Development of variable thinking. Variability of thinking of younger schoolchildren
Thinking is like a diamond: they are equally multifaceted and, when cut well, sparkle beautifully.
I would compare the well-known formulation “strong thinking skills” to a diamond, because... it combines many valuable parameters. But a diamond is not a diamond yet, right?
If you highlight the facets—varieties of thinking—and then understand what games and tasks each type develops, then working with a growing creative person will begin to resemble the work of a jeweler
I have already published selections of games for development and thinking, soon there will be a selection for systems thinking, and today we have games for variable thinking.
What it is? The ability to see many solutions, rather than focusing on one or two. This is a type of thinking that involves going beyond stereotypes and overcoming the inertia of thinking.
According to my observations, some people can easily give several answers at once, while others say one option and then fall into a stupor. But of course, like any skill, the ability to see more possibilities for solving a problem can be formed purposefully. This is what today's selection is about!
Explain the inexplicable (from 4 years old)
The pictures from the series “what the artist got mixed up” are well known. They help to see how the child navigates the world around him.
On the other hand, you can find fault here: you say, the artist made a mistake by painting snow in the middle of summer? Tell this to a resident of Surgut!
Therefore, we will practice explaining the seemingly inexplicable.
Props: pictures from the series “what did the artist mix up” (you can make such collages yourself), or plot pictures with one or two objects (a steamer is sailing, a car is driving, children are going for a walk...) + small subject pictures, the more varied the better.
Let's play!
First option. If we take a ready-made “confused” picture, then we try to find plausible explanations:
- why do buns grow on a tree (this is a decoration for the holiday),
- why is there a goose sitting in the booth (it is a special guard breed),
- why did the rooster build a nest on the roof (afraid of the goose)),
- why did such huge tomatoes grow under the tree (such is the selection nowadays))).
In the second version of the game, we attach a small one to a larger plot picture and ask: “why did the artist draw a cat on a ship?” For example, because:
“Why extra?” (from 4 years old)
Pictures from the “find the odd one out” series are often found in textbooks for preschoolers. They assume a fairly obvious answer and are again aimed at consolidating knowledge about the world around us. And we teach you to find many possible answers to a question.
Props: pictures depicting objects or figures.
Let's play!
We offer several pictures, saying that each item in turn will be “extra” so that no one is offended. You can start playing from 4 pictures.
We will compare objects with each other, for example, by color, weight, size, taste, sound, parts, habitat etc.
Here is a task for preschoolers from the distance competition “First Steps in TRIZ”, which took place in the winter of 2016:
- The fish is superfluous because it lives in water, and the rest do not.
- The elephant is superfluous because he has a trunk, while others do not.
- Cheburashka is superfluous because he is a fairy-tale hero.
- The cow is superfluous because she has horns, while others do not.
- The hare is extra because he is gray and the rest are a different color
I think the principle is clear!
Not “yes”, but “no”! (from 6 years old)
Props: imagination and ability to come up with questions
Let's play!
First you need to ask a question that you want to answer “yes”, but we will do the opposite and say “no!” And then we will discuss in what cases the answer may be negative and why.
- Do all fish swim?
- No!
- And when they don’t swim?
- When they are drawn!
Here are some more sample questions:
- Does a car always overtake a pedestrian?
- Is it always light during the day?
- Do all trees have leaves?
- Do all flowers need water?
(you will be able to come up with even more interesting questions!!!)
And, of course, all these games also wonderfully help develop a child’s speech.
Which one did you like best?
Development of variable thinking in junior schoolchildren in mathematics lessons
Under variability of thinkingIn psychology, we understand a person’s ability to find a variety of solutions. Indicators of the development of variability of thinking are its productivity, independence, originality and elaboration. Variability of thinking determines an individual’s ability to think creatively and helps to better navigate in real life. The reality around us is diverse and changeable. A modern person constantly finds himself in the situation of choosing a solution to a problem, which is optimal in a given situation. This will be done more successfully by someone who knows how to look for a variety of options and choose among a large number of solutions.
The development of variability of thinking is especially important for learning. Thus, the manifestation of this quality of thinking is required, for example, when solving problems using selection, when the student considers all possible situations, analyzes them and eliminates those that do not correspond to the conditions.
Tasks that promote the development of variability in students' thinking can be divided into several groups. These are the tasks:
1) having a single correct answer, which can be found in different ways;
2) having several answer options, and they are found in the same way;
3) having several answer options that are found in different ways.
I will give examples of tasks for each group.
Task 1 (group 1). Find expressions whose values can be calculated in different ways:
(7+20):9
(30+8)+20
(28+21):7
(10+4)*1
(60+30)-80
100:(20+5)
Answer:
(30+8)+20
(28+21):7
(10+4)*1
100:(20+5)
Task 2 (group 2). Petya lives in apartment 200. There are 3 more apartments on his floor. Write down what numbers these apartments might have.
Answer: This is a multiple-choice task. It does not indicate how Petya’s apartment is located on the floor, so all possible options are found in one way:
a) 200,201,202,203;
b) 199,200,201,202;
c) 198,199,200,201;
d) 197,198,199,200.
Task 3 (group 3). What one change needs to be made to the record so that the inequality
465 456 became correct? Consider all options.
You can complete this task in different ways, obtaining different answers. First, we can correct the inequality sign (467,456). Secondly, you can correct the first number: remove the digit in the hundreds place (67,456); change the hundreds digit (447 456, 437 456, 427 456, 417 456, 407 456). Thirdly, you can correct the second number: assign a figure indicating units of thousands (467 1456, 467 2456, etc.); change the hundreds digit (467 556, 467 656, 467 756, 467 856, 467 956); change the tens digit (467 476, 467 486, 467 496).
The tasks of the third group include combinatorial problems. When solving them by brute force, various options are made and the reasoning carried out by students may be different.
Students can be offered multi-choice tasks (which have several answers), specifically aimed at forming a certain indicator of the development of variability of thinking: productivity, originality and independence.
Tasks that contribute to the development of productivity should contain an indication of the search for various solution options. When performing them, the main thing will be the number of options the student finds. You need to start with tasks that involve a small number of options (from 2 to 4), and then you can move on to a larger number of solution options, but their number should be limited so that students do not lose interest in completing the tasks.
Task 1. Write down all possible three-digit numbers whose digits sum to four.
ANSWER: 400, 310, 301, 130, 103, 220, 202, 112, 121, 211.
Task 2. Insert action signs to make the equalities true. Give all possible options for completing the task.
a) 12…1=12;
b) 12…0=12;
c) 17…28=28…17;
d) (9…4)…2=9…(4…2);
Answer:
a) 12*1=12, 12:1=12;
b) 12+0=12, 12-0=12;
c) 17+28=28+17, 17*28=28*17;
d) (9+4)+2=9+(4+2), (9*4)*2=9*(4*2), (9+4)-2=9+(4-2), (9-4)-2=9-(4+2).
When completing this task, students rely on theoretical knowledge of arithmetic operations. You can lead students to generalizations, for example, that by rearranging two numbers only with addition and multiplication, the result will not change.
Task 3. Remember the units of various quantities. Insert names instead of dots, consider different options:
a) 1...=10...;
b) 1…=100…;
c) 1…=1000…
Answer:
a) 1cm=10mm, 1dm=10cm, 1m=10dm; 1t=10ts;
b) 1dm=100mm; 1c=100kg; 1cm =100mm; 1m=100cm, 1dm=100cm, 1m=100dm;
c) 1km=1000m, 1m=1000mm; 1kg=1000g, 1t=1000kg;
Can add:
1 ruble = 100 kopecks; 1 century = 1000 years.
The productivity indicator does not give a complete picture of the development of variability of thinking in schoolchildren. One student can give many options, but they will be similar. Another student will give only two options, but they will be fundamentally different. Therefore, it is necessary to take into account the indicator of originality.
Tasks that promote the development of originality should contain an option (or similar options) for the solution, as well as an indication of the search for options different from this one. When performing them, the degree of difference between the found options and those presented in the condition is taken into account.
TASK 1. Insert the missing length units to make the entries correct:
3…5…=35cm;
3…5…=305cm;
3…5…=350cm.
How are all the numbers after the “=” sign similar? What numbers, different from them, can appear after the “=” sign? Find them.
3…5…=…;
3…5…=…;
3…5…=… .
Answer:
3dm 5cm=35cm;
3m 5cm=305cm;
3m 5dm=350cm.
3min.5s.=185s;
3 days.5 hours=77 hours;
3 years. 5 months = 41 months.
Task 2. Insert the missing units of value so that the entries become correct:
4…-2…=38…;
4…-2…=398…;
4…-2…=3998…;
Choose units of magnitude such that the result does not end with the number 8.
Answer:
4t-2t=38t;
4ts-2kg=398kg;
4kg-2g=3998g;
4kg-2kg=2kg;
4 years - 2 months = 46 months;
4 days - 2 hours = 94 hours;
Task 3. The incorrect equality 3m-20cm=10cm was corrected by changing the result:
3m-20cm=280cm.
How else can you correct the false equality by making only one change? Consider different options.
Answer:
3dm-20cm=10cm;
3m-20cm 10cm.
In all previous tasks, the student was aimed at finding different options. But it is important that he himself strives to find out when performing tasks whether there are other solutions. It is necessary to build work on the indicator of independence of variability of thinking.
Tasks that promote the development of independence in the manifestation of variability should not contain a special instruction to search for different options. When performing them, it is not important how many options are given by the student; the main thing is that he himself, without outside prompting, began to look for different options.
At first, the wording of tasks may contain some hint of the presence of a multiple-choice answer, for example, as was done in task 1:
Task 1: What numbers can be inserted to make the equalities true?
a) 700:10= __ + __ ;
b) 5*__ = __ -400;
c) __ +8= __ :50;
d) 630: __ =70- __ .
Answer:
a) 700:10= 1+69, 700:10=2+68, etc.;
b) 5*1=405-400, 5*2=410-400, etc.;
c) 0+8=400:50, 1+8=450:50, etc.;
d) 630:9=70-7, 630:10=70-7, etc.
When completing such a task, students notice the possibility of finding different options and may ask the question: “How many options should I write down?” You can limit the time it takes to complete a task, and then each student will write down as many options as they have time.
Task 2: Subtract a two-digit number from a three-digit number. How many digits will be in the record of their difference? Give an example to support your answer.
Answer: 3 numbers: 634 – 12=621;
2 digits: 104 – 14=90;
1 digit: 100 – 99-1.
In this task, the wording no longer prompts a search for different options; students must demonstrate independence.
Task 3: Compose examples using diagrams where possible. Calculate. Where is it impossible to create an example? Explain why.
a) __ __ + __ = __ __ __ ;
b) __ __ - __ = __ __ __ ;
c) __ __ - __ = __ __ ;
d) __ __ __ - __ __ = __ __ ;
e) __ + __ + __ = __ __ __ ;
f) __ __ __ - __ - __ = __ .
Answer:
a) 99+1=100, 99+2=101, 99+3=102, etc.; 98+2=100, 98+3=101, etc.;
b) it is impossible;
c) 11-1=10, 12-2=10, etc.;
d) 100-10=90, 100-11=89, etc.; 101-10=91, 101-11=99, etc.;
e) it is impossible;
e) it is impossible.
In task 3, a more complex situation has been created in the manifestation of independent thinking, since for one part of the equations an unambiguous answer is given, and for the other a multivariate answer.
The named types of tasks should be included in training consistently.
When working to develop variable thinking, we also observe the development of such qualities as:
Logical thinking;
Ability to choose a convenient solution;
Visual perception;
Skills of analysis, synthesis, comparison, classification;
Differentiated and individual approach;
Independence of thinking (ability to make choices and decisions).
As one of the most important means of developing informed and solid knowledge in mathematics, you can use the method of varying word problems as a way of constructing educational material and as a method of organizing students’ educational activities.
I will give some methods of working on the development of variable thinking in primary school students:
- One and then two missing numeric data are inserted into the finished condition.
- Questions are posed to the prepared condition.
- The problem condition is selected for the question.
- Compiling tasks:
According to the dramatization.
Based on illustrations (picture, poster, drawing, etc.)
According to numerical data.
According to a ready-made solution.
According to the finished plan.
Preparation of similar tasks.
5. Changing the relationship between the data of the problem conditions and finding out how this change will affect the solution of the problem
6. Changing the task question.
7. Changing the conditions of the problem, introducing additional data into it or removing any data.
It is very important if, to compose problems, students use the material they “obtain” during excursions, from reference books, newspapers, magazines, etc., i.e. - from my life experience.
Here's an example of working on a task:
The distance between two bus stops is 1 km. Two buses departed from these stops. One of them walked 140 m, and the other 160 m. What was the distance between the buses? (The task contains a new subject for the child: the movement of two bodies). This movement can be of three types:
1) towards each other;
2) in opposite directions;
3) after one another.
When completing such tasks, schoolchildren not only demonstrate knowledge, skills, and abilities, but also show how developed their logical thinking is, the ability to analyze, compare, classify, and transform according to the following indicators is formulated:
a) the ability to perform any task along an independently chosen path (which allows one to judge the maturity of individual operations and the ability to use them comprehensively);
b) use of variability when performing a task;
c) the ability to switch from one search basis to another.
The use of variability characterizes the depth of the mind, since this ability manifests the ability to isolate and use the main idea in work, which allows one to systematically identify all possible options and find the most optimal one.
It is well known that, along with the formation of basic mathematical concepts, the study of the properties of numbers, and arithmetic operations in primary education, the most important place has always been occupied by the development of computational skills in schoolchildren. Today, the importance of these skills has decreased due to the widespread introduction of electronic computer technology into all spheres of human activity, the use of which undoubtedly facilitates the calculation process.
Among the studies of past years, the works of M.A. enjoy the greatest authority. Bantova, published twice in the methodological journal “Primary School”[No. 10, 1975 and No. 11, 1983].
Computing skill M.A. Bantova defined it as “a high degree of mastery of computational techniques” and identified its following characteristics - correctness, awareness, rationality, generality, automatism, strength.
Computational skill is a detailed implementation of an action in which each operation is realized and controlled. Computational skill presupposes the mastery of a computational technique. Any computational technique can be represented as a sequence of operations, the execution of each of which is associated with a specific mathematical concept or property.
Based on the specific meaning of arithmetic operations, their properties, connections and dependencies between the results and components of actions, as well as the decimal composition of numbers, methods of oral and written calculations are revealed. This approach to the study of computational techniques ensures, on the one hand, the formation of conscious skills and abilities, because students will be able to justify any computational technique, and on the other hand, with such a system, the properties of actions, their laws, etc. are better understood.
Simultaneously with the study of the properties of arithmetic operations and the corresponding methods of calculation, the connections between the components and the results of arithmetic operations are revealed on the basis of operations on sets or numbers, and observations are made of changes in the results of arithmetic operations depending on the change in one of the components.
Let us dwell in more detail on such quality of computing skill as rationality, which directlyassociated with variability.
Variability of thinking is associated with the ability to “see” several possible situations in which the essential properties of an object are preserved, but non-essential ones change.
Rationality of calculations is the selection of those computational operations from the possible ones, “the implementation of which is easier than others and quickly leads to the result of an arithmetic operation»..
Increased attention to the rationalization of calculations is associated with the practical orientation of mathematical education, which means the development of schoolchildren’s abilities to apply acquired knowledge, to act not only according to a model, but also in non-standard situations, combining known methods of solving an educational problem. Familiarity with the rationalization of calculations develops variability of thinking and shows the value of the knowledge that is used in this process. The use of the properties of arithmetic operations allows the teacher to cultivate interest in mathematics, to arouse in children the desire to learn to calculate in the fastest, easiest and most convenient ways. This approach will support the desire to use mathematical knowledge in everyday life.
The ability to rationally perform calculations is based on the conscious use of the laws of arithmetic operations, the application of these laws in non-standard conditions, and the use of artificial (universal) methods to simplify calculations.
The properties of arithmetic operations (commutative and associative properties of addition and multiplication, distributive property of multiplication relative to addition) are not a special subject of study in elementary school, but are considered in connection with the formation of oral calculation techniques. This means that in the learning process, using specific simple numerical examples, various ways of adding a number to a sum, a sum to a number are considered; subtracting a number from a sum, a sum from a number; multiplying a sum by a number, etc. in order to develop the ability to consciously choose those methods that allow the calculation process to be carried out rationally.
In the initial course of mathematics, the study of a computational technique occurs after students have mastered its theoretical basis (definitions of arithmetic operations, properties of actions and consequences arising from them). Moreover, in each specific case, students are aware of the very fact of using the corresponding theoretical principles underlying the computational technique, construct various techniques for one case of calculations, using various theoretical principles..
Mathematics textbooks present methods of rational calculations from a methodological point of view. The prevalence of model-based actions in the computing activities of younger schoolchildren in conditions of mass education determines the formation of computational stereotypes, the use of which is possible only in a familiar situation.
The problem of rational calculations has been repeatedly raised on the pages of the Elementary School magazine. . The authors of the publications describe in sufficient detail the theoretical foundations of various computational techniques, some of which can be successfully used by teachers when teaching younger schoolchildren. This is a method of grouping, multiplying and dividing by 11, 5, 50, 15, 25, etc., rounding one of the components of an arithmetic operation, etc.; their theoretical basis is the properties of arithmetic operations, which are introduced in the initial course of mathematics. Let us dwell on some of the methods of calculations that, in our opinion, are feasible for students, but are not used in the practice of teaching primary schoolchildren.
A rounding technique based on the change in the result of a calculation when one or more components change.
- Addition. To find the value of the sum, the technique of rounding one or more terms is used.
When increasing (decreasing) a term by several units, we reduce (increase) the amount by the same number of units:
- 224+48=224+(48+2)-2=(224+50)-2=274-2=272 or
- 224+48=(220+50)+4-2=270+4-2=272.
- Subtraction
- when increasing (decreasing) the one being reduced by several units, the difference is reduced (increased) by the same number of units:
397-36=(400-36)-3=364-3=361;
- when increasing (decreasing) the subtrahend by several units, the difference is increased (decreased) by the same number of units:
434-98=(434-200)+2=234+2=236;
- when increasing (decreasing) the minuend and subtrahend by several units, the difference will not change:
231-96=(231+4)-(96+4)=235-100=135.
- Multiplication
When increasing (decreasing) one of the factors by several units, multiply the resulting integer and the added (subtracted) units by another factor and subtract the second product from the first product (add the resulting products)
97x6=(100-3)x6=100x6-3x6=600-18=582.
This technique of representing one of the factors as a difference allows you to easily multiply by 9, 99, 999. To do this, just multiply the number by 10 (100, 1000) and subtract the number that was multiplied from the resulting integer: 154x9=154x10-154=1540- 154=1386.
But it’s even easier to familiarize children with the rule - “to multiply a number by 9 (99, 999), it is enough to subtract from this number the number of its tens (hundreds, thousands), increased by one, and to the resulting difference add the addition of its units digit to 10 (complement up to 100 (1000) number formed by the last two (three) digits of this number):
154x9=(154-16)x10+(10-4)=138x10+6=1380+6=1386
Schoolchildren are also interested in abbreviated multiplication methods, which include multiplication by 15, 150, 11, etc., the theoretical basis of which is multiplying a number by a sum.
For example, when multiplying by 15, if the number is odd, multiply it by 10 and add half of the resulting product: 23x15=23x(10+5)=230+115=345; if the number is even, then we proceed even simpler - we add half of it to the number and multiply the result by 10:
18x15=(18+9)x10=27x10=270.
When multiplying a number by 150, we use the same technique and multiply the result by 10, since 150 = 15x10:
24x150=((24+12)x10)x10=(36x10)x10=3600.
The theoretical basis for multiplying two-digit numbers is the rule of multiplying a sum by a number. For example, 18x16. First, the number 18 is presented as a “sum of convenient (digit) terms,” then sequential calculations are performed using the distributive law of multiplication relative to addition: (10+8)x16=10x16+8x16=160+128=288.
It is easier to find the meaning of this expression orally: to one of the numbers you need to add the number of units of the other, multiply this amount by 10 and add to it the product of the units of these numbers: 18x16=(18+6)x10+8x6= 240+48=288. Using the described method, you can multiply two-digit numbers less than 20, as well as numbers that have the same number of tens: 23x24 = (23+4)x20+4x6=27x20+12=540+12=562. The method is different from the “rational calculations” that children are taught in school.
The educational literature also describes other universal methods of quick calculation (rational calculations), which can always be justified mathematically and are based on known laws and properties of arithmetic operations.
Enumerating options when solving mathematical problems trains the variability of thinking and its mobility.
I will give examples of enumerating options.
The teacher gives an oral task from the table. This table is used only by the teacher. It has 4 columns of different numbers. Only 2 numbers that are vertically adjacent are taken.
Example of completing a task:
“What actions must be performed with the number 32 in order to obtain the next number 2?”
Students mentally go through different math operations using the number 32 to get 2. These operations may include addition, subtraction, multiplication, and division. For these numbers the following options are possible:
32:16=2 32-30=2
Then, in accordance with the table, the teacher offers to complete a new task: “What actions must be performed with the number 2 to get 60?” After going through the options, students receive:
2*30 = 60 2+58 = 60, etc.
It is advisable to gradually reduce the time to complete the task.
The previous task can be complicated by suggesting in your mind that you can solve the problem with 3 numbers using an enumeration method. The tasks are given orally by the teacher using the “Sign Finder” table.
The specified numbers are in the first column of the table. In the second column, opposite the line with the given numbers, there are 3 numbers that show the results of various actions with the given numbers. In the last column, opposite each line with specified numbers and possible results of actions with them, 3 sets of characters are given. Each set contains 2 mathematical symbols. They are located horizontally. The two signs in the first set indicate what actions must be performed with the given signs in order to obtain the result given in the first number of the result set.
For example:
Specified numbers: 11.4.7. Result: 49.8.22. Signs: - ;+-; ++.
If you perform an action with the first set of characters i.e. subtraction and multiplication, we get 49 = (11 - 4) 7.
If we perform operations with the second set of signs (addition and subtraction), we get the number 8=11+4-7.
The teacher gives the task: “Solve the problem in your mind - what actions need to be performed with the numbers 11.4.7. to get the result 49?” Students mentally go through options for actions with given numbers to get the result 49. See an example of a solution above. At first, you can allow conditions to be written down. The third character column is the key. It is intended only to facilitate the teacher's work.
The simulator is designed for solving problems with 3 numbers in your head by enumerating options for possible mathematical operations. It allows you to intensify the work to find the desired result
Thus, the use of variability characterizes the depth of the mind, since this ability manifests the ability to isolate and use the main idea in work, which allows one to systematically identify all possible options and find the most optimal one.
The variability of schoolchildren's computing skills creates interest and positive motivation for computing activities.
References:
- Bantova M.A. System for developing computing skills // Primary school. - 1993. - No. 11. - P. 38-43.
- Gelfan E.M. Arithmetic games and exercises. - M.: Education, 1968. - 112 p.
- Demidova T.E., Tonkikh A.P. Techniques of rational calculations in the initial course of mathematics // Primary school. - 2002. - No. 2. - P. 94-103.
- Zimovets N.A., Pashchenko V.P. Interesting techniques for mental calculations // Primary school. - 1990. - No. 6. - pp. 44-46.
- Faddeicheva T.I. Teaching mental calculations // Primary school. - 2003. - No. 10. - pp. 66-69.
- Chekmarev Ya.F. Method of oral calculations. - M.: Education, 1970. - 238 p.
1
1. Timofeeva N.B., Salishcheva Ya.V. Federal educational standard of the second generation - Electronic resource - access mode: http://www.scienceforum.ru/2014/761/686 (release date November 1, 2014).
2. Russian Pedagogical Encyclopedia: 2 volumes / chapter. ed. V.V. Davydov. – M.: Great Russian Encyclopedia, 1993. – T.2. – P.12.
The main tasks of a modern school are to reveal the abilities of each student, to educate a decent and patriotic person, an individual ready for life in a high-tech, competitive world. School education should be structured so that graduates can independently set and achieve serious goals and skillfully respond to different life situations. This is the state's social order for schools today.
When a child enters school, under the influence of learning, a restructuring of all his cognitive processes begins. It is the primary school age that is productive in the development of thinking. In order to educate a person capable of multivariate thinking, quickly finding a solution to a given problem, and navigating the fast modern flow, we must rely on the regulatory documents that form the basis of primary education, namely federal state standards.
In our work, we consider the problem of developing the variability of thinking of younger schoolchildren, which is reflected in the federal state standards of primary general education.
With a variable approach to learning, each student will find several ways to solve a given educational task, based on their personal characteristics and abilities, level of knowledge and mastery of the material.
The relevance of the work is due to the fact that during the period of primary school age, significant changes occur in the child’s psyche, the assimilation of new knowledge, new ideas about the world around them rebuilds the everyday concepts that children had previously developed, and school thinking, in our opinion, contributes to the development of theoretical thinking in the areas accessible to students this age forms.
The theoretical basis of the study was the work of A.D. Alferova, A.A. Lyublinskaya, R.S. Nemova, etc., dealing with the problem of developing variability of thinking in primary schoolchildren.
In our work, we analyzed the definitions of “thinking” and “variability of thinking.” We will understand thinking as “a process of human cognitive activity, characterized by a generalized and indirect reflection of objects and phenomena of reality in their essential properties, connections and relationships.” Variability of thinking - as “a person’s ability to find a variety of solutions”, which was given by E.A. Posokhova. Variability of thinking determines an individual’s ability to think creatively and helps students better navigate real life.
To identify the level of development of variability in junior schoolchildren, in our work, we used the following methods: “Questioning teachers”, “Determining the pace of implementation of indicative and operational components of thinking”, “Simple analogies”, “Excluding unnecessary things”, “Determining the level of development of variability of thinking” , the choice of which is based on the ability to obtain stable indicators, and they are also objective when interpreting the result.
The testing of the selected methods was carried out at the Municipal Educational Institution “Secondary School No. 16 named after. D.M. Karbyshev" in Chernogorsk, Republic of Khakassia, among fourth grade students, 10 primary school teachers also took part.
The results obtained using the presented methods allowed us to conclude that the ability of students to find various solutions is not fully developed in most of them. We believe that teachers need to pay more attention in mathematics lessons to working with tasks aimed at finding solutions in different ways, since by spending more time on developing the variability of thinking of younger schoolchildren, the level of other indicators in children will become higher, which will subsequently lead to fruitful learning mathematics at the level of consciousness, and not stereotyping and typicality, which can lead to stereotypes in the future.
Bibliographic link
Timofeeva N.B., Filippova Yu.S. DEVELOPMENT OF VARIABILITY OF THINKING OF JUNIOR SCHOOLCHILDREN // Modern science-intensive technologies. – 2014. – No. 12-1. – P. 92-93;URL: http://top-technologies.ru/ru/article/view?id=34849 (access date: 02/03/2020). We bring to your attention magazines published by the publishing house "Academy of Natural Sciences"
Explanatory note
Do some serious work
entertaining - that's the task
initial training.
K.D. Ushinsky.
Primary general education is designed to realize the abilities of each student and create conditions for the individual development of younger schoolchildren.
The more diverse the educational environment, the easier it is to reveal the individuality of the student’s personality, and then direct and adjust the development of the younger student, taking into account identified interests, relying on his natural activity.
Numerous studies have shown that it is in elementary school that the foundations of evidence-based thinking are laid and omissions in working with students of this age are practically irreparable. That is why it is necessary to develop a course that would ensure the formation of methods of mental activity.
The work program of the course “Development of Variable Thinking” is compiled in accordance with the requirements of the Federal State Educational Standard for Primary General Education.
Target – development of mathematical abilities, formation of methods of mental activity.
Tasks:
promote understanding of ways to solve non-standard problems, which, in turn, will allow for a new approach to solving standard word problems;
promote practical mastery of the content of logical concepts, the formation of logical skills;
contribute to the formation of interest in the subject, the desire to use mathematical knowledge in everyday life.
tasks and exercises; standard word problems that have several solutions or a non-standard solution; tasks aimed at developing logical thinking, deepening mathematical knowledge, mastering such mental operations as analysis, synthesis, comparison, classification, generalization.
Word problems are an important means of developing a system of basic mathematical concepts. Students get used to solving standard (of the same type) problems and get lost when choosing solutions to non-standard problems, the difficulty of which is determined not so much by the mathematical content as by the novelty and unusualness of the mathematical situation. When solving a problem, students should not juggle numbers, but think through the relationships between quantities and independently build and justify the course of its solution in a generalized form. The ability to analyze a task not only develops children’s thinking and speech, but also develops in them such traits as independence, the ability to think through a plan of action, and reason convincingly.
Logical exercises allow students to gain a deeper understanding of mathematical relationships and their properties, and mastering logical skills will allow them to apply logical techniques when solving problems.
General characteristics of the course.
The implementation of the task of raising an inquisitive, actively and interestedly exploring the world of a junior schoolchild, learning to solve mathematical problems of a creative and exploratory nature will be more successful if class activities are supplemented with extracurricular work. This could be the course “Development of Variable Thinking”, expanding the mathematical horizons and erudition of students, promoting the formation of cognitive universal learning activities. The proposed course is designed to develop the mathematical abilities of students, to form elements of logical and algorithmic literacy, communication skills of younger schoolchildren using collective forms of organizing classes and using modern teaching tools. Creating situations of active search in the classroom, providing the opportunity to make their own “discovery”, getting acquainted with original ways of reasoning, mastering basic research skills will allow students to realize their capabilities and gain confidence in their abilities. The content of the course “Development of Variable Thinking” is aimed at cultivating interest in the subject, developing observation, geometric vigilance, the ability to analyze, guess, reason, prove, and the ability to solve an educational problem creatively. Content can be used to show students how to apply the knowledge and skills they learn in math classes. The program provides for the inclusion of problems and assignments, the difficulty of which is determined not so much by the mathematical content as by the novelty and unusualness of the mathematical situation. This contributes to the desire to abandon the model, to show independence, to the formation of skills to work in search conditions, to the development of intelligence and curiosity. In the process of completing tasks, children learn to see similarities and differences, notice changes, identify the causes and nature of these changes, and formulate conclusions on this basis. Moving together with the teacher from question to answer is an opportunity to teach the student to reason, doubt, think, try and find a way out - the answer.
The value guidelines of the course content are: formation of the ability to reason as a component of logical literacy; mastering heuristic reasoning techniques; formation of intellectual skills related to the choice of solution strategy, situation analysis, data comparison; development of cognitive activity and independence of students; formation of the abilities to observe, compare, generalize, find the simplest patterns, use guesswork, build and test the simplest hypotheses; formation of spatial concepts and spatial imagination; involving students in the exchange of information during free communication in the classroom.
The program course is designed for 4th grade students.
Classes are held1 once a week for2 hours. Only 56 hours per year.
Expected results .
Students must:
Know the sequence of numbers within 100,000 and be able to write them;
Know the table of addition and subtraction of single-digit numbers; be able to correctly perform all four arithmetic operations with numbers within 100.
Know the rules for the order of performing actions in numerical expressions and be able to apply them in practice;
Be able to solve word problems using an arithmetic method; solve non-standard problems; solve problems related to everyday life situations (purchasing, measuring, weighing, etc.);
Be able to recognize studied geometric shapes and depict them on paper;
Compare quantities by their numerical values, express these quantities in different units;
Use acquired knowledge and skills in practical activities and everyday life to navigate the surrounding space (route planning, choosing a route of movement);
Be able to use logical techniques when solving problems.
Planned results of studying the course.
As a result of mastering the course program “Development of Variable Thinking”, the following universal educational actions are formed that meet the requirements of the Federal State Educational Standard of NEO:
Personal results:
Development of curiosity and intelligence when performing various tasks of a problematic and heuristic nature.
Development of attentiveness, perseverance, determination, and the ability to overcome difficulties - qualities that are very important in the practical activities of any person.
Fostering a sense of justice and responsibility.
Development of independent judgment, independence and non-standard thinking.
Meta-subject results:
Compare different methods of action, choose convenient methods for performing a specific task.
Model in the process of joint discussion an algorithm for solving a numerical crossword puzzle; use it during independent work.
Apply learned teaching methods and calculation techniques to work with number puzzles.
Analyze the rules of the game. Act in accordance with the given rules.
Engage in group work.
Participate in the discussion of problematic issues, express your own opinion and give reasons for it.
Carry out a trial educational action, record an individual difficulty in the trial action.
Argue your position in communication, take into account different opinions, use criteria to justify your judgment.
Compare the obtained result with a given condition.
Monitor your activities: detect and correct errors.
Analyze the text of the problem: navigate the text, highlight the condition and question, data and required numbers (quantities).
Search and select the necessary information contained in the text of the problem, in the picture or in the table, to answer the questions asked.
Simulate the situation described in the text of the problem.
Use appropriate sign-symbolic means to model the situation.
Construct a sequence of “steps” (algorithm) for solving a problem.
Explain (justify) the actions performed and completed.
Reproduce a method for solving a problem.
Compare the obtained result with a given condition.
Analyze the proposed solutions to the problem and choose the correct ones.
Choose the most effective way to solve the problem.
Evaluate the presented ready-made solution to the problem (true, false).
Participate in educational dialogue, evaluate the search process and the result of solving the problem.
Construct simple problems.
Navigate in terms of “left”, “right”, “up”, “down”.
Focus on the starting point of movement, on the numbers and arrows 1→ 1↓, etc., indicating the direction of movement.
Draw lines along a given route (algorithm).
Select a figure of a given shape in a complex drawing. Analyze the arrangement of parts (triangles, corners, matches) in the original design.
Make shapes from parts.
Determine the place of a given part in the structure.
Identify patterns in the arrangement of parts; compose parts in accordance with the given design contour.
Compare the obtained (intermediate, final) result with a given condition.
Explain the choice of parts or method of action under a given condition.
Analyze the proposed possible options for the correct solution.
Model three-dimensional figures from various materials (wire, plasticine, etc.) and from developments.
Carry out detailed control and self-control actions: compare the constructed structure with a sample.
Thematic course planning
“Development of variable thinking”
4th grade (56 hours)
№ p/p
Lesson topic
Number of hours
Lesson Objectives
date
carrying out
Introductory lesson. From the history of mathematics. "How people learned to count."
The magic of numbers. The science of numerology.
Contribute to the activation of the cognitive process.
Tree of possibilities.
Contribute to the activation of the cognitive process.
Tree of possibilities. solving combinatorial problems.
Contribute to the activation of the cognitive process.
Solving problems of finding quantities by their sum and difference
To promote the development of skills in solving problems of finding quantities by their sum and difference
Feature extraction. Similarities and differences in written multiplication by one-digit, two-digit and three-digit numbers.
For math lovers. Tournament of savvy.
Contribute to the activation of the cognitive process.
Magic circle. Comparison rules. Comparing fractions.
Reinforce the comparison of fractions using a circle as an example.
Games with numbers. Solving problems on finding a part of a number, a number from its part.
To promote the development of problem solving skills for finding parts of a number and numbers by parts.
Time machine model. Solving problems with named numbers.
Solve problems with named numbers.
Regularities in numbers and figures. Multi-digit numbers.
To promote the ability to write multi-digit numbers.
Brave traveler. Solving problems on finding speed, time and distance.
Reinforce the solution of movement problems.
Magic squares.
Finding the area of figures.
Magic square.
Finding the volume of shapes.
To promote the development of the skill of finding the area of figures and the volume of figures.
Games to develop observation skills. Estimating sums and differences when working with multi-digit numbers.
To promote the development of observation skills, the ability to find the sum and difference using the estimation method.
Solving problems to develop ingenuity and intelligence.
Promote the search for alternative ways to solve problems and examples with multi-digit numbers.
Search for alternative courses of action.
Arithmetic operations with round numbers.
Promote the search for alternative ways to solve examples with multi-digit and round numbers.
Strengthening the ability to combine. Solving complex equations.
Promote the ability to solve complex equations.
Tasks - tests.
Blitz tournament.
Drawing up algorithms and applying them in practice when solving examples.
Create a problem situation for students to create an algorithm for solving examples (multiplying a multi-digit number by a single-digit number and a two-digit number).
Actions are opposite in meaning. Using the inverse operation when solving problems, equations, examples.
To promote interest in the subject of mathematics, to activate the cognitive process.
Feature extraction. Similarities and differences in written multiplication by one-digit and two-digit numbers.
To promote interest in the subject of mathematics, to activate the cognitive process.
Mathematical puzzles.
To promote interest in the subject of mathematics, to activate the cognitive process.
Blitz tournament.
Tasks - tests.
Activate the cognitive process of students by selecting tasks from simple to complex.
Inventing by analogy. Solving problems and composing inverse problems to data.
To promote the ability to compose problems using given diagrams and mathematical expressions; create problems that are inverse to a given problem.
From the history of numbers. The use of various figures and numbers in modern life.
To promote students’ interest and ability to draw on life experience.
Developing imagination. Composing problems to find the arithmetic mean
To promote the development of students’ imagination and the ability to defend their point of view.
Magic circle. Drawing up pie charts. Solving problems using pie charts.
To promote the ability to compose tasks using this diagram.
Traveling along the number beam. Coordinates on the number line.
Expand knowledge about pie charts, number line, coordinates on number line.
Game "sea battle". Coordinates of points on the plane.
Expand knowledge about coordinates on a plane, promote the ability to play the game “Battleship”.
Summing up the training.
Review of knowledge.
Summarize the students’ knowledge acquired in the additional education course.
Short description
The purpose of the study is to solve the problem raised.
Research objectives:
1) analyze psychological, pedagogical and methodological literature in order to reveal the essence of the concepts of “thinking”, “variability of thinking”, “process of development of variability of thinking”.
2) to identify psychological and pedagogical features of the development of variability of thinking in younger schoolchildren.
Introduction………………………………………………………………………………….…3
Chapter 1. Psychological and pedagogical foundations for the development of variability of thinking in primary schoolchildren
1.1. Development of variability of thinking from the perspective of pedagogy and psychology.................................................... ........................................................ ................7
1.2. Features of the development of variability of thinking in primary school age…………………………………………………………………………………
1.3. Possibilities of mathematical tasks for developing the variability of thinking of younger schoolchildren……………………………....................................13
Conclusions on Chapter 1……………………………………….….…................15
Chapter 2. Experimental work on the problem of developing variability of thinking in primary schoolchildren in the process of performing mathematical tasks
2.1. Methodology and organization of experimental work at the stage of ascertaining experiment….……………………………………………………......19
2.2. Project of a formative experiment on the problem of developing variability of thinking in younger schoolchildren in the process of performing mathematical tasks………………………..……27
Conclusions on Chapter 2……….……………………………………………………......32
Conclusion………………………………………………………………………………34
References……………………………………………………..37
Attached files: 1 file
Introduction……………………………………………………………….…3
1.1. Development of variability of thinking from the perspective of pedagogy and psychology...................... ...................... ....... .......................... ............ ................7
1.2. Features of the development of variability of thinking in primary school age……………………………………………………………… ……
1.3. Possibilities of mathematical tasks for developing the variability of thinking of younger schoolchildren……………………………......... ..............13
Conclusions on Chapter 1……………………………………….….….......... ......15
Chapter 2. Experimental work on the problem of developing variability of thinking in younger schoolchildren in the process of performing mathematical tasks
2.1. Methodology and organization of experimental work at the stage of ascertaining experiment….……………………………………………………......19
2.2. Project of a formative experiment on the problem of developing variability of thinking in younger schoolchildren in the process of performing mathematical tasks………………………..……27
Conclusions on Chapter 2……….……………………………………………………………. ........32
Conclusion…………………………………………………… ...............34
References………………………………… …………………..37
Applications
Introduction
According to the Federal State Educational Standard for primary general education, the priority goal of education is the development of students. Issues of general development are closely connected with the development of thinking. And this is not accidental, because the process of thinking is inseparable from all other mental and mental functions: perception, memory, representation, etc.
Recently, the number of children experiencing learning difficulties has increased significantly. In every elementary school class there are many students who have learning problems. It is known that among underperforming primary school students, almost half lag behind their peers in mental development. The reason for the poor performance of students is the delay in the development of such important mental processes as perception, attention, imagination, memory and, especially, thinking, which includes such operations as analysis, synthesis, comparison, generalization. Logical thinking is the basis for the successful development of general educational skills and abilities required by the school curriculum. Students with a low level of logical thinking experience significant difficulties in solving problems, converting quantities, and mastering mental calculation techniques; when applying spelling rules in Russian language lessons, when constructing correct literate speech; when working with texts, when understanding what is read, and much more.
In teaching practice, including in primary school, children quite often have to deal with test tasks that cause difficulties, as students get lost in the proposed options and experience enormous stress. In addition, modern society requires creativity, efficiency, readiness for self-development and self-realization from a modern person. Consequently, the problem of variability and the development of variable thinking is especially relevant these days.
In psychology, the problem of the development of thinking has always occupied a special place. It was studied by such scientists as Bogoyavlensky D.N., Davydov V.V., Galperin P.Ya. Zak A.Z., Lokalova N.P., Lyublinskaya A.A., Menchinskaya N.A., Rubinstein S. L., Elkonin D.D. and others.
Many foreign (Gayson R., Inelder B., Piaget J., Tyson F., etc.) and domestic (Blonsky P.P., Velichkovsky B.M., Vygotsky L.S., Galperin P.Ya., Zinchenko P.I., Leontyev A.N., Luria A.R., Smirnov A.A., Istomina Z.M., Ovchinnikov G.S., Rubinshtein S.L., et al. ) researchers.
The reality around us is diverse and changeable. A modern person constantly finds himself in the situation of choosing a solution to a problem, which is optimal in a given situation. This will be done more successfully by someone who knows how to look for a variety of options and choose among a large number of solutions.
Many psychologists and teachers, such as Alferov A.D., Lyublinskaya A.A., Nemov R.S., have dealt with the problem of the development of variability of thinking in primary school age. and others.
These researchers understand the variability of thinking in psychology as a person’s ability to find a variety of solutions. Indicators of the development of variability of thinking are its productivity, independence, originality and elaboration. Variability of thinking determines an individual’s ability to think creatively and helps to better navigate in real life. Some of the academic subjects in elementary school that have great opportunities for developing the thinking of younger schoolchildren are “The World Around us,” “Russian Language,” and “Mathematics.” For example, the “Mathematics” course promotes the development of all types of thinking in younger schoolchildren, but to a greater extent verbal and logical, therefore the development of variability of thinking is especially important for the process of performing mathematical tasks. Thus, the manifestation of this quality of thinking is required, for example, when solving problems using selection, when the student considers all possible situations, analyzes them and eliminates those that do not correspond to the conditions.
The problem of developing the thinking of younger schoolchildren when studying mathematics and performing mathematical tasks was dealt with by such scientists as M. I. Moro, M. A. Bantova, G. V. Beltyukova, N. B. Istomina (functional development of this process) L. G. Peterson , D. B. Elkonina and V. V. Davydova (the influence of problem-based learning on the development of thinking) and others.
Thus, the problem of developing variability of thinking in mathematics lessons is relevant in modern pedagogy. It can be stated that the problem of developing verbal-logical thinking is especially actively considered in scientific works, while the analysis of pedagogical and methodological literature has shown that there is a contradiction between the need to develop the variability of thinking of younger schoolchildren in the process of performing mathematical tasks and the lack of development of the problem of developing variability of thinking younger schoolchildren in the process of performing mathematical tasks.
The problem of the research is to determine the pedagogical conditions that will contribute to the effective development of the variability of thinking of younger schoolchildren in the process of performing mathematical tasks.
The purpose of the study is to solve the problem raised.
Object of study: development of variability of thinking in junior schoolchildren.
Subject of research: pedagogical conditions for the development of variability in the thinking of younger schoolchildren in the process of performing mathematical tasks.
Research objectives:
1) analyze the psychological, pedagogical and methodological literature in order to reveal the essence of the concepts of “thinking”, “variability of thinking”, “process of development of variability of thinking”.
2) to identify psychological and pedagogical features of the development of variability of thinking in younger schoolchildren.
3) highlight the most effective methods, techniques, and means that promote the development of variability in the thinking of younger schoolchildren in the process of performing mathematical tasks;
4) develop and implement an experimental program to study this problem.
The hypothesis lies in the assumption that the development of the variability of thinking of younger schoolchildren in the process of performing mathematical tasks will be effective under the following didactic conditions:
1) systematic work on the development of variability of thinking in the conditions of problem-based learning;
2) highlighting the following procedures for the development of variability of thinking when solving educational problems as leading ones: vision of an alternative solution and its progress; vision of the structure of an object, construction of a fundamentally new method of solution, different from those known to the subject;
3) systematic use of special tasks (those having a single correct answer, which is found in different ways; having several answer options, and finding them in the same way; having several answer options, which are found in different ways).
To achieve the set goal and solve these problems, a set of scientific research methods was used.
- method of collecting information (studying literature, analyzing the products of students’ activities);
- diagnostic: questioning, ranking, observation.
- general logical methods: analysis, comparison, synthesis, generalization.
- experimental methods (ascertaining experiment).
- methods of mathematical statistics (arithmetic mean, efficiency coefficient)
Research base:
Structure of the work: this work consists of an introduction, two chapters, conclusions for each chapter, a conclusion, a list of references and an appendix. The introduction reveals the relevance of the problem, presents the methodological apparatus of the study; Chapter I defines the theoretical foundations of the study; Chapter II contains experimental work (ascertaining experiment and design of formative experiment); in conclusion, the main conclusions on the work done are presented; the bibliography contains sources; The appendix contains tables, children's work, and lesson notes.
Chapter 1. Psychological and pedagogical foundations for the development of variability of thinking in primary schoolchildren
1.1. Development of variability of thinking from the perspective of pedagogy and psychology
Objects and phenomena of reality have such properties and relationships that can be known directly, with the help of sensations and perceptions (colors, sounds, shapes, placement and movement of bodies in visible space), and such properties and relationships that can be known only indirectly and through generalization , i.e. through thinking.
Thinking is considered as the ability to reason, to think as a human property. In a broad sense, thinking is a set of mental processes that underlie cognition. Thinking includes the active side of cognition: attention and perception, the formation of evidence and judgments. In a closer sense, thinking involves the formation of judgments and conclusions through the analysis and synthesis of concepts. (D.N. Ushakov)
According to Kurbatova V.I. thinking is a rational procedure for realizing the rational existence of a person.
Ponomarev Ya.A. gives the following definition of thinking: “thinking is the highest, indirect, verbal-logical stage of cognition.”
Thinking acts as a complex activity that unfolds in the form of processes of analysis, synthesis, abstraction, and generalization. These processes are carried out at all levels of thinking, in all forms: visual-effective, visual-figurative, verbal-logical. Psychologist L.S. Vygotsky noted the intensive development of intelligence in primary school age. The development of thinking leads to a qualitative restructuring of perception and memory, their transformation into regulated, voluntary processes. “Thinking is the process of solving problems” (Afanasyev N.V.)
The difference between thinking and other mental processes of cognition is that it is always associated with an active change in the conditions in which a person finds himself. Thinking is always aimed at solving a problem. In the process of thinking, a purposeful and expedient transformation of reality is carried out. The thinking process is continuous and continues throughout life, transforming along the way due to the influence of factors such as age, social status, and stability of the living environment. The peculiarity of thinking is its indirect nature. What a person cannot know directly, directly, he knows indirectly, indirectly: some properties through others, the unknown - through the known. Thinking is distinguished by types, processes and operations. The concept of intelligence is inextricably linked with the concept of thinking. Intelligence is the general ability to understand and solve problems without trial and error i.e. "in the mind." Intelligence is considered as a level of mental development achieved by a certain age, which is manifested in the stability of cognitive functions, as well as in the degree of mastery of skills and knowledge (according to the words of Zinchenko, Meshcheryakov). Intelligence as an integral part of thinking, its component and, in its own way, a generalizing concept.
The most significant feature that distinguishes thinking from other mental processes is the focus on discovering new knowledge, i.e. its productivity. In accordance with this, a person’s capabilities for more or less independent discovery of new knowledge, determined (in the presence of other necessary conditions) by the level of development of productive thinking, form the basis, the “core” of his intellect.
Special types of thinking are distinguished - productive and reproductive.
Loading...