materiala_na_urokakh_matematiki_v_nachalnoy_shkole.html

3. Трофименко Ю.В. Разработка и практическая реализация технологии изучения геометрического материала младшими школьниками // Вестник Брянского госуниверситета, Брянск: 2016 (2). - c. 257 – 264.

4. Ивашова О.А., Останина Е.Е. Технологии начального математического образования. Часть 3 // Учебно-методическое пособие, Издательство ВВМ , 2015. – 123 с.

5. Veliyeva B. H. Psychological features of mastering geometric concepts // XIX Republican Scientific Conference of Masters, Sumgait State University, 2019. - p. 51-52.

6. Филиппова С. А. Использование геометрического материала в начальной школе. 10.06.2019. Режим доступа: https://www.prodlenka.org/metodicheskie-razrabotki/363649-ispolzovanie-geometricheskogo-materiala-na-ur

7. Kakhramanova N.M., Askerova S.S. Mathematics – 1. Textbook for the I class // Baku: “Altun kitab” Publishing House, 2010. - 128 p.

8. Kakhramanova N.M., Askerova S.S. Mathematics - 2. Textbook for II class // Baku: “Altun kitab” Publishing House, 2010. - 144 p.

9. Agaeva G.A. Computer technology as a visual tool for the formation of geometric concepts in the initial classes // XVII Republican Scientific Conference of Masters, Sumgait State University, 2017. - p. 4 – 6.