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Introduction
1. Theoretical basis formation of 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 of schoolchildren's skills to solve non-standard tasks
2.1 Non-standard tasks for elementary school students
2.2 Non-standard tasks for the main school
Conclusion
Literature
Introduction
Strategy modern education is to provide an opportunity for all students to show their talents and creativity, which implies the possibility of implementing personal plans. Therefore, today the problem of finding means for the development of mental abilities associated with the creative activity of students, both in collective and in individual forms of education, is relevant. The work of teachers T.M. is devoted to this problem. Davydenko, L.V. Zankova, A.I. Savenkov and others, which focus on determining the means of increasing the productive cognitive activity of students, organizing their creative activity.
Interest in the subject contributes to the active acquisition of knowledge, as students study by virtue of their inner attraction, of their own free will. Then educational material they learn quite easily and thoroughly. But lately, an alarming and paradoxical fact has been noted: interest in learning is decreasing from class to class, despite the fact that interest in the phenomena and events of the surrounding world continues to develop, becoming more complex in content.
Raising the interest of schoolchildren in mathematics, the development of their mathematical abilities is impossible without the use of educational process quick wit tasks, joke tasks, numerical puzzles, fairy tale tasks, etc. In this regard, there has been a tendency to use non-standard tasks as a necessary component of teaching students mathematics (S. G. Guba, 1972).
Pedagogical experience shows that “… effectively organized educational activity of students in the process of solving non-standard problems is the most important means of forming mathematical culture and the qualities of mathematical thinking; the organic combination of these qualities is manifested in the special abilities of a person, giving him the opportunity to successfully carry out creative activity.
Thus, on the one hand, it is necessary to teach students to solve non-standard tasks, since such tasks play a special role in shaping interest in the subject and in shaping creative personality On the other hand, numerous data indicate that the issue of developing the ability to solve such problems, learning how to find solutions to problems is not given due attention.
The foregoing determined the choice of the research topic: "Non-standard tasks as a means of forming students' interest in mathematics."
Object of study - the process of forming interest in mathematics among schoolchildren.
Subject of study- the formation of students' skills to solve non-standard problems for the formation of interest in mathematics.
Purpose of the study- to prove that knowledge of various methods contributes to the formation of students' skills to solve non-standard problems.
In accordance with the goal, the research objectives:
· Study of psychological-pedagogical and scientific-methodical literature and characterization of the concepts of "interest" and "non-standard task".
· Identification of types of non-standard tasks.
· Acquaintance with methods of solving non-standard problems.
· Compilation of didactic materials for students on the formation of skills 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 is of a theoretical nature, it discusses various interpretations of the concept of “interest”, highlights the role of non-standard tasks in shaping students' interest in mathematics, and provides some classifications of non-standard tasks. The second chapter presents the didactic material compiled by the author of the study, aimed at developing the skills to solve non-standard problems using different methods.
In the course of the study, a theoretical method, analysis of educational and methodological literature, and modeling were used.
1. Theoretical foundations for the formation of interest in mathematics
1.1 Essence understoodand I« interest»
There are different approaches to the concept of "interest". Various Methodists and scholars interpret it differently. So, for example, a linguist, lexicographer, doctor of philological sciences and professor Sergey Ivanovich Ozhegov gives several definitions of the concept of "interest":
1. Special attention to something, the desire to delve into the essence, to know, to understand. (Show interest in the case. Lose interest in the interlocutor. Heightened interest in everything new).
2. Amusement, significance. (The interest of the story is in its plot. The case is of public interest).
3. Numerous needs, needs. (Group interests. Protecting our 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. Amusement or significance, the importance of the matter.
Interest is the selective orientation of a person, his attention, thoughts, thoughts (S.L. Rubinshtein).
Interest is a kind of fusion of emotional-volitional and intellectual processes, which increases the activity of consciousness and human activity (L.A. Gordon).
Interest is an active cognitive orientation of a person to a particular object, phenomenon and activity, created with a positive emotional attitude towards them (V.A. Krutetsky) ".
Human interests are determined by socio-historical and individual conditions his life. With the help of interest, the connection of the subject with the objective world is established. Everything that constitutes the subject of interest is gleaned by a person from the surrounding reality. But the subject of interest for a person is far from everything that surrounds him, but only what is necessary for him, significance, value and attractiveness.
People's interests are extremely diverse. There are several classifications of interests:
material interests (Manifested in the desire for housing, 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 personal development.);
public interest (Includes interest in community service, to organizational activities.);
by direction:
broad interests (A 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 the object in all details and subtleties.);
superficial interests (Sliding on the surface of the phenomenon and there is no real interest in the object.);
By strength:
sustainable interests (Long persist, play a significant role in the life and activities of a person and are relatively fixed features of his personality.);
unstable interests (Comparatively short-term: quickly arise and quickly fade away.);
By mediation:
direct (immediate) interests (Called by the very content of a particular field of knowledge or activity, its amusement and fascination.);
indirect (mediated) interests (They are not caused by the content of the object, but by the value that it has, being associated with another object that is of direct interest to a person.);
In terms 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 in order to master the object of interest.) (G.I. Shchukina, 1988).
There is a special kind of human interests - cognitive interest.
“Cognitive interest is the selective orientation of the personality, turned 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 knowledge. 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 this subject, but also to the process of obtaining this knowledge, to cognitive activity. mathematical pedagogical student
In pedagogy, along with the term "cognitive interest", the term "learning interest" is used. The concept of "cognitive interest" is broader, since in the zone of cognitive interest there are not only knowledge, limited curricula, but also going far beyond it.
IN foreign literature the term "cognitive interest" is missing, but there is the concept of "intellectual interest". 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 connection mental processes: intellectual, strong-willed and emotional. They are very important for personal development.
In intellectual activity, proceeding under the influence of cognitive interest, the following are manifested:
· active search;
· a guess;
research approach;
readiness to solve problems.
Emotional manifestations accompanying cognitive interest:
emotions of surprise
a sense of anticipation of something new;
feeling of intellectual joy;
feeling of success.
Volitional manifestations characteristic of cognitive interest are:
search initiative;
independence in obtaining knowledge;
Promotion and setting of cognitive tasks.
So, the intellectual, volitional and emotional aspects of 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 the difficulties that lie in the way of mastering knowledge and ways to obtain it.
Psychologists and educators identify three main motives that encourage students to learn:
Interest in the subject (I study mathematics not because I pursue some goal, but because the very process of studying gives me pleasure). The highest degree of interest is passion. Passionate activities generate strength positive emotions, and the inability to practice is perceived as deprivation.
· Consciousness. (Classes on this subject are not interesting to me, but I am aware of their necessity and by an effort of will I force myself to study).
· Coercion. (I study because my parents and teachers force me). Often the compulsion is supported by the fear of punishment or the lure of reward. Various coercive measures in most cases do not give positive results (25, p. 24).
Interest in high degree improves the effectiveness of lessons. If students study because of their inner inclination, of their own free will, then they learn the educational material quite easily and thoroughly, because of this they have good grades in the subject. Most underachieving students show a negative attitude towards learning. Thus, the higher the student's interest in the subject, the more active the learning is and the better its results. The lower the interest, the more formal the training, the worse its results. Lack of interest leads to low quality of learning, quick forgetting and even complete loss of acquired knowledge, skills and abilities.
Forming the cognitive interests of students, it must be borne in mind that they cannot cover all academic subjects. Interests are selective, and one student, as a rule, can only engage in real passion in one or two subjects. But, the presence of a steady interest in a particular subject has a positive effect on academic work in other subjects, both intellectual and moral factors matter. Intensive mental development associated with in-depth study one subject, facilitates and makes more effective the teaching of the student in other subjects. On the other hand, success achieved in academic work in favorite subjects strengthens the feeling dignity student, and he strives to study diligently in general.
An important task of the teacher is to form the first two motives for learning in schoolchildren - interest in the subject and a sense of duty, 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 occurrence is associated with the appearance in children of such questions as “Why?”, “Why?”, “Why?”. Interest appears initially in the form of curiosity. To the end before school age under the influence of elders, the child develops an interest in learning at school: he not only plays 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 the teaching is formed. Over time, cognitive interests are differentiated: some like mathematics more, others like reading, etc. Children show great interest in the labor process, especially if it is performed in a team. Teaching and other types of cognition 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, pivotal interest as the basis of their life orientation and try themselves in different areas. When the interests and inclinations of adolescents are finally determined, then their abilities begin to form and manifest themselves clearly. By the end of adolescence, interests in a particular profession begin to form. In senior school age, the development of cognitive interests, the growth of a conscious attitude to learning determine further development arbitrariness of cognitive processes, the ability to manage them, consciously regulate them. At the end of senior age, 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 tasks. Let's dwell on them in more detail.
1. 2 Non-standard tasks and their types
The concept of "non-standard task" is used by many methodologists. So, Yu. M. Kolyagin reveals this concept as follows: “Under non-standard 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 the authors L.M. Fridman, E.N. Turkish: " Non-standard tasks- these are those for which there are no general rules and regulations in the course of mathematics that determine the exact program for their solution.
Do not confuse non-standard tasks with tasks of increased complexity. The conditions of problems of increased complexity are such that they allow students to quite easily select 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 implies the presence of an exploratory nature. However, if the solution of a problem in mathematics for one student is non-standard, since he is unfamiliar with the methods of solving problems of this type, then for another, the solution of the problem occurs in a standard way, since he has already solved such problems and more than one. The same task in mathematics in the 5th grade is non-standard, and in the 6th grade it is ordinary, and not even of increased complexity.
textbook analysis and teaching aids in mathematics shows that each text problem under certain conditions can be non-standard, and in others - ordinary, standard. A standard problem in one course of mathematics may be non-standard in another course.
Based on the analysis of the theory and practice of using non-standard tasks in teaching mathematics, one can establish their general and specific role. Non-standard tasks:
· teach children not only to use ready-made algorithms, but also to independently find new ways to solve problems, i.e. contribute to the ability to find original ways to solve problems;
influence the development of ingenuity, ingenuity of students;
They prevent the development of harmful clichés when solving problems, destroy incorrect associations in the knowledge and skills of students, involve not so much the assimilation of algorithmic techniques, but the discovery of new connections in knowledge, the transfer of knowledge to new conditions, and the mastery of various methods of mental activity;
create favorable conditions for increasing the strength and depth of knowledge of students, ensure the conscious assimilation of mathematical concepts.
Non-standard tasks:
should not have ready-made algorithms memorized by children;
should be accessible to all students in terms of content;
must be interesting in content;
To solve non-standard problems, students should have enough knowledge acquired by them in the program.
Solving non-standard tasks activates the activity of students. Students learn to compare, classify, generalize, analyze, and this contributes to a stronger and more 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 tasks, since this opens up the opportunity to truly differentiate the results of each participant. Such problems can also be successfully used as individual tasks for those students who easily and quickly cope with the main part. independent work in the classroom, or for those who wish as additional tasks. As a result, students receive intellectual development and preparation for active practical activity.
There is no generally accepted classification of non-standard tasks, but B.A. Kordemsky identifies the following types of such tasks:
· Tasks related to the school mathematics course, but of increased difficulty - such as tasks of mathematical Olympiads. They are intended mainly for schoolchildren with a definite interest in mathematics; thematically, these tasks are usually associated with one or another specific section of the school curriculum. The exercises related to this deepen the educational material, supplement and generalize individual provisions. school course, expand mathematical horizons, develop skills in solving difficult problems.
· Problems of the type of mathematical entertainment. They are not directly related to the school curriculum and, as a rule, do not require much mathematical preparation. This does not mean, however, that the second category of tasks includes only easy exercises. Here there are problems with a very difficult solution and such problems, the solution of which has not yet been obtained. “Non-standard tasks, presented in a fun way, bring an emotional moment to mental activities. Not connected with the need to apply learned rules and techniques every time to solve them, they require the mobilization of all accumulated knowledge, teach them to search for original, non-template methods of solving, enrich the art of solving beautiful examples, make you admire the power of the mind ".
These types of tasks include:
various numerical puzzles (“... examples in which all or some of the numbers are replaced by asterisks or letters. The same letters replace the same numbers, different letters- different numbers ".) and puzzles for ingenuity;
logical tasks, the solution of which does not require calculations, but is based on the construction of a chain of exact reasoning;
tasks, the solution of which is based on a combination of mathematical development and practical ingenuity: weighing and transfusions under difficult conditions;
mathematical sophistry is a deliberate, false conclusion that has the appearance of being correct. (Sophism is a proof of a false statement, and the error in the proof is skillfully disguised. Sophism in Greek means a cunning invention, trick, puzzle);
joke tasks;
combinatorial problems, in which various combinations of given objects that satisfy certain conditions are considered (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;
tasks that are unsolvable or unsolvable by means of a school course at a given level of knowledge of students;
Tasks that require:
conducting and using analogies, determining the differences between given objects, processes or phenomena, establishing the opposite of given phenomena and processes or their antipodes;
implementation of a practical demonstration, abstraction from certain properties of an object, process, phenomenon or concretization of one or another side of this phenomenon;
establishment of causal relationships between given objects, processes or phenomena;
construction of causal chains in an analytical or synthetic way with subsequent analysis of the resulting options;
the correct implementation of a sequence of certain actions, avoiding errors-"traps";
implementation of the 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 unified classification of non-standard tasks. There are several of them, but the author of the work used the classification proposed by I.V. Egorchenko.
1.3 Solution methodsstandard tasks
Russian philologist Dmitry Nikolaevich Ushakov in his explanatory dictionary gives such a definition of the concept of "method" - a way, a way, a 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 currently consider non-standard? Unfortunately, no one has come up with a universal recipe, given the uniqueness of these tasks. Some teachers train in template exercises. This happens as follows: the teacher shows the way to solve, and then the student repeats this when solving problems many times. At the same time, students' interest in mathematics is being killed, which is at least sad.
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, in developing creative abilities" .
Consider several methods for solving non-standard problems:
· algebraic;
· arithmetic;
enumeration method;
method of reasoning;
practical;
the method of guessing.
Algebraic Method problem solving develops creative abilities, the ability to generalize, forms abstract thinking and has such advantages as brevity of writing and reasoning when drawing up equations, saves time.
In order to solve a problem algebraic method necessary:
· to analyze the problem in order to choose the main unknown and identify the relationship between the quantities, as well as the expression of these dependencies in mathematical language in the form of two algebraic expressions;
find the basis for connecting these expressions with the sign "=" and make an equation;
find solutions to the resulting equation, organize a check of the solution of 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 what relationships these quantities can have in general, and whether they understand the real meaning of these relationships (for example, relationships expressed in the terms “later by ...”, “older by ... times " and so on.). The next step is to understand how mathematical action or, the property of the action or what connection (dependency) between the components and the result of the action can describe one or another specific relationship.
Let us give an example of solving a non-standard problem by the algebraic method.
Task. The fisherman caught a fish. When 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 body; then (1+1/2x) kg is the mass of the head. Since, by 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 solutions also require a lot of mental stress, which has a positive effect on the development of mental abilities, mathematical intuition, on the formation of the ability to foresee a real life situation.
Consider an example of solving a non-standard problem by an arithmetic method:
Task. Two fishermen were asked, "How many fish are in your baskets?"
“In my basket is half of what he has in the basket, and 10 more,” the first answered. “And I have as many in my basket as he has, and even 20,” the second one calculated. We counted, and now you count.
Let's build a diagram for the problem. Let the first segment of the diagram denote the number of fish the first fisherman has. The second segment denotes the number of fish from the second fisherman.
Due to modern man it is necessary to have an idea about the main methods of data analysis and the probabilistic patterns that play important role in science, technology and economics, elements of combinatorics, probability theory and mathematical statistics, which are easy to understand with the help of enumeration method.
The inclusion of combinatorial problems in the course of mathematics provides positive influence on the development of students. “Targeted learning to solve combinatorial problems contributes to the development of such a quality of mathematical thinking as variability. Under the variability of thinking, we mean the direction of the student's mental activity to search for various solutions to the problem in the case when there are no special instructions for this.
Combinatorial problems can be solved by 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, but the options themselves are not formed in this case.
With the “informal” method of solving, the process of compiling various options comes to the fore. And the main thing is not how much, but what options can be obtained. Such methods include enumeration method. This method is available even to younger students, and allows you to gain 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 compose all these options, and, having mastered the methods of systematic enumeration, this can be done more rationally.
Tasks are divided into three groups according to the complexity of enumeration:
1 . Tasks in which you need to make a complete enumeration of all possible options.
2. Tasks in which it is impractical to use the full enumeration technique and it is necessary to immediately exclude some options without considering them (that is, to carry out an abbreviated enumeration).
3. Tasks in which the enumeration operation is performed several times and in relation to various kinds of objects.
Here are the relevant examples of tasks:
Task. Placing the signs "+" and "-" between the given numbers 9 ... 2 ... 4, make up all possible expressions.
There is a full list of options:
a) two characters 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: large and small squares, large and small circles so that the circle is in the first place and the figures of the same shape do not stand side by side, and invites the students to guess the sequence in which these figures are arranged.
There are 24 different arrangements of these figures in total. And compose them all, and then choose the appropriate this condition impractical, so a reduced enumeration is carried out.
May be in the first place big circle, then the small one can only be in third place, while the large and small squares can be placed in two ways - in second and fourth place.
A similar reasoning is carried out if the first place is a small circle, and two options are also compiled.
Task. Three partners of the same firm keep securities in a safe with 3 locks. The companions 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 companions, 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 companion, and this does not meet the condition.
Let's assume that each companion has one key. Then if two of them come, they won't be able to open the safe.
Let's 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 come 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 can come, they will also have all the keys (1 and 3, 2 and 3).
Thus, to find the answer in this problem, you need to perform the iteration operation several times.
When selecting combinatorial problems, one should pay attention to the subject and form of presentation of these problems. It is desirable that the tasks do not look artificial, but are understandable and interesting to children, evoke positive emotions in them. Can be used to create tasks practical material from life.
There are other problems that can be solved by enumeration.
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 passed since then, if now the Cat is three times younger than its owner? The enumeration of options is represented by a table.
Age of the Marquis of Carabas and Puss in Boots
14 - 3 = 11 (years)
Answer: 11 years have passed.
At the same time, the student, as it were, experiments, observes, compares facts and, on the basis of 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 enumeration problems. In this case, the word "enumeration" is used in the sense of analyzing all possible cases that satisfy the conditions of the problem, showing that there can be no other solutions.
This problem can also be solved by an algebraic method.
Let the Cat be x years old, then the Marquis is 3x, based on the condition of the problem, we will compose the equation:
The cat is now 14 years old, then 14 - 3 = 11 (years) passed.
Answer: 11 years have passed.
reasoning method can be used to solve mathematical sophisms.
The mistakes made in sophism usually come down to the following: performing "forbidden" actions, using erroneous drawings, incorrect word usage, inaccurate wording, "illegal" generalizations, incorrect applications of theorems.
To reveal sophism means to point out an error in reasoning, based on which the external appearance of proof was created.
Analysis of sophisms, first of all, develops logical thinking, instills the skills of correct thinking. To detect an error in sophism means to recognize it, and the awareness of an error prevents it from being repeated in other mathematical reasoning. In addition to the criticality of mathematical thinking, this type of non-standard tasks reveals the flexibility of thinking. Will the student be able to “break out of the grip” of this path, which at first glance is strictly logical, break the chain of inferences at the very link that is erroneous and makes all further reasoning erroneous?
The 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 a sophism with an incorrect application of the theorem.
Let us prove that 2 2 = 5.
Let's take the following obvious equality as the initial ratio: 4: 4 = 5: 5 (1)
We take out of brackets the common factor in the left and right parts, we get:
4 (1: 1) = 5 (1: 1) (2)
The numbers in brackets are equal, so 4 = 5 or 2 2 = 5.
In the reasoning, when passing from equality (1) to equality (2), an illusion of likelihood is created on the basis of a false analogy with the distributive property of multiplication with respect to addition.
b) Sophism using "illegal" generalizations.
There are two families - Ivanovs and Petrovs. Each consists of 3 people - father, mother and son. Ivanov's father does not know Petrov's father. 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 in marital status, this does not mean that he does not know the whole family. For example, Ivanov's father may know Petrov's mother and son.
The reasoning method can also be used to solve logical problems. Sublogical tasks are usually understood as tasks that are solved using only logical operations. Sometimes their solution requires lengthy reasoning, the necessary direction of which cannot be foreseen 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 a golden key”, and on the blue one - “The green box is empty”, and on the green one - “Here sits a snake”. Tortila read the inscriptions and said: “Indeed, there is a golden key in one box, a snake in the other, and the third is empty, but all the inscriptions are wrong. 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 no snake in it, which means that the key is in the green box, the snake is in the red one, and the blue one is empty.
When solving logical problems, logical thinking is activated, and this is the ability to deduce consequences from premises, which is essential for the successful mastery of mathematics.
A rebus is a riddle, but a riddle is not quite an ordinary one. Words and numbers in mathematical puzzles are depicted using drawings, asterisks, numbers and various signs. To read what is encrypted in the rebus, you must correctly name all the objects depicted and understand which sign depicts 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 a bird, a mouse, a frog and five arrows instead of a letter to their neighbors. 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 us. We will bombard you with arrows as soon as you enter our country.”
Judging by the first letter of the sum 1), D = 1 or 2.
Suppose that D = 1. Then, Y? 5. Y \u003d 5 is excluded, because P cannot be equal to 0. Y? 6, because 6 + 6 = 12, i.e. P = 2. But such a value of P is not suitable for further verification. Likewise, U? 7.
Suppose 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 (which 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 task is original and unique in its solution. In this regard, the developed methodology for teaching search activities when solving non-standard tasks does not form skills for solving non-standard tasks, we can only talk about developing certain skills:
ability to understand the task, highlight the main (supporting) words;
the ability to identify the condition and question, known and unknown in the problem;
the ability to find a connection between the data and the desired, that is, to analyze the text of the problem, the result of which is the choice of an arithmetic operation or a logical operation to solve a non-standard problem;
the ability to record the progress of the solution and the answer to the problem;
· ability to carry out additional work on the task;
The ability to select useful information contained in the problem itself, in the process of solving it, to systematize this information, correlating it with the already existing knowledge.
Non-standard tasks develop spatial thinking, which is expressed in the ability to recreate in the mind the spatial images of objects and perform operations on them. Spatial thinking manifests itself when solving problems like: “Five dots of cream were placed on top of the edge of the round cake at the same distance from each other. Cuts were made through all pairs of points. How many pieces of cake did you get in total?
practical method can be considered for non-standard division problems.
Task. The stick needs to be cut into 6 pieces. How many cuts will be required?
Solution: Cuts will need 5.
When studying non-standard division problems, you need to understand: in order to cut a segment into P parts, you should make a (P - 1) cut. This fact must be established with children inductively, and then used in solving problems.
Task. In a three-meter bar - 300 cm. It must be cut into bars 50 cm long each. How many cuts do you need to make?
Solution: We get 6 bars 300: 50 = 6 (bars)
We argue as follows: to divide the bar in half, that is, 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 encourage students to study is interest in the subject. Interest is an active cognitive orientation of a person to 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 tasks. A non-standard task is understood as such tasks for which there are no general rules and regulations in the course of mathematics that determine the exact program for their solution. The solution of such problems allows students to actively engage in learning activities. There are various classifications of problems and methods for their solution. The most commonly used are algebraic, arithmetic, practical methods and enumeration, reasoning and conjecture.
2. Formationschoolchildrenability to solve non-standard tasks
2.1 Non-standard tasks for elementary school students
The didactic material is intended for elementary school students and teachers. It contains non-standard mathematical problems that can be used in the classroom and during extracurricular activities. Tasks are structured by solution methods: arithmetic, practical methods, enumeration, reasoning and assumptions. Tasks are presented in different types: mathematical entertainment; various numerical puzzles; logical tasks; tasks, the solution of which is based on a combination of mathematical development and practical ingenuity: weighing and transfusions under difficult conditions; mathematical sophisms; joke tasks; combinatorial tasks. Solutions and answers are given for all problems.
· Solve problems by arithmetic method:
1. Added up 111 thousand, 111 hundreds and 111 units. What was the number?
2. How much will you get if you add the numbers: the smallest two-digit, the smallest three-digit, the smallest four-digit?
3. Task:
"To the gray hat for the lesson
Seven forty arrived
And of them only 3 magpies
Prepared lessons.
How many loafers-forty
Arrived at the lesson?
4. Petya needs to go 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 for the patient, 10 tablets were bought at the pharmacy. The doctor prescribed to take medicine 3 tablets a day. How many days will this medicine last?
· Solve problems by enumeration:
6. Insert the signs "+" or "-" instead of the asterisk so that the correct equality is obtained:
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 that it turns out 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 presented with 4 books with fairy tales and poems for her birthday. There were more fairy tale books than poetry books. How many books with fairy tales were presented to Olya?
9. Vanya and Vasya decided to buy candy with all their money. Yes, that's bad luck: they had money for 3 kg of candy, and the seller had only weights of 5 kg and 2 kg. But Vanya and Vasya have "A" in mathematics, and they managed to buy what they wanted. How did they do it?
10. Three girlfriends - Vera, Olya and Tanya - went to the forest to pick berries. For picking berries they had a basket, a basket and a bucket. It is known that Olya was not with a basket and not with a basket, Vera was not with a basket. What did each of the girls take with them to pick berries?
11. In gymnastics competitions Hare, Monkey, Boa constrictor and Parrot took the first 4 places. Determine who took what place, if it is known that the Hare - 2, the Parrot did not become the winner, but he got into the prize-winners, and the Boa lost to the Monkey.
12. Milk, lemonade, kvass and water are poured in a bottle, a glass, a jug and a jar. It is known that water and milk are not in a bottle, in a jar there is neither lemonade nor water, but a vessel with lemonade stands between a jug and a vessel with kvass. A glass stands near a jar and a 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 your answer to your question. From these answers, you must guess for yourself which of us was actually 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 a Snow Maiden."
Dasha: "One of them tells the truth, and the other lies."
So which of the friends at the New Year's party was the Snow Maiden?
14. The staircase consists of 9 steps. On which step do you need to stand to be exactly in the middle of the stairs?
15. What is the middle rung of a 12 rung ladder?
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 face into two parts with a straight line so that the sums of the numbers in these parts are equal.
18. Divide the clock face with two straight lines into three parts so that, by adding the numbers, the same amounts are obtained in each part.
· Solve problems with a practical method:
19. The rope was cut in 6 places. How many parts did it make?
20. There were 5 brothers. Each brother has one sister. How many people were walking?
21. Which 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 guess method:
23. How to write the number 10 with 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 by connecting them with action signs?
27. How to draw 2 liters of water from the tap using a six-liter and four-liter vessels?
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 to fill it to the top? How many liters of water will remain in the larger vessel after this?
29. The baby elephant got sick. For his treatment, exactly 2 liters of orange juice are required, and Dr. Aibolit has only 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 to Winnie the Pooh, Piglet and Rabbit. Previously, Winnie the Pooh loved honey, Rabbit - cabbage, Piglet - acorns. But once in the enchanted forest and hungry, they found that their tastes have changed, but still everyone prefers one thing. The rabbit said, "I don't eat cabbage and acorns." Piglet was silent, and Winnie the Pooh remarked: "But I don't like cabbage." Who loves to eat?
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 “flights” 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. So, Kolya needs to go through 8 “flights”, and up to the 9th floor 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 the only way: 4 - 3 + 1. There were more books with fairy tales, so there were 3 of them.
9. Put a 5 kg weight on one scale pan, and put lollipops and a 2 kg weight on the other.
|
little basket |
||||
10. Let's put the condition of the problem in the table, and, where possible, arrange the pros and cons:
|
monkey |
|||||
It turned out that the Monkey and the Boa Constrictor were in the first and fourth places, but since, according to the condition, the Boa Constrictor lost to the Monkey, it turns out that the Monkey is in the first place, the Parrot is in the second and the Boa Constrictor is in the fourth.
11. The conditions that water is not in a bottle, milk is not in a bottle, lemonade is not in a jar, water is not in a jar will be entered in the table. From the condition that a vessel with lemonade stands between a jug and a vessel with kvass, we conclude that lemonade is not in a jug and kvass is not in a jug. And since the glass is near the jar and the vessel with milk, we can conclude that the milk is not in the jar and not in the glass. Let's arrange the "+", 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 get that both Anya and Vera were Snow Maidens, which cannot be. So, Anya's statement must be false. In this case, we get that Anya was not a Snow Maiden, Vera was not a Snow Maiden either. It remains that the Snow Maiden was Dasha.
When multiplying the number 51 by a single-digit number, we again got two-digit number. This is only possible if multiplied by 1. Hence, the second factor is 11.
13. When multiplying the first factor by 2, a four-digit number is obtained, and when multiplied by the hundreds digit and the units digit, 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. On the fifth step.
15. A ladder of 12 steps will not have a middle step, it will only have a pair of middle steps - the sixth and seventh. The solution to this problem, as well as 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
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. Dial in a six-liter, pour water from it into a four-liter, there will be 2 liters.
28. It is necessary to pour 1 liter of water, while 6 liters will remain in a larger vessel.
29. Pour 3 liters of juice into a three-liter jar, then 2 liters of juice will remain in a large jar.
30. Rabbit - honey, Winnie the Pooh - acorns, Piglet - cabbage.
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Solving an entertaining 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 rally, the Dragon King - 7-Headed Dragon counted all the heads of all those gathered.
He looked around his crowned middle head and saw 25 heads.
The king was pleased with the results of the calculations and thanked all those present for their attendance at the rally.
How many dragons came to the rally?
(a) 7; (b) 8; 9; (d) 10; (e) 11;
Solution:
Subtract from the 25 heads counted by the Dragon King, 6 heads belonging to him.
There will be 19 goals left. All remaining Dragons cannot be two-headed (19 being an odd number).
There can only be 1 7-headed Dragon (if 2, then there will be an odd number of heads for two-headed Dragons. And for three Dragons, there are not enough heads: (7 3 \u003d 21> 19).
Subtract from 19 heads 7 heads of this single Dragon 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
♦ ♦ ♦
Solving an entertaining 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 get 5 points, the loser will not get any points, and each player will get 2 points if the game ends in a draw.
They played 13 games and got 60 points together.
Alexander got three times more points for the games 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 wins (winned by Nikita)
Solution.
Each game in a draw gives 4 points to the piggy bank, and a win - 5 points.
If all the games ended in a draw, then 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 draws, which means that when he won, he scored 30 points, that is, he won 6 games.
Then Nikita won (8-6=2) 2 games.
♦ ♦ ♦
Solving an entertaining arithmetic problem
For 4 - 8 grades
How many days without food?
The Martian interplanetary ship has arrived on a visit to Earth.
The 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 stay of the Martians 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 evenings.
How many days during their visit did the Martians go without food?
(a) 0 days; (b) 1 day; 2 days; (d) 3 days; (e) 4 days; (a) 5 days;
Correct answer: 2 days (Martians went without food)
Solution.
The Martians ate 7 days, one meal a day, and the number of days they ate dinner was one more number days when they had breakfast or dinner.
Based on these data, it is possible to draw up a schedule for the Martians to eat. The likely picture is this.
The aliens had lunch on the first day, dinner on the second, breakfast on the third, lunch on the fourth, dinner on the fifth, breakfast on the sixth, and lunch on the seventh.
That is, the Martians had breakfast for 2 days, and spent 7 days without breakfast, had dinner - 2 times, and spent 7 days without dinner, had lunch 3 times, and lived 6 days without lunch.
So 7 + 2 = 9 and 6 + 3 = 9 days. So they lived on Earth for 9 days, and 2 of them went without food (9 - 7 = 2).
♦ ♦ ♦
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 for point B at a constant speed.
The cyclist arrived at point B and immediately went back and met the Pedestrian an hour after 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 the number expressing the time (in minutes) required for the 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 180 minutes is the time the Pedestrian travels from A to B)
Everything becomes clear if you draw a drawing.
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 (dotted green line)).
We get that this difference is exactly equal to the distance from point A to the second meeting.
This distance is covered by a Pedestrian in 100 minutes and by a Cyclist in 60 minutes - 40 minutes = 20 minutes. So the cyclist is going 5 times faster.
Let us denote the distance from point A to the point where 1 meeting took place as one part, and the path of the Cyclist to the 1st meeting as 5 parts.
Together, by the time they first met, they had covered double the distance between points A and B, i.e., 5 + 1 = 6 parts.
Therefore, from A to B - 3 parts. After the first meeting, the pedestrian will have to go 2 more parts to point B.
He will cover the entire distance in 3 hours or 180 minutes, since he covers 1 part in 1 hour.
The concept of "non-standard task" is used by many methodologists. So, Yu. M. Kolyagin reveals this concept as follows: “Under non-standard 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 the authors L.M. Fridman, E.N. Turkish: " Non-standard tasks- these are those for which there are no general rules and regulations in the course of mathematics that determine the exact program for their solution.
Do not confuse non-standard tasks with tasks of increased complexity. The conditions of problems of increased complexity are such that they allow students to quite easily select 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 implies the presence of an exploratory nature. However, if the solution of a problem in mathematics for one student is non-standard, since he is unfamiliar with the methods of solving problems of this type, then for another, the solution of the problem occurs in a standard way, since he has already solved such problems and more than one. The same task in mathematics in the 5th grade is non-standard, and in the 6th grade it is ordinary, and not even of increased complexity.
An analysis of textbooks and teaching aids in mathematics shows that each text task under certain conditions can be non-standard, and in others - ordinary, standard. A standard problem in one course of mathematics may be non-standard in another course.
Based on the analysis of the theory and practice of using non-standard tasks in teaching mathematics, one can establish their general and specific role. Non-standard tasks:
- · teach children not only to use ready-made algorithms, but also to independently find new ways to solve problems, i.e. contribute to the ability to find original ways to solve problems;
- influence the development of ingenuity, ingenuity of students;
- They prevent the development of harmful clichés when solving problems, destroy incorrect associations in the knowledge and skills of students, involve not so much the assimilation of algorithmic techniques, but the discovery of new connections in knowledge, the transfer of knowledge to new conditions, and the mastery of various methods of mental activity;
- create favorable conditions for increasing the strength and depth of knowledge of students, ensure the conscious assimilation of mathematical concepts.
Non-standard tasks:
- should not have ready-made algorithms memorized by children;
- should be accessible to all students in terms of content;
- must be interesting in content;
- To solve non-standard problems, students should have enough knowledge acquired by them in the program.
Solving non-standard tasks activates the activity of students. Students learn to compare, classify, generalize, analyze, and this contributes to a stronger and more 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 tasks, 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 easily and quickly cope with the main part of independent work in the lesson, or for those who wish 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 tasks, but B.A. Kordemsky identifies the following types of such tasks:
- · Tasks related to the school mathematics course, but of increased difficulty - such as tasks of mathematical Olympiads. They are intended mainly for schoolchildren with a definite interest in mathematics; thematically, these tasks are usually associated with one or another specific section of the school curriculum. The exercises related to this deepen the educational material, supplement and generalize the individual provisions of the school course, expand the mathematical horizons, and develop skills in solving difficult problems.
- · Problems of the type of mathematical entertainment. They are not directly related to the school curriculum and, as a rule, do not require much mathematical preparation. This does not mean, however, that the second category of tasks includes only easy exercises. Here there are problems with a very difficult solution and such problems, the solution of which has not yet been obtained. “Non-standard tasks, presented in a fun way, bring an emotional moment to mental activities. Not connected with the need to always apply memorized rules and techniques to solve them, they require the mobilization of all accumulated knowledge, teach them to search for original, non-standard ways of solving, enrich the art of solving with beautiful examples, make them admire the power of the mind.
These types of tasks include:
a variety of numerical puzzles ("... examples in which all or some numbers are replaced by asterisks or letters. Same letters replace the same numbers, different letters - different numbers" .) and puzzles for ingenuity;
logical tasks, the solution of which does not require calculations, but is based on the construction of a chain of exact reasoning;
tasks, the solution of which is based on a combination of mathematical development and practical ingenuity: weighing and transfusions under difficult conditions;
mathematical sophistry is a deliberate, false conclusion that has the appearance of being correct. (Sophism is a proof of a false statement, and the error in the proof is skillfully disguised. Sophism in Greek means a cunning invention, trick, puzzle);
joke tasks;
combinatorial problems, in which various combinations of given objects that satisfy certain conditions are considered (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;
- tasks that are unsolvable or unsolvable by means of a school course at a given level of knowledge of students;
- Tasks that require:
conducting and using analogies, determining the differences between given objects, processes or phenomena, establishing the opposite of given phenomena and processes or their antipodes;
implementation of a practical demonstration, abstraction from certain properties of an object, process, phenomenon or concretization of one or another side of this phenomenon;
establishment of causal relationships between given objects, processes or phenomena;
construction of causal chains in an analytical or synthetic way with subsequent analysis of the resulting options;
the correct implementation of a sequence of certain actions, avoiding errors-"traps";
implementation of the 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 unified classification of non-standard tasks. There are several of them, but the author of the work used the classification proposed by I.V. Egorchenko.
Tests and questionnaires Grade 3.
It is known that the solution of text problems presents great difficulties for students. It is also known which stage of the solution is especially difficult. This is the very first stage - the analysis of the text of the problem. Students are poorly oriented in the text of the problem, in 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 grade bought 12 carnations, and the students of the second…”, “The master made 20 parts in a shift, and his student…”.
Everything is important in the text; And characters, and their actions, and numerical characteristics. When working with a mathematical model of the 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 in the text of a mathematical problem is an important result and an important condition general development student. And you need to do this not only in mathematics lessons, but also in reading and fine arts lessons. Some tasks are good subjects for drawings. And any task is a good topic for retelling. And if there are theater lessons in the class, then some mathematical problems can be staged. Of course, all these techniques: retelling, drawing, staging - can also take place in the mathematics lessons themselves. So, work on texts math problems- an important element of the overall development of the child, an element of developmental learning.
But are the tasks that are available in the current textbooks and the solution of which is included in the mandatory minimum sufficient for this? No, not enough. The mandatory minimum includes the ability to solve problems of certain types:
about the number of elements of a certain set;
about movement, its speed, path and time;
about price and value;
about work, its time, volume and productivity.
These four themes are standard. It is believed that the ability to solve problems on these topics can teach how to solve problems in general. Unfortunately, it is not. Good students who can solve practically
any problem from the textbook on the listed topics, often fail to understand the condition of the problem on another topic.
The way out is not to be limited to any topic of text tasks, but to solve non-standard tasks, that is, tasks whose subject matter is not in itself an object of study. After all, we do not limit the plots of stories in reading lessons!
Non-standard problems need to be solved in the classroom every day. They can be scooped up in mathematics textbooks for grades 5-6 and in magazines " Primary School”, “Mathematics at school” and even “Quantum”.
The number of tasks is such that you can choose from them tasks for each lesson: one per lesson. Problems are solved at home. But very often you need to disassemble them in the classroom. Among the proposed tasks there are those that a strong student solves instantly. Nevertheless, it is necessary to demand sufficient reasoning from strong children, explaining that on easy problems a person learns the methods of reasoning that will be needed when solving difficult problems. It is necessary to educate in children a love for the beauty of logical reasoning. As a last resort, it is possible to force such reasoning from strong students, requiring them to construct an explanation that is understandable for others - for those who do not understand the quick solution.
Among the tasks there are absolutely the same type in mathematical terms. If the kids see this, great. The teacher can show it 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 the connections between tasks with a similar plot.
Not all problems need to be solved (there are more of them here than math lessons in academic year). You may want to change the order of the tasks or add a task that is not here.
