Child with early years faces the need to navigate in space. With the help of adults, he learns the simplest ideas about this: left, right, above, below, in the center, above, below, between, clockwise, counterclockwise, in the same direction, in the opposite direction, etc.
All these concepts contribute to the development spatial imagination in children. A child’s ability to imagine and predict what will happen in the near future in space lays the foundations for analysis and synthesis, logic and thinking.
Preschoolers are given the necessary primary information, and then the task is set: “What will happen if...”. The conditions under which the action must occur are formulated. The child must comprehend the data received, understand the task and make the right decision in the form of an oral or written answer.
The presented set of practical tasks will allow the preschooler to gradually, “from simple to complex,” develop his spatial imagination. The lesson is conducted in a group with older preschoolers.
Spatial orientation exercises
The teacher puts it in front of the children and asks the question: “In which corner of the square is the flower drawn?” (In the upper left.)
“I turned the square clockwise once.” (The teacher shows.)“In which corner was the flower?” (In the upper right.)
“Now I’ve turned the square counterclockwise twice.” (Turns.)“Where is the flower now?” (Bottom left.)
“I turn the square three times clockwise.” (Shows.)“In which corner is the flower?” (Bottom right.)
Next, children complete tasks individually on sheets of paper on which 4 squares are drawn.
The teacher formulates the task: “In the first square, draw a fungus in the lower left corner. Imagine that the square is rotated counterclockwise once. Where will the mushroom end up? Draw it in the second square. The second square was rotated clockwise twice. Draw in the third square where it is now.
The third square was rotated 3 times counterclockwise. In the fourth square, draw where the mushroom ended up.”
The teacher carries out the next task collectively with the whole group. The teacher puts up a poster and asks questions: “In which corner of the large square is the blue square? green square? yellow? red?".
After this, the children complete the task individually on pieces of paper on which 4 large squares are depicted. Large squares are divided into small ones. The first square is colored.
“Imagine that the first square was rotated 3 times clockwise. Where will the small squares end up? In the second square, color the small squares correctly. If the second square is rotated counterclockwise 2 times? In the third square, color the squares. And now the third square is rotated 4 times clockwise. Where will each square end up now? Color them in the fourth square."
The following tasks are carried out similarly:
After this, preschoolers complete tasks individually on sheets of paper:
4. The task of moving windows left and right is carried out in a similar way.
5. The pyramid was assembled in different ways. Color all the details of the assembled pyramids.
Look carefully at the picture. How many circles are colored? What color is the painted flower? |
Where will the red circle be if you move it 3 circles to the right and 1 circle up? Color it. |
Where will the red circle be if it moves 1 circle to the right, 1 circle up, 3 circles to the right and 1 circle down? Color it.
14. Color square A1 red, A2 blue, B2 yellow, B3 green, B1 brown, B2 purple.
15. Place a dot in square A2 and a cross in A3. In square B1 draw a circle, in square B4 - a triangle, in square B5 - an oval. In square B2 draw small square, in B3 - a rectangle, in B5 - a polygon. |
CHAPTER 1. THEORETICAL FOUNDATIONS OF THE PROCESS OF FORMING SPATIAL IMAGINATION OF SCHOOLCHILDREN WHEN STUDYING STEREOMETRY USING INFORMATION TECHNOLOGY
1.1 Analysis of literature on the research problem
Associationists (A. Behn, D. Hartley) considered thinking as a reproductive process, a process as a result of which nothing fundamentally new arises, but only a recombination of initial elements occurs. Currently, this approach has found its expression in behaviorism (A. Weiss, B. Skinner).
In the works of Soviet psychologists, productivity appears as the most characteristic, specific feature of thinking, distinguishing it from other mental processes, and at the same time, its contradictory connection with reproduction is considered.
Among the works devoted to the development of spatial thinking in teaching mathematics, noteworthy are the works of V. A. Krutetsky, D. Polya, L. M. Fridman, E. N. Turetsky, B. G. Ananyev, P. Ya. Galperin, A. V. Zaporozhets, A. N. Leontyev, N. A. Menchinskaya and many others. Much attention was paid to the problem of developing students’ spatial thinking when teaching mathematics and other subjects in studies on mathematics methods in the 1950s-70s (N.F. Chetverukhin, A.I. Fetisov, G.G. Maslova, A.M. Lopovok, Kh.B. Abugova, R.S. Cherkasov, etc.). Each of the researchers offered his own, new, view on the problem under consideration, thereby expanding and deepening it. The research results were introduced into teaching practice and successfully used by teachers. However, the strengthening of the logical component of the geometry course and the desire to build the course on a strictly deductive basis led to the fact that the problem of developing spatial thinking faded into the background, which negatively affected the results of teaching geometry and, first of all, stereometry.
Various aspects of computerization in the field of education were studied in the works of I.N. Antipova, G.A. Bortsovsky, Ya.A. Vagramenko, D.Kh. Jonassena, A.P. Ershova, I.G. Zakharova, M.P. Lapchika, E.I. Mashbitsa, N.Yu. Talyzina and others. The problem of using information technology in teaching geometry in secondary and higher schools dedicated to the publication of Yu.S. Branovsky, V.A. Dalingera, Yu.A. Drobysheva, A.I. Azevich, T.A. Matveeva, I.V. Robert, M.A. Nikiforova and others. The main attention in these studies is paid not only to the creation of software and pedagogical tools, the conditions for their use, but also to the development of appropriate computer-oriented methods for studying individual topics and sections of the school geometry course. Due to a number of circumstances, information technologies acquire particular importance in the process of developing spatial concepts of schoolchildren. There are two main motives for their use. The first is associated with the widespread use of information methods in geometric science; the second – with increasing the quality of learning material.
The problem of using computer mathematical systems in the process of teaching mathematics to students in secondary and higher schools is addressed in the publications of I.N. Antipov, E.V. Ashkinuse, G.A Bordovsky, Yu.S. Branovsky, B.B. Besedina, G.D. Glaser, Yu.G. Guzun, V.A. Dalinger, Yu.A. Drobysheva, I.V. Drobysheva, A.P. Ershova, S.A. Zhdanova, V.A. Izvozchikova, A.A Kuznetsova, E.I. Kuznetsova, M.P. Lapchik, V.M. Monakhova, M.N. Maryukova, I.V. Robert, A.V. Yakubov and others.
Analyzing domestic and foreign experience use information technology as a means of teaching and forming spatial representations of schoolchildren when studying geometry, we can conclude that certain experience has been accumulated on this problem; profound results of theoretical and practical significance were obtained. Research into the problems of computer support for teaching mathematical disciplines in secondary and higher schools has been particularly intensive recently. Research is being conducted in various directions. The publications of E.V. are dedicated to them. Ashkinuse, B.B. Besedina, Yu.S. Branovsky, Yu.G. Guzuna, V.A. Dalingera, Yu.A. Drobysheva, I.V. Drobysheva, V.L. Matrosov, M.N. Maryukova, I.V. Robert, A.V. Yakubov and others. The main attention in these studies is paid not only to the creation of software and pedagogical tools for educational purposes with methods of their application, but also to the development of appropriate computer-oriented methods for studying individual topics and sections of school and university mathematics courses. Analysis of these studies allows us to conclude that the use of information technology in mathematics courses has great potential. Much that has been done in this area deserves attention, and many positive things prevail.
1.2 Psychological patterns of development of spatial imagination
Spatial imagination is a type of mental activity that ensures the creation of spatial images and manipulation of them in the process of solving various practical and theoretical problems. Spatial imagination is a psychological formation that is formed in various types of activities (practical and theoretical). Productive forms of activity are of great importance for its development: design, visual (graphic). In the course of mastering them, the ability to represent the results of one’s actions in space and embody them in a drawing, drawing, construction, or craft is purposefully formed. Mentally modify them and create new ones on this basis, in accordance with the created image, plan the results of your work, as well as the main stages of its implementation, taking into account not only the temporal, but also the spatial sequence of their implementation.
Spatial imagination in its developed form operates with images, the content of which is the reproduction and transformation of the spatial properties and relationships of objects: their shape, size, relative position of parts. Operating with spatial images in visible or imaginary space is the content of spatial imagination. Isolating spatial dependencies from an object of perception is often difficult due to the complexity of its design. Many features (eg internal structure) are hidden from direct observation. Therefore, it is often necessary to highlight the spatial dependencies inherent in an object indirectly, through comparison and juxtaposition of various parts and elements of the structure. The general thing that characterizes any spatial image is the reflection in it of the objective laws of space. Spatial properties and relationships are inseparable from concrete things and objects - their carriers, but they appear most clearly in geometric objects (volumetric bodies, planar models, drawings, diagrams, etc.), which are original abstractions from real objects. It is no coincidence that geometric objects (their various combinations) serve as the basic material on which spatial images are created and manipulated.
In modern psychology, the concept of spatial representations is associated with the concept of the image of an object or phenomenon that arises as a result of perception. In this case, much attention is paid to visual images, since their information capacity is especially large. They allow you to instantly grasp the relationship between the real and the imagined situation. Spatial representations are holistic subjective images of spatial objects or phenomena that are reflected and consolidated in memory based on the perception of visual material in the process of activity. Then the formation and development of spatial representations can be considered as a process of creating images and operating with them.
This view of spatial representations was taken as a basis by many methodological scientists when developing methods for the formation and development of students’ spatial representations. By spatial representations they most often understand the image of one or another spatial (geometric) figure, the relationship between its elements. The process of formation and development of spatial representations is characterized by the ability to mentally construct spatial images or schematic configurations of the objects being studied and to perform mental operations on them that correspond to those that must be performed on the objects themselves.
The cognitive nature of ideas is revealed in the fact that they are an intermediate link in the transition from sensation to thought. Clear and distinct ideas about geometric objects, consistently formed in the minds of students, are a solid basis for mastering scientific knowledge. Presentation like important element cognition is designed to connect images of objects and phenomena with the meaning and content of the concept of them. But, in turn, the formation of ideas requires mastery of the concept, since the concept determines the content of the image. Spatial concepts in relation to thinking are the initial basis, a condition for development, but, at the same time, the formation of ideas requires prior mastery of concepts and facts. We can say that the process of forming spatial ideas about geometric objects takes place on the basis of knowledge about them.
Based on the foregoing, we can conclude that the content of spatial representations should be considered as an image of a reflected object or phenomenon, in conjunction with knowledge about the object, extracted in the process of its perception. It is the result of spatial imagination, which combines the interrelated components (spatial and logical) of thinking.
So, by the spatial representation formed in the process of teaching geometry, we will understand a generalized image of a geometric object that develops as a result of processing (analysis) of information about it coming through the senses.
The scientific heritage of the outstanding Swiss scientist J. Piaget has been of interest to psychologists around the world for decades. His research, "dedicated to the development of children's cognition - perception and especially thinking - constitutes, according to P.Ya. Galperin and D.B. Elkonin, one of the most significant, if not the most significant phenomenon of modern foreign psychology."
Recognizing the formal-logical approach used by J. Piaget as a possible description of the patterns of development of a child’s thinking, many domestic and foreign scientists still note its limitations and try to consider mental activity as a kind of new mental reality that is formed at certain stages of development (P.Ya. Galperin , V.V. Davydov, L.F. Obukhova, D.B. Elkonin, M. Donaldson, R.V. In particular, trying to explain the mental mechanisms underlying the famous phenomena of J. Piaget, P.Ya. Galperin and D.B. Elkonin hypothesized that their reason lies in the absence of a clear consistent differentiation of some objective characteristics of objects, such as length, shape, weight, etc.
The next productive step in this direction was taken by N.I. Chuprikova. She managed to connect the indicated hypothesis of P.Ya. Galperin and D.B. Elkonin with research that argued that, firstly, the differentiation of cognitive structures and processes constitutes a relevant component of intellectual development (H. Werner, H.A. Witkin) and, secondly, that the child’s ability to differentiate various signs and relationships of objects is the core line during the transition from direct sensory cognition to abstract thinking(G. Hegel, I.M. Sechenov, J. Miller, N.I. Chuprikova). Based on these and a number of other results of theoretical and experimental work, N.I. Chuprikova set the task of substantiating the connection between the phenomena of non-conservation of J. Piaget and the insufficient differentiation of the reflection of various properties of objects. In the process of solving it, the author put forward and confirmed a hypothesis according to which, behind the very different, at first glance, methods of developing the ability to solve conservation problems in children with appropriate capabilities, there always lies the process of developing a differentiated reflection of the various properties of objects.
According to the facts described by J. Piaget, S.L. Rubinstein, N.N. Poddyakov, F.N. Shemyakin, a series of experiments conducted by I.S. Yakimanskaya and under her guidance, the child differentiates spatial characteristics in the objects around him. regarding symbols. 3. Ability to apply skills acquired at..., contribute formation spatial imagination. Besides...
People's Theater How means education schoolchildren
Abstract >> PedagogyFolklore theater How means formation personality, and... are engaged in the development imagination and... temporary and spatial segments, ... with development information technologies at various... at using folk elements in working with modern ones schoolchildren ...
Information era: economy, society and culture
Book >> SociologyYears at using new information technologies. ... of the Internet, in imagination people...first move formation funds mass media... Even schoolchildren know... spatial dispersion and concentration through information technologies. How ...
Theory and technologies training. Collection of texts
Book >> PedagogyTemporary and spatial. Organizational form... schoolchildren joys of knowledge How funds formation ... schoolchildren. At ... imagination, personalities) younger schoolchildren ... technologies requires a search for adequate telecommunications funds And information technologies ...
Popova O.N.
mathematics teacher at Municipal Educational Institution Gymnasium No. 1 in Lipetsk
It is no secret that many students do not have sufficiently developed spatial imagination. The problem is old, but relevant. If a teacher does not solve it even when he is teaching primary and secondary classes, then after a few years his stereometry lessons with the same students will lose most of their effectiveness.
All mental processes, including spatial imagination, are improved as a result of activity. This activity must be stimulated and directed by something, that is, a system of exercises is necessary.
This article offers non-standard and entertaining tasks for the development of spatial imagination. The answers are given in square brackets short solutions, instructions.
To solve many of these problems you do not need special knowledge, i.e. they can be offered already in the 5th grade, and some - in elementary school. The solution is most complex tasks can be rewarded with a mark.
The first series of tasks can be called “entering space.”
These are oral tasks in which, it would seem, nothing is said about space. On the contrary, the mention of triangles in task 2 and the arrangement of coins in task 3 (students immediately think that the coins should lie on a plane) imposes “planar” images. Need to overcome this is to “bring” the thought “into space” in order to correctly complete the proposed tasks.
1. Divide the round cheese into 8 pieces using three cuts. [Answer in Fig. 1].
2. From six matches, make four regular triangle so that each side has a whole match. [Triangular pyramid with an edge equal to a match].
3. Arrange 5 identical coins so that each of them touches the other four. [Answer in fig. 2].
4. Is it possible to arrange 6 identical pencils so that each one touches the other five? [You can, the answer is in Fig. 3].
5. Cut out the same shape from a whole sheet of paper as in Fig. 4a. [Cut the rectangular sheet into pieces A,b, With(Fig. 4b), rotate the shaded part about the straight line l
180°]. 
Rice. 1 Fig. 2 Fig. 3 
Rice. 4
It is often advised to accompany the study of the axioms of stereometry and their consequences with images of polyhedra, solving problems on constructing sections, etc. But students must “see” this polyhedron. Therefore, even before studying stereometry, it is necessary to offer students problems with a cube, parallelepiped and some other figures. This series of tasks is related to illusions and impossible objects.
In Fig. 5 any mathematician sees a cube, and not just two squares, the vertices of which are connected in pairs. But the squares are still drawn...
Rice. 5 Fig. 6
A well-developed spatial imagination allows us to see a cube. But it’s surprising: one time we see this cube as if from above and to the right (Fig. 6a), and the other time - from below and to the left (Fig. 6b). These are already incidents of illusion that you need to be able to manage, subordinating your imagination to the reality that is discussed in a specific task. But many students take a long time to learn this. It is necessary to help them master this skill in middle school by offering exercises 6–10.
Cover the front face of the cube with a sheet of colored paper and describe your impressions. [A cube like the one in Fig. is more clearly visible. 6a.]
Cover the back face of the cube with a sheet of colored paper and try to convey your impressions with a drawing. What does the drawing look like: a locker? shelf?
What do you see in Fig. 7? [A block with a recess (the back wall of the recess is plane AB), or a block with a protruding tenon, where AB is its front edge, or the open part of an empty box with a brick adjacent to the walls from the inside].
In Fig. 8a the figure is not completed (the upper part of the image is covered with a sheet of paper.) Complete it.
Explain whether it can exist not on paper, but in life the figure shown in Fig. 9.
Rice. 9 Fig. 10
The third series of tasks uses developments of a cube and a cylinder.
11. How many sides does a hexagonal pencil have? [Eight if the pencil is not sharpened. The answer is often “six”].
12. A cube was glued together from paper. It is clear that it can be cut into six equal squares. Is it possible to cut it into twelve squares? [It is not difficult to prove that a figure consisting of a union of triangles A And IN in Fig. 10 located in the same plane is a square].
13. In Fig. 11 on the left shows a scan of a cube. Which cubes from those given on the right in the same figure can be folded from this development? [Cubes in Fig. 11, b, With,f].
14. In Fig. 12a
shows a cube with the numbers 1, 2, 3, 4, 5, 6 written on its faces. (We see only the first three numbers.) The sum of the numbers on the opposite faces is 7. On four scans of the cube (Fig. 12b)
write five numbers - one has already been written - so that it corresponds to our cube.
15. In Fig. 13a shows a piece of paper. Is it possible to paste over some cube in one layer with this piece of paper without cutting it? [It is possible if the face of the cube is the same as the shaded one in Fig. 13b].
16. Which of the eight drawings (see Fig. 14) did the painter apply to the wall with the roller shown right there? [The sixth drawing is “knurled.”]
Rice. 14
Tasks on the projection of figures.
17. What shape does the shadow of a cube have on a plane perpendicular to its diagonal from a beam of light rays parallel to this diagonal? [Regular hexagon].
18. In Fig. Figure 15a shows figures bent from wire with a thick line. Draw three of their projections: on the front face of the cube, on its side face and on the top face. [Answers in Fig. 15b under the images of the corresponding figures].
Rice. 15
19. Bend a figure from soft wire, when parallelly projected onto different planes, the letters are obtained: S, L, O, G. [See. rice. 16. There are other solutions if you fit a wire figure into a cube].
20. In Fig. 17a shows a board with various holes. Find the single plug that covers the three holes. [Answer in fig. 17b].
Many of the tasks listed here are valuable because the items they talk about can be made by students themselves. It is not difficult to bend the wire and use it to check your solutions to problems 18 and 19. Making paper cube developments, which are discussed in problems 12 – 15, will not cause any technical difficulties.
The board with holes for problem 20 can also be viewed in real life - cut out of cardboard, plywood or foam plastic.
However, in all cases, it is advisable to make models after the solution, and not for the solution. If the teacher begins to consider the proposed tasks with models, then it is the students’ imagination that is not involved and the incentive for its development is weak.
In conclusion, I note that the originality of tasks arouses students’ interest both when working in class and in extracurricular activities, and this is one of the necessary conditions successful study of the subject.
The uniqueness of geometry, which distinguishes it from other branches of mathematics, and all sciences in general, lies in the inextricable organic combination of living imagination with strict logic. Geometry in its essence is spatial imagination, permeated and organized by strict logic.
In any truly geometric sentence, be it an axiom, theorem or definition, these two elements are inextricably present: a visual picture and a strict formulation, a strict logical conclusion. Where one of the two sides is missing, there is no true geometry.
Visualization and imagination belong more to art, strict logic is the privilege of science. The dryness of a precise conclusion and the vividness of a visual picture - “ice and fire are not so different from each other.” So geometry combines these two opposites. This is how it should be studied, combining vivid imagination with logic, visual pictures with strict formulations and evidence.
Therefore, the basic rule for studying geometry is that when encountering a definition, theorem or problem, you must first of all imagine and understand their content: visualize, draw or, even better, although more difficult, imagine what it is about we're talking about, and at the same time understand how it is exactly expressed.
It is no secret that many students do not have sufficiently developed spatial imagination. The problem is old, but relevant. If a teacher does not solve it even when he is teaching primary and secondary classes, then after a few years his stereometry lessons with the same students will lose most of their effectiveness.
All psychological processes, including spatial imagination, develop and improve as a result of activity. This activity must be stimulated and directed by something, i.e. a system of exercises is needed.
Over the years of working at school, I came to the conclusion that students’ spatial imagination should be developed from the first mathematics lessons in the fifth grade.
Currently, various systems have been developed for the development of spatial imagination in junior schoolchildren, including computer ones. For a number of years, I have been using a simpler system, which I call the “Introduction to Geometry” course, designed for teaching in grades 5–6. Its goal is to prepare students to master a systematic course in geometry.
When determining the content of the “Introduction,” it was necessary to understand what exactly is most difficult for children at the beginning of a systematic course. This course is dogmatic. He has almost no motivation, his logic is hidden from children. In fact, it starts with points and lines, then angles, then triangles, etc. But the students do not know what will happen ahead; they do not know about cylinders or pyramids.
The separation of planimetry and stereometry is a very harmful feature of the course. Students' spatial imagination is suppressed. The latest editions of the textbook “Geometry”, grades 10 – 11, authors L.S. Atanasyan, V.F. Butuzov and others, try to smooth out the transition from planimetry to stereometry, depicting volumetric bodies in color, but when students move from the textbook to workbooks, this attempt fades away. The image of the figure in the notebook becomes colorless, and students have difficulty reading and drawing such drawings. (Don’t force high school students to draw with colored pencils!)
In search of overcoming this shortcoming, it is appropriate to turn to the origins of geometry. The initial geometric information that has reached us is contained in Egyptian papyri and Babylonian cuneiform tables dating back more than four thousand years. Obtaining new geometric facts using reasoning (proof) dates back to the 6th century. BC and is associated with the name of the ancient Greek mathematician Thales, who first used movements: bending a drawing, rotating part of a figure, etc. Gradually, geometry becomes a deductive science, i.e. a science in which the vast majority of facts are established through inference and evidence. The pinnacle of ancient Greek geometry was the book “Elements”, written by Euclid (III century BC), containing the properties of parallelograms and trapezoids, the similarity of polygons, the Pythagorean theorem, etc.
In the current course, only the Euclidean stage of the history of geometry is presented, and the pre-Euclidean stage is not considered at all. It does not reflect the time when scientists did not yet master the methods of rigorous proof, but already knew almost everything that is included in current school geometry. Why not introduce students to all the objects of study before a systematic course, using for this part of the hours allocated for repeating the material studied in grades 5–6. Then in 7th grade you can clearly set the task - to organize already familiar material so that you can prove the validity of already known facts and others still unknown. With this formulation of the question, dogmatism is eliminated, and those skills that can be developed in grades 5–6 make further study of geometry not so difficult.
The measurement of lengths is known from primary school, and when studying the measurement of areas, volumes and angles, it is easier to explain the practical need for measuring volumes. Therefore, it is convenient to begin the introduction to geometry with the manufacture of a liter container - a cube with an edge of 1 dm. At the same time, students’ attention is drawn to the fact that to make this cube you need to have six squares with a side of 1 dm and when gluing them they need to be applied to each other in a certain way. Students gain very important experience, which is unattainable in current conditions, because the measurement of volumes is studied in the course of stereometry of grades X - XI. (You can’t force high school students to glue cubes!) Already in this example, certain skills are visible: children measure, draw, cut out, and glue. In the future, calculations using formulas are added.
The next question is measuring the volume of a half-liter container, which is very common in trade and in everyday life. You can cut a liter cube in half with a horizontal (vertical) plane passing through the middle of the sides, or with a vertical (horizontal) plane passing along the diagonals of the bases.
In the first case, we divided the height of the cube in half, but did not touch the base. In general, if you do not change the base, but change the height, then the volume will change by the same amount. In the second case, we did not touch the height, but halved the area of its base. This is how we come to an explanation of the formula for the volume of a prism. Students apply the acquired knowledge when performing practical work.
Note that solving many problems does not require special knowledge, i.e. they can be offered to students as early as the fifth grade.
The first series of tasks can be conditionally called “entering space.” These are oral tasks in which, it would seem, nothing is said about space. On the contrary, the mention of triangles in problem 2 and the location of coins in problem 3 (the reader immediately thinks that the coins should lie on a plane) imposes “planar” images. You need to overcome this, “bring” your thought “into space” in order to correctly complete the proposed tasks.
For example:
1. Divide the round cheese into eight pieces using three cuts.
2. From six matches, fold four regular triangles so that the side of each is a whole match.
3. Arrange five identical coins so that each of them touches the other four.
4. Is it possible to arrange six identical pencils so that each one touches the other five? (See Appendix 1 for answers)
It is often necessary to accompany the study of the axioms of stereometry and their consequences by depicting polyhedra, solving problems on constructing sections, etc. But students must “see” this polyhedron. Therefore, even before studying stereometry, it is appropriate to propose problems with a cube, parallelepiped, and some other geometric bodies. This group of tasks is related to illusions and impossible objects.
In this picture<Рисунок1>any mathematician sees a cube, and not just two squares whose vertices are connected in pairs. But squares are still drawn... A well-developed spatial imagination allows us to see a cube. But it’s surprising: once we see this cube as if from above and to the right<Рисунок2>, and the other - below and to the left<Рисунок3>. These are already incidents of illusion that you need to be able to manage, subordinating your imagination to the reality that is discussed in a specific task.
 |
 |
But many students cannot immediately learn to see in flat figure convex bodies. Our task is to help them in middle school. By offering a series of planar drawings, we will try to overcome the difficulties of perception.
For example:
5. Cover the front face of the cube with a sheet of colored paper and describe your impressions. (A cube like the one in Figure 2 is more clearly visible)
6. Cover the back face of the cube with a sheet of colored paper and try to convey your impressions with a drawing. What does your drawing look like: a locker? shelf?
7. Try to imagine, looking at the drawing, first the corridor<Рисунок4>(pipe<Рисунок5>, along which you move, then an inverted children's bucket, which you look at from above. (In the first case, the larger square (circle) is closer to us, in the second - further away).
 |
The third series of tasks uses the development of a cube, prism, cylinder and cone.
8. How many sides does a hexagonal pencil have? (Eight, if the pencil is not sharpened. Often the answer is “six”).
9. A cube was glued together from paper. It is clear that it can be cut into six equal squares. Is it possible to cut it into twelve squares? (It is not difficult to prove that a figure consisting of the union of triangles of the front and top faces located in the same plane is a square).<Рисунок6>
10. The picture shows a piece of paper. Is it possible to paste over some cube in one layer with this piece of paper without cutting it? (It is possible if the face of the cube is the same as the one highlighted in color).<Рисунок7>
The next series of tasks are projection tasks. Children very often play, depicting various shadows on the wall, table, etc. As an example, I will give the following task:
11. What shape does the shadow of a cube have on a plane perpendicular to its diagonal from a beam of light rays parallel to this diagonal? (Regular hexagon).
In tasks on the projection of figures, tasks on the image of figures bent from wire, when a beam of light is directed at a cube at different angles, can be widely used. These tasks are valuable because the objects they talk about can be made by students themselves. The production of paper cube developments will not cause any technical difficulties. However, it should be noted: in all cases, it is advisable to make models after decisions, not For solutions. If we start considering the proposed tasks with models, then it is the students’ imagination that is not involved and the incentive for its development is weak.
A special place in the development of thinking is occupied by learning to compare, in particular, comparing a fact expressed verbally with its interpretation in a drawing. The drawing can serve as a refutation of some general statement. By learning to refute incorrect statements, schoolchildren gradually get used to evidence. And this is a necessary type of activity when studying geometry.
So, versatile work with drawing and drawing not only contributes to the general mental development of schoolchildren, but develops spatial imagination, providing a more complete and productive study of geometry, and this work must begin in grades 5–6 when studying mathematics.
Spatial thinking is an important element of human mental activity. It is responsible for orientation in space, the ability to solve problems in geometry, and the ability to represent objects in three dimensions. Violation of this type of thinking leads to global disorientation of a person.
From a psychological point of view, this is a process that creates spatial images and determines the relationships between them.
From a practical point of view, spatial thinking allows a person to more easily solve problems in geometry, chemistry, physics, drawing, and even better cope with the process of studying literature. With the help of three-dimensional thinking, it is possible to form dynamic pictures in the mind, which makes the process of reading or studying something exciting and interesting. This kind of thinking reaches high level development in the sports profession associated with orientation in space.
In psychology, there is also an opinion that orientation in space and the thinking associated with it manifests itself differently in residents of different areas. Research showed that people living in the mountains can easily determine the size of an object located below than directly or above. Those living in valleys tend to correctly determine the size and distance on the plain. And this feature is not a violation of the functioning of spatial perception, this observation suggests the following:
Spatial thinking and the skills associated with it are developed and reach a higher level through a person's receipt of similar experiences.
Components
The characteristics of spatial intelligence include several stages that have a number of specific features:
- Analysis is the division of an object or task into its constituent parts.
- Synthesis is the opposite process of analysis - combining an object or task into a single whole.
- Abstraction is the definition of several stages of a task that should be in it. At this stage, concepts are formed.
- Generalization – definition and highlighting significant parts object or item that needs to be compared with each other.
- Concretization is the reverse process of generalization - identifying stages characteristic of a task that are not related to the stages of solutions.
At the end of the article, a test is presented to determine the level of development of spatial intelligence, which is built on these stages. Basically, the test consists of determining the relationship and sequence of various figures.
Development methods
The development of spatial thinking is best started in early childhood, because adolescence its formation is considered to be completely completed. However, in psychology there are methods and exercises that contribute to the development of a higher level at a more mature age. From a psychological point of view, a minor violation in the structure of three-dimensional thinking can be corrected by also using exercises and games, the list of which is presented below:
- Origami, puzzles
The formation of shapes in the head occurs in the process of folding puzzles and various paper objects. This happens due to the fact that before you put the figure together, you need to imagine it in your head. Methods of construction activities are also suitable for studying subjects at school - they facilitate the study of literature, switching children to practical actions.
- Shape Manipulation
To do this, you need to take several shapes - for example, a square, circle, cube, etc. You need to try to superimpose them on each other and imprint the result in your mind. To complicate this exercise, try to do the same thing mentally - imagine the figure in a three-dimensional format, name its sides, connection points, what the figure will look like and its characteristics will change if another is superimposed on it, etc.
- Redrawing figures
Methods for studying geometry and drawing are the basis of this exercise. This technique has several variants of complexity:
- Simple redrawing: the layout of the figure must be transferred to paper.
- Redrawing with changes: the figure is copied onto paper, but either a few cm or another figure must be added to it.
- Redrawing with changing scale. The essence of the exercise is to copy an object with a change in size, for example, twice as large or smaller.
- Redrawing from memory. The figure must be imagined in the mind and then transferred to paper.
From a psychological point of view, the tasks from this exercise contribute to the formation of not only three-dimensional thinking, but also drawing and memorization skills.
- Performances.
It is better to operate with lines and segments, for example: imagine several lines, connect them into one whole and then draw a figure on paper, or put a cube on several segments and reproduce what comes out of it.
- Schemes and drawings.
This includes any objects and items, figures, details or apartment plan. You can depict them either according to the layout or based on your own ideas. Creation of diagrams and drawings is available online.
- Game "Guess the object".
This technique is suitable for the youngest and takes place in the format of a game: the child closes his eyes and is given an object for tactile study. Exploring an object should take no more than one minute; peeking and hints are a violation of the rules of the game. The child’s task is to guess what kind of object it is and describe its characteristics.
- Game "Fly".
Games for adults will also help develop spatial intelligence. This is intended for a company of 3 people - two are directly involved, the third monitors the game process and monitors possible violations of the rules. Two players imagine a grid of 9 squares long and 9 wide. There is a fly in the very top right corner. Players take turns taking steps, moving the fly to different squares. The grid diagram, shown on paper, is available to the third participant, where he notes all the actions of the players. He then says “stop” and the participants say where they think the fly is. The one who names the correct square wins.
How is your spatial thinking developed?
Forming a high level of spatial intelligence makes our life easier in many ways. The activities of some professions are directly related to this skill. For example, you will never be able to become successful in the profession of a designer, artist, engineer, constructor, or logistician without the ability to perceive three-dimensionally.
IN everyday life spatial thinking will allow you to systematize the space in your apartment, house, navigate while driving and do without a navigator.
A test consisting of 10 questions will help determine the degree of development of this type of mental activity. This test can be taken online.
