Research work “Development of critical thinking of high school students in literature lessons” material on literature on the topic. Peculiarities of thinking of high school students and teenagers

In the context of the Modernization of Russian education, an important place is given to specialized schools. An integral part of all high school education are elective courses.

Working on the research topic of the formation and development of combinatorial-logical thinking of high school students, we offer a series of elective courses in mathematics, which are not only aimed at obtaining subject knowledge, but also have a main function within the framework of experimental research, namely, aimed at developing combinatorial -logical thinking.

By the development of combinatorial-logical thinking we will understand thinking aimed at the development of logical laws and operations with a finite variability of the phenomena and concepts under consideration.

The new form of final certification of school students - the Unified State Examination form - also convinces us of the importance of this kind of thinking. Section “A” in mathematics of the unified state exam requires choosing the correct answer. The need to search for new effective means of developing combinatorial-logical thinking in schoolchildren is due to its importance for the further self-realization of the individual in modern society.

The formation of combinatorial-logical thinking involves the process of obtaining subjectively new knowledge, which can be carried out in various ways of organizing educational activities related to the study of extracurricular material.

The means of forming the elements of such activity for students are the materials we have developed, which take into account:

1) increased level of difficulty through a system of tasks, through the structure of tasks (L.V. Zankov);

2) development of students’ thinking in the “zone of proximal development” (L.S. Vygodsky);

3) the theory of the gradual formation of mental actions, expressing modern principles of learning theory (P..Ya. Galperin);

4) the concept of educational activity based on changing the content of education (V.V. Davydov-D.V. Elkonin);

5) stages of the creative process (V.P. Zinchenko).

Let's clarify each of them.

The theory of the relationship between training and development, developed by L.V. Zankov and his followers, as a starting point, asserts an objective connection between the structure of education and the nature of the general development of schoolchildren.

Didactic principles play a specific and regulating role:

training at a high level of difficulty;
training with the leading role of theoretical knowledge;
studying program material at a fast pace;
students' awareness of the learning process.

Developmental education is the type of education that is focused on the “zone of proximal development” (L. S. Vygotsky). Therefore, training should be conducted at the maximum level of difficulty corresponding to the real capabilities of the student (“difficult, but feasible”), and, therefore, the tasks presented to students, if possible, should be individualized so that training has the maximum developmental effect.

P.Ya. Halperin distinguishes four types of action:

physical action. “The peculiarity and limitation of a physical action is that in the inorganic world the mechanism producing the action is indifferent to its results, and the result does not have any, other than random, influence on the preservation of the mechanism that generated it”;
level of physiological action. At this stage, “we find organisms that not only perform actions in the external environment, but are also interested in certain results of these actions, and, consequently, in their mechanisms”;
level of action of the subject. “New, more or less changed values of objects are used without fixing them, only for one time. But on the other hand, each time the procedure can be easily repeated, the action can be adapted to individual, individual circumstances”;
level of individual action. “Here the subject of action takes into account not only his perception of objects, but also the knowledge about them accumulated by society and not only their natural properties and relationships, but also their social meaning and social forms of attitude towards them.” P.Ya.

Halperin notes that “each higher stage of development of action necessarily includes the previous ones”

V.V. Davydov argues that “the basis of developmental education is its content, from which the methods (or ways) of organizing training are derived.” This understanding of learning is also typical for L.S. Vygotsky, D.B. Elkonina. As a result of educational activities, schoolchildren reproduce “the real process of people creating concepts, images, values and norms” As noted by E.V. Ilyenkov, “in a condensed, abbreviated form reproduced the actual historical process of the birth and development of ... knowledge”

"A. Emergence of a theme. At this stage there is a feeling of need to begin work, a feeling of directed tension that mobilizes creative forces.

B. Perception of the topic, analysis of the situation, awareness of the problem. At this stage, an integral holistic image of the problem situation is created, an image of what is and a premonition of the future of the whole...

B. At this stage, often painful work is carried out to solve the problem. There is a feeling that the problem is me, and I am the problem...

D. The emergence of an idea (equally an image-eidos) of a solution (insight). There are countless indications of the presence and decisive importance of this stage, but there are no meaningful descriptions, and its nature remains unclear.

D. The executive, essentially technical stage.”

Let's consider a system of elective courses that can be implemented both individually and in a single chain (it all depends on the desire and degree of development of combinatorial and logical abilities of high school students):

– “Mathematics of Reasoning”, elective course, designed for 17 hours. This course develops initial skills in the variability of logical reasoning, teaches how to build similar versions of mathematical and logical problems and search for their solutions.

- “Four typical problems of combinatorial-logical thinking” , an elective course lasting 17 hours, allowing students to master the main types of problems aimed at developing combinatorial logical thinking.

- “Basic methods for solving mathematical problems”, 17-hour elective course.

The purpose of the system of elective courses is to increase the level of creative thinking aimed at the formation and development of combinatorial-logical thinking, the formation of a sustainable interest in mathematics.

Objectives of the elective course system:

expand the scope of students’ knowledge in the fields of mathematics, logic, and combinatorics;
to develop in students the skills of making final choices when searching for solutions to both mathematical problems and “life” ones, which help to make the right choice, including the choice of an individual trajectory of professional growth;
develop skills in the variability of logical reasoning;
to form students’ ideas about scientific and logical methods for solving mathematical problems;
develop skills in collective decisions, public speaking, and project activities.

Structure of the system of elective courses.

In our opinion, 17 hours each should be allocated to studying the system of courses on the formation and development of combinatorial-logical thinking, which will allow a combinatorial approach to their implementation. Depending on the preparedness of students, it will be possible to vary the options for choosing courses. In addition, we propose to implement a propaedeutic course “Logical methods of proof” (17 hours) at the stage of pre-professional training of students, which will allow students to gain initial skills in constructing logical reasoning.

In the system of elective courses we presented, it is advisable to distribute the proposed number of hours as follows:

“Mathematics of Reasoning”, 17 hours:

entrance testing (1 hour);
pedagogical workshop for building knowledge “Oh, how many wonderful discoveries we have…” (motivational stage, 2 hours);
logical exercises based on mathematics (6 hours);
educational project “Tree of solutions for mathematical problems” (5 hours);
solving mathematical problems using various solution methods (2 hours);
selection of individual projects within the framework of the elective course topic (1 hour);

“Four typical problems of combinatorial-logical thinking”, 17 hours.

When developing new content, in the intertwining of logic and combinatorics, we propose to consider four options for educational tasks:

logical problems that involve several possible solutions. Finding ways to solve and developing similar problems at this stage for the student will be the leading learning activity (2 hours);
combinatorial problems of practical orientation (combinatorial plot problems), considering situations of choice that the student will encounter in the near future (4 hours);
tasks of combinatorial-logical content, for the solution of which it will be necessary to go through all stages of the creative process (V.P. Zinchenko) (2 hours);
problems of mathematical content, the solution of which uses combinatorial and logical methods of solution (6 hours);
selection of individual projects within the framework of the elective course topic (1 hour).

Note: it is advisable to start studying this elective course with a motivational pedagogical workshop “Finding an approach to solving a problem (the art of asking questions)”, 2 hours.

“Basic methods for solving mathematical problems”, 17-hour elective course:

pedagogical workshop “Wanderings: searching for an approach”, 2 hours;
general scientific methods for solving mathematical problems (8 hours):

Analysis in its various forms (ascending, descending, analysis in the form of dissection);

Analogy;

Generalization;

Specification;

logical methods for solving mathematical problems (4 hours):

Induction (complete and incomplete);

Deduction (direct and indirect evidence, in the latter case - methods of proof by contradiction, alternative indirect evidence, reduction to absurdity).

educational project “Combinatorial methods for solving problems” (2 hours);
final testing, summing up (1 hour).

Let's consider one example of the tasks of the typology we presented:

Problems of combinatorial-logical content

To solve this type of problem, it will be necessary to go through all stages of the creative process (V.P. Zinchenko).

Task No. 1

5 students take the swimming test. The test is passed if the student swims 100 meters (any time). If the student has to be caught, then the test is not passed. In how many ways can the swim end?

A. Emergence of a theme.

The teacher offers students the text of the problem.

B. Perception of the topic, analysis of the situation, awareness of the problem.

At this stage, students independently or with the help of a teacher identify the conditions of the problem, its conclusion, and carry out reasoning to find a solution.

B. At this stage, often painful work is carried out to solve the problem. There is a feeling that the problem is in me, and I am in the problem...

At this stage, students, working in groups, develop strategic ways to solve a given problem.

There is a discussion of the solution options developed by each group and a more rational solution is selected.

D. The executive, essentially technical stage.”

Formulation of the solution to the problem.

Let's introduce designations for 5 students based on the first letter of their fictitious names.

And we will consider various options for the success or failure of the swim for each of them in the form of a table. “1” will denote a successful swim, “0” will indicate an unsuccessful one.

When solving, we will use the brute force method already known to us.

A shorter solution is also possible, since the problem ultimately came down to considering the following situation: how many sequences of length 5 can be made from the numbers 0 and 1? The problem can be solved by using the product rule, since at each place in the sequence we have a choice of two possibilities. Thus, the total number of outcomes is

After solving this problem, students are asked to compose the text of similar problems with a different number of elements.

Having considered similar problems, students come to the conclusion that “If a set N contains n elements, then it has subsets.”

Note: the teacher clarifies that the generalized form of such a formula (for n-elements) requires proof. And for this there is a special method of proof - the method of mathematical induction.

When implementing any elective course, an important role is played not only by the modified content, but also by the technology of implementation. At one of the main stages - motivational, we will use one of the innovative pedagogical technologies, the basis of which is dialogue - “Technology of pedagogical workshops” .

The organization of collective creative activity in the workshop has its own patterns, its own algorithm, which allows you to consistently move towards the goal.

We draw your attention to the fact that the workshop, as one of the dialogue technologies, requires constant discussion of a particular situation, a problem proposed or independently identified, and therefore requires the mandatory use of a group form of work. Groups can be formed either chaotically or according to the algorithm provided in the workshop scenario. For example, students enter the classroom and draw chips of different colors from a bag, and groups are formed according to the chosen color.

Algorithm for building a workshop”:

Inductor– “guidance” on a topic (key words or phrases, a photograph or a set of photographs, an object, music, illustration, model, etc.).
Self-construction- a simple, accessible task. Each group member must complete a task that is feasible for himself: draw, write, draw, sculpt, come up with a script, etc.
(individual activity, not discussed with other group members). Socioconstruction
– comparison of one’s experience with the experience of another (In pairs, in groups). Socialization
– the whole group of workshop participants discuss, reflect, develop a mini-project, a small performance, etc. Advertising
– presentation of the results of the group’s activities. Discussion.
A prerequisite is that you cannot evaluate the ideas of others. The slogan of this stage and the entire workshop: “Every point of view has the right to exist, no matter how paradoxical and unsuccessful it may be.”

Reflection.

The task of the master (workshop organizer) is to explain, send to reference literature, give an additional “portion” of material, etc.

At the stages of studying new material and developing knowledge, the most important place will be given to project technology, or as the often described method of projects.

Born from the idea of free education, the project method is currently becoming an integrated component of the education system.

The essence remains the same - to stimulate children’s interest in certain problems that require possession of a certain amount of knowledge and, through project activities, to show the practical application of the acquired knowledge.

The project method is based on the development of students’ cognitive skills, the ability to independently construct their knowledge and navigate the information space, and the development of critical thinking.

The project method is always focused on students’ independent activities - individual, pair, group, which are carried out over a certain period of time.

In the experimental work, three stages were defined: ascertaining, formative, generalizing.

At the ascertaining stage, research was carried out to determine the level of combinatorial logical thinking, philosophical, psychological, methodological, and special literature on the study of the issue under consideration was studied. In addition, more than 30 abstracts and dissertations were reviewed and analyzed, which presented the latest discoveries on this issue.

The objectives of the ascertaining stage were:

studying philosophical, psychological, methodological, special literature on the research problem;
research into the organization and methodological support of the educational process aimed at the formation of combinatorial-logical thinking;
determination of the level of development of combinatorial-logical thinking of students.

At the second, formative stage, the didactic model of the development of combinatorial-logical thinking was tested against the background of specially created pedagogical conditions.

Tasks of the formative stage:

methodically ensure the development of combinatorial-logical thinking through the implementation of elective courses based on specially selected technologies and techniques that maximally contribute to solving the problem posed;
experimentally test the selection of pedagogical conditions that promote the formation of combinatorial-logical thinking;
experimentally confirm the effectiveness of the influence of the developed electives on the development of combinatorial-logical thinking of students;
experimentally confirm the influence of the developed pedagogical conditions on the formation of combinatorial-logical thinking of students;

The third stage is generalizing. At this stage, the results of the previous stages are summed up. Theoretical and practical conclusions were carried out, and the research results were introduced into the practice of the secondary school. Methods of observation and mathematical statistics were used.

Main conclusions

1. General indicators of the development of combinatorial-logical thinking of high school students are uneven; they reflect the features of the individual development of each child and the choice of a major.

The ability for combinatorial and logical reasoning is clearly expressed in students inclined to the exact sciences.

More than half of high school students, and in classes of physics, mathematics and information technology, more than 70% demonstrate the normatively expected level.

2. A necessary condition for the formation and development of combinatorial-logical thinking is the system of elective courses we have developed.

3. To successfully master the skills of combinatorial-logical thinking, we have proposed a special system of tasks, a system of lessons, and developed methodological recommendations for teachers.

4. The positive impact on the overall development of a high school student of the proposed methodology for the formation of combinatorial-logical thinking has been experimentally proven: R. Amthauer’s intelligence test, J. Guilford’s tasks for assessing divergent thinking.

5. Thanks to the mastery of combinatorial-logical actions, students freely carried out the transfer of various intellectual, practical, “life” tasks into similar and even non-standard situations.

Literature

Galperin P.Ya. Introduction to Psychology, Moscow University Publishing House, 1976.
Gusev V.A. Psychological and pedagogical foundations of teaching mathematics - M.: LLC Publishing House “Verbum-M”, LLC Publishing Center “Academy”, 2003.
Davydov V.V. Problems of developmental education: experience of theoretical and experimental research M., Pedagogy, 1986, p. 111.
Zinchenko V.P. Psychological foundations of pedagogy (Psychological and pedagogical foundations for building a developmental education system by D.B. Elkonina - V.V. Davydov): Textbook. Benefit. - M.: Gardariki, 2002.- 431 p., p. 110-111).
The concept of specialized training at the senior level of general education. Approved by Order of the Minister of Education No. 2783 of July 18, 2002, Moscow 2002.
Kuzmin O.V. Combinatorial methods for solving logical problems: textbook, M.: Drofa, 2006
Kuzmin O.V. Enumerative combinatorics: textbook. M.: Bustard, 2005
Okunev A.A. How to teach without teaching. - St. Petersburg: Peter Press, 1996.
Popova T.G. Pedagogical workshop in mathematics lessons. Collection of scientific works “Issues of teaching mathematics and computer science at school and university”, branch of ISPU, 2005, 5 pages.
Erdniev P.M., Erdniev B.P. Teaching mathematics at school/ Integrating didactic units. Book for teachers - 2nd edition. corr. and additional - M.: JSC “Stoletie”, 1996.

Thinking plays a truly enormous role in cognition. It expands the boundaries of knowledge, makes it possible to go beyond the immediate experience of sensations and perceptions, to know and judge what a person does not directly observe or perceive. It allows us to foresee the occurrence of phenomena that do not currently exist. Thinking processes information contained in sensations and perceptions, and the results of mental work are tested and applied in practice (8).

The difference between thinking and other psychological processes is also that it is almost always associated with the presence of a problem situation, a task that needs to be solved, and an active change in the conditions in which this task is given. Thinking, unlike perception, goes beyond the limits of the sensory data and expands the boundaries of knowledge. In thinking based on sensory information, certain theoretical and practical conclusions are made. It reflects existence not only in the form of individual things, phenomena and their properties, but also determines the connections that exist between them, which most often are not given directly to man in his very perception. The properties of things and phenomena, the connections between them are reflected in thinking in a generalized form, in the form of laws and entities.

In practice, thinking as a separate mental process does not exist; it is invisibly present in all other cognitive processes: perception, attention, imagination, memory, speech. The highest forms of these processes are necessarily associated with thinking, and the degree of its participation in these cognitive processes determines their level of development.

A specific result of thinking can be a concept - a generalized reflection of a class of objects in their most general and essential features (16).

1.1.2. Peculiarities of thinking of high school students

More complex content and methods of teaching high school students require from them a higher level of independence, activity, organization, and the ability to apply thinking techniques and operations in practice. Thinking becomes deeper, more complete, more versatile and more and more abstract; in the process of becoming familiar with new techniques of mental activity, old ones mastered at previous stages of training are modernized. Mastery of higher forms of thinking contributes to the development of the need for intellectual activity, ultimately leading to an understanding of the importance of theory and the desire to apply it in practice.

For older schoolchildren, the importance of the teaching itself, its tasks, goals, content and methods is important. A high school student first tries to understand the significance of a method of mental activity, and then master it, if it is truly significant. The motives for teaching also change, because they acquire important life meaning for a high school student.

Abstract thinking takes the leading role in the thinking of a high school student, but the role of concrete thinking is by no means diminished: acquiring a generalized meaning, concrete thinking appears in the form of technical images, diagrams, drawings, etc., it becomes the bearer of the general, and the general acts as an exponent of the concrete. Mastery of abstract and theoretical knowledge leads to a change in the very flow of the thought process in high school students. Their mental activity is distinguished by a high level of generalization and abstraction; students strive to establish cause-and-effect relationships and other patterns between the phenomena of the surrounding world, demonstrate critical thinking, the ability to argue judgments, and more successfully transfer knowledge and skills from one situation to another. In the course of mastering educational material, high school students strive to independently reveal the relationship between the general and the specific, highlight the essential, and then formulate definitions of scientific concepts.

All of the above speaks of a high degree of development of theoretical thinking, a multifaceted and deep manifestation of inner speech, and “proving” thinking. The thinking of boys and girls becomes dialectical: they not only realize the subject and content of mental activity and consider phenomena, events, processes in continuous movement, changes and transformations, but also begin to understand some of the patterns of their thinking, consciously use operations and thinking techniques and try to improve them in the process of educational activities.

However, some studies also note shortcomings in the thinking of high school students. Thus, a considerable number of them show a tendency to unfounded reasoning, speculative philosophizing, operating with abstract concepts in isolation from their real content, and putting forward original ideas arising from vague associations or fantastic inventions and conjectures. There are often cases when the essential is assessed as less significant than the unimportant, the transfer of knowledge is not always carried out correctly or widely, there is poor development of speech, and a tendency to an uncritical attitude towards the acquired knowledge. There are well-performing students who exaggerate their mental abilities and therefore become complacent. But all this, as the authors usually point out, concerns only a minority of high school students or their individual representatives, while the majority reaches a fairly high level of development of mental abilities and is well prepared for further educational and cognitive activities (21).

1.1.3. Definition of learning activities

Activity can be defined as a specific type of human activity aimed at knowledge and creative transformation of the surrounding world, including oneself and the conditions of one’s existence. In activity, a person creates objects of material and spiritual culture, transforms his abilities, preserves and improves nature, builds society, creates something that would not exist in nature without his activity (16).

People's activities are diverse, but at the same time they can be reduced to three main types: educational, work and play.

Educational activity is a process as a result of which a person acquires new or changes his existing knowledge, skills and abilities, improves and develops his abilities. Such activity allows him to adapt to the world around him, navigate in it, and more successfully and more fully satisfy his basic needs for intellectual growth and personal development (17).

Studying is an activity aimed at acquiring knowledge, skills and abilities necessary for broad education and subsequent work. The student's educational activities are carried out under the guidance of the teacher. The student actively acquires knowledge and actively acquires skills. Assimilation of knowledge is a manifestation of the student’s active mental work. Mastering the material requires the indispensable ability to analyze it, compare, generalize, highlight the main, essential, find similarities and differences. Knowledge acquisition is associated with the application of knowledge in practice. A student's knowledge is considered acquired only when he knows how to apply it in practice.

When we start talking about any pedagogical element, a logical question arises: is it necessary to introduce it throughout the entire course of study at once, or is it worth defining the framework within which it will work? In this section we will try to show that the psychological characteristics of senior school age children allow us to easily teach them the basics of mathematical logic.

First of all, let's try to understand what logical thinking is. Modern Russian psychologist V.P. Zinchenko wrote that “the classification of types of thinking still remains quite vague due to the fact that there are no clear boundaries between them and in fact there is only a living process of thinking, in which all its varieties are represented in different parts.” According to the classification he presented, thinking is divided into: concrete-figurative thinking, a type of which is visual; verbal intelligence or verbal-logical, discursive thinking; sign-symbolic And mythological thinking.

In this work, we consider verbal-logical thinking, which is otherwise simply called logical.

At different times, various aspects of the mental activity of schoolchildren have been studied.

This was done by researchers such as, for example, S.L. Rubinstein (1946), P.P. Blonsky (1979), Ya.A. Ponomarev (1967), Yu.A. Samarin (1962), M.N. Shardakov (1963). Abroad, the same question was raised by J. Piaget (1969), G.A. Austin (1956), M.I. Goldschmid (1976), K.W. Fischer (1980), R.J. Sternberg (1982). One of the researchers, Russian psychologist R.S. Nemov wrote that thinking, unlike other processes, occurs in accordance with a certain logic. Thus, in the structure of thinking, he identified the following logical operations:, comparison, analysis, synthesis And abstraction.

In addition to these types and operations, R.S. Nemov also highlighted the processes of thinking. He referred to them judgment, inference, definition of concepts, induction, deduction. Judgment- is a statement containing a certain thought. Inference is a series of logically related statements from which new knowledge is derived. Definition of concepts is considered as a system of judgments about a certain class of objects (phenomena), highlighting their most general characteristics. Induction and deduction are methods of producing inferences that reflect the direction of thought from the particular to the general or vice versa. Induction involves the derivation of a particular judgment from a general one, and deduction- derivation of a general judgment from particular ones.

In general, the issue of psychological characteristics of the development of a child’s thinking has been examined by many researchers. Soviet sociologist I.S. Cohn wrote, following the famous foreign researcher J. Piaget, that during adolescence, the ability to abstract mental operations from the objects on which these operations are performed matures in a teenager. The tendency to theorize becomes, to some extent, an age-related feature. The general decisively prevails over the particular. Another feature of the youthful psyche according to I.S. Konu is a change in the relationship between the categories of possibility and reality. A child thinks first of all about reality; for a young man, the category of possibility comes to the fore. Logical thinking operates not only with real, but also with imaginary objects; mastering this style of thinking inevitably gives rise to intellectual experimentation, a kind of game with concepts, formulas, etc. Hence the peculiar egocentrism of youthful thinking: by assimilating the entire world around him into his universal theories, the young man leads behave as if the world had to obey systems, and not systems - reality.

R.S. Nemov confirmed this hypothesis, considering the most important intellectual acquisition of late adolescence to be the ability to operate with hypotheses. He wrote that by high school age, students acquire many scientific concepts and learn to use them in the process of solving various problems. This means that they have developed theoretical or verbal-logical thinking.

R.S. Nemov also argued that a person masters logical processes and operations as they grow older. In the lower grades, many of the thinking processes are still inaccessible to the child, while in the older grades the development of cognitive processes reaches such a level that high school students are practically ready to perform all types of mental work of an adult, including the most complex. The cognitive processes of high school students acquire qualities that make them perfect and flexible, and the development of the means of cognition is somewhat ahead of the personal development of boys and girls.

In general, following a fairly detailed generalization by R.S. Nemov regarding the thinking of high school students, we can say that young men can already think logically, engage in theoretical reasoning and self-analysis. They have the ability to draw general conclusions based on particular premises and, on the contrary, move to particular conclusions based on general premises, i.e. the ability to induction And deduction.

Adolescence is different and increased intellectual activity that is stimulated not only by the natural age-related curiosity of adolescents, but also by the desire to develop, demonstrate to others their abilities, and receive high appreciation from them. In this regard, young men in public strive to take on the most difficult and prestigious tasks, often demonstrating not only highly developed intelligence, but also extraordinary abilities. They are characterized by an emotionally negative affective reaction to too simple tasks. Such tasks do not attract them, and they refuse to perform them for reasons of prestige. Behind all this one can see the natural interest and increased curiosity of students of this age. The questions that a high school student asks adult children, teachers and parents are often quite deep and go to the very core of things.

Young men can formulate hypotheses, reason speculatively, explore and compare different alternatives when solving the same problems. V.A. Krutetsky argued that this ability of senior schoolchildren means that their logical thinking is developed, which is a significant difference between senior students and middle and junior students.

Thus Based on the works of various researchers, we can say that it is the senior school age that is best suited for the development of logical thinking. First of all, this is due to the fact that at this age logical thinking has already been formed, and development as a process of improving skills and abilities is impossible without the formation of the fundamental principle. As noted in the paragraph, primary schoolchildren do not have the proper level of abstract thinking, therefore mathematical logic is not the best tool for instilling in them a logical culture. Only at high school age does a student begin to think on an equal footing with an adult, so teaching him logical constructions is completely justified.

Development of spatial thinking in high school students

Senior secondary school students have a poor understanding of figures in space, the location of straight lines and planes.

The ability to navigate in space plays a significant role in all areas of human activity. A person’s orientation in time and space is a necessary condition for his social existence, a form of reflection of the surrounding world, a condition for successful cognition and active transformation of reality.

Free handling of spatial images unites different types of educational and work activities and is one of the professionally important qualities, therefore secondary schools, vocational schools, universities, along with the formation of professional skills in students, set the task of developing spatial thinking in them.

Spatial thinking is an essential component in preparing for practical activities in many specialties.

The importance of spatial thinking in educational and professional activities.

INstructurein the general mental development of a person, a special place is occupied by imaginative thinking, which ensures the formation of generalized ideas about the surrounding world and its social values. The ability to create images and operate with them is a distinctive feature of human intelligence. It consists in the ability to arbitrarily update images based on given visual material, modify them under the influence of various conditions, freely transform and, on this basis, create new images that are significantly different from the original ones.

Spatial thinking as a type of figurative thinking plays an important role not only in mastering knowledge of the fundamentals of science, but also in many areas of work activity.

In the educational and work activities of schoolchildren, the formation of their thinking is significantly influenced by the use of various sign systems.This occurs when mastering the fundamentals of science, as well as when mastering technical knowledge, labor skills and abilities. The ability to create spatial images and operate with them largely determines success in artistic, graphic and constructive-technical activities, when it acts as an independent activity. Students develop a strong interest and inclination towards those types of activities where this ability is most fully realized.

1) In science and technology, graphic modeling is widely used, which is closely related to the mathematization and formalization of many areas of knowledge. There are two ways to use graphical modeling:

first - creation of a visual system in which the shape of selected signs or any other means of display resembles the displayed objects. However, in many cases, due to the diversity and differences in the content of specific objects, this turns out to be difficult to achieve;

second way - reflection of the properties of objects through conventional signs that do not in any way resemble the displayed objects, but make it possible to identify their most significant connections and dependencies hidden from direct observation.

Graphic modeling is widely used in mastering technical knowledge. Drawings, graphs, electrical diagrams, instruction cards are used to describe various technical objects and technological processes. Drawing is the language of technology. Being a visual image, it models various properties and relationships inherent in technical objects. Operating with images of technical objects is carried out, as a rule, based on spatial diagrams, which is the most important feature of technical thinking.

To operate in a technical way means not only to have an idea of a specific object that is in a static state in space, but also to see it in movement, change, interaction with other technical objects, i.e. in dynamics. Any graphic model is a planar image from which it is necessary to recreate the spatial position of a real technical object.

2) In many industries (instrument-making, electrical and radio engineering) the tendency to schematize and formalize images is noticeably increasing. When designing technological documentation, the idea is put forward of replacing descriptions of typical technological operations with conventional signs and designations, which makes it possible to create a unified system of graphic images in all technical and technological documentation.

3) In drawing There is a desire to combine the subject content of images with the widespread use of iconic models, which conditionally replace the subject of the image and have lost any visual analogy with it. More universal methods of depiction are being introduced, making it possible to indicate the structural features of objects hidden from direct observation, simplifying the methods of their depiction.

All of the above is reflected incontent and methods of learning school knowledge. When mastering knowledge in many academic subjects in modern schools, along with visual images of specific objects, conventional images in the form of spatial diagrams, graphs, diagrams, etc. are widely used.

Mastering modern scientific knowledge and successful work in many types of theoretical and practical activities are inextricably linked with operating with spatial images.

In the assimilation of knowledge, the role of graphic material has increased: the scope of its application has expanded significantly, its functions have changed significantly, and new means of visualization have been introduced. Many of the images used are not just an auxiliary, illustrative tool, but an independent source of acquiring new knowledge. Instead of various formulations, verbal explanations, and definitions, graphic models of the processes and phenomena being studied are widely used in the form of various spatial diagrams and mathematical expressions, which makes it possible to more accurately and economically describe the processes and phenomena being studied.

Thus, the verbal form of knowledge transfer has ceased to be universal. Along with it, a system of conventional symbols and signs, various spatial schemes, which are specific “linguistic” material, are widely used as an independent system.

Changes in the content of acquired knowledge are reflected inteaching methods.

Currently, the scope of application of this method of assimilation, in which the formation of a system of concepts occurs through the gradual generalization of specific individual facts, has been significantly narrowed. The most widely applicable is another way, when the basic patterns underlying the acquired material are first revealed, and then specific material is analyzed on their basis.

The psychological and pedagogical foundations of this path of assimilation were most fully developed by V.V. Davydov in his concept of meaningful abstraction and fruitfully developed in the works of his collaborators: L.I. Aidarova, A.K. Markova, G.G. Mikulina, L.M. Friedman and others. They proposed and experimentally developed a way of learning in which students first master the natural connections and relationships identified on the basis of theoretical analysis, and then explore their manifestation in specific situations of the reality they are studying. This significantly changes the principles of constructing educational material and developing exercises. With thisway of teaching The formation of generalizations is based not on the comparison of particular individual cases, but on the identification in the material to be learned of its original “cell” - general theoretical dependencies. These dependencies are clearly recorded by a unique spatial-functional model, which is a symbolic image.

In this paragraph, the importance of spatial thinking in various types of educational and professional activities was considered. Increasing the theoretical content of knowledge, using the modeling method and structural analysis in the study of phenomena of objective reality - all this leads to the fact that a person constantly creates spatial images in the process of activity, which characterizes spatial thinking.

The structure of spatial thinking

Spatial thinking is considered as a multi-level, hierarchical whole, multifunctional at its core.

Creation images andoperating they are closely interconnected processes. At the heart of each of them is the activity of presentation.

When creating any image, the visual basis on which the image arises is subject to mental transformation. When operating with an image, the image already created on this basis is mentally modified, often in conditions of complete abstraction from it.

Underspatial thinking This implies free manipulation of spatial images created on various visual bases, their transformation taking into account the requirements of the task.

Indicators of the development of spatial thinking:

The main indicator of the development of spatial thinking is takentype of image operation . In order for this indicator to be reliable, two more closely related indicators are used, namelybreadth of operation Andcompleteness of the image .

Type of surgery image there is a way available to the student to transform the created image.

The creation of images ensures the accumulation of ideas, which in relation to thinking are the initial basis, a necessary condition for its implementation. The richer and more varied the stock of spatial representations, the more advanced the methods for creating them, the easier the process of operating with them will be.

The whole variety of cases of operating with spatial images can be reduced to three main ones: leading to a change in the position of an imaginary object (type I), a change in its structure (type II) and a combination of these transformations (type III). Let us dwell on the description of each type of operation.

First type operating is characterized by the fact that the initial image, already created on a graphical visual basis, in the process of solving the problem is mentally modified in accordance with the conditions of the problem. These changes mainly concernspatial position and do not affect the structural features of the image. Typical cases of such operations are various mental rotations and movements of an already created image.

Second type operation is characterized by the fact that the original image under the influence of the task is transformed mainlyby structure . This is achieved through various transformations of the original image by mental regrouping of its constituent elements using various techniques of superposition, combination, addition, etc. With the second type of operation, the image changes so much that it becomes little similar to the original one. The degree of novelty of the created image in this case is much higher than that observed in the first type of operation, since the original image here undergoes a more radical transformation.

Third type operation is characterized by the fact that transformations of the original image are carried out for a long time and repeatedly. They represent a whole series of mental actions, successively replacing each other and aimed at transforming the original image simultaneously both in spatial position and in structure.

A comparative analysis of three types of operating with spatial images shows that the operation can be carried out in relation to different elements in the structure of the image: its shape, position, and their combinations.

The identified types of operating with spatial images and their accessibility to students are considered as one of the important and very

reliable indicators characterizing the level of development of spatial thinking.

As studies have shown, the type of surgery available to a student is sustainable. It manifests itself in the process of solving problems of various contents, when operating with different graphic images (visual, projection, conditionally symbolic), when choosing a method for solving a problem, etc.

In accordance with three types of surgery, there arethree levels development of spatial thinking (low, medium, high).

Breadth of operation is the degree of freedom to manipulate the image, taking into account the graphic basis on which the image was originally created.

The ease and speed of transition from one image to another, the number of exercises required, the nature and extent of assistance are indicators of the breadth of image manipulation.

The use of indicators such as breadth and type of image manipulation makes it possible to measure the level of development of spatial thinking in two different directions: longitudinal (horizontal) and transverse (vertical).

Operating in a spatial manner assumes that students mentally transform a given graphical visualization in three closely interrelated directions: shape, size, spatial position. The reflection of these signs in the image, mentally transformed, characterizes the completeness of the image.

Completeness of the image characterizes its structure, i.e. a set of elements, connections between them, their dynamic relationship. The image reflects not only the composition of the elements included in its structure (shape, size), but also their spatial arrangement (relative to a given plane or relative position of the elements).

The completeness of the image is an important indicator of the development of representational activity. That is why the type, breadth of operation and completeness of the image are accepted as the main indicators of the development of spatial thinking.

The ability to isolate spatial relationships and operate with them does not directly depend on the acquisition of knowledge.

In ontogenesis, sensory activity, on the basis of which spatial thinking is formed, has several stages. First, children learn to distinguish individual objects by their shape and size, and to carry out operations of comparison, generalization, and classification on this basis. By highlighting one or another spatial feature as a leading one, they generalize objects in accordance with the highlighted feature. So, for example, they distribute objects according to their geometric shape (round, square, rectangular, mixed, etc.), assessing the ratio of their sides and angles; make quantitative estimates of quantities, on the basis of which they form ideas: “more, less, different in size”; “higher, lower, different in height”; “longer-shorter-different in length”; “wider-already-different in width”; “thicker, thinner, different in thickness.” Often, the analysis of objects is carried out simultaneously according to a number of parameters, since their totality (combination) determines the qualitative originality of the object.

During ontogenesis, children continue to navigate in space for a very long time, distributing surrounding objects relative to the position of their own body.

Psychological research confirms that by the time children enter school they are ready to master geometric space. Moreover, the very nature of children's perception determines the possibility of arbitrarily changing observation positions.

During ontogenesis, spatial thinking develops in the depths of those forms of thinking that reflect the natural stages of general intellectual development. First, it is formed in the system of visual-effective thinking. Then, in the most developed and independent forms, it appears in the context of figurative thinking.

Tasks that form spatial thinking

The transition from planimetry to the study of stereometry causes great difficulties for students, and they are associated with the fact that this course lacks algorithms and with the fact that schoolchildren have undeveloped spatial concepts.

The tasks that should be used to form spatial concepts in schoolchildren should be of two types: a) tasks to create spatial images;

b) tasks for operating with spatial images.

1. The relative position of lines in space.

1) Straight And located in different half-planes And . How is the line located? relatively straight ?

2) How is the straight line located? relatively straight cubed ?

3) Plane And intersect in a straight line .Through point A of the plane and a point in the plane a direct line was drawn (points A, B do not lie on a straight line). How is the line located? relatively ?

2. Parallelism of a straight line and a plane.

1) A straight line is parallel to two given planes. What can be said about the relative position of these planes?

2) Two lines are parallel to the plane. Are they parallel to each other? Is there a line in the plane parallel to both given lines?

3) A straight line intersects two sides of a triangle. Does it intersect its plane?

3. Parallelism of planes.

1) Are there any unnecessary words in the formulation below: “If two intersecting lines of one plane are respectively parallel to two intersecting lines of another plane, then the planes are parallel”?

2) The height and base of the triangle are respectively parallel to the two sides of the rectangle: the planes of the figures do not coincide. How is the plane of the triangle located relative to the plane of the rectangle?

4. Perpendicularity of a straight line and a plane.

1) Line p is perpendicular to two sides of the triangle. Is it perpendicular to its height?

2) An infinite number of lines intersect a lineqat right angles. Do these lines belong to the same plane?

3) The straight line is not perpendicular to the plane. Is it inclined to this plane?

5. Other tasks:

1) Find the error:

ABC - line of intersection of two intersecting planes And .

2) The pictures show pyramids. DirectS.A.AndS.K.respectively perpendicular to the planes of their bases. Name:

a) the faces of the pyramid perpendicular to the plane of the base;

b) flat right angles.

3) Are they straightM.C.AndP.K.parallel in space?

4) It is useful to propose tasks for recognizing spatial objects in non-standard situations. So, for example: “Is there a quadrangular pyramid whose two opposite faces are perpendicular to the base of the pyramid?”

5) Development tasks. For example: from the proposed configurations, indicate which are the cube scans?

During lessons, it is advisable to look at different images of the same body. For example:

a) various images of a cube;

B) various images of the tetrahedron.

6) Complete the image of the cube:

These problems can be used in elective geometry classes at school.

We can conclude that the verbal form of knowledge transfer has ceased to be universal. Along with it, a system of conventional symbols and signs, various spatial schemes, which are specific “linguistic” material, are widely used as an independent system.

Literature:

1. Atanasyan L.S., Bazylev V.T. Geometry in 2 parts. Part 1. M.: Education, 1986.

2. Age and individual characteristics of students’ imaginative thinking / Ed. I.S. Yakimanskaya. M.: Education, 1989.

3. Dalinger V.A. Methods for developing spatial thinking in students when teaching geometry: a textbook. Omsk 1992.

4. Dalinger V.A. A drawing teaches you to think // Mathematics at school No. 4, 1990.

5. Kaplunovich I.Ya. Development of the structure of spatial thinking // Issue. Psycho. No. 1 1986

6. Mukhin Yu.N., Tolstopyatov V.P. Analytical stereometry: met. resolution Sverdlovsk 1991

7. Yakimanskaya I.S. Development of spatial thinking in schoolchildren. M.: Education, 1986.

First of all, thinking is the highest cognitive process. It represents the generation of new knowledge, an active form of creative reflection and transformation of reality by man. Thinking generates a result that does not exist either in reality itself or in the subject at a given moment in time. Thinking (in elementary forms it is also present in animals) can also be understood as the acquisition of new knowledge, the creative transformation of existing ideas.

Thinking is the movement of ideas that reveals the essence of things. Its result is not an image, but a certain thought, an idea. A specific result of thinking can be a concept - a generalized reflection of a class of objects in their most general and essential features.

Thinking is a special kind of theoretical and practical activity that involves a system of actions and operations included in it of an indicative, research, transformative and cognitive nature.

Let's consider the types of thinking:

Theoretical conceptual thinking is such thinking, using which a person, in the process of solving a problem, turns to concepts, performs actions in the mind, without directly dealing with the experience gained through the senses. He discusses and searches for a solution to a problem from beginning to end in his mind, using ready-made knowledge obtained by other people, expressed in conceptual form, judgments, and inferences. Theoretical conceptual thinking is characteristic of scientific theoretical research.

Theoretical figurative thinking differs from conceptual thinking in that the material that a person uses here to solve a problem is not concepts, judgments or inferences, but images. They are either directly retrieved from memory or creatively recreated by the imagination. This kind of thinking is used by workers in literature, art, and in general people of creative work who deal with images. In the course of solving mental problems, the corresponding images are mentally transformed so that a person, as a result of manipulating them, can directly see the solution to the problem that interests him.

Both types of thinking considered - theoretical conceptual and theoretical figurative - in reality, as a rule, coexist. They complement each other quite well, revealing to a person different but interconnected aspects of existence. Theoretical conceptual thinking provides, although abstract, but at the same time the most accurate, generalized reflection of reality. Theoretical figurative thinking allows us to obtain a specific subjective perception of it, which is no less real than the objective-conceptual one. Without one or another type of thinking, our perception of reality would not be as deep and versatile, accurate and rich in various shades as it actually is.

A distinctive feature of the following type of visual thinking is that the thought process in it is directly related to the thinking person’s perception of the surrounding reality and cannot be accomplished without it. Thoughts are visual and figurative, a person is tied to reality, and the images themselves necessary for thinking are presented in his short-term and operative memory (in contrast, images for theoretical figurative thinking are extracted from long-term memory and then transformed).

The last type of thinking is visual-effective. Its peculiarity lies in the fact that the thinking process itself is a practical transformative activity carried out by a person with real objects. The main condition for solving the problem in this case is the correct actions with the appropriate objects. This type of thinking is widely represented among people engaged in real production work, the result of which is the creation of a specific material product.

Let us note that the listed types of thinking also act as levels of its development. Theoretical thinking is considered more perfect than practical thinking, and conceptual thinking represents a higher level of development than figurative thinking.

Senior school age is characterized by the continuing development of children’s general and special abilities on the basis of the main leading activities: learning, communication and work. The study develops general intellectual abilities, especially conceptual theoretical thinking. This occurs through the assimilation of concepts, improving the ability to use them, and reason logically and abstractly. A significant increase in subject knowledge creates a good basis for the subsequent development of skills in those types of activities where this knowledge is practically necessary.

In adolescence and early adolescence, the formation of cognitive processes, and above all thinking, is completed. During these years, thought is finally combined with the word, as a result of which inner speech is formed as the main means of organizing thinking and regulating other cognitive processes. Intelligence in its highest manifestations becomes verbal, and speech becomes intellectualized. Full-fledged theoretical thinking arises. Along with this, there is an active process of forming scientific concepts that contain the foundations of a person’s scientific worldview within the framework of the sciences that are studied at school. Mental actions and operations with concepts, based on the logic of reasoning and distinguishing verbal-logical, abstract thinking from visual-effective and visual-figurative, acquire their final forms. Is it possible to speed up all these processes, and if so, how to do this?

It seems that from the point of view of the psychological and pedagogical development opportunities that middle and high school students have, from the standpoint of improving teaching and learning, this question should be answered in the affirmative. The intellectual development of children can be accelerated in three directions: conceptual structure of thinking, verbal intelligence and internal action plan. The development of thinking in high school can be facilitated by this type of activity, which is still, unfortunately, poorly represented in secondary schools, as rhetoric, understood as the ability to plan, compose and deliver public speeches, conduct a discussion, and skillfully answer questions. Various forms of written presentation of thoughts, used not only in language and literature classes (in the form of traditional presentation or essay), but also in other school subjects, can be of great benefit. They can well be used in mathematics classes, in particular in stereometry, when solving a construction problem at the stage of analyzing the problem conditions and at the stage of researching possible solutions. It is important to evaluate not only the content, but also the form of presentation of the material.

Accelerated formation of scientific concepts can be achieved in classes in special subjects, where relevant concepts are introduced and studied. When introducing any concept, including a scientific one, to a student, it is important to pay attention to the following points:

a) almost every concept, including scientific ones, has several meanings;

b) ordinary words from everyday language that are used to define scientific concepts are polysemantic and accurate enough to determine the scope and content of a non-scientific concept. Therefore, any definitions of concepts through words of ordinary language can only be approximate;

c) the noted properties allow, as a completely normal phenomenon, the existence of different definitions of the same concepts that completely coincide with each other, and this applies even to the most exact sciences, such as mathematics and physics. A scientist who uses the corresponding concepts is usually clear about what he is talking about, and therefore he does not always care that the definitions of all scientific concepts without exception are the same;

d) for the same person as he develops, as well as science and the scientists representing it as they penetrate into the essence of the phenomena being studied, the volume and content of concepts naturally change. When we pronounce the same words over a significant period of time, we usually give them slightly different meanings that change over time. It follows from this that in middle and high school students should not mechanically learn and repeat rigid definitions of scientific concepts. Rather, we should ensure that students themselves find and define these concepts. This will undoubtedly speed up the process of developing the conceptual structure of thinking in high school students. The formation of an internal plan of action can be helped by special exercises aimed at ensuring that the same actions are performed as often as possible not with real, but with imaginary objects, that is, in the mind. For example, in mathematics classes, students should be encouraged to count not on paper or using a calculator, but to themselves, to find and clearly formulate the principle and successive steps in solving a certain problem before they practically begin to implement the solution found. We must adhere to the rule: until the decision is fully thought through in the mind, until a plan for the actions included in it has been drawn up and until it has been verified for logic, one should not begin to implement the decision in practice. These principles and rules can be used in classes in all school subjects without exception, and then students will form an internal plan of action faster.

A characteristic feature of adolescence is the readiness and ability for many different types of learning, both in practical terms (labor skills) and theoretical ones (the ability to think, reason, use concepts). Another trait that is fully revealed for the first time precisely in adolescence is the tendency to experiment, which manifests itself, in particular, in a reluctance to take everything for granted. Teenagers exhibit broad cognitive interests associated with the desire to double-check everything on their own and personally verify the truth. By the beginning of adolescence, this desire decreases somewhat, and instead there appears more trust in other people's experience, based on a reasonable attitude towards its source.

Adolescence is characterized by increased intellectual activity, which is stimulated not only by the natural age-related curiosity of adolescents, but also by the desire to develop, demonstrate to others their abilities, and receive high appreciation from them. In this regard, teenagers in public strive to take on the most difficult and prestigious tasks, often demonstrating not only highly developed intelligence, but also extraordinary abilities. They are characterized by an emotionally negative affective reaction to too simple tasks. Such tasks do not attract them, and they refuse to perform them for reasons of prestige.

The increased intellectual and labor activity of adolescents is based not only on the above motives. Behind all this one can see the natural interest and increased curiosity of children of this age. The questions that a teenager asks adult children, teachers and parents are often quite deep and go to the very essence of things.

Teenagers can formulate hypotheses, reason speculatively, explore and compare different alternatives when solving the same problems. The sphere of cognitive, including educational, interests of adolescents goes beyond the boundaries of school and takes the form of cognitive initiative - the desire to search and acquire knowledge, to develop useful skills. Teens find activities and books that suit their interests and provide intellectual satisfaction. The desire for self-education is a characteristic feature of both adolescence and early adolescence.

The thinking of a teenager is characterized by the desire for broad generalizations. Independence of thinking is manifested in the independence of choosing a method of behavior. Teenagers and especially young men accept only what they personally think is reasonable, appropriate and useful.