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Non-standard tasks. Non-standard tasks

Tests and questionnaires 3rd grade.

Solving word problems is known to be very difficult for students. It is also known which stage of the solution is especially difficult. This is the very first stage - analysis of the task text. Students are poorly oriented in the text of the problem, its conditions and requirements. The text of the problem is a story about some life facts: “Masha ran 100 m, and towards her ...”,

“The students of the first class bought 12 carnations, and the students of the second...”, “The master made 20 parts during the shift, and his student...”.

Everything in the text is important; and the characters and their actions, and numerical characteristics. When working with a mathematical model of a problem ( numerical expression or equation), some of these details are omitted. But we are precisely teaching the ability to abstract from some properties and use others.

The ability to navigate the text of a mathematical problem is an important result and important condition general development student. And this needs to be done not only in mathematics lessons, but also in reading and fine arts lessons. Some problems make good subjects for drawings. And any task - good topic for retelling. And if there are theater lessons in the class, then some math problems can be staged. Of course, all these techniques: retelling, drawing, dramatization - can also take place in the mathematics lessons themselves. So, working on the texts of mathematical problems is an important element of the child’s overall development, an element of developmental education.

But are the tasks that are in current textbooks and the solution of which is included in the mandatory minimum sufficient for this? No, not enough. The required minimum includes the ability to solve certain types of problems:

about the number of elements of a certain set;

about movement, its speed, path and time;

about price and cost;

about work, its time, volume and productivity.

The four topics listed are standard. It is believed that the ability to solve problems on these topics can teach one to solve problems in general. Unfortunately, it is not. Good students who can solve practically

any problem from a textbook on the listed topics, they are often unable to understand the conditions of a problem on another topic.

The way out is not to limit yourself to any topic of word problems, but to solve non-standard problems, that is, problems whose topics are not in themselves the object of study. After all, we don’t limit the plots of stories in reading lessons!

Non-routine problems need to be solved in class every day. They can be found in mathematics textbooks for grades 5-6 and in the magazines “Primary School”, “Mathematics at School” and even “Quant”.

The number of tasks is such that you can choose tasks from them for each lesson: one per lesson. Problems are solved at home. But very often you need to sort them out in class. Among the proposed problems there are those that a strong student solves instantly. Nevertheless, it is necessary to require sufficient argumentation from strong children, explaining that from easy problems a person learns the methods of reasoning that will be needed when solving difficult problems. We need to cultivate in children a love for the beauty of logical reasoning. As a last resort, you can force such reasoning from strong students by requiring them to construct an explanation that is understandable for others - for those who do not understand the quick solution.

Among the problems there are completely similar ones in mathematical terms. If children see this, great. The teacher can show this himself. However, it is unacceptable to say: we solve this problem like that one, and the answer will be the same. The fact is that, firstly, not all students are capable of such analogies. And secondly, in non-standard problems the plot is no less important than the mathematical content. Therefore, it is better to emphasize connections between tasks with a similar plot.

Not all problems need to be solved (there are more of them here than there are math lessons in academic year). You may want to change the order of tasks or add a task that is not here.

The concept of “non-standard task” is used by many methodologists. Thus, Yu. M. Kolyagin explains this concept as follows: “Under non-standard is understood task, upon presentation of which students do not know in advance either how to solve it or how educational material decision is based."

The definition of a non-standard problem is also given in the book “How to Learn to Solve Problems” by authors L.M. Fridman, E.N. Turetsky: “ Non-standard tasks- these are those for which there is no mathematics in the course general rules and provisions defining the exact program for their solution."

Non-standard tasks should not be confused with tasks of increased complexity. The conditions of problems of increased complexity are such that they allow students to quite easily identify the mathematical apparatus that is needed to solve a problem in mathematics. The teacher controls the process of consolidating the knowledge provided by the training program by solving problems of this type. But a non-standard task presupposes a research character. However, if solving a problem in mathematics for one student is non-standard, since he is unfamiliar with methods for solving problems of this type, then for another, solving the problem occurs in a standard way, since he has already solved such problems and more than one. The same problem in mathematics in the 5th grade is non-standard, but in the 6th grade it is ordinary, and not even of increased complexity.

Analysis of textbooks and teaching aids in mathematics shows that each word problem in certain conditions can be non-standard, and in others - ordinary, standard. Standard task in one mathematics course may be non-standard in another course.

Based on an analysis of the theory and practice of using non-standard problems in teaching mathematics, it is possible to establish their general and specific role. Non-standard tasks:

· teach children to use not only ready-made algorithms, but also to independently find new ways to solve problems, i.e. promote the ability to find original ways to solve problems;
· influence the development of ingenuity and intelligence of students;
· prevent the development of harmful cliches when solving problems, destroy incorrect associations in the knowledge and skills of students, imply not so much the assimilation of algorithmic techniques, but rather the finding of new connections in knowledge, the transfer of knowledge to new conditions, and the mastery of various techniques of mental activity;
· create favorable conditions for increasing the strength and depth of students’ knowledge, ensure conscious assimilation of mathematical concepts.

Non-standard tasks:

· should not have ready-made algorithms that children have memorized;
· the content must be accessible to all students;
· must be interesting in content;
· To solve non-standard problems, students must have enough knowledge acquired by them in the program.

Solving non-standard problems activates students' activities. Students learn to compare, classify, generalize, analyze, and this contributes to a more durable and conscious assimilation of knowledge.

As practice has shown, non-standard problems are very useful not only for lessons, but also for extracurricular activities, for Olympiad tasks, since this opens up the opportunity to truly differentiate the results of each participant. Such tasks can also be successfully used as individual tasks for those students who can easily and quickly cope with the main part independent work in class, or for those interested as additional assignments. As a result, students receive intellectual development and preparation for active practical work.

There is no generally accepted classification of non-standard problems, but B.A. Kordemsky identifies the following types of such tasks:

· Problems related to the school mathematics course, but increased difficulty- type of problems of mathematical olympiads. Intended mainly for schoolchildren with a definite interest in mathematics; thematically, these tasks are usually related to one or another specific section of the school curriculum. The exercises related here deepen the educational material, complement and generalize individual provisions of the school course, expand mathematical horizons, and develop skills in solving difficult problems.
· Problems such as mathematical entertainment. Directly related to school curriculum do not have and, as a rule, do not require extensive mathematical training. This does not mean, however, that the second category of tasks includes only light exercises. There are problems with very difficult solutions and problems for which no solution has yet been obtained. “Unconventional problems, presented in an exciting way, bring an emotional element to mental exercises. Not associated with the need to always apply memorized rules and techniques to solve them, they require the mobilization of all accumulated knowledge, teach people to search for original, non-standard solutions, enrich the art of solving with beautiful examples, and make one admire the power of the mind.”

This type of task includes:

various number puzzles (“... examples in which all or some numbers are replaced by asterisks or letters. The same letters replace the same numbers, different letters- different numbers.”) and puzzles for ingenuity;

logical problems, the solution of which does not require calculations, but is based on building a chain of precise reasoning;

tasks whose solution is based on a combination of mathematical development and practical ingenuity: weighing and transfusion under difficult conditions;

mathematical sophisms are a deliberate, false conclusion that has the appearance of being correct. (Sophism is proof of a false statement, and the error in the proof is skillfully disguised. Sophistry translated from Greek means a clever invention, trick, puzzle);

joke tasks;

combinatorial problems in which various combinations of given objects are considered that satisfy certain conditions (B.A. Kordemsky, 1958).

No less interesting is the classification of non-standard problems given by I.V. Egorchenko:

· tasks aimed at finding relationships between given objects, processes or phenomena;
· problems that are insoluble or cannot be solved by means of a school course at a given level of knowledge of students;
tasks that require:

drawing and using analogies, determining the differences between given objects, processes or phenomena, establishing the opposition of given phenomena and processes or their antipodes;

implementation of practical demonstration, abstraction from certain properties of an object, process, phenomenon or specification of one or another aspect of a given phenomenon;

establishing cause-and-effect relationships between given objects, processes or phenomena;

constructing analytically or synthetically cause-and-effect chains with subsequent analysis of the resulting options;

correct implementation of a sequence of certain actions, avoiding “trap” errors;

making a transition from a planar to a spatial version of a given process, object, phenomenon, or vice versa (I.V. Egorchenko, 2003).

So, there is no single classification of non-standard problems. There are several of them, but the author of the work used in the study the classification proposed by I.V. Egorchenko.

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Introduction

1. Theoretical basis developing interest in mathematics

1.1 The essence of the concept of “interest”

1.2 Non-standard tasks and their types

1.3 Methods for solving non-standard problems

2. Formation in schoolchildren of the ability to solve non-standard problems

2.1 Non-standard tasks for elementary school students

2.2 Non-standard tasks for primary school

Conclusion

Literature

Introduction

Strategy modern education is to provide all students with the opportunity to demonstrate their talents and creative potential, implying the possibility of realizing personal plans. Therefore, today the problem of finding means of developing thinking abilities associated with the creative activity of students in both collective and individual forms of education is relevant. The work of teachers T.M. is devoted to this problem. Davydenko, L.V. Zankova, A.I. Savenkova et al., which focus on identifying means to increase productive cognitive activity students, organizing their creative activities.

The active acquisition of knowledge is facilitated by interest in the subject, since students study due to their inner attraction, according to at will. Then they learn the educational material quite easily and thoroughly. But in Lately An alarming and paradoxical fact is noted: interest in learning decreases from class to class, despite the fact that interest in the phenomena and events of the surrounding world continues to develop and becomes more complex in content.

Cultivating schoolchildren's interest in mathematics, developing their mathematical abilities impossible without the use of intelligence tasks, joke problems, number puzzles, fairy tale problems, etc. in the educational process. In this regard, there has been a tendency to use non-standard problems as a necessary component of teaching students mathematics (S. G. Guba, 1972).

Pedagogical experience shows that “...effectively organized educational activities of students in the process of solving non-standard problems is the most important means formation of mathematical culture and qualities of mathematical thinking; the organic combination of these qualities manifests itself in a person’s special abilities, giving him the opportunity to successfully carry out creative activities.”

Thus, on the one hand, it is necessary to teach students to solve non-standard problems, since such tasks play a special role in the formation of interest in the subject and in the formation of a creative personality, on the other hand, numerous data indicate that the issue of developing the ability to solve such problems, teaching methods for finding solutions to problems is not given due attention.

The above determined the choice of the research topic: “Non-standard problems as a means of developing interest in mathematics among students.”

Object of study - the process of developing interest in mathematics among school students.

Subject of study-developing students’ ability to solve non-standard problems to develop interest in mathematics.

Purpose of the study- to prove that knowledge of various methods contributes to the development of students’ skills in solving non-standard problems.

In accordance with the goal, the research objectives:

· Study of psychological, pedagogical and scientific-methodological literature and characterization of the concepts of “interest” and “non-standard task”.

· Identification of types of non-standard tasks.

· Familiarization with methods for solving non-standard problems.

· Compilation of didactic materials for students to develop the ability to solve non-standard problems using different methods.

This work consists of an introduction, two chapters, a conclusion and a list of references. The first chapter wears theoretical nature, it examines various interpretations of the concept of “interest”, highlights the role of non-standard problems in the formation of interest in mathematics among students, and provides some classifications of non-standard problems. The second chapter presents a study compiled by the author didactic material aimed at developing the ability to solve non-standard problems using different methods

During the study we used theoretical method, analysis of educational and methodological literature, modeling.

1. Theoretical foundations for developing interest in mathematics

1.1 The essence is understoodand I« interest»

Exist different approaches to the concept of "interest". Different methodologists and scientists interpret it differently. For example, linguist, lexicographer, doctor of philological sciences and professor Sergei Ivanovich Ozhegov gives several definitions of the concept “interest”:

1. Particular attention to something, the desire to get to the bottom of it, find out, understand. (Show interest in the matter. Lose interest in the interlocutor. Increased interest in everything new).

2. Entertaining, significant. (The interest of a story is in its plot. The case is of public interest.)

3. Numerous needs, requirements. (Group interests. Protect your interests. Spiritual interests. It is not in our interests).

4. Benefit, self-interest (colloquial). (He has his own interest here. Play for interest - for money) (S.I. Ozhegov, 2009).

The Russian scientist and writer Vladimir Ivanovich Dal, who became famous as the author of the “Explanatory Dictionary of the Living Great Russian Language,” gives the following definition:

"Interest - benefit, benefit, profit; interest, growth on money; sympathy for someone or something, participation, care. Interest or significance, importance of the matter.

Interest is the selective focus of a person, his attention, thoughts, thoughts (S.L. Rubinstein).

Interest is a kind of alloy of emotional-volitional and intellectual processes that increases the activity of human consciousness and activity (L.A. Gordon).

Interest is a person’s active cognitive focus on a particular object, phenomenon and activity, created with a positive emotional attitude towards them (V.A. Krutetsky).”

A person’s interests are determined by the socio-historical and individual conditions of his life. With the help of interest, a connection between the subject and the objective world is established. Everything that constitutes a subject of interest is drawn by a person from the surrounding reality. But the subject of interest for a person is not everything that surrounds him, but only what has necessity, significance, value and attractiveness for him.

People's interests are extremely diverse. There are several classifications of interests:

material interests (Manifested in the desire for housing amenities, gastronomic products, clothing, etc.);

spiritual interests (These are cognitive interests in mathematics, physics, chemistry, biology, philosophy, psychology, etc., interests in literature and different types arts (music, painting, theater). Characterize high level personality development.);

public interests (Includes interest in social work, to organizational activities.);

by direction:

broad interests (Variety of interests in the presence of a main, central interest.);

narrow interests (The presence of one or two limited and isolated interests with complete indifference to everything else.);

deep interests (The need to thoroughly study an object in all its details and subtleties.);

superficial interests (Sliding along the surface of a phenomenon and no real interest in the object.);

by strength:

sustainable interests (maintain for a long time, play significant role in a person’s life and activity and are relatively fixed features of his personality.);

unstable interests (Comparatively short-term: they arise quickly and quickly fade away.);

· by indirectness:

direct (immediate) interests (Caused by the very content of a particular area of knowledge or activity, its interestingness and fascination.);

indirect (mediated) interests (Caused not by the content of the object, but by the meaning that it has, being associated with another object that is directly of interest to a person.);

by level of effectiveness:

passive interests;

contemplative interests (When a person is limited to the perception of an object of interest.);

active interests;

effective interest (When a person is not limited to contemplation, but acts with the goal of mastering the object of interest.) (G. I. Shchukina, 1988).

There is a special type of human interest - cognitive interest.

“Cognitive interest is a selective orientation of the individual, addressed to the field of knowledge, to its subject side and the very process of mastering knowledge.”

Cognitive interest can be broad, extending to obtaining information in general, and in-depth in a specific area of cognition. It is aimed at mastering the knowledge that is presented in school subjects. At the same time, it is addressed not only to the content of a given subject, but also to the process of obtaining this knowledge, to cognitive activity. math teacher student

In pedagogy, along with the term “cognitive interest,” the term “learning interest” is used. The concept of “cognitive interest” is broader, since the zone of cognitive interest includes not only knowledge limited by the curriculum, but also that goes far beyond its limits.

IN foreign literature the term “cognitive interest” is missing, but the concept of “intellectual interest” exists. This term also does not include everything that is included in the concept of “cognitive interest,” since cognition includes not only intellectual processes, but also elements of practical actions related to cognition.

Cognitive interest is a combination of mental processes: intellectual, volitional and emotional. They are very important for personal development.

In intellectual activity, occurring under the influence of cognitive interest, the following are manifested:

· active search;

· a guess;

· research approach;

· Willingness to solve problems.

Emotional manifestations accompanying cognitive interest:

· emotions of surprise;

· feeling of expectation of something new;

· feeling of intellectual joy;

· feeling of success.

Volitional manifestations characteristic of cognitive interest are:

· search initiative;

· independence in acquiring knowledge;

· putting forward and setting cognitive tasks.

So, intellectual, strong-willed and emotional side cognitive interest act as a single interconnected whole.

The originality of cognitive interest is expressed in in-depth study, in the constant and independent acquisition of knowledge in the area of interest, in the active acquisition of the necessary methods for this, in the persistent overcoming of difficulties that lie in the way of mastering knowledge and methods of obtaining it.

Psychologists and teachers identify three main motives that encourage schoolchildren to study:

· Interest in the subject (I study mathematics not because I am pursuing some goal, but because the process of learning itself gives me pleasure). The highest degree of interest is passion. Exercising with passion generates strong positive emotions, and the inability to exercise is perceived as deprivation.

· Consciousness. (Classes on this subject I’m not interested, but I realize their necessity and force myself to study with an effort of will).

· Coercion. (I study because my parents and teachers force me). Often compulsion is supported by fear of punishment or the temptation of reward. Various coercive measures in most cases do not produce positive results (25, p. 24).

Interest greatly increases the effectiveness of lessons. If students study due to their inner attraction, of their own free will, then they learn the educational material quite easily and thoroughly, and therefore have good grades in the subject. Most underperforming students a negative attitude towards teaching is revealed. Thus, the higher the student’s interest in the subject, the more active the learning and the better the results. The lower the interest, the more formal the training, the worse its results. Lack of interest leads to low quality of learning, rapid forgetting and even complete loss of acquired knowledge, skills and abilities.

When forming cognitive interests among students, we must keep in mind that they cannot cover everyone educational subjects. Interests are selective, and one student, as a rule, can study with real passion only in one or two subjects. But, the presence of a stable interest in a particular subject has a positive effect on academic work in other subjects; both intellectual and moral factors are important here. Intensive mental development associated with in-depth study of one subject facilitates and makes the student’s learning in other subjects easier and more effective. On the other hand, the success achieved in academic work in favorite subjects strengthens the feeling self-esteem student, and he strives to study diligently in general.

An important task of the teacher is to form in schoolchildren the first two motives for learning - interest in the subject and a sense of duty and responsibility in learning. Their combination will allow the student to achieve good results in educational activities.

The formation of cognitive interests begins long before school, in the family; their emergence is associated with the appearance in children of such questions as “Why?”, “Why?”, “Why?”. Interest appears initially in the form of curiosity. Towards the end before school age under the influence of elders, the child develops an interest in learning at school: he not only plays at school, but also makes successful attempts to master reading, writing, counting, etc.

IN primary school cognitive interests deepen. A consciousness of the vital significance of teaching is formed. Over time, cognitive interests differentiate: some like mathematics more, others like reading, etc. Children show great interest in the labor process, especially if it is done in a team. Learning and other types of knowledge come into conflict, since the new interests of schoolchildren are not sufficiently satisfied at school. The scattered and unstable interests of adolescents are also explained by the fact that they “grope” for their main, central, core interest as the basis of their life orientation and try themselves in different areas. When the interests and inclinations of adolescents are finally determined, their abilities begin to form and clearly manifest themselves. By the end of adolescence, interests in a particular profession begin to form. At high school age, the development of cognitive interests and the growth of a conscious attitude towards learning determine the further development of the arbitrariness of cognitive processes, the ability to manage them, and consciously regulate them. At the end of the senior years, students master their cognitive processes, subordinate their organization to certain tasks of life and activity.

One of the means of developing interest in mathematics is non-standard problems. Let's look at them in more detail.

1. 2 Non-standard tasks and their types

The concept of “non-standard task” is used by many methodologists. Thus, Yu. M. Kolyagin explains this concept as follows: “Under non-standard is understood task, upon presentation of which students do not know in advance either the method of solving it or what educational material the solution is based on.”

The definition of a non-standard problem is also given in the book “How to Learn to Solve Problems” by authors L.M. Fridman, E.N. Turetsky: “ Non-standard tasks- these are those for which the mathematics course does not have general rules and regulations that determine the exact program for their solution.”

Analysis of textbooks and teaching aids in mathematics shows that each word problem in certain conditions can be non-standard, and in others - ordinary, standard. A standard problem in one mathematics course may be non-standard in another course.

Based on an analysis of the theory and practice of using non-standard problems in teaching mathematics, it is possible to establish their general and specific role. Non-standard tasks:

· teach children to use not only ready-made algorithms, but also to independently find new ways to solve problems, i.e. promote the ability to find original ways to solve problems;

· influence the development of ingenuity and intelligence of students;

· prevent the development of harmful cliches when solving problems, destroy incorrect associations in the knowledge and skills of students, imply not so much the assimilation of algorithmic techniques, but rather the finding of new connections in knowledge, the transfer of knowledge to new conditions, and the mastery of various techniques of mental activity;

· create favorable conditions for increasing the strength and depth of students’ knowledge, ensure conscious assimilation of mathematical concepts.

Non-standard tasks:

· should not have ready-made algorithms that children have memorized;

· the content must be accessible to all students;

· must be interesting in content;

· To solve non-standard problems, students must have enough knowledge acquired by them in the program.

Solving non-standard problems activates students' activities. Students learn to compare, classify, generalize, analyze, and this contributes to a more durable and conscious assimilation of knowledge.

As practice has shown, non-standard tasks are very useful not only for lessons, but also for extracurricular activities, for olympiad assignments, since this opens up the opportunity to truly differentiate the results of each participant. Such tasks can be successfully used as individual tasks for those students who can easily and quickly cope with the main part of independent work in class, or for those who wish to do so as additional tasks. As a result, students receive intellectual development and preparation for active practical work.

There is no generally accepted classification of non-standard problems, but B.A. Kordemsky identifies the following types of such tasks:

· Problems related to the school mathematics course, but of increased difficulty - such as problems of mathematical olympiads. Intended mainly for schoolchildren with a definite interest in mathematics; thematically, these tasks are usually related to one or another specific section of the school curriculum. The exercises related here deepen the educational material, complement and generalize individual provisions of the school course, expand mathematical horizons, and develop skills in solving difficult problems.

· Problems such as mathematical entertainment. They are not directly related to the school curriculum and, as a rule, do not require extensive mathematical training. This does not mean, however, that the second category of tasks includes only light exercises. There are problems with very difficult solutions and problems for which no solution has yet been obtained. “Unconventional problems, presented in an exciting way, bring an emotional element to mental exercises. Not associated with the need to always apply memorized rules and techniques to solve them, they require the mobilization of all accumulated knowledge, teach people to search for original, non-standard solutions, enrich the art of solving with beautiful examples, and make one admire the power of the mind.”

This type of task includes:

various number puzzles (“... examples in which all or some numbers are replaced by asterisks or letters. The same letters replace the same numbers, different letters - different numbers.”) and puzzles for ingenuity;

logical problems, the solution of which does not require calculations, but is based on building a chain of precise reasoning;

tasks whose solution is based on a combination of mathematical development and practical ingenuity: weighing and transfusion under difficult conditions;

joke tasks;

combinatorial problems in which various combinations of given objects are considered that satisfy certain conditions (B.A. Kordemsky, 1958).

No less interesting is the classification of non-standard problems given by I.V. Egorchenko:

· tasks aimed at finding relationships between given objects, processes or phenomena;

· problems that are insoluble or cannot be solved by means of a school course at a given level of knowledge of students;

tasks that require:

drawing and using analogies, determining the differences between given objects, processes or phenomena, establishing the opposition of given phenomena and processes or their antipodes;

implementation of practical demonstration, abstraction from certain properties of an object, process, phenomenon or specification of one or another aspect of a given phenomenon;

establishing cause-and-effect relationships between given objects, processes or phenomena;

constructing analytically or synthetically cause-and-effect chains with subsequent analysis of the resulting options;

correct implementation of a sequence of certain actions, avoiding “trap” errors;

making a transition from a planar to a spatial version of a given process, object, phenomenon, or vice versa (I.V. Egorchenko, 2003).

So, there is no single classification of non-standard problems. There are several of them, but the author of the work used in the study the classification proposed by I.V. Egorchenko.

1.3 Methods for solving problemsandart tasks

Russian philologist Dmitry Nikolaevich Ushakov in his explanatory dictionary gives the following definition of the concept “method” - path, method, technique theoretical research or the practical implementation of something (D. N. Ushakov, 2000).

What are the methods of teaching solving problems in mathematics, which we consider in this moment non-standard? Unfortunately, no one has come up with a universal recipe, given the uniqueness of these tasks. Some teachers teach in formulaic exercises. This happens in the following way: the teacher shows a solution, and then the student repeats this many times when solving problems. At the same time, students' interest in mathematics is killed, which is sad, to say the least.

In mathematics there are no general rules that allow solving any non-standard problem, since such problems are to some extent unique. A non-standard task in most cases is perceived as “a challenge to the intellect, and gives rise to the need to realize oneself in overcoming obstacles and in developing creative abilities.”

Let's consider several methods for solving non-standard problems:

· algebraic;

· arithmetic;

· brute force method;

method of reasoning;

· practical;

· guessing method.

Algebraic method problem solving develops Creative skills, the ability to generalize, forms abstract thinking and has such advantages as brevity of recording and reasoning when composing equations, saves time.

In order to solve the problem using the algebraic method, you need to:

· analyze the problem in order to select the main unknown and identify the relationship between quantities, as well as express these dependencies in mathematical language in the form of two algebraic expressions;

· find the basis for connecting these expressions with the “=” sign and create an equation;

· find solutions to the resulting equation, organize verification of the solution to the equation.

All these stages of solving the problem are logically interconnected. For example, we mention the search for a basis for connecting two algebraic expressions with an equal sign as a special stage, but it is clear that at the previous stage these expressions are not formed arbitrarily, but taking into account the possibility of connecting them with the “=” sign.

Both the identification of dependencies between quantities and the translation of these dependencies into mathematical language require intense analytical and synthetic mental activity. Success in this activity depends, in particular, on whether students know in what relationships these quantities can generally exist, and whether they understand the real meaning of these relationships (for example, relationships expressed by the terms “later by ...”, “older by ... times” " and so on.). Next, we need to understand what kind of mathematical action or property of the action or what kind of connection (dependence) between the components and the result of the action can describe this or that specific relationship.

Let us give an example of solving a non-standard problem using the algebraic method.

Task. The fisherman caught the fish. When he was asked: “What is its mass?”, he replied: “The mass of the tail is 1 kg, the mass of the head is the same as the mass of the tail and half of the body. And the mass of the body is the same as the mass of the head and tail together.” What is the mass of the fish?

Let x kg be the mass of the torso; then (1+1/2x) kg is the mass of the head. Since, according to the condition, the mass of the body is equal to the sum of the masses of the head and tail, we compose and solve the equation:

x = 1 + 1/2x + 1,

4 kg is the mass of the body, then 1+1/2 4=3 (kg) is the mass of the head and 3+4+1=8 (kg) is the mass of the whole fish;

Answer: 8 kg.

Arithmetic method solving also requires a lot of mental effort, which has a positive effect on the development of mental abilities, mathematical intuition, and the formation of the ability to foresee a real life situation.

Let's consider an example of solving a non-standard problem using the arithmetic method:

Task. Two fishermen were asked: “How many fish are in your baskets?”

“My basket contains half of what is in his basket, plus 10 more,” answered the first one. “And I have as much in my basket as he has, and 20 more,” the second one counted. We have counted, now you count.

Let's build a diagram for the problem. Let us denote by the first segment of the diagram the number of fish the first fisherman has. The second segment denotes the number of fish the second fisherman has.

Due to to modern man it is necessary to have an understanding of the basic methods of data analysis and probabilistic patterns that play an important role in science, technology and economics; elements of combinatorics, probability theory and mathematical statistics are introduced into the school mathematics course, which are convenient to understand with the help of brute force method.

The inclusion of combinatorial problems in a mathematics course has a positive impact on the development of schoolchildren. “Targeted training in solving combinatorial problems contributes to the development of such a quality of mathematical thinking as variability. By variability of thinking we understand the focus of the student’s mental activity on finding different solutions to a problem in the case when there are no special instructions for this.”

Combinatorial problems can be solved using various methods. Conventionally, these methods can be divided into “formal” and “informal”. With the “formal” solution method, you need to determine the nature of the choice, select the appropriate formula or combinatorial rule (there are sum and product rules), substitute numbers and calculate the result. The result is the number of possible options; the options themselves are not formed in this case.

With the “informal” solution method, the process of drawing up various options comes to the fore. And the main thing is not how many, but what options can be obtained. Such methods include brute force method. This method is accessible even to primary schoolchildren, and allows them to accumulate experience in the practical solution of combinatorial problems, which serves as the basis for the introduction of combinatorial principles and formulas in the future. In addition, in life a person has to not only determine the number of possible options, but also directly compile all these options, and, knowing the techniques of systematic enumeration, this can be done more rationally.

Tasks based on the complexity of enumeration are divided into three groups:

1 . Problems in which you need to perform a complete search of all possible options.

2. Problems in which it is impractical to use the exhaustive search technique and you need to immediately exclude some options without considering them (that is, carry out a reduced search).

3. Problems in which the enumeration operation is performed several times and in relation to various types of objects.

Here are the corresponding examples of tasks:

Task. By placing the signs “+” and “-” between the given numbers 9...2...4, make up all possible expressions.

A full selection of options is carried out:

a) two signs in the expression can be the same, then we get:

9 + 2 + 4 or 9 - 2 - 4;

b) two signs can be different, then we get:

9 + 2 - 4 or 9 - 2 + 4.

Task. The teacher says that he drew 4 figures in a row: a large and a small square, a large and a small circle so that the circle is in the first place and the figures of the same shape are not next to each other, and invites the students to guess in what sequence these figures are arranged.

There are a total of 24 different arrangements of these figures. And it is impractical to compile them all and then select those that correspond to a given condition, so an abbreviated search is carried out.

A large circle can be in the first place, then a small one can only be in third place, while large and small squares can be placed in two ways - in second and fourth place.

A similar reasoning is carried out if a small circle is in the first place, and two options are also drawn up.

Task. Three partners of one company store securities in a safe with 3 locks. The partners want to distribute the keys to the locks among themselves so that the safe can only be opened in the presence of at least two partners, but not one. How can I do that?

First, all possible cases of key distribution are enumerated. Each companion can be given one key, or two different keys, or three.

Let's assume that each companion has three different keys. Then the safe can be opened by one partner, and this does not meet the condition.

Let's assume that each partner has one key. Then, if two of them come, they will not be able to open the safe.

We will give each companion two different keys. The first - 1 and 2 keys, the second - 1 and 3 keys, the third - 2 and 3 keys. Let's check when any two companions arrive to see if they can open the safe.

The first and second companions can come, they will have all the keys (1 and 2, 1 and 3). The first and third companions can come, they will also have all the keys (1 and 2, 2 and 3). Finally, the second and third companions may come, they will also have all the keys (1 and 3, 2 and 3).

Thus, to find the answer to this problem, you need to perform the enumeration operation several times.

When selecting combinatorial problems, you need to pay attention to the topic and form of presentation of these problems. It is desirable that the tasks do not look artificial, but are understandable and interesting to children, and evoke positive emotions in them. You can use practical material from life to compose problems.

There are other problems that can be solved by brute force.

As an example, let’s solve the problem: “Marquis Karabas was 31 years old, and his young energetic Puss in Boots was 3 years old, when the events known from the fairy tale took place. How many years have happened since then, if now the Cat is three times younger than his owner? Let's present the list of options in a table.

Age of Marquis Karabas and Puss in Boots

14 - 3 = 11 (years)

Answer: 11 years have passed.

At the same time, the student experiments, observes, compares facts and, based on particular conclusions, makes certain general conclusions. In the process of these observations, his real-practical experience is enriched. This is precisely the practical value of search problems. In this case, the word “brute force” is used in the sense of analyzing all possible cases that satisfy the conditions of the problem, showing that there cannot be other solutions.

This problem can also be solved using the algebraic method.

Let the Cat be x years old, then Marquis is 3x, based on the conditions of the problem, we will create the equation:

The cat is now 14 years old, then 14 - 3 = 11 (years) have passed.

Answer: 11 years have passed.

Method of reasoning can be used to solve mathematical sophisms.

Mistakes made in sophism usually boil down to the following: performing “forbidden” actions, using erroneous drawings, incorrect word usage, inaccurate formulations, “illegal” generalizations, and incorrect applications of theorems.

To reveal sophistry means to indicate an error in reasoning, based on which the external appearance of proof was created.

Analysis of sophisms, first of all, develops logical thinking and instills correct thinking skills. To discover an error in sophism means to realize it, and awareness of the error prevents it from being repeated in other mathematical reasoning. In addition to the criticality of mathematical thinking, this type of non-standard problems reveals the flexibility of thinking. Will the student be able to “break out of the clutches” of this strictly logical at first glance path, break the chain of conclusions at the very link that is erroneous and makes all further reasoning erroneous?

Analysis of sophisms also helps the conscious assimilation of the material being studied, develops observation and a critical attitude towards what is being studied.

a) Here, for example, is sophistry with misuse theorems.

Let's prove that 2 2 = 5.

Let us take the following obvious equality as the initial ratio: 4: 4 = 5: 5 (1)

Let's take the common factor on the left and right sides out of brackets, and we get:

4 (1: 1) = 5 (1: 1) (2)

The numbers in brackets are equal, which means 4 = 5 or 2 2 = 5.

In the reasoning, when moving from equality (1) to equality (2), an illusion of plausibility is created on the basis of a false analogy with the distributive property of multiplication relative to addition.

b) Sophistry using “illegal” generalizations.

There are two families - the Ivanovs and the Petrovs. Each consists of 3 people - father, mother and son. Father Ivanov does not know Father Petrov. Ivanov's mother does not know Petrova's mother. The only son of the Ivanovs does not know the only son of the Petrovs. Conclusion: not a single member of the Ivanov family knows a single member of the Petrov family. Is this true?

If a member of the Ivanov family does not know a member of the Petrov family equal to himself in family status, this does not mean that he does not know the whole family. For example, father Ivanov may know the mother and son of the Petrovs.

The reasoning method can also be used to solve logical problems. Sublogical problems are usually understood as those problems that can be solved using logical operations alone. Sometimes solving them requires lengthy discussions, required direction which cannot be predicted in advance.

Task. They say that Tortila gave the golden key to Pinocchio not as simply as A.N. Tolstoy said, but in a completely different way. She brought out three boxes: red, blue and green. On the red box it was written: “Here lies the golden key,” and on the blue box, “The green box is empty,” and on the green box, “Here lies a snake.” Tortila read the inscriptions and said: “Indeed, in one box there is a golden key, in another there is a snake, and the third is empty, but all the inscriptions are incorrect. If you guess which box contains the golden key, it’s yours.” Where is the golden key?

Since all the inscriptions on the boxes are incorrect, the red box does not contain a golden key, the green box is not empty and there is not a snake in it, which means there is a key in the green box, a snake in the red box, and the blue box is empty.

When solving logical problems, logical thinking is activated, and this is the ability to derive consequences from premises, which is extremely necessary for the successful mastery of mathematics.

A rebus is a riddle, but it is not an ordinary riddle. Words and numbers in math puzzles depicted using pictures, stars, numbers and various symbols. To read what is encrypted in the rebus, you need to correctly name all the depicted objects and understand which sign represents what. People used puzzles even when they couldn’t write. They composed their letters from objects. For example, the leaders of one tribe once sent their neighbors, instead of a letter, a bird, a mouse, a frog and five arrows. This meant: “Can you fly like birds and hide in the ground like mice, jump through swamps like frogs? If you don’t know how, then don’t try to fight with us. We will shower you with arrows as soon as you enter our country."

Judging by the first letter of the sum 1), D = 1 or 2.

Let's assume that D = 1. Then, Y? 5. We exclude Y = 5, because P cannot be equal to 0. Y? 6, because 6 + 6 = 12, i.e. P = 2. But this value of P is not suitable for further verification. Likewise, U? 7.

Let's assume that Y = 8. Then, P = 6, A = 2, K = 5, D = 1.

A magic (magic) square is a square in which the sum of the numbers vertically, horizontally and diagonally is the same.

Task. Arrange the numbers from 1 to 9 so that vertically, horizontally and diagonally you get the same sum of numbers equal to 15.

Although there are no general rules for solving non-standard problems (that is why these problems are called non-standard), we have tried to give a number of general guidelines - recommendations that should be followed when solving non-standard problems of various types.

Each non-standard problem is original and unique in its solution. In this regard, the developed teaching methodology search activity when solving non-standard problems, it does not develop skills for solving non-standard problems; we can only talk about practicing certain skills:

· ability to understand the task, highlight the main (support) words;

· the ability to identify conditions and questions, known and unknown in a problem;

· the ability to find a connection between the given and the desired, that is, to analyze the text of the problem, the result of which is the choice of an arithmetic operation or logical operation to solve a non-standard problem;

· ability to record the progress of solving and answering a problem;

· ability to conduct extra work over the task;

· ability to select useful information contained in the problem itself, in the process of solving it, systematize this information, correlating it with existing knowledge.

Non-standard tasks develop spatial thinking, which is expressed in the ability to recreate spatial images of objects in the mind and perform operations on them. Spatial thinking manifests itself when solving problems like: “On top of the edge of a round cake, 5 dots of cream were placed at the same distance from each other. Cuts were made through all pairs of points. How many pieces of cake were there in total?

Practical method may be considered for non-standard division problems.

Task. The stick needs to be cut into 6 parts. How many cuts will be required?

Solution: 5 cuts will be required.

When studying non-standard division problems, you need to understand: in order to cut a segment into P parts, you must make (P - 1) cuts. This fact must be established inductively with children and then used when solving problems.

Task. A three-meter block has 300 cm. It must be cut into bars 50 cm long each. How many cuts should be made?

Solution: We get 6 bars 300: 50 = 6 (bars)

We reason like this: to divide a block in half, i.e. into two parts, you need to make 1 cut, into 3 parts - 2 cuts, and so on, into 6 parts - 5 cuts.

So, you need to make 6 - 1 = 5 (cuts).

Answer: 5 cuts.

So, one of the main motives that encourages schoolchildren to study is interest in the subject. Interest is a person’s active cognitive focus on a particular object, phenomenon and activity, created with a positive emotional attitude towards them. One of the means of developing interest in mathematics is non-standard problems. A non-standard problem is understood as a problem for which the mathematics course does not have general rules and regulations that define the exact program for solving it. Solving such problems allows students to actively engage in learning activities. There are various classifications of problems and methods for solving them. The most commonly used are algebraic, arithmetic, practical and enumeration methods, reasoning and assumptions.

2. Formationamong schoolchildrenskills to solve non-standard problems

2.1 Non-standard tasks for elementary school students

Didactic material is intended for primary school students and teachers. It contains non-standard mathematical problems that can be used in lessons and in extracurricular activities. The problems are structured by solution methods: arithmetic, practical methods, brute-force methods, reasoning and assumptions. Problems are presented in different types: mathematical entertainment; various number puzzles; logical tasks; tasks whose solution is based on a combination of mathematical development and practical ingenuity: weighing and transfusion under difficult conditions; mathematical sophisms; joke tasks; combinatorial problems. Solutions and answers are provided for all problems.

· Solve problems using the arithmetic method:

1. Added 111 thousand, 111 hundreds and 111 ones. What number did you get?

2. How much do you get if you add up the numbers: smallest two-digit, smallest three-digit, smallest four-digit?

3. Task:

"To the gray hat for class

Seven forty arrived

And of them only 3 are magpies

We have prepared our lessons.

How many quitters - forty

Arrived for class?

4. Petya needs to climb 4 times more steps than Kolya. Kolya lives on the third floor. What floor does Petya live on?

5. According to the doctor's prescription, 10 tablets were bought at the pharmacy for the patient. The doctor prescribed me to take 3 tablets a day. How many days will this medicine last?

· Solve problems using brute-force methods:

6. Insert “+” or “-” signs instead of the asterisk so that you get the correct equality:

a) 2 * 3 * 1 = 6;

b) 6 * 2 * 3 = 1;

c) 2 * 3 * 1 = 4;

d) 8 * 1 * 4 = 5;

e) 7 * 2 * 4 = 5.

7. There are no “+” and “-” signs between the numbers. It is necessary to arrange the signs as quickly as possible in such a way as to make 12.

a) 2 6 3 4 5 8 = 12;

b) 9 8 1 3 5 2 = 12;

c) 8 6 1 7 9 5 = 12;

d) 3 2 1 4 5 3 = 12;

e) 7 9 8 4 3 5 = 12.

8. Olya was given 4 books with fairy tales and poems for her birthday. There were more books with fairy tales than books with poetry. How many books with fairy tales were given to Olya?

9. Vanya and Vasya decided to buy candy with all their money. But here’s the problem: they had money for 3 kg of candy, but the seller only had 5 kg and 2 kg weights. But Vanya and Vasya got an “A” in mathematics, and they managed to buy what they wanted. How did they do it?

10. Three girlfriends - Vera, Olya and Tanya - went into the forest to pick berries. To collect berries they had a basket, a basket and a bucket. It is known that Olya was not with a basket or a basket, Vera was not with a basket. What did each of the girls take with them to pick berries?

11. In the gymnastics competition, the Hare, Monkey, Boa Constrictor and Parrot took the first 4 places. Determine who took what place, if it is known that the Hare was 2, the Parrot did not become a winner, but was a prize-winner, and the Boa constrictor lost to the Monkey.

12. Milk, lemonade, kvass and water are poured into a bottle, glass, jug and jar. It is known that water and milk are not in a bottle, neither lemonade nor water are in a jar, but a vessel with lemonade stands between a jug and a vessel with kvass. The glass stands next to the jar and the vessel with milk. Determine which liquid is which.

13. At the New Year's party, three friends, Anya, Vera and Dasha, were active participants, one of them was the Snow Maiden. When their friends asked which of them was the Snow Maiden, Anya told them: “Each of us will give our own answer to your question. Based on these answers, you should guess for yourself which of us was really the Snow Maiden. But know that Dasha always tells the truth.” “Okay,” the friends answered, “let’s listen to your answers. It’s even interesting.”

Anya: “I was the Snow Maiden.”

Vera: “I was not the Snow Maiden.”

Dasha: “One of them is telling the truth, and the other is lying.”

So which of the friends at the New Year's party was the Snow Maiden?

14. The staircase consists of 9 steps. Which step do you need to stand on to be right in the middle of the stairs?

15. What is the middle step of a 12-step staircase?

16. Anya told her brother: “I am 3 years older than you. How many years older will I be than you in 5 years?”

17. Divide the clock dial into two parts with a straight line so that the sums of the numbers in these parts are equal.

18. Divide the clock dial into three parts with two straight lines so that, by adding the numbers, the same sums are obtained in each part.

· Solve problems using a practical method:

19. The rope was cut in 6 places. How many parts did you get?

20. 5 brothers were walking. Each brother has one sister. How many people were there in total?

21. What is heavier: a kilogram of cotton wool or half a kilogram of iron?

22. A rooster, standing on one leg, weighs 3 kg. How much will a rooster weigh standing on two legs?

· Solve problems by assumption method:

23. How to write the number 10 using five identical numbers, connecting them with action signs?

24. How to write the number 10 with four different numbers, connecting them with action signs?

25. How can the number 5 be written as three identical numbers, connecting them with action signs?

26. How can the number 1 be written as three different numbers, connecting them with action signs?

27. How can you get 2 liters of water from the tap using a six-liter and a four-liter vessel?

28. A seven-liter vessel is filled with water. There is a five-liter vessel nearby, and it already contains 4 liters of water. How many liters of water must be poured from the larger vessel into the smaller one so that it is filled to the top? How many liters of water will remain in the larger vessel after this?

29. The baby elephant got sick. To treat it, exactly 2 liters of orange juice are required, and Dr. Aibolit only has a full five-liter jar of juice and an empty three-liter jar. How can Aibolit measure exactly 2 liters of juice?

30. An incredible story happened with Winnie the Pooh, Piglet and Rabbit. Winnie the Pooh used to love honey, Rabbit loved cabbage, and Piglet loved acorns. But once they got into the enchanted forest and got hungry, they discovered that their tastes had changed, but everyone still preferred one thing. The rabbit said: “I don’t eat cabbage and acorns.” Piglet remained silent, and Winnie the Pooh remarked: “I don’t like cabbage.” Who began to love eating?

Answers and solutions

1. 111000 + 11100 + 111 = 122211.

2. 10 + 100 + 1000 = 110.

4. Petya lives on the 9th floor. Kolya lives on the third floor. There are 2 “spans” to the third floor: from the first to the second, from the second to the third. Since Petya needs to go through 4 times more steps, then 2 4 = 8. This means that Kolya needs to go through 8 “flights”, and to the 9th floor there are 8 “flights”.

5. 3+3+3+1=10. On the fourth day, only 1 tablet will remain.

a) 2 + 3 - 1 = 4;

b) 2 + 3 + 1 = 6;

c) 6 - 2 - 3 = 1;

d) 8 + 1 - 4 = 5;

e) 7 + 2 - 4 = 5.

a) 2 + 6 - 3 + 4 - 5 + 8 = 12;

b) 9 + 8 + 1 - 3 - 5 + 2 = 12;

c) 8 - 6 - 1 + 7 + 9 - 5 = 12;

d) 3- 2 - 1 + 4 + 5 + 3 = 12;

e) 7 + 9 + 8 - 4 - 3 - 5 = 12.

8. The number 4 can be represented as the sum of two different terms in the only way: 4 - 3 + 1. There were more books with fairy tales, which means there were 3 of them.

9. Place a 5 kg weight on one cup of the scale, and lollipops and a 2 kg weight on the other.

	Basket

10. Let's put the problem conditions in the table, and, where possible, put the pros and cons:



Monkey

It turned out that Monkey and Boa constrictor are in first and fourth place, but since, according to the condition, Boa constrictor lost to Monkey, it turns out that Monkey is in first place, Parrot is in second and Boa constrictor is in fourth.

11. We will put the conditions that water is not in a bottle, milk is not in a bottle, lemonade is not in a can, water is not in a can in the table. From the condition that the vessel with lemonade stands between the jug and the vessel with kvass, we conclude that the lemonade is not in the jug and the kvass is not in the jug. And since the glass is standing next to the jar and the vessel with milk, we can conclude that the milk is not in the jar or in the glass. Let’s arrange “+”, and as a result we get that milk is in a jug, lemonade is in a bottle, kvass is in a jar and water is in a glass.

12. From Dasha’s statement we get that among the statements of Anya and Vera, one is true and the other is false. If Vera’s statement is false, then we will get that both Anya and Vera were Snow Maidens, which cannot be. This means that Anya’s statement must be false. In this case, we get that Anya was not the Snow Maiden, and neither was Vera. It remains that Dasha was the Snow Maiden.

When multiplying the number 51 by single digit number got it again two-digit number. This is only possible if it is multiplied by 1. This means that the second factor is 11.

13. When you multiply the first factor by 2, you get a four-digit number, and when you multiply by the hundreds digit and the units digit, you get three-digit numbers. We conclude that the second factor is 121. The first digit of the first factor is 7, and the last is 6. We get the product of the numbers 746 and 121. The 1st digit in the 1st factor is 7, the last is 6.

14. To the fifth step.

15. The staircase will not have 12 steps middle stage, it only has a couple of middle steps - the sixth and seventh. The solution to this problem, like the previous one, can be checked by drawing.

16. For 3 years.

17. You need to draw a line between the numbers 3 and 4 and between 10 and 9.

18. 11, 12, 1, 2; 9, 10, 3, 4: 5, 6, 7, 8.

19. You will get 7 parts.

20. 6 people 5 brothers and 1 sister.

21. Kilogram of cotton wool

22. 3 kg.

23. 2 + 2 + 2 + 2 + 2 = 10.

24. 1 + 2 + 3 + 4 = 10

25. 5 + 5 - 5 = 5

26. 4 - 2 - 1; 4 - 1 - 2; 5 - 3 - 1; 6 - 4 - 1; 6 - 2 - 3, etc.

27. Fill a six-liter container, pour water from it into a four-liter container, 2 liters will remain.

28. It is necessary to pour 1 liter of water, while 6 liters will remain in the larger vessel.

29. Pour 3 liters of juice into a three-liter jar, then 2 liters of juice will remain in the large jar.

30. Rabbit - honey, Winnie the Pooh - acorns, Piglet - cabbage.

...

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NON-STANDARD TASKS IN MATHEMATICS LESSONS

Teacher primary classes Shamalova S. V.

Each generation of people makes its own demands on school. An ancient Roman proverb says: “We study not for school, but for life.” The meaning of this proverb is still relevant today. Modern society dictates to the education system an order to educate an individual who is ready to live in constantly changing conditions, to continue education, and who is capable of learning throughout his life.

Among the spiritual abilities of man, there is one that has been the subject of close attention of scientists for many centuries and which, at the same time, is still the most difficult and mysterious subject of science. This is the ability to think. We constantly encounter it in work, in learning, in everyday life.

Any activity of a worker, schoolchild and scientist is inseparable from mental work. In any real matter, it is necessary to rack your brains, to stretch your mind, that is, in the language of science, you need to carry out a mental action, intellectual work. It is known that a problem can be solved or not solved, one person will cope with it quickly, the other thinks for a long time. There are tasks that are feasible even for a child, and some have been worked on by entire teams of scientists for years. This means there is the ability to think. Some are better at it, others worse. What kind of skill is this? In what ways does it arise? How to buy it?

No one will argue that every teacher should develop the logical thinking of students. This is stated in methodological literature, in explanatory notes to curriculum. However, we teachers do not always know how to do this. This often leads to the development logical thinking largely spontaneously, so most students, even high school students, do not master the initial techniques of logical thinking (analysis, comparison, synthesis, abstraction, etc.).

According to experts, the level of logical culture of schoolchildren today cannot be considered satisfactory. Experts believe that the reason for this lies in the lack of work on the targeted logical development of students in the early stages of education. Most modern manuals for preschoolers and primary schoolchildren contain a set of various tasks that focus on such techniques of mental activity as analysis, synthesis, analogy, generalization, classification, flexibility and variability of thinking. In other words, the development of logical thinking occurs largely spontaneously, so most students do not master thinking techniques even in high school, and these techniques need to be taught to younger students.

In my practice I use modern educational technology, various shapes organizations educational process, a system of developmental tasks. These tasks should be developmental in nature (teach certain thinking techniques), they should take into account age characteristics students.

In the process of solving educational problems, children develop the ability to be distracted from unimportant details. This action is given to younger schoolchildren with no less difficulty than highlighting the essential. Younger schoolchildren as a result of studying at school, when it is necessary to regularly complete tasks in mandatory, learn to manage their thinking, to think when necessary. First, logical exercises accessible to children are introduced, aimed at improving mental operations.

In the process of performing such logical exercises, students practically learn to compare various objects, including mathematical ones, to build correct judgments on what is available, and to carry out simple proofs using their life experience. Logic exercises gradually become more complex.

I also use non-standard developmental logical tasks in my practice. There is a significant variety of such problems; Especially a lot of such specialized literature has been published in recent years.

In the methodological literature, the following names have been assigned to developmental tasks: tasks for intelligence, tasks for ingenuity, tasks with a “twist”. In all their diversity, we can distinguish into a special class such tasks, which are called tasks - traps, provoking tasks. The conditions of such tasks contain various kinds of references, instructions, hints that encourage the choice of an erroneous solution path or an incorrect answer. I will give examples of such tasks.

Problems that impose one, very definite answer.

Which of the numbers 333, 555, 666, 999 is not divisible by 3?

Tasks that encourage you to make an incorrect choice of answer from the proposed correct and incorrect answers.

One donkey is carrying 10 kg of sugar, and the other is carrying 10 kg of popcorn. Who had the heavier luggage?

Tasks whose conditions push you to perform some action with given numbers, whereas there is no need to perform this action at all.

The Mercedes car traveled 100 km. How many kilometers did each of its wheels travel?

Petya once said to his friends: “The day before yesterday I was 9 years old, and next year I will turn 12 years old.” What date was Petya born?

Solving logical problems using reasoning.

Vadim, Sergey and Mikhail study various foreign languages: Chinese, Japanese, Arabic. When asked what language each of them was studying, one replied: “Vadim is studying Chinese, Sergei is not studying Chinese, and Mikhail is not studying Arabic.” Subsequently, it turned out that only one statement in this statement is true. What language is each of them studying?

The shorties from Flower City planted a watermelon. Watering it requires exactly 1 liter of water. They only have two empty 3 liter cans. And 5 l. How to use these cans. Collect exactly 1 liter from the river. water?

How many years did Ilya Muromets sit on the stove? It is known that if he had stayed in prison 2 more times, his age would have been the largest two-digit number.

Baron Munchausen counted the number of magical hairs in the beard of old man Hottabych. It turned out to be equal to the sum of the smallest three-digit number and the largest two-digit number. What is this number?

When learning to solve non-standard problems, I observe the following conditions:V first of all , tasks should be introduced into the learning process in a certain system with a gradual increase in complexity, since an impossible task will have little effect on the development of students;V o secondly , it is necessary to provide students with maximum independence when searching for solutions to problems, give them the opportunity to go to the end along the wrong path in order to be convinced of the mistake, return to the beginning and look for another, correct path of solution;Thirdly , we need to help students understand some ways, techniques and general approaches to solving non-standard arithmetic problems. Most often, the proposed logical exercises do not require calculations, but only force children to make correct judgments and provide simple proofs. The exercises themselves are entertaining in nature, so they contribute to the emergence of children’s interest in the process of mental activity. And this is one of the cardinal tasks of the educational process at school.

Examples of tasks used in my practice.

Find the pattern and continue the garlands

Find a pattern and continue the series

a B C D E F, …

1, 2, 4, 8, 16,…

The work began with the development in children of the ability to notice patterns, similarities and differences as tasks gradually became more complex. For this purpose I selectedtasks to identify patterns, dependencies and formulate generalizationswith a gradual increase in the level of difficulty of tasks.Work on the development of logical thinking should become the object of serious attention of the teacher and be systematically carried out in mathematics lessons. For this purpose, logic exercises should always be included in oral work in class. For example:

Find the result using this equality:

3+5=8

3+6=

3+7=

3+8=

Compare the expressions, find the commonality in the resulting inequalities, formulate a conclusion:

2+3*2x3

4+4*3x4

4+5*4x5

5+6*5x6

Continue the series of numbers.

3. 5, 7, 9, 11…

1, 4, 7, 10…

Come up with a similar example for each given example.

12+6=18

16-4=12

What do the numbers on each line have in common?

12 24 20 22

30 37 13 83

Numbers given:

23 74 41 14

40 17 60 50

Which number is the odd one in each line?

In elementary school math lessons, I often use counting stick exercises. These are problems of a geometric nature, since during the solution, as a rule, there is transfiguration, the transformation of some figures into others, and not just a change in their number. They cannot be solved in any previously learned way. In the course of solving each new problem, the child is involved in an active search for a solution, while striving for the final goal, the required modification of the figure.

Exercises with counting sticks can be combined into 3 groups: tasks on composing a given figure from a certain number of sticks; tasks for changing figures, to solve which you need to remove or add a specified number of sticks; tasks, the solution of which consists in rearranging sticks in order to modify, transform a given figure.

Exercises with counting sticks.

Tasks on making figures from a certain number of sticks.

Make two different squares using 7 sticks.

Problems involving changing a figure, where you need to remove or add a specified number of sticks.

Given a figure of 6 squares. You need to remove 2 sticks so that 4 squares remain."

Problems involving rearranging sticks for the purpose of transformation.

Arrange two sticks to make 3 triangles.

Regular exercise is one of the conditions for the successful development of students. First of all, from lesson to lesson it is necessary to develop the child’s ability to analyze and synthesize; short-term teaching of logical concepts does not give effect.

Solving non-standard problems develops in students the ability to make assumptions, check their accuracy, and justify them logically. Speaking for the purpose of evidence contributes to the development of speech, the development of skills to draw conclusions, and build conclusions. In the process of using these exercises in lessons and in extracurricular work in mathematics, a positive dynamics of the influence of these exercises on the level of development of students’ logical thinking appeared.

No wonder that entertaining mathematics has become entertainment “for of all times and peoples." To solve such problems, no special knowledge is required - one guess is enough, which, however, is sometimes more difficult to find than methodically solving a standard school problem.

Solving a fun arithmetic problem.
For 3 – 5 grades

How many dragons?

2-headed and 7-headed dragons gathered for a rally.
At the very beginning of the meeting, the Dragon King, the 7-headed Dragon, counted everyone gathered by their heads.

He looked around his crowned middle head and saw 25 heads.
The king was pleased with the results of the calculations and thanked everyone present for their attendance at the meeting.

How many dragons came to the rally?

(a) 7; (b) 8; 9; (d) 10; (e) 11;
Solution:

Let us subtract 6 heads belonging to him from the 25 heads counted by the Dragon King.

There will be 19 goals left. All remaining Dragons cannot be two-headed (19 is an odd number).

There can only be 1 7-headed Dragon (if 2, then for two-headed Dragons there will be an odd number of heads left. And for three Dragons there are not enough heads: (7 · 3 = 21 > 19).

Subtract 7 heads of this single Dragon from 19 heads and get the total number of heads belonging to two-headed Dragons.

Therefore, 2-headed Dragons:
(19 - 7) / 2 = 6 Dragons.

Total: 6 +1 +1 (King) = 8 Dragons.

Correct answer:b = 8 Dragons

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Solving a fun math problem

For 4 - 8 grades

How many wins?

Nikita and Alexander are playing chess.
Before the game started, they agreed

that the winner of the game will receive 5 points, the loser will receive no points, and each player will receive 2 points if the game ends in a draw.

They played 13 games and got 60 points together.
Alexander received three times more points for those games that he won than for those that were drawn.

How many victories did Nikita win?

(a) 1; (b) 2; 3; (d) 4; (e) 5;
Correct answer: (b) 2 victories (Nikita won)

Solution.

Each draw game gives 4 points, and each win gives 5 points.
If all the games ended in a draw, the boys would score 4 · 13 = 52 points.
But they scored 60 points.

It follows that 8 games ended with someone winning.
And 13 - 5 = 5 games ended in a draw.

Alexander scored 5 · 2 = 10 points in 5 draw games, which means that if he won, he scored 30 points, that is, he won 6 games.
Then Nikita won (8-6=2) 2 games.

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Solving a fun arithmetic problem

For 4 - 8 grades

How many days without food?
The Martian interplanetary spacecraft arrived on a visit to Earth.
Martians eat at most once a day, either in the morning, at noon, or in the evening.

But they only eat when they feel hungry. They can go without food for several days.
During the Martians' stay on Earth, they ate 7 times.
We also know that they went without food 7 times in the morning, 6 times at noon and 7 times in the evening.
How many days did the Martians spend without food during their visit?

(a) 0 days; (b) 1 day; 2 days; (d) 3 days; (e) 4 days; (a) 5 days;
Correct answer: 2 days (the Martians spent without food)

Solution.
The Martians ate for 7 days, once a day, and the number of days they ate lunch was one more number days when they had breakfast or dinner.

Based on these data, it is possible to create a food intake schedule for Martians. This is the probable picture.

The aliens had lunch on the first day, had dinner on the second day, had breakfast on the third, had lunch on the fourth, had dinner on the fifth, had breakfast on the sixth, and had lunch on the seventh.

That is, the Martians ate breakfast for 2 days, and spent 7 days without breakfast, ate dinner 2 times, and spent 7 days without dinner, ate lunch 3 times, and lived without lunch for 6 days.

So 7 + 2 = 9 and 6 + 3 = 9 days. This means they lived on Earth for 9 days, and 2 of them went without food (9 - 7 = 2).

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Solving an entertaining non-standard problem

For 4 - 8 grades

How much time?
The cyclist and the pedestrian left point A at the same time and headed to point B at a constant speed.
The cyclist arrived at point B and immediately set off on the way back and met the Pedestrian an hour later from the moment they left point A.
Here the Cyclist turned around again and they both began to move in the direction of point B.

When the cyclist reached point B, he turned back again and met the Pedestrian again 40 minutes after their first meeting.
What is the sum of the digits of a number expressing the time (in minutes) required for a Pedestrian to get from point A to point B?
(a) 2; (b) 14; 12; (d) 7; (e)9.
Correct answer: e) 9 (the sum of the digits of the number is 180 minutes - this is how long the Pedestrian travels from A to B)

Everything becomes clear if you draw a drawing.
Let's find the difference between the two paths of the Cyclist (one path is from A to the first meeting (solid green line), the second path is from the first meeting to the second (dashed green line)).

We find that this difference is exactly equal to the distance from point A to the second meeting.
A pedestrian covers this distance in 100 minutes, and a cyclist travels in 60 minutes - 40 minutes = 20 minutes. This means the cyclist travels 5 times faster.

Let us denote the distance from point A to the point at which 1 meeting occurred as one part, and the Cyclist’s path to the 1st meeting as 5 parts.

Together, by the time of their first meeting, they had covered double the distance between points A and B, i.e. 5 + 1 = 6 parts.

Therefore, from A to B there are 3 parts. After the first meeting, the pedestrian will have to walk another 2 parts to point B.

He will cover the entire distance in 3 hours or 180 minutes, since he covers 1 part in 1 hour.

Non-standard tasks. Non-standard tasks

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