Formation of logical thinking of junior schoolchildren. A set of tasks for developing the logical thinking of younger schoolchildren. Developing the logical thinking of schoolchildren means

Let us recall that the various cognitive processes that provide the child’s diverse activities do not function in isolation from each other, but represent a complex system, each of them is connected with all the others. This connection does not remain unchanged throughout childhood: at different periods, one of the processes acquires leading importance for general mental development.

Psychological research shows that during this period it is thinking that largely influences the development of all mental processes.

Depending on the extent to which the thought process is based on perception, idea or concept, three main types of thinking are distinguished:

1. Subject-effective (visual-effective).
2. Visual-figurative.
3. Abstract (verbal-logical).

Subject-active thinking is thinking associated with practical, direct actions with the subject; visual-figurative thinking - thinking that is based on perception or representation (typical for young children). Visual-figurative thinking makes it possible to solve problems in a directly given, visual field. The further path of development of thinking is the transition to verbal-logical thinking - this is thinking in terms of concepts devoid of direct clarity inherent in perception and representation. The transition to this new form of thinking is associated with a change in the content of thinking: now these are no longer specific ideas that have a visual basis and reflect the external characteristics of objects, but concepts that reflect the most essential properties of objects and phenomena and the relationships between them. This new content of thinking at primary school age is determined by the content of the leading educational activity.

Younger schoolchildren, as a result of studying at school, when it is necessary to regularly complete tasks without fail, learn to manage their thinking, to think when necessary. In many ways, the formation of such voluntary, controlled thinking is facilitated by the teacher’s assignments in class, which encourage children to think.

When communicating in primary school, children develop conscious critical thinking. This happens due to the fact that in the class, ways to solve problems are discussed, various solution options are considered, the teacher constantly asks students to justify, tell, and prove the correctness of their judgment. A junior schoolchild regularly logs into the system when he needs to reason, compare different judgments, and make inferences.

In the process of solving educational problems, children develop such operations of logical thinking as analysis, synthesis, comparison, generalization and classification. Let us recall that analysis as a mental action presupposes the decomposition of the whole into parts, the selection by comparison of the general and the particular, the distinction between the essential and the inessential in objects and phenomena.

Mastery of analysis begins with the child’s ability to identify various properties and characteristics in objects and phenomena. As you know, any subject can be viewed from different points of view. Depending on this, one or another feature or properties of the object come to the fore. The ability to identify properties is given to younger schoolchildren with great difficulty. And this is understandable, because the child’s concrete thinking must do the complex work of abstracting a property from an object. As a rule, out of an infinite number of properties of any object, first-graders can identify only two or three. As children develop, their horizons expand and they become familiar with various aspects of reality, this ability certainly improves. However, this does not exclude the need to specifically teach younger schoolchildren to see their different sides in objects and phenomena and to identify many properties.

In parallel with mastering the technique of isolating properties by comparing different objects (phenomena), it is necessary to derive the concept of general and distinctive (particular), essential and non-essential features, using such thinking operations as analysis, synthesis, comparison and generalization. The inability to identify the general and the essential can seriously hamper the learning process. In this case, typical material: subsuming a mathematical problem under an already known class, highlighting the root in related words, a brief (highlighting only the main) retelling of the text, dividing it into parts, choosing a title for a passage, etc. The ability to highlight the essential contributes to the formation of another skill - to be distracted from unimportant details. This action is given to younger schoolchildren with no less difficulty than highlighting the essential.

During the learning process, tasks become more complex: as a result of identifying the distinctive and common features of several objects, children try to divide them into groups. Here, such a thinking operation as classification is necessary. In elementary school, the need to classify is used in most lessons, both when introducing a new concept and at the consolidation stage. In the process of classification, children analyze the proposed situation, identify the most significant components in it, using the operations of analysis and synthesis, and make a generalization for each group of objects included in the class. As a result, objects are classified according to essential characteristics. As can be seen from the above facts, all operations of logical thinking are closely interconnected and their full formation is possible only in a complex. Only their interdependent development contributes to the development of logical thinking as a whole.

Techniques of logical analysis, synthesis, comparison, generalization and classification are necessary for students already in the 1st grade; without mastering them, the educational material cannot be fully mastered. These data show that it is precisely at primary school age that it is necessary to carry out targeted work to teach children the basic techniques of mental activity. A variety of psychological and pedagogical exercises can help with this.

Formation of logical thinking of junior schoolchildren

Shapochnikova Natalya Aleksandrovna, tutor at the Municipal Educational Institution “Gymnasium No. 18” in the city of Magnitogorsk.
This material will be useful to primary school teachers, primary school tutors, teachers of extended day groups in extracurricular activities, psychologists, and parents of primary schools.
Target: to form the logical thinking of younger schoolchildren.
The relevance of the problem of thinking development is explained by the fact that the success of any activity largely depends on the characteristics of thinking development. It is precisely at primary school age, as special studies show, that logical thinking should develop quite intensively. Thinking plays a huge role in cognition. It expands the boundaries of knowledge, makes it possible to go beyond the immediate experience of sensations and perception. Thinking makes it possible to know and judge what a person does not directly observe or perceive.
Since the subject of our research is the formation of logical thinking in younger schoolchildren, we will dwell in more detail on the characteristics of this term. But first, let's give a general definition of the concept of thinking.
So, thinking is a process of cognitive activity, characterized by a generalized and indirect reflection of reality, thanks to which a person reflects objects and phenomena in their essential characteristics and reveals their relationships.
And logical thinking is a type of thinking in which the reflection of objects and phenomena of the surrounding reality, their connections and relationships is carried out with the help of concepts and logical constructs. Logical thinking is a kind of thinking in which actions are mainly internal, carried out in speech form, and the material for them is concepts.
Human logical thinking is the most important moment in the process of cognition. All methods of logical thinking are inevitably used by the human individual in the process of understanding the surrounding reality, in everyday life. The ability to think logically allows a person to understand what is happening around him, to reveal significant aspects, connections in objects and phenomena, to draw conclusions, solve various problems, check these decisions, prove, refute, in a word, everything that is necessary for the life and successful activity of any person.
Let us dwell on the characteristics of the forms of thinking of children of primary school age. As you know, primary school age is an extremely important and rewarding period of learning. The possibilities inherent in it are associated with the development of cognitive abilities and the assimilation of intellectual aspects of activity.
When developing logical thinking, it is necessary to lead children to identify common essential features in different subjects. By generalizing them and abstracting from all secondary features, the child masters the concept. In such work, the most important is:
1) observations and selection of facts demonstrating the concept being formed;
2) analysis of each new phenomenon (object, fact) and identification of essential features in it that are repeated in all other objects classified in a certain category;
3) abstraction from all secondary features, for which objects with varying non-essential features are used while preserving the essential ones;
4) inclusion of new items in known groups, designated by familiar words.
Such complex mental work is not immediately possible for a child. He does this job, making a number of mistakes. Some of them can be considered characteristic. After all, to form a concept, a child must learn to generalize, relying on the commonality of essential features of different objects. But, firstly, he does not know this requirement, secondly, he does not know which features are essential, and thirdly, he does not know how to isolate them in the whole object, abstracting from all other features, often much more striking. In addition, the child must know the word denoting the concept.
Practice shows that by the time children enter fourth grade, they are usually freed from the influence of individual, often clearly given, features of an object and begin to indicate all possible features in a row, without distinguishing the essential and general ones from the particular ones. Thus, when explaining the concept of “wild animals,” many third-grade students, along with highlighting the main feature – lifestyle, also name such insignificant ones as “covered with fur,” “claws on their paws,” or “sharp teeth.” Analyzing the animals, most of the students in grades I and II classified the whale and dolphin as a group of fish, highlighting the habitat (water) and the nature of movement (swim) as the main and essential features.
As for the word, this only form of existence of a concept, the introduction of the corresponding terms showed not only the accessibility of their assimilation by children aged 7 - 10 years, but also their high efficiency.
Next, we will give a description of the mental operations of younger schoolchildren. It should be noted that the peculiarities of logical thinking of younger schoolchildren are clearly manifested both in the very course of the thought process and in each of its individual operations. Let's take an operation such as comparison. This is a mental action aimed at establishing similarities and differences in two (or more) comparing objects. The difficulty of comparison for a child is that, firstly, at first he does not know what “comparing” is, and secondly, he does not know how to use this operation as a method of solving the task assigned to him. The children's answers speak to this. Here, for example: “Is it possible to compare an apple and a ball?” “No, you can’t,” the child answers. “You can eat an apple, but a ball rolls, and another one flies if you let go of the thread.”
Another way to pose the question: “Take a good look at the orange and the apple and say: how are they similar?” - “They are both round, you can eat them.” “Now tell me: how are they different from each other? What is different about them? - “An orange has a thick peel, and an apple has a thin peel. An orange is red, but an apple is green, sometimes it’s red and the taste is not the same.”
This means we can lead children to the correct use of comparison. Without guidance, a child usually picks out any feature, most often some catchy one or one that is most familiar to him and, therefore, significant for him. Among the latter, the purpose of the object and its use by humans are most often indicated. To master the operation of comparison, a person must learn to see similarities in different things and different things in similar things. This will require a clearly targeted analysis of both (or three) objects being compared, a constant comparison of the distinguished features in order to find homogeneous and different ones. It is necessary to compare form with form, the purpose of an object with the same quality of another.
Research has shown that the thinking of younger schoolchildren is characterized by a feature - unilinear comparison, i.e. they establish either only differences, without seeing similarities, or only general and similar, without establishing differences. Mastering the comparison operation is of great importance in the mental activity of primary schoolchildren.
After all, most of the content learned in the lower grades is based on comparison. This operation underlies the classification of phenomena and their systematization. Without comparison, a child cannot acquire systematic knowledge.
Peculiarities of children's thinking often appear in children's judgments about the actions and goals of people they hear or read about. These same features are clearly revealed in guessing riddles, in explaining proverbs, and in other forms of working with verbal material that require logical thinking.
For example, children are given a riddle: “I know everything, I teach everyone, but I myself am always silent. To make friends with me, you need to learn to read and write” (Book).
Most children in grades I and II give a confident answer: “Teacher” (“She knows everyone, teaches everyone”). And although the text says: “But I myself am always silent,” this most important element, without being emphasized, is simply omitted. In this riddle, the accented element of the whole was the words “I teach everyone,” which immediately caused an erroneous answer.
The illogicality is “visible” in various judgments of children, and in many questions that they ask adults and each other, in disputes and evidence. For example: “Is the fish alive or not?” - “Alive.” "Why do you think so?" - “Because she swims and opens her mouth.” “And the log? It's alive! Why? After all, it also floats in water? - “Yes, but the log is made of wood.”

Here children do not distinguish between cause and effect or change their places. They use the words “because” not to designate causal dependencies, but to list facts side by side, to designate the whole.
The development of thinking in primary school age is largely associated with the improvement of mental operations: analysis and synthesis, comparison, generalization, systematization, classification, and with the assimilation of various mental actions. To create optimal conditions for the development of thinking, it is necessary to know these characteristics of the child. A number of scientists have identified psychological characteristics and conditions for the development of thinking in learning. The theory of developmental learning, developed by D. B. Elkonin and V. V. Davydov, has received the greatest fame and recognition not only in domestic but also in world science.
D. B. Elkonin and V. V. Davydov not only declared the need for logic and change in connection with this method and technique of teaching, but also laid down its principles in the structure of educational subjects and their content. Naturally, they made logical thinking a key link in the chain of mental development of schoolchildren.
Our gymnasium works according to the developmental education program of D. B. Elkonin and V. V. Davydov. In our work we adhere to the main goal and principles of developmental education.
Let us recall that the main goal of developmental education by D. B. Elkonin and V. V. Davydov is to provide optimal conditions for the development of a child as a subject of educational activity, interested in self-change and capable of it, the formation of mechanisms that allow children to set themselves the next task and find means and methods for solving it.
In my work, I use the following principles of developmental education by D. B. Elkonin and V. V. Davydov:
1. Search principle. In work, knowledge is not given ready-made. Finding a way to solve a new problem is the basis of the desire and ability to learn.
2. The principle of setting the problem. The need to find a way to solve a new problem is not dictated by the requirements of the teacher. When children discover that a problem cannot be solved using the methods they already know, they themselves declare the need to find new ways of acting. (Solving puzzles)
3. Modeling principle. The universal attitude that children discover when transforming the object of study does not have sensory clarity. It needs a model method of representation. The model, acting as a product of mental analysis, can then itself become a means of human mental activity.
4. The principle of correspondence between content and form. In order for children to be able to discover a new way of action through search activities, special forms of organizing the joint activities of children and the teacher are necessary. The basis of this organization is a general discussion in which each proposal made is evaluated by the other participants. Children participate in the development of control and evaluation criteria along with the teacher. Thanks to this, they develop the ability to self-control and self-esteem.
In the process of developing the logical thinking of children aged 7-10 years, perhaps the most important thing is to teach children to make, albeit small, but their own discoveries, which as a result contributes to their development and strengthening of formal logical connections. For this purpose, I have developed a series of classes united by a common idea - solving logical problems. The most typical tasks are solving anagrams, puzzles, identifying common features and identifying unnecessary objects in the proposed series, words, etc., that do not correspond to the found pattern; classification according to one or more characteristics, etc. Let us note the main features of our approach:
1. Fairytale-game nature of the tasks. The tests that are offered to the child must correspond to his spirit, be interesting and exciting. The series of developed activities represents a journey through the Magic Land of “Rebus Mania”, “Match Carousel”.
2. Consistent complication of the nature of completing tasks from lesson to lesson, while the formulation of the tasks may remain the same. For example,
Another option for complicating tasks is to increase the number of features characterizing the objects under consideration. For example, the pattern of placing objects can be based only on color, but performing a more complex task involves taking into account not only color, but also shape, size, etc.
3. Lack of strictly fixed time for completing tasks. The main goal of the proposed tasks is not to state a certain level of thinking skills, but to develop logical thinking, provide opportunities for finding new ways to solve problems, and children's discoveries.
4. The active role of the child in the process of completing tasks. He should not just choose the desired figure from those proposed, but try to draw it, paint it in the desired color, identifying a pattern. During the decision process, the teacher should no longer give any hints. All the necessary accents are placed by him at the stage of setting the task. By being observant, students can determine the solution key themselves.
5. Collective analysis of task completion. At the end of the lesson, you should have a reserve of time (10-15 minutes) so that schoolchildren can talk about their “discoveries”, while success is psychologically consolidated, which is especially important for children 7-10 years old. In the process of collective analysis, schoolchildren learn to control the correctness of assignments, compare their reasoning and results with the results of a friend, and evaluate the answer of another student. When summing up, it is important to communicate not only the finished result, but also the method for obtaining it. Children learn to justify their answer, highlight what is essential in a task, and draw conclusions. It is very important for the teacher to organize the discussion in such a way as to bring children’s thought processes out into the open, using them to show the nature of the emergence of guesses.
It is useful to discuss different approaches to completing tasks and compare them. Collective discussion allows you to take into account answers that were not initially provided by the teacher. If the child has logically substantiated his result, then it must be considered correct. For example, when solving the anagram ETLO, the possible answers are SUMMER and BODY.
The idea of a collective discussion of not only a ready-made solution, but also a search for a solution was implemented during the testing process in the final lesson, where the most difficult tasks were proposed. It took place in the form of a “Tournament of Thinkers”, a meeting of the “Club of Intellectuals”, where two teams competed. Children solved problems within their group, with opponents receiving the same tasks. The solution to each task was submitted to the jury, after which it had to be argued. The teams did this in turns, and the opponents could ask questions to clarify the decision, or point out an error.
We tested the students in our class as follows: the experiment began when the children were in second grade, and the end of the experiment occurred when the children completed fourth grade. The work was carried out with each individual, and based on these results, general trends were derived. The experiment was carried out over three years from 2013 to 2015. At the final stage of the experiment, we conducted final testing.
As a result of an experimental study of the problem of interest to us, we obtained the data presented in Table 1.
Table 1
Quantitative composition of students by level of mastery of logical operations of thinking at the beginning of the experiment

table 2
2 "A" classes at the beginning of the experiment

Analysis of the data shows that 35% of students have the ability to identify the essential at an above-average level, 57% at an average level, and 8% at a below-average level. Such a logical operation as comparing objects and concepts is proficient at an above-average level by 13% of students, at an average level by 61%, at a below-average level by 18%, and at a low level by 8% of the students surveyed. 35% of students can analyze relationships and concepts at an above-average level and 65% at an average level. The operation “generalization” is proficient by 27% of students at a high level, 30% - at an above-average level, 27% of students at an average level, 8% - at a below-average level, 8% - at a low level. 20 people (87%) are proficient in theoretical analysis, 3 people (13%) are not proficient.
Analysis of the data shows that the average indicators of the development of logical thinking of students in grade 2 “a” at the beginning of the experiment are as follows: 9% of students have a high level of development of logical thinking, above average - 26%, average level - 52%, below average - 9%, low - 4%.
In this regard, to develop students’ ability to identify what is essential, we conducted the following games and exercises: “What is the main thing?”, “What cannot exist without?”
To develop the comparison operation among students, the following games and exercises were used: “Compare the object”, “How are they similar, how are they different?”
To develop the generalization operation, the following games and exercises were carried out: “Name what is common between...”, “What is superfluous?”, “Name the common features.”
To consolidate the ability to analyze concepts, the following exercises were used: “Complete the definition”, “Fill in the blanks”, “Choose a concept”.
To develop logical thinking and maintain interest in classes, in addition to the above-mentioned exercises and games, students were offered non-traditional tasks, exercises, and logical problems: for example, “Encrypted Word”, “Attention - Guess”, puzzles, charades, crosswords. Classes were held for the “Thinkers” circle, the “Lucky Chance” quiz and “Tournament of Thinkers” were held, where non-traditional tasks were used.
As for the results of determining the levels of mastery of logical operations of thinking at the end of the experiment, they are presented in Table 3.
Table 3
Quantitative composition of students by level of mastery of logical operations of thinking at the end of the experiment

Table 4
Average indicators of development of logical thinking of students
4 “A” grades at the end of the experiment

Table 5
Average indicators of development of logical thinking of students
at the beginning and end of the experiment

Analysis of the data at the end of the experiment shows that 17% of students have the ability to identify the essential at a high level, 43% of students have it at an above-average level, and 40% have it at an average level. Such a logical operation as comparing objects and concepts is proficient at a high level by 4% of students, at an above-average level by 57% of students, at an average level by 35%, and at a low level by 4% of the students surveyed. 22% of students can analyze relationships and concepts at a high level, 51% can analyze relationships and concepts at an above-average level, and 27% of students can analyze them at an average level. The “generalization” operation is performed by 27% of students at a high level, 47% at an above-average level, 22% of students at an average level, and 4% at a low level. 20 people (87%) are proficient in theoretical analysis, 3 people (13%) are not proficient.
Analysis of the data shows that the average indicators of the development of logical thinking of students in grade 4 “A” at the end of the experiment are as follows: 18% of students have a high level of development of logical thinking, above average - 48%, average level - 30%, below average - 0%, low - 4%.
Having analyzed the data obtained at the end of the experiment, we concluded that the number of students with a high level of development of logical thinking increased from 9% to 18%, students with an above average level increased from 26% to 48%, students with an average level decreased from 52% to 30%, there were no students with a level below average, students with a low level of development of logical thinking remained at the same level of 4%. It was found that children of primary school age, mastering the material, are able to master knowledge that reflects the natural, essential relationships of objects and phenomena; skills that allow one to independently obtain such knowledge and use it in solving a variety of specific problems, and skills that are manifested in the wide transfer of mastered actions to various practical situations. It was established, therefore, that with the acquisition of knowledge, skills and abilities of the noted nature, already at primary school age, children form the foundations of logical thinking.
Well-developed logical thinking of students allows them to apply acquired knowledge in new conditions, solve atypical problems, find rational ways to solve them, take a creative approach to any activity, and actively and with interest participate in their own learning process.
The problem of developing a child’s logical thinking is one of the most important tasks, the solution of which determines the improvement of the entire educational process of the school, aimed at the formation of productive thinking, internal needs and the ability to independently acquire knowledge, the ability to apply the existing knowledge in practice, in creative transformation reality.
The research we conducted and the results obtained during diagnostics prove the need for the formation of logical thinking in younger schoolchildren. Determining the prospects for the research, we note that the work performed does not pretend to be an exhaustive development of the problem of developing logical thinking in primary schoolchildren. It seems relevant to further work with students on the formation of logical thinking.
In conclusion, I would like to hope that our experience will be of interest to primary school teachers and will give them an impetus for their own creativity and new experiments. The fairytale-playful nature of the material will allow it to be used not only for clubs at school, but can also serve as a good basis for family activities.

Introduction 3

Chapter I. Philosophical – psychological – pedagogical features of the development of thinking of younger schoolchildren

Thinking as a philosophical - psychological - pedagogical category 4

Features of logical thinking of junior schoolchildren 11

Word problems as a means of developing logical thinking 16

Chapter II. A set of tasks for the development of logical thinking in junior schoolchildren:

2.1. Problems - jokes, wit (simple) 21

2.2. Problems in verses, simple - compound 23

2.3. Historical problems 27

2.4. Puzzles, crosswords, charades 29

2.5. Geometry problems 32

Conclusion 33

References 35

Introduction

The social transformations taking place in Russia today have created certain conditions for perestroika processes in the field of education, including in first-level schools. Modern concepts of primary education are based on the priority of developing the student’s personality on the basis of leading activities. It was precisely this understanding of the goals of primary school that prompted the introduction of the term “developmental education” into didactics.

It cannot be said that the idea of developmental education is new, that previously the problems of child development in the learning process were not raised and solved.

Primary education at the present stage is not closed, but is considered as a link in the system of basic education, and it is the foundation on which the links of this system are built. In this regard, primary schools have a special responsibility.

The relevance lies in the fact that in modern times children learn using developmental technologies, where logical thinking is the basis. From the beginning of training, thinking moves to the center of mental development (L.S. Vygotsky) and becomes decisive in the system of other mental functions, which, under its influence, become intellectualized and acquire an arbitrary character. Numerous observations of teachers and research by psychologists have convincingly shown that a child who has not learned to study, who has not mastered the techniques of mental activity in the primary grades of school, usually goes into the category of underachievers in the middle grades.

The study of thinking and the process of mental development was carried out by such prominent scientists as G. Eysenck, F. Galton, J. Ketell, K. Meili, J. Piaget, C. Spearman and others. In domestic science, S.L. Rubinstein, L.S. Vygotsky, N.A. Podgoretskaya, P.P. Blonsky, A.V. Brushlinsky, V.V. Davydov, A. made their contribution to the study of this issue. V. Zaporozhets, G.S. Kostyuk, A.N. Leontyev and others.

One of the important directions in solving this problem is the creation in primary classes of conditions that ensure the full mental development of children, associated with the formation of stable cognitive processes, skills of mental activity, quality of mind, creative initiative and independence in the search for solutions tasks. However, such conditions are not yet fully provided in primary education, since a still common technique in teaching practice is the teacher’s organization of students’ actions according to a model: too often teachers offer children training-type exercises based on the content and not requiring manifestation of invention and initiative.

The formation of independence in thinking, activity in finding ways, and achieving a set goal involves children solving atypical, non-standard problems, which sometimes have several solutions, although correct, but to varying degrees optimal.

The above determined the topic of the study: “Development of logical thinking of younger schoolchildren when solving word problems in mathematics lessons.”

Object of study: educational activities of junior schoolchildren.

Subject of study: logical thinking of junior schoolchildren.

Purpose of the study: identify the development of students’ logical thinking in mathematics lessons.

To achieve the research goal, it is necessary to solve the following tasks:

To reveal the essence of logical thinking and the peculiarities of its formation in a primary school student;

Compose a set of tasks (tasks) to develop the logical thinking of a primary school student;

ChapterI. Philosophical – psychological – pedagogical feature of the development of thinking of younger schoolchildren

Thinking as a philosophical – psychological – pedagogical category

Information received by a person from the surrounding world allows a person to imagine objects in their absence, to foresee their changes over time, to rush with thought into unimaginable distances and micro-worlds. All this is possible thanks to the thinking process. In psychology, thinking is understood as the process of cognitive activity of an individual, characterized by a generalized and indirect reflection of reality. Thinking expands the boundaries of our knowledge due to its nature, which allows us to reveal indirectly - by inference - what is not given indirectly - by perception.

What is thinking in philosophy? There is a statement that a person is always thinking about something, even when it seems to him that he is not thinking about anything. A thoughtless state, as psychologists say, is a state that is essentially maximally relaxed, but still thinking, at least not thinking about anything. From sensory knowledge, from establishing facts, the dialectical path of knowledge leads to logical thinking. Thinking is a purposeful, indirect and generalized reflection by a person of the essential properties and relationships of things. Creative thinking is aimed at obtaining new results in practice, science, and technology. Thinking is an active process aimed at posing problems and solving them. Inquisitiveness is an essential sign of a thinking person. The transition from sensation to thought has its objective basis in the bifurcation of the object of knowledge into internal and external, essence and its manifestation, into separate and general.

The special structure of our sense organs and their small number do not set an absolute limit to our knowledge because they are joined by the activity of theoretical thinking. “The eye sees far, but the thought even further,” says a popular saying. Our thought, overcoming the appearance of phenomena, their external appearance, penetrates into the depths of the object, into its essence. Based on the data of sensory and empirical experience, thinking can actively correlate the readings of the senses with all the existing knowledge in the head of each individual, moreover, with all the total experience and knowledge of mankind, and to the extent that they have become the property of a given person, and solve practical and theoretical problems, penetrating through phenomena into the essence of an increasingly deeper order.

Logical - this means subordinate to rules, principles and laws, according to which thought moves towards truth, from one truth to another, deeper one. Rules, laws of thinking constitute the content of logic as a science. These rules and laws are not something immanently inherent in thinking itself. Logical laws are a generalized reflection of the objective relationships of things based on practice. The degree of perfection of human thinking is determined by the degree of correspondence of its content to the content of objective reality. Our mind is disciplined by the logic of things, reproduced in the logic of practical actions and all by the system of spiritual culture. The real process of thinking unfolds not only in the head of an individual, but also in the bosom of the entire history of culture. The logicalness of a thought with the reliability of the starting points is, to a certain extent, a guarantee not only of its correctness, but also of truth. This is the great power of logical thinking.

The first essential feature of thinking is that it is a process of indirect cognition of objects. This mediation can be very complex and multi-stage. Thinking is mediated, first of all, by the sensory form of cognition, often by the symbolic content of images, and by language. Based on the visible, audible and tangible, people penetrate into the unknown, inaudible and intangible. It is on such indirect knowledge that science is built.

What is the possibility of indirect cognition based on? The objective basis of the mediated process of cognition is the presence of indirect connections in the world. For example, cause-and-effect relationships make it possible to draw a conclusion about the cause based on the perception of the effect, and to foresee the effect based on knowledge of the cause. The indirect nature of thinking also lies in the fact that a person cognizes reality not only on the basis of his personal experience, but also takes into account the historically accumulated experience of all mankind.

In the process of thinking, a person draws into the flow of his thoughts threads from the fabric of the general stock of knowledge available in his head about a wide variety of things, from all the experience accumulated in life. And often the most incredible comparisons, analogies and associations can lead to the solution of an important practical and theoretical problem. Theorists can successfully extract scientific results about things they may never have seen.

In life, not only “theorists” think, but also practitioners. Practical thinking is aimed at solving specific specific problems, while theoretical thinking is aimed at finding general patterns, if theoretical thinking is focused primarily on the transition from sensation to thought, idea, theory, then practical thinking is aimed primarily at the implementation of thoughts, ideas, theories in life. Practical thinking is directly included in practice and is constantly subject to its controlling influence. Theoretical thinking is subject to practical testing not in every link, but only in the final results. The rational content of the thinking process is clothed in historically developed logical forms. The main forms in which thinking arose, develops and is carried out are concepts, judgments and inferences.

A concept is a thought that reflects the general, essential properties, connections of objects and phenomena. Concepts not only reflect the general, but also dissect things, group, classify them in accordance with their differences. Unlike sensation, perception and ideas, concepts are devoid of clarity or sensuality. A concept arises and exists in a person’s head only in a certain connection, in the form of judgments. To think means to judge something, to identify certain connections and relationships between various aspects of an object and between objects.

A judgment is a form of thought that, through the connection of concepts, confirms (or denies) something about something. Judgment exists where we find affirmation or negation, falsity or truth, as well as something conjectural.

Thinking is not just judgment. In the real process of thinking, concepts or judgments do not stand alone. They are included as links in a chain of more complex mental actions - in reasoning. A relatively complete unit of reasoning is inference. From existing judgments it forms a new conclusion. From existing judgments it forms a new one - a conclusion. It is the derivation of new judgments that is characteristic of inference as a logical operation. The propositions from which the conclusion is drawn are premises. Inference is a thinking operation during which a new judgment is derived from a comparison of a number of premises.

The discovery of relationships, connections between objects is an essential task of thinking: this determines the specific path of thinking to an ever deeper knowledge of existence.

The task of thinking is to identify significant, necessary connections based on real dependencies, separating them from random coincidences.

In the detailed process of thinking in the course of solving a complex problem that cannot be determined by an unambiguous algorithm, several main stages or phases can be distinguished. The beginning of the thought process is seen in the creation of a problem situation. Already this stage is not within the power of everyone - those who are not used to thinking take the world around them for granted. The more knowledge, the more problems a person sees. You need to have the mindset of I. Newton to see a problem in an apple falling to the ground. A problem situation, as a rule, contains a contradiction and does not have a clear solution.

The main mental operations are analysis, synthesis, comparison, abstraction, concretization, generalization.

Analysis- this is the mental decomposition of the whole into parts or the mental isolation of the whole of its sides, actions, relationships. In its elementary form, analysis is expressed in the practical decomposition of objects into their component parts.

Synthesis – This is the mental unification of parts, properties, actions into a single whole. The operation of synthesis is the opposite of analysis. In its process, the relationship of individual objects or phenomena as elements or parts to their complex whole, object or phenomenon is established. Synthesis is not a mechanical connection of parts and therefore cannot be reduced to their sum.

Comparison– establishing similarities or differences between objects and phenomena or their individual characteristics. In practice, comparison can be one-sided (incomplete according to one characteristic) and multilateral (complete, according to all characteristics); superficial and deep; unmediated and indirect.

Abstraction- consists in the fact that the subject, isolating any properties, signs of the object being studied, is distracted from the rest. Abstraction is usually carried out as a result of analysis. It was through abstraction that abstract, abstract concepts of length, latitude, quantity, equality, value, etc. were created. Abstraction is a complex process that depends on the uniqueness of the object being studied and the goals facing the study. Thanks to abstraction, a person can escape from the single, concrete.

Specification– involves the return of thought from the general and abstract to the specific in order to reveal the content. Concretization is turned to in the event that the expressed thought turns out to be incomprehensible or it is necessary to show the manifestation of the general in the individual.

Generalization– mental association of objects and phenomena according to their essential and common characteristics.

All of these operations cannot occur in isolation, without connection with each other. On their basis, more complex operations arise, such as classification, systematization, etc. Human thinking not only includes various operations, but also occurs in aggregate and allows us to talk about the existence of different types of thinking.

We can distinguish creative (productive), reproducing (reproductive), theoretical, practical, objective-effective, visual-figurative, verbal-logical thinking.

Creative thinking is aimed at creating new ideas; its result is the discovery of something new or the improvement of a solution to a particular problem.

It is necessary to distinguish between the creation of an objectively new thing, i.e., something that has not yet been created, and a subjectively new one for a given specific person.

Unlike creative thinking, reproductive thinking is the application of ready-made knowledge and skills.

The features of objectively effective thinking are manifested in the fact that problems are solved with the help of a real, physical transformation of the situation, testing the properties of objects. This form of thinking is most typical for children under 3 years of age.

Visual-figurative thinking is associated with operating with images. This type of thinking is spoken of when a person, solving a problem, analyzes, compares, generalizes various images, ideas about phenomena and objects. Visually-imaginative thinking most fully recreates the whole variety of various factual characteristics of an object. The image can simultaneously capture the vision of an object from several points of view. In this capacity, visual-imaginative thinking is practically inseparable from imagination.

Verbal-logical thinking functions on the basis of linguistic means and represents the latest stage in the historical and ontogenetic development of thinking. Verbal-logical thinking is characterized by the use of concepts and logical constructions that do not have a direct figurative expression (for example, cost).

It should be noted that all types of thinking are closely interconnected. Separate types of thinking constantly flow into each other. Thus, it is almost impossible to separate visual-figurative and verbal-logical thinking when the content of the task is diagrams and graphs. Practically effective thinking can be both intuitive and creative. Therefore, when trying to determine the type of thinking, you should remember that this process is always relative and conditional.

Thus, logical thinking is the ability to operate with abstract concepts, this is controlled thinking, this is thinking through reasoning, this is strict adherence to the laws of inexorable logic, this is the impeccable construction of cause and effect relationships.

Features of logical thinking of a junior schoolchild

By the beginning of primary school age, the child’s mental development reaches a fairly high level. All mental processes: perception, memory, thinking, imagination, speech - have already gone through quite a long path of development, since the child’s curiosity is constantly aimed at understanding the world around him and building the world around him. The child, while playing, experiments, tries to establish cause and effect relationships. He himself, for example, can find out which objects will sink and which will float.

Various cognitive processes that provide a variety of child activities do not function in isolation from each other, but represent a complex system, each of them is connected with all the others. This connection does not remain unchanged throughout childhood: at different periods, one of the processes acquires leading importance for overall mental development.

Depending on the extent to which the thought process is based on perception, idea or concept, three main types of thinking are distinguished:

1. Subject-effective (visual-effective).

2. Visual-figurative.

3. Abstract (verbal-logical).

Subject-active thinking is thinking associated with practical, direct actions with an object; visual-figurative thinking – thinking that is based on perception or representation (typical for young children). An example is the game “Postman”, used in a mathematics lesson: The game involves three students - postmen. Each of them needs to deliver a letter to three houses. Each house depicts one of the geometric figures. The postman's bag contains letters - 10 geometric shapes cut out of cardboard. At the teacher's signal, the postman looks for the letter and carries it to the appropriate house. The winner is the one who delivers all the letters to the houses faster - by arranging geometric shapes.

Visual-figurative thinking makes it possible to solve problems in a directly given, visual field. The further path of development of thinking is the transition to verbal-logical thinking - this is thinking in terms of concepts devoid of direct clarity inherent in perception and representation. The transition to this new form of thinking is associated with a change in the content of thinking: now these are no longer specific ideas that have a visual basis and reflect the external characteristics of objects, but concepts that reflect the most essential properties of objects and phenomena and the relationships between them. This new content of thinking at primary school age is determined by the content of the leading educational activity. For example, you can use tasks such as: make 2 squares out of 7 sticks; continue the pattern and others.

Verbal-logical, conceptual thinking is formed gradually throughout primary school age. At the beginning of this age period, visual-figurative thinking is dominant, therefore, if in the first two years of schooling children work a lot with visual examples, then in the following grades the volume of this type of activity is reduced. As the student masters educational activities and masters the fundamentals of scientific knowledge, he gradually becomes familiar with the system of scientific concepts, his mental operations become less connected with specific practical activities or visual support. Verbal-logical thinking allows the student to solve problems and draw conclusions, focusing not on visual signs of objects, but on internal, essential properties and relationships. During training, children master the techniques of mental activity, acquire the ability to act “in their minds” and analyze the process of their own reasoning. The child develops logically correct reasoning: when reasoning, he uses the operations of analysis, synthesis, comparison, classification, and generalization. When developing verbal-logical thinking through solving logical problems, it is necessary to select tasks that would require inductive (from individual to general), deductive (from general to individual) and traductive (from individual to individual or from general to general, when premises and conclusion are judgments of the same generality) inferences. Traductive reasoning can be used as the first stage of learning the ability to solve logical problems. These are tasks in which, based on the absence or presence of one of two possible features in one of the two objects under discussion, a conclusion follows about, respectively, the presence or absence of this feature in the other object. For example, “Natasha’s dog is small and fluffy, Ira’s is big and fluffy. What is the same about these dogs? What is different?”

Younger schoolchildren, as a result of studying at school, when it is necessary to regularly complete tasks without fail, learn to control their thinking, to think when necessary.

In many ways, the formation of such voluntary, controlled thinking is facilitated by the teacher’s assignments in the lesson, which encourage children to think.

When communicating in primary school, children develop conscious critical thinking. This happens due to the fact that in the class, ways to solve problems are discussed, various solution options are considered, the teacher constantly asks students to justify, tell, and prove the correctness of their judgment. A junior schoolchild regularly joins the system when he needs to reason, compare different judgments, and make inferences.

In the process of solving educational problems, children develop such operations of logical thinking as analysis, synthesis, comparison, generalization and classification.

Let us recall that analysis as a mental action presupposes the decomposition of the whole into parts, the selection by comparison of the general and the particular, the distinction between the essential and the inessential in objects and phenomena.

Mastery of analysis begins with the child’s ability to identify various properties and characteristics in objects and phenomena. As you know, any subject can be viewed from different points of view. Depending on this, one or another feature or properties of the object come to the fore. The ability to identify properties is given to younger schoolchildren with great difficulty. And this is understandable, because the child’s concrete thinking must do the complex work of abstracting a property from an object. As a rule, out of an infinite number of properties of any object, first-graders can single out only two or three. As children develop, their horizons expand and they become familiar with various aspects of reality, this ability certainly improves. However, this does not exclude the need to specifically teach younger schoolchildren to see their different sides in objects and phenomena and to identify many properties.

In parallel with mastering the technique of isolating properties by comparing different objects (phenomena), it is necessary to derive the concept of general and distinctive (particular), essential and non-essential features, using such thinking operations as analysis, synthesis, comparison and generalization. The inability to distinguish between the general and the essential can seriously hamper the learning process. In this case, typical material: subsuming a mathematical problem under an already known class. The ability to highlight the essential contributes to the formation of another skill - to be distracted from unimportant details. This action is given to younger schoolchildren with no less difficulty than highlighting the essential.

In the process of classification, children analyze the proposed situation, identify the most significant components in it, using the operations of analysis and synthesis, and make a generalization for each group of objects included in the class. As a result, objects are classified according to essential characteristics.

As can be seen from the above facts, all operations of logical thinking are closely interconnected and their full formation is possible only in a complex. Only their interdependent development contributes to the development of logical thinking as a whole. Techniques of logical analysis, synthesis, comparison, generalization and classification are necessary for students already in the 1st grade; without mastering them, the educational material cannot be fully mastered.

These data show that it is precisely at primary school age that it is necessary to carry out targeted work to teach children the basic techniques of mental activity.

Word problems as a means of developing logical thinking

The term “task” is one of the most common in science and educational practice in terms of frequency of use.

A cognitive task is the subject of research in many scientific fields, so the definition of this concept reflects the specifics of each of them.

In psychology, the term “task” is used to designate objects related to three different criteria: 1) the goal of the subject’s actions, the requirements set before the subject; 2) to a situation that includes, along with the goal, the conditions in which it must be achieved; 3) to the verbal formulation of this situation.

Some authors view the concept of “task” as indefinable and, in the broadest sense, meaning something that requires the execution of a decision. There are attempts to clarify the content of the task through the generic concept of “learning phenomenon” and specific differences: to be a way of organizing and managing educational and cognitive activities; a bearer of actions adequate to the content of training; a means of purposeful formation of knowledge, skills and abilities; act as a form of teaching methods; serve as a means of connecting theory with practice.

The latter interpretation covers the entire range of subject problems presented in textbooks, as well as those that can take their place in them. These are non-standard research tasks in their formulation.

The multiplicity of points of view on the content of the concept of “task”, their classification, and the priority of one or another type is due to the dynamics of changes in the role and place of tasks in student learning. The study of this phenomenon leads to the conclusion that the attitude to the tasks depended on the status of education, teaching methods, various pedagogical concepts, in particular the concepts of learning content, etc.

In the history of using tasks, the following stages can be distinguished:

theory is studied with the aim of learning to solve problems;

teaching the subject is accompanied by problem solving;

learning through problem solving;

problem solving as the basis of the educational process

The peculiarity of the first stage is clearly visible from the preface to “Arithmetic” by L.F. Magnitsky, where it was stated that mathematics should be “tested” to solve problems.

Today, methodologists are searching for didactic techniques, the use of which helps schoolchildren master the ability to apply knowledge to solve problems of a certain type

The second stage, at which teaching the subject is accompanied by problem solving, is due to the fact that the formation of skills to apply theoretical material is declared as one of the main goals of training. Mastering a theory comes down to memorizing it and reproducing it when solving problems. In the depths of this stage, the idea of expanding the functions of tasks arises. So, S.I. Shokhor-Troitsky, in his work “The Purpose and Means of Teaching Lower Mathematics from the Point of View of the Requirements of General Education,” noted that tasks should serve as a starting point for teaching, and not as a means of training students in a certain direction.

This view of the role of tasks constituted the content of the new (III) stage: teaching a subject by solving problems. These thoughts were reflected in official documents. Thus, the resolution of the International Congress of Mathematicians (1966 Moscow) emphasizes that problem solving is the most effective form of not only the development of mathematical activity, but also the acquisition of knowledge, skills, methods and applications of mathematics.

However, despite such documented claims, the role of tasks in learning is limited to using them as a means of developing and applying theory. This can be confirmed by the teaching scheme presented, for example, in the book “Pedagogy of Mathematics” by A.A. Stolyara: "Tasks - theory - tasks" (M., 1986)

In this scheme, the role of tasks in the assimilation of theory continues to be correlated with its memorization and reproduction. Knowledge is still identified with educational information.

Since the second half of the 20th century, publications have appeared that discuss advanced task functions. For example, K.I. Neshkov and A.D. Samushin distinguishes the following groups of tasks:

with didactic functions;

with cognitive functions;

with developmental functions.

The problems of the first group are intended for mastering theoretical material; in the process of solving problems of the second type, students deepen their knowledge of the theory and methods of solving them. The content of problems of the third type may “deviate” from the main course and complicate as much as possible some of the previously studied questions of the course. Of course, it is advisable to widely use tasks in teaching, but one cannot agree that developmental functions are inherent only in tasks, the content of which “deviates” from the compulsory course, expanding it.

Research into the function of tasks has contributed to understanding their role and place in learning. All scientists are unanimous that tasks serve both the acquisition of knowledge and skills and the formation of a certain style of thinking (logical thinking). It is already becoming clear that the formation of knowledge (concepts, judgments, theories) cannot be carried out outside of activity.

Research by educators has led to new thinking about the content of education. If previously the content was composed of subject knowledge, now, in addition to it, methods of activity are included in the form of various actions included in the content of learning through tasks. This is a completely new turn: from a means of developing skills, tasks begin to turn into a multifaceted learning phenomenon. They become the bearer of actions adequate to the content of training; a means of purposeful formation of knowledge, skills and abilities; the way of organizing and managing the educational and cognitive activities of students; one of the forms of implementation of teaching methods; a link between theory and practice.

Solving problems should ensure mastery of the following skills: recognize objects belonging to the concept; draw consequences from the belonging of an object to a concept, move from the definition of a concept to its characteristics; rethink objects in terms of different concepts, etc.

With the change in the role and place of tasks in learning, the content of the tasks itself is updated. If previously the requirement of a problem was expressed by the words: “find”, “construct”, “calculate”, “prove”, now - “explain”, “choose the most optimal from various methods of solution”, “predict various methods of solution”, “is it correct solution?", "explore".

Some scholars have attempted to define a criterion basis for selecting an aesthetically pleasing task.

For example, E.T. Bell, performing similar studies on a mathematical object, identifies the following signs of attractiveness:

universality of use in various branches of mathematics;

productivity or the possibility of stimulating influence on further advancement in a given field based on abstraction and generalization;

maximum coverage capacity of objects of the type in question.

That is, now is a new stage in the use of tasks, when they serve as the basis for the education, development and upbringing of students. We need tasks whose solution requires students to integrate knowledge from various educational fields.

In fact, everyday human activity consists of solving problems in all the diversity of their content.

In the course of theoretical foundations of mathematics and in teaching mathematics to younger schoolchildren, text and plot problems predominate. These tasks are formulated in natural language (that's why they are called text tasks); they usually describe the quantitative side of some phenomena or events (therefore they are often called plot ones). They are problems of searching for what is being sought and boil down to calculating the unknown value of a certain quantity (that’s why they are sometimes called computational). By problems (in the school course) we mean equations, and finding the value of a numerical expression, etc., because according to the structure (there is a condition - known, there is a requirement - the sought), therefore, these are problems. Moreover, “data” is a sufficient condition, “sought” is necessary, i.e. there is a logical consequence, and this shows that the problem is being solved.

That is, word problems in a mathematics course, like the entire mathematics course, develop the logical thinking of students of any age. For this development to proceed successfully, it is necessary to start from the first grade, but for this, primary school teachers must know the essence of logical reasoning and be able to teach their students to think logically.

ChapterII. A set of tasks for the development of logical thinking in junior schoolchildren

2.1. Problems - jokes, a matter of ingenuity

There were 40 magpies sitting on one tree. A hunter passed by, shot and killed 6 magpies. How many magpies are left on the tree? (Not a single one (the magpies got scared of the shot and flew away)).

How many ends does a stick have? - Two. How many ends do two and a half sticks have? (Six)

The two approached the river. There is only one boat near the shore. How can they get across to the other side if the boat can only take one person? (The travelers approached the opposite banks of the river.)

How many ends does thirty and a half sticks have? (62 ends)

One fifth grader wrote about himself this way: “I have twenty-five fingers on one hand, the same number on the other, and 10 on both feet.” How come? You need to put punctuation marks correctly: “I have twenty fingers: five on one hand, the same on the other, and 10 on both feet.”

The shepherd was chasing geese. One walks in front of the three, one pushes the three, and two walk in the middle. How many geese did he have? (Four)

The shepherd was asked how many geese he had. He replied: “One walks in front of the two, one pushes the two, one walks in the middle.” How many geese did the shepherd herd? (Three)

There are months that end with the number 30 or 31. And in which months does the number 28 appear? (In all)

A team of three horses traveled 60 km. How many kilometers did each horse gallop? (60 km)

An airplane flies the distance from city A to city B in 1 hour 20 minutes. However, he makes the return flight in 80 minutes. How do you explain this? (80 min = 1 hour 20 min)

Two trains left Leningrad and Moscow at the same time. The speed of the Leningrad one is 2 times higher than the Moscow one. Which train will be further from Moscow when they meet? (Both trains will be at the same distance from Moscow).

When can a person race at the speed of a racing car? (When he's in this car)

Is it possible to throw a ball so that, after flying for some time, it stops and starts moving in the opposite direction? (The ball must be thrown up)

Two fathers and two sons divided three oranges among themselves so that each got one orange. How could this happen? (They were grandfather, father and grandson)

A boy has as many sisters as brothers, and his sister has half as many sisters as brothers. How many brothers and sisters are there in this family? (1 sister and 2 brothers)

How many ends does 72 and a half sticks have? (146 ends)

A cyclist rode from a city to a village, the distance between them being 32 km, at a speed of 12 km/h. A pedestrian left the village for the city at the same time at a speed of 4 km/h. Which one will be further from the city in 2 hours? (In 2 hours they will be the same distance from the city)

Someone decided to enter the protected area and began to watch the gatekeeper. The first visitor was asked the question: "Twenty-two?" He answered: “Eleven,” and was allowed through the gate. The second one was asked: “Twenty-eight?” After the answer: “Fourteen,” they let him through. “How simple,” someone thought and walked up to the gate. He was asked: "Forty-eight?" He said, “Twenty-four,” and was arrested.
How should he answer to be allowed through? (He must answer: “Eleven,” since the answer password was the number of letters in the number that the gatekeeper asked).

2.1. Problems in verse, simple - compound

Problems in verse

Apples fell from the branch to the ground.

They cried, they cried, they shed tears
Tanya collected them in a basket.
I brought it as a gift to my friends
Two for Seryozhka, three for Antoshka,
Katerina and Marina,
Ole, Sveta and Oksana,
The biggest thing is for mom.
Speak quickly,
How many are Tanya's friends? (7 friends)

P challenging tasks:

The turtle crawled for 3 minutes at a speed of X m/min. Which way did she crawl?

What values can X take?

Maybe 1000m?

More or less? (less than 5 m)

How far will it crawl if X = 5 m/min?

5 ∙ 3 = 15 (m.)

Answer: 15 m.

There were 18 candies, we ate 2/9. How many candies did you eat?

18: 9 ∙ 2 = 4 (k)

Answer: ate 4 candies.

For 6 kg of apples they paid d rubles. What is the price of apples?

What values does the variable d take?

d = 60, 120, 66, 72.

At what values of d will the price be expressed in kopecks? (77, 62, 123, 67).

Two flies compete in a race. They run from the floor to the ceiling and back. The first fly runs in both directions at the same speed. The second runs down twice as fast as the first, and up twice as slow as the first. Which fly will win?

Answer: The first fly reaches the ceiling when the second one is halfway there; the first returns to the floor when the second reaches the ceiling. The first one wins.

Component tasks:

Four hobbits were traveling along a large road. Each carried 24 kg of provisions. How many days will this food last if the hobbits eat 6 kg every day?

(24 ∙ 4) : 6 = 16 (in.)

Answer: there will be enough provisions for 16 days.

A family of crocodiles was walking down the street: a grandfather, two fathers and two sons. All together were 90 years old. How many crocodiles were walking down the street? How old is everyone if each father is 25 years older than his son?

1)90 – 25 – 25 – 25 = 15 (l.) – three parts

2) 15: 3 = 5 (l.) – grandson

3) 5 + 25 = 30 (l.) – dad

4) 30 + 25 = 55 (l.) – grandfather

Answer: grandson is 5 years old, father is 30 years old, grandfather is 55 years old.

Robinson and Friday have 11 nuts together. Robinson and his Parrot have 13 nuts. Parrot and Friday have 12 nuts. How many nuts do Robinson, Friday and Parrot have in total?

The Parrot has 7 op.

Friday has 5 ops.

Robinson has 6 op.

P + Pyat = 11

Pop + Heel = 12

2P + 2Pyat + 2Pop = 36

P + Pt + Pop = 18 (op.) – total

Answer: All together there are 18 nuts.

“Ah - ah, from the Earth to the Moon is only 384,400 km!” - exclaimed the Hare. He loaded 15,800 kg of equipment onto the spacecraft and began the flight to the Moon. "Wait for it!" - said the Wolf. He loaded 6480 kg of equipment onto the spaceship, less than a hare, and flew in pursuit. He caught up with the hare at a distance of 105,600 km from the Earth. Which of the following questions can be answered based on the problem statement?

How many kilograms does the Hare weigh?

How many kilograms of equipment did Wolf load onto the spaceship?

At what distance from the Moon did the Wolf catch up with the Hare?

How many kilometers from the Moon to the Earth?

2) 15800 – 6480 = 9320 (kg) – loaded by the Wolf

4) 384400 – 105600 = 278800 (km) – from the Moon

The average age of the eight people in the room was 12 years. When 1 person left the room, the average age became 11 years. How old was the person who left the room?

12 ∙ 8 = 96 (l.) – that was all

11 ∙ 7 = 77 (l.) – became the remaining 7

96 – 77 = 19 (l.) – was the one who came out.

Answer: The man who left was 19 years old.

2.3. Historical tasks

On October 4, 1956, the first artificial Earth satellite weighing 84 kg was launched in the Soviet Union. Calculate the mass of the second Earth satellite along with the equipment and the dog Laika (which launched in the USSR on November 3, 1957), if its mass was 425 kg greater than the mass of the first satellite. How many full years, months and days have passed since the launch of the first satellite in the Soviet Union to the present day? (until March 20, 2004)

84 + 425 = 509 (kg) – mass of the second satellite

1956 9 months 3 days

46 l. 5 months 16 days

Orenburg was founded on April 30, 1733. How many years, months and days does the city of Orenburg exist (as of March 20, 2004)

2003 2 months 19 days

1742 3 months 29 days

260 l. 10 months 19 days

The peasant needs to transport a wolf, a goat and a cabbage across the river. The boat is small: a peasant can fit in it, and with him only a goat, or only a wolf, or only a cabbage. But if you leave a wolf with a goat, then the wolf will eat the goat, and if you leave a goat with cabbage, then the goat will eat the cabbage. How did the peasant transport his cargo?

Answer: We'll have to start everything with a goat. The peasant, having transported the goat, returns and takes the wolf, which he transports to the other bank, where he leaves it, but he also takes the goat and takes it back to the first bank. Here he leaves her and transports the cabbage to the wolf. Following this, returning, he transports the goat, and the crossing ends safely.

It is said that two fathers and two sons found three rupees (silver coins) on the road leading to Bombay and quickly divided them among themselves, each getting a coin. How did they manage to cope with the task?

Answer: The travelers were able to divide the find equally, because there were three of them: grandfather, father and son (or in another way: two fathers, two sons).

While passing through a small town, one merchant stopped by a restaurant for a snack and then decided to get a haircut. There were only two hairdressers in the town, and in each there was only one hairdresser, who was also the owner. In one, the barber was unkemptly shaved and had a bad haircut, and in the other, he was clean-shaven and had an excellent haircut. The merchant decided to get his hair cut at the first barbershop. Do you think he made the right choice?

Answer: The merchant correctly reasoned that since there are only two barbers in the city, they probably cut each other’s hair. This means you have to go get a haircut from someone who has a bad haircut.

A peasant woman came to the market to sell eggs. The first customer bought half of all the eggs from her and another half of the eggs. The second customer purchased half the remaining eggs and another half egg. The third bought only one egg. After this, the peasant woman had nothing left. How many eggs did she bring to the market?

Answer: After the second buyer purchased half of the remaining eggs and another half egg, the peasant woman had only one egg left. This means that one and a half eggs constitute the second half of what is left after the first sale. It is clear that the total remainder is three eggs. By adding half an egg, we get half of what the peasant woman originally had. So, the number of eggs she brought to the market is seven.

2.4. Puzzles, crosswords, charades

Rebuses

Guess 4 names:

(Seva, Seryozha, Nastya, Vova)

What closed the question?

(Number 1, because the upper fish are the minuend, the lower ones are the subtrahend, and the number is the difference between the obtained numbers)

Crosswords

TO Rossword No. 1

Vertically:

1. Division action component. (Dividend)

2. Largest remainder when divided by five. (Four)

3. To find out how many times one number is greater than another, you need to perform the action...? (Subtraction)

4. Multiplication action component. (Factor)

Horizontally:

5. A dividend that is completely divisible by some number.

TO Rossword No. 2

Horizontally:

There are ten in one meter... (Decimeter)

This unit of mass measures the weight of a person. (Kilogram)

There are ten in one decimeter... (Centimeter)

A record made up of numbers, letters and arithmetic symbols. (Expression)

A device made of transparent material with which you can measure the area of a figure. (Palette)

Vertically :

Read the keyword. What does it mean? (Ton is the name of various units of mass).

Charades

You measure the area
Remember first -
Her you are at school,
Undoubtedly, they studied it.
Five letters
Those who follow are inspired,
They won't survive
Without dance, music and stage.
To the exhibits
Weapon gazing
You will find the answer
In the historical museum. (Ar - ballet)

The number and note next to it,

Yes, add a consonant to the letter,

But in general, there is only one master,

He makes beautiful furniture. (One hundred - la - r)

He has a high title and rank.

And the whole word is a designation,

Dividing training into doses. (Pair - Count)

In the dance you will find the first syllable,

And give an excuse.

In general - the one who protects

Glory, honor of the native country,

He knows no fear in battle

And in work - a hero of work. (Pa – three – from).

2.5. Geometric problems

“Friend! You are given a figure of 5 squares: 4 small and one large. You need to remove several matches so that 2 squares (of any size) remain.” How many matches, at least, do you think need to be removed so that instead of five squares there are two? (2 matches will need to be removed).

Five Little Chefs decided to share a large rectangular chocolate bar among themselves.

But it fell to the floor and when they unwrapped it, they saw that the chocolate bar had broken into 7 pieces. Nikolai ate the largest piece. Sveta and Masha ate the same amount of chocolate, but Sveta ate three pieces, and Masha only one piece. Bella ate 1/7 of the whole chocolate bar, and Katya ate the rest. Which piece of chocolate did Katya get? (Nikolai ate the sixth. Sveta ate 7, 5, 4, and Masha ate the third. Bella ate the first. So Katya ate the second.)

Conclusion

The development of logical thinking as a pedagogical process must be carried out in accordance with the laws of development of the child’s body, in unity and harmony with the intellectual development of the child.

Since logical thinking can be considered as a new priority direction of pedagogical theory and practice, its content today is at the stage of formation, revision of the object of study, determination of methodological approaches, that is, the problem is relevant.

This problem was studied by: G. Eysenck, F. Galton, J. Ketell, K. Meili, J. Piaget, C. Spearman, S. L. Rubinstein, L. S. Vygotsky, N. A. Podgoretskaya and others. According to these researchers, logical thinking is a purposeful, indirect and generalized reflection by a person of the essential properties and relationships of things aimed at obtaining new results in practice, science, and technology.

Having determined the main tasks for the development of logical thinking in younger schoolchildren, you need to think about what general foundations and principles its content should be based on. For they largely determine the effectiveness of teaching, education and development of schoolchildren in intellectual development. The formation of initial logical techniques in mathematics lessons is carried out through the operations of logical thinking:

Identification of the basis, properties in the objects under study, and their comparison

Familiarity with the signs of necessary and sufficient

Classification of objects and concepts

Analysis and synthesis of tasks and assignments

Generalization, i.e. logical conclusion.

A mathematics lesson provides a unique opportunity to ensure the relationship between the pedagogical process and the child’s process of mastering non-standard tasks, which simultaneously interact with the basic concepts of mathematics.

The system of classes conducted in mathematics lessons, on solving problems, is the optimal form of work with younger schoolchildren on the formation of logical thinking.

One of the most important tasks facing a primary school teacher is the development of independent logic of thinking, which would allow children to build conclusions, provide evidence, express judgments that are logically related to each other, justify their judgments, draw conclusions, and, ultimately, independently acquire knowledge. Logical thinking is not innate, so it can and should be developed. Solving logical problems in elementary school is precisely one of the techniques for developing thinking. In many ways, the role of teaching mathematics in the development of thinking is due to modern developments in the field of modeling and design techniques, especially in objectively oriented modeling and design, based on inherently human conceptual thinking.

Of course, the problem raised is quite deep and voluminous and requires more than one year of painstaking work.

Literature

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A necessary condition for the qualitative renewal of our society is the increase in its intellectual potential. The solution to this problem largely depends on the general education school as the basic link in the system of lifelong education. The intellectual level of an individual is generally characterized by two main parameters: erudition (the amount of acquired information) and intellectual development (the ability to use this information to solve various kinds of problem situations that arise in the process of activity). The modern student needs to be conveyed not so much information as a collection of ready-made answers, but rather a method for obtaining them, analysis and forecasting of development, that is, to form the student’s general logical thinking skills.

In the conditions of the modern education system, the problem of the development of logical thinking (thinking in the form of concepts, judgments and conclusions according to the rules and laws of logic (formal), carried out consciously and fully in speech and with its help) becomes particularly relevant. It is necessary to carry out specially organized work to form and improve the mental activity of students, equipping them with “logical literacy” - fluency in a complex of elementary logical concepts and actions that make up the ABC of logical thinking and the necessary basis for its development.

The logic of thinking is not given to a person from birth; he masters it in the process of life, in learning. In psychological and pedagogical research and practice of logical training of children in the 19th-20th centuries. The ideas are theoretically developed and experimentally proven that, under certain conditions, primary logical skills can be successfully formed in younger schoolchildren. There is a large number of works (A.K. Artemov, I.Ya. Lerner, I.L. Nikolskaya, A.A. Stolyar, K.O. Ananchenko, V.S. Ablova, T.A. Kondrashenkova, L.N. Udovenko, N.G. Salmina, V.N. Sokhina, T.K. Kamalova, E.P. Malanyuk, O.V. Alekseeva, I.V. Titova, etc.) devoted to this problem.

The system for the development of logical thinking is an integral unity of target, procedural, technological, organizational, and content components of students’ logical training.

The goal of developing logical thinking is certainty, consistency, and evidence of thought. The objectives are knowledge and mastery of basic mental operations; knowledge and ability to identify the structure of logical forms of thinking; carry out the transfer of methods of mental activity from one area of knowledge to another.

The technology for developing logical thinking in junior schoolchildren is a combination of the following forms of work:

1) introduction of elements of logic into preschool mathematical training (programs “Development” (directed by L.A. Verner), “Childhood” (directed by T.N. Doronova, L.I. Ivanova), “Rainbow” (E.V. .Solovyova), intellectual training by E.A. Ponomareva);

4) introduction of elements of logic into the study of basic subjects, especially mathematics as the most theoretical science of all studied in school, a science built in accordance with logic. This approach is the most preferable, since it can provide universal initial logical training, its organic connection with the subject content of school courses and continuity between primary and secondary schools.

For the greatest effectiveness, the organization of logical training of junior schoolchildren (in mathematics lessons) should be built on the following principles:

2) continuity between primary and secondary schools;

4) a gradual increase in the level of abstraction of the proposed material and methods of operating with it (from actions with real objects to operating with their models and verbal descriptions of logical relationships);

5) revealing the general significance of logical skills and actions, their independence from the specific content of the material, the ability to transfer methods of mental activity from the field of mathematical knowledge to others;

6) practical mastery of logical skills without using special terminology, without memorizing rules;

7) scientific character;

8) accessibility.

I. Isolating features of objects and operating with them.

1. Identification of features of objects (concrete and abstract).

2. Comparison of two or more items:

A) identification of common features (properties) of two, three or more objects;

B) identifying the distinctive features of two, three or more objects;

3. Identifying the common property of a group of objects:

A) selection of a common name (collective name) for a group of objects;

B) identifying an extra item in a given group;

C) finding the missing item in a given group;

D) comparison of groups of objects.

4. Identification of patterns in the arrangement of objects in a row or matrix.

5. Recognizing objects by their characteristics.

6. Description of objects according to their characteristics.

II. Classification.

1. Verbal description of classes in the finished classification.

2. Division into classes according to a given basis. Assigning objects to a class.

3. Introduction of the basis for independently carried out classification.

4. Checking the results of the classification.

III. Understanding and correct use of logical words (and, or, all, some and others).

In modern technologies for teaching mathematics in primary grades, focused on the intellectual education of the individual, the development of logical thinking is one of the main tasks. At the moment, there is no single program for the implementation of logical training throughout the entire period of study in primary school. But the logical component is represented to one degree or another in the programs of all teams of authors, and each of them in its own way determines the content aspect and sequence of formation of logical skills . Authors of curricula and manuals need to pay special attention to the above principles of construction and the content of logical training for younger schoolchildren, since education in primary school should not be based on formal training. but on a substantive basis.

I. Introduction.

Primary general education is designed to help the teacher realize the abilities of each student and create conditions for the individual development of younger schoolchildren.

The more diverse the educational environment, the easier it is to reveal the individuality of a student’s personality, and then direct and adjust the development of a younger student, taking into account identified interests, relying on his natural activity.

The ability to solve various problems is the main means of mastering a mathematics course in secondary school. This is also noted by G.N. Dorofeev. He wrote: “The responsibility of mathematics teachers is especially great, since there is no separate subject “logic” in school, and the ability to think logically and build correct conclusions must be developed from the first “touch” of children with mathematics. And how we can introduce this process into various school programs will depend on which generation will come to replace us.”

Schoolchildren begin to develop a sustainable interest in mathematics at the age of 12–13. But in order for middle and high school students to take math seriously, they must first understand that thinking about difficult, nonroutine problems can be fun. Problem solving skills

is one of the main criteria for the level of mathematical development.

At primary school age, as psychological research shows, the further development of thinking becomes of primary importance. During this period, a transition occurs from visual-figurative thinking, which is basic for a given age, to verbal-logical, conceptual thinking. Therefore, the development of theoretical thinking takes on leading importance for this age.

V. Sukhomlinsky devoted significant attention to the issue of teaching logical problems to younger schoolchildren in his works. The essence of his thoughts boils down to the study and analysis of the process of solving logical problems by children, while he empirically identified the peculiarities of children’s thinking. He also writes about work in this direction in his book “I Give My Heart to Children”: “There are thousands of tasks in the world around us. They were invented by the people, they live in folk art as stories - riddles."

Sukhomlinsky observed the progress of children’s thinking, and observations confirmed “that, first of all, it is necessary to teach children to cover with their mind’s eye a number of objects, phenomena, events, and to comprehend the connections between them.

Studying the thinking of slow-witted people, I became increasingly convinced that the inability to comprehend, for example, a task is a consequence of the inability to abstract, to be distracted from the concrete. We need to teach the kids to think in abstract concepts.”

The problem of introducing logical problems into the school mathematics course was dealt with not only by researchers in the field of pedagogy and psychology, but also by mathematicians and methodologists. Therefore, when writing the work, I used specialized literature, both the first and second directions.

The facts stated above determined the chosen topic: “Development of logical thinking of junior schoolchildren when solving non-standard problems.”

The purpose of this work– consider different types of tasks for developing the thinking of younger schoolchildren.

Chapter 1. Development of logical thinking in younger schoolchildren.

1. 1. Features of logical thinking of younger schoolchildren.

Various cognitive processes that provide diverse types of child activity do not function in isolation from each other, but represent a complex system, each of them is connected with all the others. This connection does not remain unchanged throughout childhood: at different periods, one of the processes acquires leading importance for general mental development.

Psychological research shows that during this period it is thinking that largely influences the development of all mental processes.

Depending on the extent to which the thought process is based on perception, idea or concept, three main types of thinking are distinguished:

Subject-effective (visually effective)
Visual-figurative.
Abstract (verbal-logical)

Younger schoolchildren, as a result of studying at school, when it is necessary to regularly complete tasks without fail, learn to control their thinking, think when necessary.

In many ways, the formation of such voluntary, controlled thinking is facilitated by teacher assignments in the classroom, encouraging children to think

When communicating in primary school, children develop conscious critical thinking. This happens due to the fact that in the class, ways to solve problems are discussed, various solution options are considered, the teacher constantly asks students to justify, tell, and prove the correctness of their judgment. The younger student regularly logs into the system. When he needs to reason, compare different judgments, and make inferences.

In the process of solving educational problems, children develop such operations of logical thinking as analysis, synthesis, comparison, generalization and classification.

In parallel with mastering the technique of isolating properties by comparing different objects (phenomena), it is necessary to derive the concept of general and distinctive (particular), essential non-essential features, using such thinking operations as analysis, synthesis, comparison and generalization. The inability to identify the general and the essential can seriously hamper the learning process. The ability to highlight the essential contributes to the formation of another skill - to be distracted from unimportant details. This action is given to younger schoolchildren with no less difficulty than highlighting the essential.

From the above facts it is clear that all operations of logical thinking are closely interconnected and their full formation is possible only in a complex. Only their interdependent development contributes to the development of logical thinking as a whole. It is at primary school age that it is necessary to carry out targeted work to teach children the basic techniques of mental activity. A variety of psychological and pedagogical exercises can help with this.

1. 2. Psychological prerequisites for the use of logical problems in mathematics lessons in elementary school

Logical and psychological research in recent years (especially the works of J. Piaget) revealed the connection between some “mechanisms” of children’s thinking and general mathematical and general logical concepts.

In recent decades, the issues of the formation of children's intelligence and the emergence of their general ideas about reality, time and space have been studied especially intensively by the famous Swiss psychologist J. Piaget and his colleagues. Some of his works are directly related to the problems of developing a child’s mathematical thinking. Let us consider the main provisions formulated by J. Piaget in relation to the issues of constructing a curriculum.

J. Piaget believes that psychological research into the development of arithmetic and geometric operations in the child’s mind (especially those logical operations that carry out preconditions in them) makes it possible to accurately correlate the operator structures of thinking with algebraic structures, order structures and topological ones.

The structure of order corresponds to such a form of reversibility as reciprocity (rearrangement of order). In the period from 7 to 11, a system of relationships based on the principle of reciprocity leads to the formation of a structure of order in the child’s mind.

These data indicate that traditional psychology and pedagogy did not sufficiently take into account the complex and capacious nature of those stages of a child’s mental development that are associated with the period from 7 to 11 years.

J. Piaget himself directly correlates these operator structures with basic mathematical structures. He argues that mathematical thinking is possible only on the basis of already established operator structures. This circumstance can be expressed in this form: it is not “familiarity” with mathematical objects and the assimilation of methods of acting with them that determine the formation of operator mental structures in a child, but the preliminary formation of these structures is the beginning of mathematical thinking, the “isolation” of mathematical structures.

Consideration of the results obtained by J. Piaget allows us to draw a number of significant conclusions in relation to the design of a mathematics curriculum. First of all, factual data on the formation of a child’s intellect from 7 to 11 years old indicate that at this time not only are the properties of objects described through the mathematical concepts of “relationship-structure” not “alien” to him, but the latter themselves organically enter into the child’s thinking . (12-15s.)

Traditional objectives of the primary school mathematics curriculum do not take this circumstance into account. Therefore, they do not realize many of the opportunities hidden in the process of a child’s intellectual development. In this regard, the practice of introducing logical problems into the initial mathematics course should become normal.

2. Organization of various forms of work with logical tasks.

It has been repeatedly stated above that the development of logical thinking in children is one of the important tasks of primary education. The ability to think logically and make inferences without visual support is a necessary condition for the successful assimilation of educational material.

Having studied the theory of the development of thinking, I began to include tasks in lessons and extracurricular work in mathematics related to the ability to draw conclusions using the techniques of analysis, synthesis, comparison and generalization.

To do this, I selected material that was entertaining in form and content.

To develop logical thinking, I use didactic games in my work.

Didactic games stimulate, first of all, visual-figurative thinking, and then verbal-logical thinking.

Many didactic games set children the task of rationally using existing knowledge in mental actions, finding characteristic features in objects, comparing, grouping, classifying according to certain characteristics, drawing conclusions and generalizing. According to A.Z. Zak, with the help of games, the teacher teaches children to think independently and use the acquired knowledge in various conditions.

For example, she offered ancient and non-standard problems, the solution of which required students to be quick-witted, the ability to think logically, and to look for unconventional solutions. (Appendix No. 2)

The plots of many problems were borrowed from works of children's literature, and this contributed to the establishment of interdisciplinary connections and increased interest in mathematics.

In my previous editions, only guys with pronounced mathematical abilities could cope with such tasks. For other children with an average and low level of development, it was necessary to assign tasks with obligatory support on diagrams, drawings, tables, and keywords that allow them to better understand the content of the task and choose a recording method.

It is advisable to start working on the development of logical thinking with classes in the preparatory group. (Appendix No. 3)

We learn to identify essential features
We teach the child to compare.
We learn to classify objects.
"What common?"
"What's extra?"
“What unites?”

3. Methods of using logical problems in mathematics lessons in elementary school.

I will supplement the general idea about the importance of widespread introduction of non-standard problems into school mathematics lessons with a description of the corresponding methodological guidelines.

In the methodological literature, special names have been assigned to developmental tasks: problems of reasoning, “tasks with a twist,” tasks of ingenuity, etc.

In all their diversity, we can distinguish into a special class such tasks that are called trap tasks, “deceptive” tasks, provoking tasks. The conditions of such tasks contain various kinds of references, instructions, hints, hints, and encouragement to choose an erroneous solution path or an incorrect answer.

Provoking tasks have high development potential. They contribute to the development of one of the most important qualities of thinking - criticality, teach them to analyze perceived information, its comprehensive assessment, and increase interest in mathematics classes.

Type I Tasks that explicitly impose one very definite answer.

1st subtype. Which of the numbers 333, 555, 666, 999 is not divisible by 3?

Since 333 = 3x111, 666 = 3x222, 999 = 3*333, many students, when answering the question, name the number 555.

But this is incorrect, since 555=3*185. Correct answer: None.

2nd subtype. Tasks that encourage you to make an incorrect choice of answer from the proposed correct and incorrect answers. What is easier: a pound of fluff or a pound of iron?

Many people believe that a pound of fluff is lighter, since iron is heavier than fluff. But this answer is incorrect: a pound of iron has a mass of 16 kg and a pound of fluff also has a mass of 16 kg.

II type. Problems whose conditions push the solver to perform some action with given numbers or quantities, while performing this action is not required at all.

1. Three horses galloped 15 km. How many kilometers did each horse gallop?

I would like to do the division 15:3 and then the answer is: 5 km. In fact, there is no need to do the division at all, since each horse has galloped the same amount as the three.

2. (Old problem) A man was walking to Moscow, and 7 praying mantises were walking towards him, each of them had a bag, and in each bag there was a cat. How many creatures were heading to Moscow?

The Decider can hardly restrain himself from saying: “15 creatures, since 1+7+7=15”, but the answer is incorrect, you do not need to find the sum. After all, one man was going to Moscow.

III type. Problems whose conditions allow for the possibility of “refuting” a semantically correct solution with a syntactic or other non-mathematical solution

1. Three matches are laid out on the table so that there are four. Could this have happened if there were no other objects on the table?

The obvious negative answer is refuted by the drawing

2. (Old problem) A peasant sold three goats at the market for three rubles. The question is: “Where did each goat go?”

The obvious answer is: "One ruble at a time"- is refuted: goats don’t walk on money, they walk on the ground.

Experience has shown that non-standard problems are very useful for extracurricular activities as Olympiad tasks, since this opens up opportunities to truly differentiate the results of each student.

Such tasks can be successfully used as additional individual tasks for those students who easily and quickly cope with the main tasks during independent work in class, or for those wishing to do it as homework.

The variety of logical problems is very large. There are also many solutions. But the most widely used methods for solving logical problems are:

Tabular;
Through reasoning.

Problems solved by compiling a table.

When using this method, the conditions contained in the problem and the results of reasoning are recorded using specially compiled tables.

1. The shorties from the flower town planted a watermelon. Watering it requires exactly 1 liter of water. They only have 2 empty cans with a capacity of 3L and 5L. Using these cans, how can you collect exactly 1 liter of water from the river?

Solution: Let's present the solution in a table.

Let's make an expression: 3*2-5=1. It is necessary to fill a three-liter vessel twice and empty a five-liter vessel once.

Solving non-standard logical problems using reasoning.

This method solves simple logical problems.

Vadim, Sergey and Mikhail are studying various foreign languages: Chinese, Japanese and Arabic. When asked what language each of them was studying, one answered: “Vadim is studying Chinese, Sergey is not studying Chinese, and Mikhail is not studying Arabic.” Subsequently, it turned out that in this answer only one statement is true, and the other two are false. What language is each young person learning?

Solution. There are three statements:

Vadim is studying Chinese;
Sergey does not study Chinese;
Mikhail does not study Arabic.

If the first statement is true, then the second is also true, since young men learn different languages. This contradicts the statement of the problem, so the first statement is false.

If the second statement is true, then the first and third must be false. It turns out that no one studies Chinese. This contradicts the condition, so the second statement is also false.

Answer: Sergey is studying Chinese, Mikhail is studying Japanese, Vadim is studying Arabic.

Conclusion.

In the process of writing the work, I studied a variety of literature regarding the content of developmental tasks and tasks in it. Developed a system of exercises and tasks to develop logical thinking.

Solving non-standard problems develops in students the ability to make assumptions, check their accuracy, and justify them logically. Speaking for the purpose of proof contributes to the development of students’ speech, the development of the ability to draw conclusions from premises, and build conclusions.

When performing creative tasks, students analyze the conditions, highlight what is essential in the proposed situation, correlate the data with what they are looking for, and highlight the connections between them.

Solving non-standard problems increases learning motivation. For this purpose, I use developmental tasks. These are crosswords, rebuses, puzzles, labyrinths, ingenuity tasks, joke tasks, etc.

In the process of using these exercises in lessons and extracurricular activities in mathematics, the positive dynamics of the influence of these exercises on the level of development of logical thinking of my students and improving the quality of knowledge in mathematics was revealed.